Download - Stochastic Dynamics of Financial Markets
Stochastic Dynamics of Financial Markets
A thesis submitted to the University of Manchester for the degree of
Doctor of Philosophy
in the Faculty of Humanities
2013
Mikhail Valentinovich Zhitlukhin
School of Social Sciences/Economic Studies
Contents
Abstract 5
Declaration 6
Copyright Statement 7
Acknowledgements 8
List of notations 9
Introduction 10
Chapter 1. Multimarket hedging with risk 24
1.1. The model of interconnected markets and the hedging principle 25
1.2. Conditions for the validity of the hedging principle . . . . . . . . . . . . . 31
1.3. Risk-acceptable portfolios: examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.4. Connections between consistent price systems and equivalent
martingale measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5. A model of an asset market with transaction costs and portfolio
constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.6. Appendix 1: auxiliary results from functional analysis . . . . . . . . . . 52
1.7. Appendix 2: linear functionals in non-separable πΏ1 spaces . . . . . . 55
Chapter 2. Utility maximisation in multimarket trading 57
2.1. Optimal trading strategies in the model of interconnected asset
markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2. Supporting prices: the definition and conditions of existence. . . . 62
2.3. Supporting prices in an asset market with transaction costs and
portfolio constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.4. Examples when supporting prices do not exist. . . . . . . . . . . . . . . . . . 70
2.5. Appendix: geometric duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2
Chapter 3. Detection of changepoints in asset prices 75
3.1. The problem of selling an asset with a changepoint . . . . . . . . . . . . . 76
3.2. The structure of optimal selling times . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3. The proof of the main theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4. Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References 96
Word count: 31991.
3
List of Figures
Fig. 1. Stopping boundaries for the ShiryaevβRoberts statistic . . . . . . . . . . 93
Fig. 2. Stopping boundaries for the posterior probability process . . . . . . . 93
Fig. 3. Value functions for the ShiryaevβRoberts statistic . . . . . . . . . . . . . . . 94
Fig. 4. Value functions for the posterior probability process . . . . . . . . . . . . 94
Fig. 5. Sample paths of the random walk with changepoints . . . . . . . . . . . . 95
Fig. 6. Paths of the ShiryaevβRoberts statistic . . . . . . . . . . . . . . . . . . . . . . . . . 95
4
The University of Manchester
Mikhail Valentinovich Zhitlukhin
PhD Economics
Stochastic Dynamics of Financial Markets
September 2013
Abstract
This thesis provides a study on stochastic models of financial marketsrelated to problems of asset pricing and hedging, optimal portfolio managingand statistical changepoint detection in trends of asset prices.
Chapter 1 develops a general model of a system of interconnected stochas-tic markets associated with a directed acyclic graph. The main result ofthe chapter provides sufficient conditions of hedgeability of contracts in themodel. These conditions are expressed in terms of consistent price systems,which generalise the notion of equivalent martingale measures. Using thegeneral results obtained, a particular model of an asset market with trans-action costs and portfolio constraints is studied.
In the second chapter the problem of multi-period utility maximisationin the general market model is considered. The aim of the chapter is toestablish the existence of systems of supporting prices, which play the role ofLagrange multipliers and allow to decompose a multi-period constrained util-ity maximisation problem into a family of single-period and unconstrainedproblems. Their existence is proved under conditions similar to those ofChapter 1.
The last chapter is devoted to applications of statistical sequential meth-ods for detecting trend changes in asset prices. A model where prices aredriven by a geometric Gaussian random walk with changing mean and vari-ance is proposed, and the problem of choosing the optimal moment of timeto sell an asset is studied. The main theorem of the chapter describes thestructure of the optimal selling moments in terms of the ShiryaevβRobertsstatistic and the posterior probability process.
5
Declaration
No portion of the work referred to in the thesis has been submitted in
support of an application for another degree or qualification of this or any
other university or other institute of learning.
6
Copyright Statement
The author of this thesis (including any appendices and/or schedules to
this thesis) owns certain copyright or related rights in it (the Copyright)
and he has given The University of Manchester certain rights to use such
Copyright, including for administrative purposes.
Copies of this thesis, either in full or in extracts and whether in hard or
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has from time to time. This page must form part of any such copies made.
The ownership of certain Copyright, patents, designs, trade marks and
other intellectual property (the Intellectual Property) and any reproductions
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tions), which may be described in this thesis, may not be owned by the
author and may be owned by third parties. Such Intellectual Property and
Reproductions cannot and must not be made available for use without the
prior written permission of the owner(s) of the relevant Intellectual Property
and/or Reproductions.
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tion and commercialisation of this thesis, the Copyright and any Intellec-
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striction declarations deposited in the University Library, The University Li-
braryβs regulations (see http://www.manchester.ac.uk/library/aboutus/
regulations) and in The Universityβs policy on presentation of Theses.
7
Acknowledgements
I owe my deepest gratitude to my supervisors, Professor Igor Evstigneev
and Professor Goran Peskir, for their continuous support of my research, for
the immense knowledge and inspiration they gave me. It was my sincere
pleasure to work with them and I deeply appreciate their help during all the
time of the research and writing the thesis.
I am grateful to Professor Albert Shiryaev, whom I consider my main
teacher in stochastic analysis and probability. It is impossible to overestimate
the importance of his continuous attention to my research and my career.
I am indebted to Professor William T. Ziemba and Professor Sandra
L. Schwartz, who motivated me to study applications of sequential statis-
tical methods to stock markets.
I thank Professor Hans Rudolf Lerche and Dr Pavel Gapeev for the valu-
able discussions of the mathematical problems related to my research.
My study at The University of Manchester was funded by the Economics
Discipline Area Studentship from The School of Social Sciences, which I
greatly acknowledge.
An acknowledgement is also made to the Hausdorff Research Institute
for Mathematics (Bonn, Germany), where I attended trimester program
βStochastic Dynamics in Economics and Financeβ in summer 2013. The
hospitality of HIM and the excellent research environment were invaluable
for the preparation of the thesis. In particular, I thank the organisers of the
program, Professor Igor Evstigneev, Professor Klaus Reiner Schenk-Hoppe
and Professor Rabah Amir.
8
List of notations
R the set of real numbers
R+ the set of non-negative real numbers
πΏ1 the space of all integrable functions on a measure space
πΏβ the space of all essentially bounded functions on a measure
space
π* the positive dual cone of a set π in a normed space
P | F the restriction of a measure P to a π-algebra F
π₯+ max{π₯, 0}
π₯β βmin{π₯, 0}
I{π΄} the indicator of a statement π΄ (I{π΄} = 1 if π΄ is true,
I{π΄} = 0 if π΄ is false)
π―π‘+ the set of all successors of a node π‘ in a graph π―
π―π‘β the set of all predecessors of a node π‘ in a graph π―
π―β the set of all nodes in a graph π― with at least one successor
π―+ the set of all nodes in a graph π― with at least one predecessor
N (π, π2) Gaussian distribution with mean π and variance π2
Ξ¦ the standard Gaussian cumulative distribution function
(Ξ¦(π₯) = 1β2π
β« π₯
ββ πβπ¦2/2ππ¦)
a.s. βalmost surelyβ (with probability one)
i.i.d. independent and identically distributed (random variables)
9
Introduction
This thesis provides a study on stochastic models of financial markets.
Questions of derivatives pricing and hedging, optimal portfolio managing
and detection of changes in asset prices trends are considered.
The first range of questions β hedging and pricing of derivative securities β
has been studied in the literature since 1960s. The celebrated BlackβScholes
formula [3] for the price of a European option was one of the first fundamental
results in this direction. Derivative securities (or contingent claims) play an
important role in modern finance as they allow to implement complex trading
strategies which reduce the risk from indeterminacy of future asset prices.
One of the main questions in derivatives trading consists in determining
the fair price of a derivative, which satisfies both the seller and the buyer.
The BlackβScholes formula provides an explicit answer to this question in the
model when asset prices are modelled by a geometric Brownian motion. Later
this result was extended to a wider class of market models. The development
of the derivatives pricing theory has resulted in that nowadays the volume
of derivatives traded is much higher than the volume of basic assets [41].
The second range of questions considered in the thesis concerns consump-
tion-investment problems, where a trader needs to manage a portfolio of as-
sets choosing how much to consume in order to maximise utility over a period
of time. Questions of this type were originally studied in relation to models
of economic growth, where the objective is to find a trade-off between goods
produced and consumed with the aim of the optimal development of the econ-
omy. One of the first results in the financial context was obtained by Merton
[42], who provided an explicit solution of the consumption-investment prob-
lem for a model when asset prices are driven by a geometric Brownian motion.
There have been several extensions of Mertonβs result which include factors
like transaction costs, possibility of bankruptcy, general classes of stochastic
processes describing asset prices, etc. (see e. g. [4, 10, 36, 44]).
10
The third part of the thesis contains applications of sequential methods
of mathematical statistics to detecting changes in asset prices trends. The
mathematical foundation of the corresponding statistical methods β the the-
ory of changepoint detection (or disorder detection) β was laid in the papers
by W. Shewhart, E. S. Page, S.W. Roberts, A.N. Shiryaev, and others in
1920-1960s, and initially was applied in questions of production quality con-
trol and radiolocation. Recently, these methods have gained attention in
finance.
The main results of the thesis generalise the classical theory to advanced
market models. We obtain results that broaden the models available in the
literature and reflect several important features of real markets that have
not been studied earlier in the corresponding fields.
The rest of the introduction provides a detailed description of the prob-
lems considered in the thesis.
Asset pricing and hedging
Consider the classical model of a stochastic financial market, which op-
erates at discrete moments of time π‘ = 0, 1, . . . , π , and where π assets are
traded. The stochastic nature of the market is represented by a filtered
probability space (Ξ©,F , (Fπ‘)ππ‘=0,P), where each π-algebra Fπ‘ in the filtra-
tion F0 β F1 β . . .Fπ describes random factors that might affect the
market at time π‘.
The prices of the assets at time π‘ are given by Fπ‘-measurable strictly
positive random variables π1π‘ , π
2π‘ , . . . , π
ππ‘ . Asset 1 is assumed to be riskless
(e. g. cash deposited with a bank account) with the price π1π‘ β‘ 1 (after being
discounted appropriately), while assets π = 2, . . . , π are risky with random
prices.
An investor can trade in the market by means of buying and selling assets.
A trading strategy is a sequence π₯0, π₯1, . . . , π₯π of random π -dimensional
vectors, where each π₯π‘ = (π₯1π‘ , . . . , π₯ππ‘ ) is Fπ‘-measurable and specifies the
11
portfolio held by the investor between the moments of time π‘ and π‘+1. The
coordinate π₯ππ‘ (π = 1, . . . , π) is equal to the amount of physical units of asset
π in the portfolio.
An important class of trading strategies consists of self-financing trading
strategies, which have no exogenous inflow or outflow of money. Namely, a
trading strategy (π₯π‘)π‘6π is called self-financing if
π₯π‘β1ππ‘ = π₯π‘ππ‘ for each π‘ = 1, . . . , π,
where the left-hand side is the value of the old portfolio (established βyester-
dayβ), and the right-hand side is the value of the new portfolio (established
βtodayβ). The equality is understood to hold with probability one.
The central question of the derivatives pricing and hedging theory con-
sists in finding fair prices of derivative securities (or contingent claims). A
derivative is a financial instrument which has no intrinsic value in itself,
but derives its value from underlying basic assets [15]. Derivative securities
include options, futures, swaps, and others (see e. g. [31]).
As an example, consider a standard European call option on asset π, which
is a contract that allows (but does not oblige) its buyer to buy one unit of
asset π at a fixed time π in the future for a fixed price πΎ. The seller incurs
a corresponding obligation to fulfil the agreement if the buyer decides to
exercise the contract, which she does if the spot price πππ is greater than πΎ
(thus receiving the gain πππ βπΎ). In order to obtain the option, the buyer
pays the seller some premium, the price of the option, at time π‘ = 0 .
Another example is a European put option, which is a contract giving its
buyer the right to sell asset π at a fixed time π for a fixed price πΎ. If the
buyer exercises the option (which happens whenever πΎ > πππ ), she receives
the gain πΎ β πππ .
Mathematically, contracts of this type can be identified with random
variables π representing the payoff of the seller to the buyer at time π . For
example, π = (ππ βπΎ)+ for a European call option, and π = (πΎ β ππ )+
for a European put option.
It is said that a self-financing trading strategy (π₯π‘)π‘6π (super)hedges a
12
derivative π if
π₯πππ > π a.s.,
i. e. the seller who follows the strategy (π₯π‘)π‘6π can fulfil the payment asso-
ciated with the derivative with probability one. The minimal value π₯ of the
initial portfolio π₯0 is called the (upper) hedging price of π and is denoted
by π(π):
π(π) = inf{π₯ : there exists (π₯π‘)π‘6π superhedging π such that π₯0π0 = π₯}.
The price π(π) is the minimal value of the initial portfolio that allows
the seller to fulfil her obligations (provided that the infimum in the definition
is attained; see [61, Ch. VI, S 1b-c]). On the other hand, if she could sell the
derivative for a higher price π > π(π), it would be possible to find a trading
strategy which delivers her a free-lunch β a non-negative and non-zero gain
by time π , for which the buyer has no incentive to agree.
The central result of the asset pricing and hedging theory states that in
a market without arbitrage opportunities the price of a contingent claim can
be found as the supremum of its expected value with respect to equivalent
martingale measures.
It is said that a self-financing trading strategy (π₯π‘)π‘6π realises an arbitrage
opportunity in the market if
π₯0π0 = 0, π₯πππ > 0 and P(π₯πππ > 0) > 0.
A probability measure P, equivalent to the original measure P (P βΌ P),
is called an equivalent martingale measure (EMM), if the price sequence π is
a P-martingale, i. e. EP(ππ
π‘ | Fπ‘β1) = πππ‘β1 for all π‘ = 1, . . . , π , π = 1, . . . , π .
The set of all EMMs is denoted by P(P).
These two notions express some form of market efficiency. The absence
of arbitrage opportunities means that there is no trading strategy with zero
initial capital, which allows to obtain a non-zero gain without downside
risk (a free lunch) at time π . The existence of an equivalent martingale
measure allows to change the underlying measure P, preserving the sets of
zero probability, in a way that the assets have zero return rates.
13
Theorem. Equivalent martingale measures exist in a market if and only
if there is no arbitrage opportunities.
In a market without arbitrage opportunities, the price of a contingent
claim π such that EP|π| <β for any P β P(P) can be found as
π(π) = supPβP
EPπ.
Remark. In the case when there is only one equivalent martingale mea-
sure (the case of a complete market), π(π) = EPπ. It turns out that a
complete market has a simple structure β the π-algebra Fπ is purely atom-
istic with respect to P and consists of no more that ππ atoms. Note that in
continuous time, however, there exist examples of complete markets where
Fπ is not purely atomistic (e. g. the model of geometric Brownian motion).
The above theorem is referred to as the Fundamental Theorem of Asset
Pricing and the Risk-Neutral Pricing Principle (see e. g. [24, 61]). It con-
stitutes the core of the classical derivatives pricing theory. However it does
not take into account several important features of real markets, which are
necessary to consider when applying the theory in practice. The present the-
sis addresses these issues and develops a model that includes the following
improvements.
1. Transaction costs and portfolio constraints. The act of buying or
selling assets in a real market typically reduces the total wealth of a trader
(due to brokerβs commission, differences in bid and offer prices, etc.). As
a result, an investor may need to limit the number of trading operations in
order not to lose too much money on transaction costs. Real markets also set
constraints on admissible portfolios in order to prevent market participants
from using too risky trading strategies. For example, such constraints can
be expressed in a form of a margin requirement, which obliges investors to
choose only those strategies that allow to liquidate their portfolios if prices
move unfavourably.
Both transaction costs and portfolio constraints limit investor abilities,
and thus, generally, increase hedging prices. These aspects have already been
14
considered in the literature, but for the most part separately. One can point,
e.g., to the monograph by Kabanov and Safarian [33] discussing transaction
costs, the papers by Jouini, Kallal [32] and Evstigneev, Schurger, Taksar [22]
dealing with portfolio constraints, and the references therein.
It turns out that under the presence of transaction costs and portfolio
constraints, the problem of pricing contingent claims has a solution simi-
lar to the classical model. Namely, the price of a contingent claim can be
found as the supremum of its expected value with respect to consistent price
systems, which are vector analogues of equivalent martingale measures (see
Chapter 1 for details). However, the existence of consistent price systems
in a market with transaction costs and portfolio constraints becomes a con-
siderably difficult question, and, generally, requires conditions stronger than
the absence of arbitrage opportunities. Several stronger conditions have been
introduced in the literature which guarantee the existence of consistent price
systems, thus allowing to price contingent claims. Their formulations can be
found in e.g. [22, 33].
2. Hedging with risk. The classical superhedging condition requires
that the seller of a derivative chooses a trading strategy that covers the
payment with probability one, i. e. without any risk of non-fulfilling her
obligation. However, it may be acceptable for the seller to guarantee the
required amount of payment only with some (high) level of confidence β for
example, in unfavourable outcomes she may use exogenous funds, which is
compensated by a higher gain in favourable outcomes.
This is especially important when the volume of trading is large, so not
the result of every single deal is important, but only the average result of a
large number of them. Weakening the superhedging criterion can possibly
reduce derivatives prices and lead to a potential gain while maintaining an
acceptable level of risk of unfavourable situations.
An approach of hedging with risk, used in the thesis, is based on replac-
ing the superhedging condition with a general principle requiring that the
difference between the required payment and the portfolio used to cover it
belongs to a certain set of acceptable portfolios (the exact formulation will
15
be given in Chapter 1). The superhedging condition is a particular case of
this model.
This approach has already been used in the literature. Especially, much
attention has been devoted to hedging with respect to coherent and convex
risk measures (see e. g. the papers [7, 9], where it is called the no good deals
pricing principle).
3. Multimarket trading. The framework developed in the thesis is
capable of modelling a system of interconnected markets associated with
nodes of a given acyclic directed graph. The nodes of the graph represent
different trading sessions that may be related to different moments of time
and/or different asset markets. In particular, the standard model of a single
market can be represented by the graph being the linearly ordered set of
moments of time, while the general case allows to consider the problem of
distributing assets between several markets, which may operate at the same
or different moments of time.
We also consider contracts with payments at arbitrary trading sessions
(not only the terminal ones), which broadens the range of possible financial
instruments. This also makes the model potentially applicable not only in
finance, but in other areas, e.g. it can be used in insurance, where an insurer
receives a premium at time π‘ = 0 and needs to manage a portfolio in order
to be able to cover claims occurring randomly.
The above features of real markets have been studied in the literature
for the most part separately. In the thesis a new general model is proposed,
which incorporates all of them. The central result of the first chapter for the
new model is the hedging criterion formulated in terms of consistent price
systems β direct analogues of equivalent martingale measures. We prove their
existence and show that a contract is hedgeable if and only if its value with
respect to any consistent price system is non-negative. In order to obtain
the result, we systematically use the idea of margin requirements which limit
allowed leverage of admissible portfolios. This differs from the standard
approach based on the absence of arbitrage. However, margin requirements
always present in one form or another in any real market, which makes our
16
approach fully justified from the applied point of view.
The model is based on the framework of von Neumann β Gale dynamical
systems introduced by von Neumann [68] and Gale [25] for deterministic
models of a growing economy, and later extended by Dynkin, Radner and
their research groups to the stochastic case. A model of a financial market
based on von Neumann β Gale systems was also proposed in the paper [11]
for the case of a discrete probability space (Ξ©,F ,P) and a linearly ordered
set of moments of time.
In the thesis this framework is extended to financial market models, which
have several important distinctions from models of growing economies.
Optimal trading strategies
In the second chapter the problem of finding trading strategies that max-
imise the utility function of an investor over a period of time is studied.
In the financial literature, problems of this type are commonly referred
to as consumption-investment problems, and there exists a large number of
results in this area. The subject of our research is the optimal investment
problem for the general model proposed in Chapter 1. The goal is to obtain
conditions for the existence of supporting prices, which allow to reduce a
multi-period constrained maximisation problem of investorβs utility function
to a family of single-stage unconstrained problems. Supporting prices play
the role similar to that of Lagrange multipliers.
The problem of utility maximisation and existence of supporting prices
plays a central role in the von Neumann β Gale framework of economic
growth. The setting of the problem and the main results there consist in the
following.
A von Neumann β Gale system is a sequence of pairs of random vectors
(π₯π‘, π¦π‘) π‘ = 0, 1, . . . , π , such that (π₯π‘, π¦π‘) β π΅π‘ for some given sets π΅π‘, and
π₯π‘ 6 π¦π‘β1. In the financial interpretation, a vector π₯π‘ can be regarded as
a portfolio of assets held before a trading session π‘, and π¦π‘ as a portfolio
17
obtained during the session. The inequality π₯π‘ 6 π¦π‘β1 means that there is
no exogenous infusion of assets (but disposal of assets is allowed). The sets
π΅π‘ can describe the self-financing condition, portfolio constraints, etc.
In models of economic growth vectors π₯π‘ describe amounts of input com-
modities for a production process with output commodities π¦π‘. The sets π΅π‘
consist of all possible production processes and the condition π₯π‘ 6 π¦π‘β1 re-
flects the requirement that the input of any production process should not
exceed the output of the previous one.
Suppose that with each set π΅π‘ a real-valued utility function π’π‘(π₯π‘, π¦π‘) is
associated and interpreted as the utility from the production process with
input commodities π₯π‘ and output commodities π¦π‘. Let π₯0 be a given vector of
initial resources. Then the problem consists in finding a production process
π = (π₯π‘, π¦π‘)π‘6π , represented by a von Neumann β Gale system, such that
π₯0 = π₯0 and which maximises the utility
π’(π) :=πβπ‘=0
π’π‘(π₯π‘, π¦π‘).
This is a constrained maximisation problem of the function π’(π) over
all sequences π = (π₯π‘, π¦π‘)π‘6π with (π₯π‘, π¦π‘) β π΅π‘ satisfying the constraints
π₯π‘ 6 π¦π‘β1. Under some assumptions the solution π* of the problem exists,
and, moreover, there exist random vectors ππ‘, π‘ = 0, . . . , π + 1 such that
π’(π*) >πβπ‘=0
(π’π‘(π₯π‘, π¦π‘) + E[π¦π‘ππ‘+1 β π₯π‘ππ‘]
)+ Eπ₯0π0
for any sequence (π₯π‘, π¦π‘)π‘6π with (π₯π‘, π¦π‘) β π΅π‘. Thus π* solves the uncon-
strained problem of maximising the right hand side of the above inequality.
Moreover, in order to maximise the right-hand size it is sufficient to max-
imise each term π’π‘(π₯π‘, π¦π‘) + E[π¦π‘ππ‘+1 β π₯π‘ππ‘] independently. Gale [26] noted
the great importance of results of this type by saying that it is βthe single
most important tool in modern economic analysis both from the theoretical
and computational point of viewβ.
The aim of the second chapter is to obtain similar results for our model
of interconnected financial markets. The main mathematical difficulty here
18
consists in that in our model portfolios (π₯π‘, π¦π‘) may have negative coordi-
nates (corresponding to short sales), unlike commodities vectors in models
of economic growth. The key role in establishing the main results of the
second chapter will be played by the assumption of margin requirements.
The existence of supporting prices will be proved under a condition on the
size of the margin.
Detection of trend changes in asset prices
The third part of the thesis studies statistical methods of change detection
in trends of asset prices. We consider a model, where prices initially rise, but
may start falling at a random (and unknown) moment of time. The aim of
an investor is to detect this change in the prices trend and to sell the asset
as close as possible to its highest price.
It will be assumed that the price of an asset is represented by a geometric
Gaussian random walk π = (ππ‘)ππ‘=0 defined on a probability space (Ξ©,F ,P),
whose drift and volatility coefficients may change at an unknown time π:
π0 > 0, logππ‘
ππ‘β1=
β§β¨β©π1 + π1ππ‘, π‘ < π,
π2 + π2ππ‘, π‘ > π,for π‘ = 1, 2, . . . , π,
where π1, π2, π1, π2 are known parameters, ππ‘ βΌ N (0, 1) are i.i.d. normal
random variables with zero mean and unit variance, and π is the moment of
time when the probabilistic character of the price sequence changes.
In order to model the uncertainty of the moment π, it will be assumed
that π is a random variable defined on (Ξ©,F ,P), but an investor can observe
only the information included in the filtration F = (Fπ‘)ππ‘=0, Fπ‘ = π(ππ’;π’ 6
π‘), generated by the price sequence, and cannot observe π directly. The
distribution law of π is known, and π takes values 1, 2, . . . , π with known
probabilities ππ‘ > 0, so thatπβπ‘=1
ππ‘ 6 1. The quantity ππ+1 = 1βπβπ‘=1
ππ‘ is the
probability that the change of the parameters does not occur until the final
19
time π and π1 is the probability that the logarithmic returns already follow
N (π2, π2) since the initial moment of time.
By definition, a moment π when one can sell the asset should be a stopping
time of the filtration F, which means that {π : π(π) 6 π‘} β Fπ‘ for any
0 6 π‘ 6 π . The notion of a stopping time expresses the idea that the
decision to sell the asset at time π‘ should be based only on the information
available from the price history up to time π‘, and does not rely on future
prices. The class of all stopping times π 6 π of the filtration F is denoted
by M.
The problem we consider consists in maximising the power or logarithmic
utility from selling the asset. Namely, let ππΌ(π₯) = πΌπ₯πΌ for πΌ = 0 and
π0(π₯) = log π₯. For an arbitrary πΌ β R we consider the optimal stopping
problem
ππΌ = supπβM
EππΌ(ππ ).
The problem consists in finding the value ππΌ, which is the maximum expected
utility one can obtain from selling the asset, and finding the stopping time
π *πΌ at which the supremum is attained (we show that it exists).
The study of methods of detecting changes in probabilistic structure
of random sequences and processes (called disorder detection problems or
changepoint detection problems) began in the 1950-1960s in the papers by
E. Page, S. Roberts, A.N. Shiryaev and others (see [45, 46, 49, 57β59]); the
method of control charts proposed by W.A. Shewhart in the 1920s [55] is
also worth mentioning.
A financial application of changepoint detection methods was considered
in the paper [2] by M. Beibel and H.R. Lerche, who studied the problem
of choosing the optimal time to sell the asset in continuous time, when the
asset price process π = (ππ‘)π‘>0 is modelled by a geometric Brownian motion
whose drift changes at time π:
πππ‘ = ππ‘[π1I(π‘ < π) + π2I(π‘ > π)]ππ‘+ πππ‘ππ΅π‘, π0 > 0,
where π΅ = (π΅π‘)π‘>0 is a standard Brownian motion on a probability space
(Ξ©,F ,P), and π1, π2, π are known real parameters. The paper [2] assumes
20
that π is an exponentially distributed random variable with a known param-
eter π > 0 and is independent of π΅. An investor looks for the stopping time
π * of the filtration generated by the process π that maximises the expected
gain Eππ (the time horizon in the problem is π = β, i. e. π can be un-
bounded). By changing the parameters π1, π2, π one can solve the problem
of maximising EππΌπ for any πΌ except πΌ = 0.
Beibel and Lerche show that if π1, π2, π satisfy some relation, then the
optimal stopping time π * can be found as the first moment of time when the
posterior probability process π = (ππ‘)π‘>0, ππ‘ = P(π 6 π‘ | Fπ‘), exceeds some
level π΄ = π΄(π1, π2, π, π):
π * = inf{π‘ > 0 : ππ‘ > π΄}.
In other words, the optimal stopping time has a very clear interpretation:
one needs to sell the asset as soon as the posterior probability that the change
has happened exceeds a certain threshold. An explicit representation of ππ‘
through the observable process ππ‘ is available (see e. g. [47, Section 22]).
In the paper [14] the conditions on π1, π2, π were relaxed and it was
shown that the result holds for all possible values of the parameters (except
some trivial cases). Also, the optimal threshold was found explicitly as
π΄ = π΄β²/(1 + π΄β²), where the constant π΄β² = π΄β²(π1, π2, π, π) is the unique
positive root of the (algebraic) equation
2
β« β
0
πβππ‘π‘(π+πΎβ3)/2(1 + π΄β²π‘)πΎβπ+1)/2ππ‘
= (πΎ β π+ 1)(1 + π΄β²)
β« β
0
πβππ‘π‘(π+πΎβ1)/2(1 + π΄β²π‘)(πΎβπβ1)/2ππ‘
with the parameters
π =2π
π2, π =
2
π
(π
πβ π
), πΎ =
β(πβ 1)2 + 4π,
where
π =π2 β π1
π, π =
2(πβ π2)
π2.
The paper [56] studied the problem of maximising the logarithmic utility
21
from selling the asset with finite time horizon π (but still assuming that π
is exponentially distributed, so the parameters do not change until the end
of the time horizon with positive probability). The solution was based on
an earlier result of the paper [28]. It was shown that the optimal stopping
time can be expressed as the first moment of time when ππ‘ exceeds some
time-dependent threshold:
ππ‘ = inf{π‘ > 0 : ππ‘ > π*(π‘)},
where π*(π‘) is a function on [0, π ], dependent on π, π1, π2, π. The authors
showed that it can be found as a solution of some nonlinear integral equation.
They also briefly discussed the optimal stopping problem for the linear utility
function with a finite time horizon, and reduced it to some two-dimensional
optimal stopping problem for the process ππ‘, but did not provide its explicit
solution.
The problem on a finite time horizon was solved in the paper [69] by
M.V. Zhitlukhin and A.N. Shiryaev for the both logarithmic and linear
utility functions, provided that π is uniformly distributed on [0, π ] (however,
the solution can be generalised to a wide class of prior distributions of π). In
each problem, the optimal stopping time can be expressed as the first time
when the value of ππ‘ exceeds some function π*(π‘) characterised by a certain
integral equation. These equations can be solved numerically by βbackward
inductionβ as demonstrated in the paper.
This result was used by A.N. Shiryaev, M.V. Zhitlukhin and W.T.
Ziemba in the research [66] on stock prices bubbles of Internet related com-
panies. The method of changepoint detection was applied to the daily closing
prices of Apple Inc. in 2009β2012 and the daily closing values of NASDAQ-
100 index in 1994β2002. These two assets had spectacular runs from their
bottom values and dramatic falls after reaching the top values, thus being
good candidates to be modelled by processes with changepoints in trends.
For specific dates of entering the market, the method provided exit points at
approximately 75% of the maximum value of the NASDAQ-100 index, and
90% of the maximum price of Apple Inc. stock.
22
The aim of the third chapter of the thesis is to solve the optimal stopping
problem for a geometric Gaussian random walk with a changepoint in dis-
crete time and a finite time horizon for an arbitrary prior distribution of π.
It will be shown that the optimal stopping time can be expressed as the first
moment of time when the sequence of the ShiryaevβRoberts statistic (which
is obtained from the posterior probability sequence by a simple transforma-
tion) exceeds some time-dependent level. This result is similar to the results
available in the literature for the case of continuous time. However it allows
to consider any prior distribution of π (not only exponential or uniform) and
can be used in models where also the volatility coefficient π changes.
A backward induction algorithm for computing the optimal stopping level
is described in the chapter. Using it, we present numerical simulations of
random sequences with changepoints and obtain the corresponding optimal
stopping times.
23
Chapter 1
Multimarket hedging with risk
The results of this chapter extend the classical theory of asset pricing and
hedging in several directions. We develop a general model including transac-
tion costs and portfolio constraints and consider hedging with risk, which is
βsofterβ than the classical superreplication approach. These aspects of the
modelling of asset markets have already been considered in the literature,
but for the most part separately. One can point, e.g., to the monograph by
Kabanov and Safarian [33] discussing transaction costs, the papers by Jouini
and Kallal [32] and Evstigneev, Schurger and Taksar [22] dealing with portfo-
lio constraints and the studies by Cochrane and Saa-Requejo [9] and Cherny
[7] involving hedging with risk. However, up to now no general model re-
flecting all these features of real financial markets has been proposed.
Another novel aspect of this study is that, in contrast with the conven-
tional theory, we consider asset pricing and hedging in a system of intercon-
nected markets. These markets (functioning at certain moments of discrete
time) are associated with nodes of a given acyclic directed graph. The model
involves stochastic control of random fields on directed graphs. Control prob-
lems of this kind were considered in the context of modelling economies with
locally interacting agents in the series of papers by Evstigneev and Taksar
[16β20].
In the case of a single market β when the graph is a linearly ordered set
of moments of time β the model extends the one proposed by Dempster,
Evstigneev and Taksar [11]. The approach of [11] was inspired by a paral-
lelism between dynamic securities market models and models of economic
growth. The underlying mathematical structures in both modelling frame-
works are related to von Neumann-Gale dynamical systems (von Neumann
[68], Gale [25]) characterised by certain properties of convexity and homo-
24
geneity. This parallelism served as a conceptual guideline for developing the
model and obtaining the results.
The main results of the chapter provide general hedging criteria stated
in terms of consistent price systems, generalising the notion of an equivalent
martingale measure. Existence theorems for such price systems are counter-
parts of various versions of the well-known Fundamental Theorem of Asset
Pricing (Harrison, Kreps, Pliska and others). However, the assumptions we
impose to obtain hedging criteria are substantially distinct from the standard
ones. We systematically use the idea of margin requirements on admissible
portfolios, setting limits for the allowed leverage. Such requirements are
present in one form or another in all real financial markets. Being fully jus-
tified from the applied point of view, they make it possible to substantially
broaden the frontiers of the theory.
The chapter is organised as follows. In Section 1.1 we introduce the gen-
eral model. In Section 1.2 we state and prove the main results. Section 1.3
contains examples of general hedging conditions, and Section 1.4 explains the
connection between consistent price systems and equivalent martingale mea-
sures. By using the general results obtained, we study a specialised model
of a stock market in Section 1.5. Several auxiliary results from functional
analysis are assembled in Sections 1.6 and 1.7.
A shortened version of this chapter was published in the paper [23].
1.1 The model of interconnected markets and the hedging
principle
Let (Ξ©,F ,P) be a probability space and πΊ a directed acyclic graph with
a finite set of nodes π― . The nodes of the graph represent different trading
sessions that may be related to different moments of time and/or different
asset markets. With each π‘ β π― a π-algebra Fπ‘ β F is associated describing
random factors that might affect the trading session π‘.
A measurable space (Ξ,J , π) with a finite measure π is given, whose
25
points represent all available assets. If Ξ is infinite, the model reflects the
idea of a βlargeβ asset market (cf. Hildenbrand [30]). For each π‘ β π― , a real-
valued (Pβ π)-integrable (Fπ‘ β J )-measurable function π₯ : Ωà Πβ R is
interpreted as a portfolio of assets that can be bought or sold in the trading
session π‘. The space of all (equivalence classes of) such functions with the
norm βπ₯β =β«|π₯|π(Pβ π) is denoted by πΏ1
π‘ (Ξ©ΓΞ) or simply πΏ1π‘ . The value
π₯(π, π) of a function π₯ represents the number of βphysical unitsβ of asset π
in the portfolio π₯. Positive values π₯(π, π) are referred to as long positions,
while negative ones as short positions of the portfolio.
If π is an integrable real-valued measurable function defined on some
measure space, by Eπ we denote the value of its integral over the whole space
(when the measure is a probability measure, Eπ is equal to the expectation
of π).
For a subset π β π― we denote by πΏ1π (Ξ©ΓΞ), or simply πΏ1
π , the space of
functions π¦ : π ΓΞ©ΓΞ β π with finite norm βπ¦β =βπ‘βπ
β«|π¦(π‘, π, π)|π(Pβπ).
Where it is convenient, we represent such functions as families π¦ = (π¦π‘)π‘βπ
of functions π¦π‘(π, π) = π¦(π‘, π, π).
The symbols π―π‘β and π―π‘+ will be used to denote, respectively, the sets of
all direct predecessors and successors of a node π‘ β π― , and π―β, π―+ will stand,
respectively, for the set of all nodes having at least one successor and the
set of all nodes having at least one predecessor (so that π―β =βπ‘βπ―
π―π‘β and
π―+ =βπ‘βπ―
π―π‘+). For the convenience of further notation we define πΏ1π‘+ = πΏ1
π―π‘+
for each π‘ β π―β, and πΏ1π‘β = πΏ1
π―π‘β for each π‘ β π―+.In a trading session π‘ β π―β one can buy and sell assets and distribute
them between trading sessions π’ β π―π‘+. This distribution is specified by a
set of portfolios π¦π‘ = (π¦π‘,π’)π’βπ―π‘+ β πΏ1π‘+, where π¦π‘,π’ is the portfolio delivered
to the session π’.
Trading constraints in the model are defined by some given (convex)
cones π΅π‘ β πΏ1π‘ β πΏ1
π‘+, π‘ β π―β. A trading strategy π is a family of functions
π = (π₯π‘, π¦π‘)π‘βπ―β such that
(π₯π‘, π¦π‘) β π΅π‘ for each π‘ β π―β.
26
Each function π₯π‘ represents the portfolio held before one buys and sells assets
in the session π‘. For π‘ β π―β, the function π¦π‘ = (π¦π‘,π’)π’βπ―π‘+ specifies the
distribution of assets to the sessions π’ β π―π‘+.Let π¦π‘β =
βπ’βπ―π‘β
π¦π’,π‘ denote the portfolio of assets delivered to a trading
session π‘ from other sessions (π¦π‘β := 0 if π‘ is a source, i.e. has no predecessors).
In each session π‘ β π― , the portfolio π¦π‘β β π₯π‘ can be used for the hedging of a
contract (we define π₯π‘ = 0 if π‘ is a sink, i.e. π‘ /β π―β). By definition, a contract
πΎ is a family of portfolios
πΎ = (ππ‘)π‘βπ― , ππ‘ β πΏ1π‘ ,
where ππ‘ stands for the portfolio which has to be delivered β according to
the contract β at the trading session π‘. The value of ππ‘(π, π) can be negative;
in this case the corresponding amount of asset π is received rather than
delivered. The notion of a contract encompasses contingent claims, derivative
securities, insurance contracts, etc.
Assume that a non-empty closed cone π β πΏ1π― is given. We say that a
trading strategy π = (π₯π‘, π¦π‘)π‘βπ―β hedges a contract πΎ = (ππ‘)π‘βπ― if
(ππ‘)π‘βπ― β π, where ππ‘ = π¦π‘β β π₯π‘ β ππ‘. (1.1)
Each ππ‘ represents the difference between the portfolio π¦π‘β β π₯π‘ delivered at
the session π‘ by the strategy π and the portfolio ππ‘ that must be delivered
according to the contract πΎ. The cone π is interpreted as the set of all
risk-acceptable families of portfolios. If π is the cone of all non-negative
functions, then π is said to superhedge (superreplicate) πΎ. General cones πmake it possible to consider hedging with risk.
A contract is called hedgeable if there exists a trading strategy hedging it.
The main aim of our study is to characterise the class of hedgeable contracts.
Let πΏβπ‘ = πΏβ
π‘ (Ξ©ΓΞ) denote the space of all essentially bounded Fπ‘βJ -
measurable functions π : Ξ©ΓΞ β R, and π+ = πβ(βπ) stand for the set of
strictly risk-acceptable families of portfolios (if πΌ β π+, then πΌ is acceptable
but βπΌ is not acceptable).
The characterisation of hedgeable contracts will be given in terms of (mar-
27
ket) consistent price systems that are, by definition, families π = (ππ‘)π‘βπ― of
functions ππ‘ β πΏβπ‘ (Ξ©ΓΞ) satisfying the properties
Eπ₯π‘ππ‘ >βπ’βπ―π‘+
Eπ¦π‘,π’ππ’ for each (π₯π‘, π¦π‘) β π΅π‘ and π‘ β π―β, (1.2)
βπ‘βπ―
Eππ‘ππ‘ > 0 for each πΌ = (ππ‘)π‘βπ― β π+. (1.3)
Property (1.2) means that it is impossible to obtain strictly positive ex-
pected profit E(β
π’βπ―π‘+π¦π‘,π’ππ’ β π₯π‘ππ‘), which is computed in terms of the prices
ππ‘, in the course of trading, as long as the trading constraints are satisfied.
Condition (1.3) is a non-degeneracy assumption, saying that the expected
value of any strictly risk-acceptable family of portfolios is strictly positive.
Throughout the chapter, we suppose that the cone π satisfies the follow-
ing assumption.
Assumption (A). π+ = β and there exists π = (ππ‘)π‘βπ― , ππ‘ β πΏβπ‘ ,
satisfying (1.3).
The assumption states that there exists an πΏ1π― -continuous linear func-
tional strictly positive on π+ = β . Its existence for any closed π with
π+ = β can be established, e.g., when πΏ1π― (Ωà Ξ) is separable (see Remark
1.6 in Section 1.6).
Our main results, given in the next section, provide conditions guaran-
teeing that the following principle holds.
Hedging principle. The class of consistent price systems is non-empty,
and a contract (ππ‘)π‘βπ― is hedgeable if and only ifβπ‘βπ―
Eππ‘ππ‘ 6 0 for all consis-
tent price systems (ππ‘)π‘βπ― .
The hedging principle states that a contract is hedgeable if and only if its
value in any consistent price system is non-positive. This principle extends
various hedging (and pricing) results available in the literature (cf. e.g. [61,
Ch. V.5, VI.1], [33, Ch. 2.1, 3.1β3.3]).
Note that unlike the classical frictionless asset pricing and hedging theory,
we do not aim to find the price of a contract β in the general model it may be
unclear what can be called the price of a contract (for example, there may be
28
no βbasicβ asset in terms of which the price can be expressed, or, due to the
possible presence of transaction costs, a portfolio may have different bid and
offer prices). However, if one chooses a particular method of computing the
price of a portfolio, then the price of a contract can be found as the infimum
over the set of the prices of the initial portfolios of trading strategies hedging
this contract. Consequently, the problem of establishing the validity of the
hedging principle can be considered to be more general than the problem of
finding contracts prices.
We conclude this section by several remarks about the properties of con-
sistent price systems.
Remark 1.1. It is useful to observe that condition (1.3) implies that
the price of any risk-acceptable family of portfolios is non-negative with
respect to any consistent price system, i. e. EπΌπ :=βπ‘βπ―
Eππ‘ππ‘ > 0 for any
consistent price system π = (ππ‘)π‘βπ― and any πΌ = (ππ‘)π‘βπ― β π. Indeed,
suppose the contrary: EπΌπ < 0 for some π and πΌ. Consider any πΌβ² β π+.
Then ππΌ+ πΌβ² β π+ for any real π > 0, while E(ππΌ+ πΌβ²)π < 0 for all π large
enough, which contradicts condition (1.3).
Remark 1.2. The interpretation of a consistent price system (ππ‘)π‘βπ― as a
system of prices is justified if each ππ‘ is strictly positive (ππ‘ > 0, Pβπ-a.s.). Asimple sufficient condition for that is when the cone π contains all sequences
(ππ‘)π‘βπ― of non-negative functions ππ‘ and does not contain any sequence of
non-positive ππ‘ except the zero one. Another mild condition is provided by
the following proposition.
Proposition 1.1. Each ππ‘ is strictly positive if the following two condi-
tions hold:
(a) for any π‘ β π―β and 0 6 π₯π‘ β πΏ1π‘ , P(π₯π‘ = 0) > 0, there exists 0 6 π¦π‘ β
πΏ1π―π‘+, P(π¦π‘ = 0) > 0, such that (π₯π‘, π¦π‘) β π΅π‘;
(b) for any π‘ β π― β π―β and 0 6 ππ‘ β πΏ1π‘ , P(ππ‘ = 0) > 0, we have (πβ²π’)π’βπ― β
π+ if πβ²π‘ = ππ‘ and πβ²π’ = 0 for π’ = π‘.
In other words, (a) means that it is possible to distribute a non-negative
non-zero portfolio π₯π‘ into non-negative portfolios π¦π‘,π’ at least one of which
29
is non-zero; and (b) means that a sequence of portfolios with only one non-
zero portfolio ππ‘, where π‘ is a sink node, is strictly risk-acceptable if ππ‘ is
non-negative.
Proof. Denote by π― π the set {π‘ β π― : π (π‘) = πΎ β π}, where πΎ is the
maximal length of a directed path in the graph and π (π‘) is the maximal
length of a directed path emanating from a node π‘. The sets π― 0, π― 1, . . . , π― πΎ
form a partition of π― such that if there is a path from π‘ β π― π to π’ β π― π,
then π < π. Also, note that π― β = π― 0 βͺ π― 1 βͺ . . . βͺ π― πΎβ1.
The proposition is proved by induction over π = πΎ,πΎβ1, . . . , 0. For any
π‘ β π― πΎ and ππ‘ as in (b), from (1.3) we have Eππ‘ππ‘ > 0, so ππ‘ > 0. Suppose
ππ > 0 for any π ββπ>π
π― π. Then for arbitrary π‘ β π― πβ1, according to (a), for
any 0 6 π₯π‘ β πΏ1π‘ , P(π₯π‘ = 0) > 0, we can find 0 6 π¦π‘ β πΏ1
π―π‘+ such that P(π¦π‘ =0) > 0 and (π₯π‘, π¦π‘) β π΅π‘. Inequality (1.2) implies Eπ₯π‘ππ‘ >
βπ’βπ―π‘+
Eπ¦π‘,π’ππ’ > 0,
and hence ππ‘ > 0.
Remark 1.3. Some comments on the relations between the above frame-
work and the von Neumann β Gale model of economic growth [1, 21, 25, 68]
are in order.
In the latter, elements (π₯π‘, π¦π‘) in the cones π΅π‘, π‘ = 1, 2, . . ., are interpreted
as feasible production processes, with input π₯π‘ and output π¦π‘. Coordinates of
π₯π‘, π¦π‘ represent amounts of commodities. The cones π΅π‘ are termed technology
sets. The counterparts of contracts in that context are consumption plans
(ππ‘)π‘. Sequences of π§π‘ = (π₯π‘, π¦π‘) β π΅π‘, π‘ = 1, 2, . . ., are called production plans.
The inequalities π¦π‘β1 β π₯π‘ > ππ‘, π‘ = 1, 2, . . . (analogous to the hedging condi-
tion (1.1)) mean that the production plan (π§π‘) guarantees the consumption
of ππ‘ at each date π‘. Consistent price systems are analogues of sequences of
competitive prices in the von Neumann β Gale model.
30
1.2 Conditions for the validity of the hedging principle
This section contains the formulations and the proofs of the general results
related to the model described in the previous section.
We will use the notation π₯+ = max{π₯, 0}, π₯β = βmin{π₯, 0}. We write
βa.s.β if some property holds for Pβ π-almost all (π, π). We say that a set
π β πΏ1 is closed with respect to πΏ1-bounded a.s. convergence if for any
sequence πΌπ = (πππ‘)π‘βπ― β π such that supπ βπΌπβ < β and πΌπ β πΌ a.s., we
have πΌ β π. In particular, this implies the closedness of π in πΏ1 because
from any sequence converging in πΏ1 it is possible to extract a subsequence
converging with probability one.
Theorem 1.1. The hedging principle holds if the cones π and π΅π‘, π‘ βπ―β, are closed with respect to πΏ1-bounded a.s. convergence and there exist
functions π 1π‘ β πΏβπ‘ , π 2π‘,π’ β πΏβ
π‘ , π‘ β π―β, π’ β π―π‘+, with values in [π , π ], where
π > 0, π > 1, and a constant 0 6 π < 1 such that for all π‘ β π―β, π’ β π―π‘+,(π₯π‘, π¦π‘) β π΅π‘, and (ππ)πβπ― β π, the following conditions are satisfied:
(a) Eπ₯π‘π 1π‘ > Eπ¦π‘,π’π
2π‘,π’;
(b) πEπ¦+π‘,π’π 2π‘,π’ > Eπ¦βπ‘,π’π
2π‘,π’;
(c) πEπ₯+π‘ π 1π‘ > Eπ₯βπ‘ π
1π‘ ;
(d) Eππ‘π 1π‘ > 0.
The functions π 1π‘ (π, π) and π 2π‘,π’(π, π) can be interpreted as some systems of
asset prices. Condition (a) means that in the course of trading the portfolio
value cannot increase βtoo muchβ, at least on average. In specific examples,
this assumption follows from the condition of self-financing. Conditions (b)
and (c) express a margin requirement, saying that the total short position
of any admissible portfolio should not exceed on average π times the total
long position (cf. e.g. [29]). Condition (d) states that the expectation of the
value (in terms of the price system π 1π‘ ) of any portfolio in a risk-acceptable
family is non-negative.
The proof of the theorem is based on a lemma. Below we denote by βthe set of all hedgeable contracts.
31
Lemma 1.1. For any π = (ππ‘)π‘βπ― , ππ‘ β πΏβπ‘ , such that EπΌπ > 0 for every
πΌ β π+, the following conditions are equivalent:
(i) π is a consistent price system;
(ii) EπΎπ 6 0 for any contract πΎ β β.
Proof. (i)β(ii). Suppose πΎ = (ππ‘)π‘βπ― β β and consider a trading strat-
egy (π₯π‘, π¦π‘)π‘βπ― β hedging the contract πΎ.
Then (ππ‘)π‘βπ― β π, where ππ‘ = π¦π‘β β π₯π‘ β ππ‘ (recall that π¦π‘β := 0 if π‘ is a
source and π₯π‘ := 0 if π‘ is a sink), and so the following formula is valid:
0 6βπ‘βπ―
Eππ‘ππ‘ =βπ‘βπ―
E
[ βπ’βπ―π‘β
π¦π’,π‘ππ‘ β π₯π‘ππ‘
]ββπ‘βπ―
ππ‘ππ‘
=βπ‘βπ―β
E
[ βπ’βπ―π‘+
π¦π‘,π’ππ’ β π₯π‘ππ‘
]β
βπ‘βπ―
ππ‘ππ‘ 6 ββπ‘βπ―
Eππ‘ππ‘,
where we put π¦π’,π‘ := 0 if π―π‘β = β , i. e. π‘ is a source node. In the above chain
of relations, the first inequality holds by virtue of (1.3) and Remark 1.1,
the second equality obtains by changing the order of summation (and inter-
changing βπ‘β and βπ’β),βπ‘βπ―
βπ’βπ―π‘β
Eπ¦π’,π‘ππ‘ =βπ’βπ―β
βπ‘βπ―π’+
Eπ¦π’,π‘ππ‘ =βπ‘βπ―β
βπ’βπ―π‘+
Eπ¦π‘,π’ππ’,
and the last inequality follows from (1.2). Consequently, (ii) holds.
(ii)β(i). Fix π‘ β π―β and suppose (π₯π‘, π¦π‘) β π΅π‘ for some arbitrary (π₯π‘, π¦π‘),
where π¦π‘ = (π¦π‘,π’)π’βπ―π‘+.
Consider the trading strategy π β² = (π₯β²π‘, π¦β²π‘)π‘βπ―β with π₯β²π’ = π¦β²π’ = 0 for π’ = π‘
and π₯β²π‘ = π₯π‘, π¦β²π‘ = π¦π‘. Define a contract πΎ = (ππ‘)π‘βπ― by
ππ‘ = βπ₯π‘, ππ’ = π¦π‘,π’ for π’ β π―π‘+, ππ’ = 0 for π’ /β {π‘} βͺ π―π‘+.
Then we have π¦β²π£β β π₯β²π£ β ππ£ = 0 for all π£ β π― . Indeed, for π£ = π‘, we have
π¦β²π‘β β π₯β²π£ β ππ£ = 0 β π₯π‘ + π₯π‘ = 0. If π£ = π’ β π―π‘+, then π¦β²π£β β π₯β²π£ β ππ£ =
π¦π‘,π’ β 0 β π¦π‘,π’ = 0. If π£ = π‘ and π£ /β π―π‘+, then π¦β²π£β = π₯β²π£ = ππ£ = 0.
32
Consequently, π β² hedges πΎ, and so πΎ β β. By virtue of (ii), we have
0 >βπ£βπ―
Eππ£ππ£ = βEπ₯π‘ππ‘ +βπ’βπ―π‘+
Eπ¦π‘,π’ππ’,
which implies that π is a consistent price system.
Proof of Theorem 1.1. In order to prove the theorem, we first show
that β is closed in πΏ1π― and β β© π+ = β , and then apply a version of the
Kreps-Yan theorem and its corollary (Propositions 1.4 and 1.5 in Section 1.6)
to the cones β and π.
Step 1. Let us show that there exists a constant πΆ such that for any
(π₯π‘, π¦π‘) β π΅π‘, π‘ β π―β, π’ β π―π‘+, it holds that
βπ₯π‘β 6 πΆEπ₯π‘π 1π‘ and βπ¦π‘,π’β 6 πΆβπ₯π‘β.
Define πΆ = (1 +π)/(1βπ). Then we have
πΆEπ₯π‘π 1π‘ = πΆ[ 1πΆ Β· Eπ₯+π‘ π 1π‘ +(1β 1πΆ
)Β· Eπ₯+π‘ π 1π‘ β Eπ₯βπ‘ π
1π‘
]> πΆ[ 1πΆ Β· Eπ₯+π‘ π 1π‘ +
(1
π
(1β 1πΆ
)β 1
)Β· Eπ₯βπ‘ π 1π‘
]> Eπ₯+π‘ π
1π‘ + Eπ₯βπ‘ π
1π‘ = βπ₯π‘π 1π‘β = E|π₯π‘||π 1π‘ |
> βπ₯π‘βπ ,
(1.4)
where the first inequality follows from the fact that Eπ₯+π‘ π 1π‘ β₯ πβ1Eπ₯βπ‘ π
1π‘
according to (c), the second holds because πΆ((1β 1/ πΆ)/πβ 1) = 1, and the
last is valid because π 1π‘ β₯ π .
Further, we have
π βπ¦π‘,π’β 6 πΆEπ¦π‘,π’π 2π‘,π’ 6 πΆEπ₯π‘π 1π‘ β€ πΆβπ₯π‘π 1π‘β β€ πΆβπ₯π‘βπ ,where the first inequality is proved similarly to the one for π₯π‘ (replace in the
above argument π₯π‘ by π¦π‘,π’, π 1π‘ by π 2π‘,π’ and use (b) instead of (c)), and the
second inequality follows from (a). Consequently, the sought-for constant
πΆ can be defined as
πΆ =πΆ Β· π π
.
33
Step 2. Let us prove that β is closed in πΏ1π― . Consider a sequence of
hedgeable contracts πΎπ = (πππ‘)π‘βπ― β β, π = 1, 2, . . ., such that πΎπ β πΎ in πΏ1π―
as πβ β, where πΎ = (ππ‘)π‘βπ― . We have to show πΎ β β.
Let π― π = {π‘ β π― : π (π‘) = πΎ β π} be the sets introduced in the proof of
Proposition 1.1 (i. e. πΎ denotes the maximal length of a directed path in the
graph and π (π‘) is the maximal length of a directed path emanating from π‘).
Let π π = (π₯ππ‘, π¦ππ‘)π‘βπ―β be trading strategies hedging πΎπ. We will prove the
following assertion:
supπ
βπ₯ππ‘β <β and supπ
βπ¦ππ‘,π’β <β, π’ β π―π‘+, (1.5)
for each π‘ β π―β.To this end we will prove by induction with respect to π = 0, . . . , πΎ β 1
that (1.5) is valid for all π‘ β π― π. We first note that if supπ βπ¦ππ‘ββ < βfor some node π‘ of the graph, then (1.5) is true for this node. Indeed, put
πππ‘ = π¦ππ‘β β πππ‘ β π₯ππ‘. Then (πππ‘)π‘βπ― β π because π π hedges πΎπ, and so Eπππ‘π 1π‘ > 0
by virtue of (d). Therefore
βπ₯ππ‘β 6 πΆEπ₯ππ‘π 1π‘ = πΆE(π¦ππ‘β β πππ‘)π
1π‘ β πΆEπππ‘π
1π‘ 6 πΆπ Β· (βπ¦ππ‘ββ+ βπππ‘β),
and so supπ βπ₯ππ‘β <β and supπ βπ¦ππ‘,π’β 6 πΆ supπ βπ₯ππ‘β <β.
Having this in mind, we proceed by induction. For π‘ β π― 0 we have π¦ππ‘β = 0
since any π‘ β π― 0 is a source. Thus (1.5) is valid for all π‘ β π― 0. Suppose we
have established (1.5) for all π‘ in each of the sets π― 0, π― 1, . . . , π― π. Consider
any π‘ β π― π+1, where π + 1 < πΎ. We have supπ βπ¦ππ‘ββ < β because all the
predecessors of the node π‘ belong to one of the sets π― 0, π― 1, . . . , π― π. This
implies, as we have demonstrated, the validity of (1.5) for the node π‘. Thus
(1.5) holds for all π‘ β π―β.Step 3. By the Komlos theorem (Proposition 1.6), there exists a subse-
quence π π1, π π2, . . . Cesaro-convergent a.s. to some π = (π₯π‘, π¦π‘)π‘βπ―β, i.e. for
each π‘ β π―β, π’ β π―π‘+ we have π₯ππ‘ := πβ1(π₯π1π‘ + . . . + π₯πππ‘ ) β π₯π‘ β πΏ1
π‘ a.s. andπ¦ππ‘,π’ := πβ1(π¦π1π‘,π’ + . . .+ π¦πππ‘,π’) β π¦π‘,π’ β πΏ1
π’ a.s. Then π is trading strategy since
π΅π‘ are closed with respect to πΏ1-bounded a.s. convergence.
Moreover, π hedges πΎ because πππ‘ := π¦ππ‘ββπ₯ππ‘ βπππ‘ (with πππ‘ = πβ1(ππ11 + . . .+
34
ππππ‘ )) converge a.s. to ππ‘ = π¦π‘β β π₯π‘ β ππ‘, and supπ βπππ‘β < β for each π‘ β π― ,
which implies (ππ‘)π‘βπ― β π since π is closed with respect to πΏ1-bounded a.s.
convergence. Thus β is a closed cone in πΏ1π― .
Step 4. Let us show that
β β©π+ = β ,
which can be interpreted as the absence of arbitrage opportunities.
Suppose there exists πΎ = (ππ‘)π‘βπ― β β β© π. Consider a trading strategy
π = (π₯π‘, π¦π‘)π‘βπ―β hedging the contract πΎ. We claim that in this case π₯π‘ =
π¦π‘,π’ = 0 for all π‘ β π―β and π’ β π―π‘+.Indeed, proceeding by induction over π― 0, . . . , π― πΎβ1, suppose π¦π‘β = 0 for
each π‘ β π― π. By virtue of Step 1,
βπ₯π‘β 6 πΆEπ₯π‘π 1π‘ 6 πΆE(π¦π‘β β ππ‘)π
1π‘ = βπΆEππ‘π 1π‘ β€ 0
for any π‘ β π― π. Hence βπ₯π‘β = 0 for each π‘ β π― π. Furthermore, βπ¦π‘,π’β 6
πΆβπ₯π‘β = 0 for each π’ β π―π‘+, and so π¦π‘β = 0 for each π‘ β π― π+1. Therefore,
π₯π‘ = π¦π‘,π’ = 0, which implies πΎ β (βπ) by the definition of hedging, meaning
that β β©π+ = β .Step 5. Applying Proposition 1.4 to the cones β and π in the space πΏ1
π― ,
we obtain the existence of a family π = (ππ‘)π‘βπ― , ππ‘ β πΏβπ‘ , such that EπΌπ β€ 0
for any πΌ β β and EπΎπ > 0 for any πΎ β π+. According to Lemma 1.1, π is
a consistent price system, so the class of such price systems is non-empty.
Step 6. If a contract πΎ is hedgeable, we have EπΎπ 6 0 for any consistent
price system π according to the implication (i)β(ii) in Lemma 1.1.
To prove the converse, suppose EπΎπ 6 0 for some contract πΎ and any
consistent price system π. Observe that any π = (ππ‘)π‘βπ― , ππ‘ β πΏβπ‘ , such
that EπΌπ > 0 for any πΌ β π+ and EπΎβ²π 6 0 for any πΎβ² β β is a consistent
price system according to the implication (ii)β(i) in Lemma 1.1. Therefore,
EπΎπ β€ 0 for any such π and the given πΎ. By virtue of Proposition 1.5, this
implies πΎ β β.
The next result provides a version of Theorem 3.1 with other assumptions.
35
Theorem 1.2. The hedging principle holds if the cones π and π΅π‘, π‘ β π―β,are closed in πΏ1, for any (ππ‘)π‘βπ― β π we have ππ’ = 0, π’ β π―β, and there
exist functions π 1π‘ β πΏβπ‘ , π 2π‘,π’ β πΏβ
π‘ , π‘ β π―β, π’ β π―π‘+, with values in [π , π ]
(where π > 0, π > 1), and a constant 0 6 π < 1 satisfying conditions
(aβ²) π₯π‘π 1π‘ > π¦π‘,π’π
2π‘,π’ a.s.;
(bβ²) ππ¦+π‘,π’π 2π‘,π’ > π¦βπ‘,π’π
2π‘,π’ a.s.
Remark 1.4. The condition ππ’ = 0, π’ β π―β, for any (ππ‘)π‘βπ― β π, means
that a contract should be hedged exactly at all the intermediate trading
sessions and it should be hedged with risk (in a risk-acceptable manner) at
all the terminal trading sessions β those trading sessions that are represented
by sink nodes of the graph.
Proof of Theorem 1.2. The proof is conducted along the lines of the
proof of Theorem 1.1. First we show that |π¦π‘,π’| 6 πΆ|π₯π‘| a.s. for all π‘ β π―β,π’ β π―π‘+, where the constant πΆ = πΆ Β· (π /π ) with πΆ = (1 +π)/(1βπ).
Indeed, with probability one it holds that
π |π¦π‘,π’| 6 πΆπ¦π‘,π’π 2π‘,π’ 6 πΆπ₯π‘π 1π‘ 6 πΆ|π₯π‘π 1π‘ | 6 πΆ|π₯π‘|π ,where the first inequality is proved similarly to (1.4), and the second inequal-
ity follows from (aβ²).
In order to prove the closedness of β, consider any πΎπ = (πππ‘)π‘βπ― β βconverging in πΏ1
π― to πΎ = (ππ‘)π‘βπ― . For trading strategies π π = (π₯ππ‘, π¦ππ‘)π‘βπ―β
hedging πΎπ, using the induction over π― π, we prove that supπ βπ₯ππ‘β < β,
supπ βπ¦ππ‘,π’β <β and the sequences π₯ππ‘, π¦ππ‘,π’ are uniformly integrable.
Indeed, if supπ βπ¦ππ‘ββ < β and π¦ππ‘β is uniformly integrable for any π‘ β π― π
then π₯ππ‘ = π¦ππ‘β β πππ‘ is uniformly integrable and supπ βπ₯ππ‘β <β. Consequently,
supπ βπ¦ππ‘ββ < β and π¦ππ‘β is uniformly integrable for any π‘ β π― π+1, because
|π¦ππ‘,π’| 6 πΆ|π₯ππ‘| a.s.By using the Komlos Theorem, we find a subsequence π ππ Cesaro-conver-
gent a.s. to some π = (π₯π‘, π¦π‘)π‘βπ―β, and hence convergent in πΏ1 because π₯ππ‘ and
π¦ππ‘,π’ are uniformly integrable. Since π΅π‘ are closed in πΏπ‘, π is a trading strategy.
It hedges πΎ because πππ‘ := π¦ππ‘ββ π₯ππ‘ βπππ‘ converge in πΏ1 to ππ‘ = π¦π‘ββπ₯π‘β ππ‘ as
36
they are uniformly integrable and a.s. convergent, and the cone π is closed
in πΏ1π― . Thus β is closed in πΏ1
π― .
Furthermore, β β© π+ = β because if there exists πΎ = (ππ‘)π‘βπ― β β β© πthen for a trading strategy (π₯π‘, π¦π‘)π‘βπ―β hedging πΎ, we have π₯π‘ = π¦π‘,π’ = 0
for all π‘ β π―β, π’ β π― +π‘ (this is proved by induction over π― π using that
π₯π‘ = π¦π‘β β ππ‘, ππ‘ = 0 for each π‘ β π―β, and |π¦π‘,π’| 6 πΆ|π₯π‘| a.s.). This implies
πΎ β (βπ) by the definition of hedging. To complete the proof it remains to
apply Propositions 1.4 and 1.5 to the cones β and π.
Remark 1.5 (on the no-arbitrage hypothesis). The central role in the
classical theory of asset pricing and hedging is played by the no-arbitrage
hypothesis, which postulates that arbitrage opportunities do not exist. This
assumption is rather natural since it is thought that in real markets arbi-
trage opportunities are quickly eliminated by market forces. Perhaps, it is
even more important from the point of view of constructing mathematical
models of financial markets, as the absence of arbitrage is equivalent to the
existence of equivalent martingale measures (in the frictionless framework),
which allows to apply the risk-neutral principle for pricing contingent claims.
In the frictionless model, the validity of the no-arbitrage hypothesis de-
pends on the probabilistic structure of the random process (or sequence)
which describes the evolution of asset prices. Consequently, processes that
allow arbitrage opportunities are usually considered as inadequate models of
a market.
However, as follows from the results of this chapter, the presence of arbi-
trage opportunities can also be caused by inadequate trading constraints in a
model. Indeed, the classical frictionless approach allows unlimited short sales
and borrowings from the bank account, which is certainly impossible in real
trading. On the other hand, the introduction of margin requirements elimi-
nates arbitrage opportunities under the conditions of Theorems 1.1 and 1.3
(see Step 4 in the proof of Theorem 1.1) and implies the validity of the
hedging principle.
This fact has important implications for modelling financial markets since
the introduction of margin requirements allows to consider price processes
37
that describe important market features, but allow arbitrage.
One example of such a process is fractional geometric Brownian motion.
Unlike the standard geometric Brownian motion, it exhibits long-range de-
pendence β a property typically observed in real asset prices. However, it
is well-known that this process admits arbitrage opportunities in the fric-
tionless model [51, 60]. Moreover, even its approximation by binary random
walks constructed on a finite probability space is not arbitrage-free [67].
Thus, consideration of market models with margin requirements can pos-
sibly allow to use a wider class of stochastic processes describing asset prices
and broaden the frontiers of the theory.
1.3 Risk-acceptable portfolios: examples
In this section we provide examples of cones π of risk-acceptable families
of portfolios defined in terms of their liquidation values.
Suppose for each π‘ β π― there is an operator ππ‘ : πΏ1π‘ (Ξ© Γ Ξ) β πΏ1(Ξ©),
where ππ‘(ππ‘)(π) is interpreted as the liquidation value of the portfolio ππ‘ in
the trading session π‘, and a closed cone π΄π‘ β πΏ1π‘ (Ξ©) interpreted as a cone of
risk-acceptable liquidation values.
Define
π = {(ππ‘)π‘βπ― β πΏ1π― : ππ‘(ππ‘) β π΄π‘ for all π‘ β π― }.
According to this definition, a family of portfolios is acceptable if the liqui-
dation value of each portfolio is acceptable. In order to guarantee that π is a
closed cone in πΏ1, it is sufficient to assume that for each π‘ β π― the following
conditions are satisfied:
(i) ππ‘(πππ‘) = πππ‘(ππ‘), ππ‘(ππ‘ + πβ²π‘) > ππ‘(ππ‘) + ππ‘(πβ²π‘) for any real π > 0 and
ππ‘, πβ²π‘ β πΏ1
π‘ ,
(ii) the cone π΄π‘ contains all non-negative Fπ‘-measurable integrable random
variables,
(iii) there exists a constant π such that βππ‘(ππ‘) β ππ‘(πβ²π‘)β 6 πβππ‘ β πβ²π‘β for
any ππ‘, πβ²π‘ β πΏ1
π‘ .
38
The coneπ is closed with respect to πΏ1π― (Ξ©ΓΞ)-bounded a.s. convergence,
if, additionally, the following two conditions hold:
(iv) ππ‘(πππ‘ ) β ππ‘(ππ‘) a.s. whenever π
ππ‘ β ππ‘ a.s., πβ β, and πππ‘ , ππ‘ β πΏ1
π‘ ;
(v) the cone π΄π‘ is closed with respect to πΏ1π‘ (Ξ©)-bounded a.s. convergence.
A natural example of a liquidation value is
ππ‘(ππ‘)(π) =
β«Ξ
ππ‘(π, π)ππ‘(π, π)π(ππ),
where the function ππ‘ β πΏβπ‘ (Ξ© Γ Ξ) represents asset prices at the trading
session π‘. In other words, the liquidation value of a portfolio is equal to its
value in terms of the prices ππ‘.
More generally, one can define
ππ‘(ππ‘)(π) =
β«Ξ
[π+π‘ (π, π)ππ‘(π, π)β πβπ‘ (π, π)ππ‘(π, π)
]π(ππ),
where ππ‘, ππ‘ β πΏβπ‘ (Ξ© Γ Ξ), ππ‘ 6 ππ‘, are bid and ask asset prices. It is
easy to see that this liquidation value operator satisfies above conditions (i)
and (iii). If the asset space (Ξ,J , π) is finite, it also satisfies (iv).
We provide three examples of cones π΄π‘ of risk-acceptable liquidation val-
ues and show that they are closed with respect to πΏ1(Ξ©)-bounded a.s. con-
vergence or in πΏ1, and Eπ£ > 0 for each π£ β π΄π‘ β {0}, i.e. any non-zero
risk-acceptable liquidation value is strictly positive on average.
1. Superhedging. Define
π΄1 = {π£ β πΏ1 : π£ β₯ 0 a.s.},
i. e. each risk-acceptable value is non-negative with probability one, which
is the classical approach to hedging (see e. g. [24, 61]). Clearly, π΄1 is closed
under πΏ1-bounded a.s.-convegence and Eπ£ > 0 for any π£ β π΄1 β {0}.
Superhedging is a comparatively strong assumption, which requires to
fulfil contract obligations with probability one. In the next two examples we
deal with weaker approaches to hedging: the cones π΄2 and π΄3 introduced
below are larger than the cone π΄1.
39
2. Acceptable Sharpe ratio. The Sharpe ratio of a non-constant
random variable π£ β πΏ2 is defined by Eπ£/βVar π£. For a given number π > 0
define the cone
π΄2 = {π£ β πΏ1 : βπ’ β πΏ2, π’ 6 π£, Eπ’ > πβVar π’}.
In other words, a liquidation value is acceptable if it exceeds a random vari-
able with the Sharpe ratio not less than π.
Clearly Eπ£ > 0 for any π£ β π΄2 β {0}. Let us show that π΄2 is closed with
respect to πΏ1-bounded a.s. convergence. Suppose π£π β π΄2, E|π£π| 6 πΌ < βand π£π β π£ a.s. Take π’π β πΏ2 such that π’π 6 π£π, Eπ’π > π
βVar π’π. Then
πβVar π’π 6 πΌ and the sequence π’π is bounded in the πΏ2-norm. Since the
ball in πΏ2 is weakly compact, we can find a subsequence π’ππ β π’ weakly in
πΏ2. Then we obtain
Eπ’ = limπββ
Eπ’ππ > π lim infπββ
βVar π’ππ > π
βVar π’,
where the last inequality follows from the weak lower semi-continuity of
the norm. Using the Komlos theorem, we find a subsequence π’πππ such
that π’π β‘ πβ1(π’ππ1 + . . . + π’πππ ) β π’ a.s. Then π£ > π’ and π’ = π’, since
EIΞπ’ = limπ EIΞπ’π = EIΞπ’ for each measurable set Ξ, where the former is
valid because π’ππ β π’ weakly and the latter is true because the sequence π’πis uniformly integrable (bounded in πΏ2) and converges to π’ a.s. Thus π£ β π΄2.
3. Acceptable average value at risk. The average value at risk at a
level π of a random variable π£ β πΏ1 is defined by the formula
AV@Rπ(π£) = supπβππ
E(βππ£),
where ππ is the set of all random variables 0 6 π 6 1/π such that Eπ = 1.
In other words, AV@Rπ(π£) is the maximal expected value of (βπ£) under
all probability measures absolutely continuous with respect to the original
measure such that the density does not exceed 1/π (see [24, Ch. 4] for details).
Consider the cone
π΄3 = {π£ β πΏ1 : AV@Rπ(π£) 6 0},
40
where π β (0, 1) is a given number. According to this definition of π΄3, the
average value at risk at the level π of each acceptable liquidation value is
non-positive (i. e. such a liquidation value is not risky in terms of AV@Rπ).
To show that Eπ£ > 0 for any π£ β π΄3 we use the representation
AV@Rπ(π£) = ββ« 1
0
ππ£(ππ )ππ ,
where ππ£(π ) = inf{π : P(π£ β€ π) > π } is the quantile function of π£ (see [24,
Ch. 4]). If π£ is non-constant, we have
AV@Rπ(π£) > ββ« 1
0
ππ£(π )ππ = βEπ£,
and so Eπ£ > 0 for a non-constant π£ β π΄3 β {0}. But if π£ β π΄3 β {0} is
constant, then necessarily π£ > 0, and so Eπ£ > 0.
Finally observe that the cone π΄3 is closed in πΏ1. Indeed, if π£π β π΄3 and
π£π β π£ in πΏ1, then E(βππ£) = limπ E(βππ£π) 6 0 for any random variable
0 6 π 6 1/π such that Eπ = 1, consequently, AV@Rπ(π£) β€ 0.
1.4 Connections between consistent price systems and
equivalent martingale measures
The notion of a consistent price system generalises the notion of an equiv-
alent martingale measure β the cornerstone of the classical stochastic finance.
To see this, consider the model of a frictionless market of π assets rep-
resented by a linear graph πΊ with nodes π‘ = 0, 1, . . . , π , where asset 1 is a
riskless asset with the discounted price π1π‘ β‘ 1 at each trading session π‘, and
assets π = 2, . . . , π are risky with the discounted prices πππ‘ > 0 being Fπ‘-
measurable random variables (the space of assets (Ξ,J , π) here is simply
Ξ = {1, 2, . . . , π}, J = 2Ξ, π{π} = 1 for each π β Ξ).
It is assumed that the π-algebras Fπ‘ form a filtration, i. e. Fπ‘ β Fπ‘+1 for
each π‘ = 0, . . . , π β 1, and each Fπ‘ is completed by all F -measurable sets of
measure 0.
41
Denoting ππ‘ = (π1π‘ , . . . , π
ππ‘ ), introduce the cones
π΅π‘ =
{(π₯π‘, π¦π‘) β πΏ1
π‘ β πΏ1π‘ :
πβπ=1
π₯ππ‘πππ‘ >
πβπ=1
π¦ππ‘πππ‘ a.s.
},
π =
{(0, . . . , 0, ππ ) : ππ β πΏ1
π ,
πβπ=1
ππππππ > 0 a.s.
}.
The cones π΅π‘ define self-financing trading strategies (with free disposal),
i. e. the values of their portfolios do not increase in the course of trading.
According to the definition of π, a contract (π0, . . . , ππ ) is hedgeable, if there
exists a trading strategy that pays exactly ππ‘ at each trading session π‘ < π
and pays not less than ππ at the session π (the superhedging approach).
Since it is assumed that Fπ‘ form a filtration and are complete, we have
πΏπ‘ β πΏπ‘+1 for each π‘ = 0, . . . , π β 1, so this model is a particular case of the
general model (in the general model π¦π‘ β‘ π¦π‘,π‘+1 β πΏπ‘+1).
Recall that a probability measure P defined on (Ξ©,F ) and equivalent to
the original probability measure P (P βΌ P) is called an equivalent martingale
measure if the sequence π0, π1, . . . , ππ is a P-martingale, which means that
EP(ππ
π‘ | Fπ‘β1) = πππ‘β1 a.s. for all π‘ = 1, . . . , π , π = 1, . . . , π .
Consistent price systems and equivalent martingale measures in the model
at hand are connected by the following properties.
Proposition 1.2. 1) If P is an equivalent martingale measure, then the
sequence of ππ‘ = ππ‘ππ‘, where ππ‘ = E(πP/πP | Fπ‘), is a consistent price
system, provided that ππ‘ β πΏβπ‘ .
2) If (π0, . . . , ππ‘) is a consistent price system, then the measure P defined
by πP = (π1π/Eπ1π )πP is an equivalent martingale measure.
Proof. To keep the notation concise, let us denote by π₯π‘ππ‘ and π¦π‘ππ‘ the
corresponding scalar products of the vectors π₯π‘, π¦π‘, ππ‘.
1) If (π₯π‘, π¦π‘) β π΅π‘ then π₯π‘ππ‘ > π¦π‘ππ‘ and consequently EPππ‘π₯π‘ > EPππ‘π¦π‘.
Also EPππ‘π¦π‘ = EPππ‘+1π¦π‘, because the sequence ππ‘ is a P-martingale as follows
from the formula (see e. g. [63, Chapter II, S 7])
EP(ππ‘ππ‘ |Fπ‘β1) = ππ‘β1EP(ππ‘ |Fπ‘β1). (1.6)
42
Thus EPππ‘π₯π‘ > EPππ‘+1π¦π‘. It is also clear that Eππππ > 0 for any ππ such
that ππππ > 0 and P(ππππ = 0) > 0, so (π0, . . . , ππ ) satisfies the definition
of a consistent price system.
2) Observe that the following relations hold:
E(ππ‘+1 | Fπ‘) = ππ‘ for π‘ = 0, . . . , π β 1, ππ‘ = π1π‘ππ‘ for π‘ = 0 . . . , π.
The first one follows from (1.2) with (π₯π‘, π₯π‘) β π΅π‘ for π₯π‘ = (0, . . . ,Β±IΞ, . . . , 0)
and an arbitrary Ξ β Fπ‘. The second one for π‘ = 0, . . . , π β 1 follows from
(1.2) taking (π₯π‘, 0) β π΅π‘, π₯π‘ = (Β±πππ‘IΞ, 0, . . . ,βIΞ, . . . , 0) with an arbitrary
Ξ β Fπ‘, and for π‘ = π it follows from (1.3) taking π₯π β πΏ1π of the same form
such that (0, . . . , π₯π ) β π. Using (1.6) with ππ‘ = π1π‘ , we obtain that ππ‘ is aP-martingale, so P is an equivalent martingale measure.
1.5 A model of an asset market with transaction costs and
portfolio constraints
We consider a market where π assets are traded at trading sessions π‘ β π― .
Assets π = 2, . . . , π represent stock and asset π = 1 cash (deposited with a
bank account). The model we deal with in this section is a special case of
the general one where Ξ = {1, 2, . . . , π} and π(π) = 1 for each π β Ξ. We
assume that the π-algebras Fπ‘ associated with the nodes of the graph are
such that Fπ‘ β Fπ’ for any π‘ β π―β and π’ β π―π‘+ (i. e. the βinformationβ does
not decrease along the paths in the graph), and each Fπ‘, π‘ β π― , is completed
by all F -measurable sets of measure 0.
For each trading session π‘ β π― , we are given Fπ‘-measurable essentially
bounded non-negative random variables πππ‘ 6 π
π
π‘, π = 2, . . . , π , representing
bid and ask stock prices, and π·ππ‘ β€ π·
π
π‘, π = 2, . . . , π , representing dividend
rates for long and short stock positions1. The bid and ask price vectors are
1It is assumed that a short seller pays the dividends on the amount of stock sold short to thelender of stock. Dividend rates for long and short positions may be different β for example, whensome assets pay dividends in a currency different from asset 1 and there is a bid-ask spread inthe exchange rates.
43
denoted by ππ‘ = (1, π2π‘ , . . . , π
ππ‘ ) and ππ‘ = (1, π
2
π‘ , . . . , ππ
π‘ ). We assume each
πππ‘ is bounded away from zero (i.e. ππ
π‘ β₯ π for a constant π > 0).
For each π‘ β π―β, Fπ‘-measurable essentially bounded non-negative random
variables π π‘,π’ 6 π π‘,π’, π’ β π―π‘+, are given, which describe interest rates for
lending and borrowing cash between the sessions π‘ and π’.
We write π₯π‘(π) = (π₯1π‘ (π), . . . , π₯ππ‘ (π)) for portfolios of assets represented
by functions π₯π‘(π, π) in πΏ1π‘ (Ξ©ΓΞ), where π₯ππ‘(π) = π₯π‘(π, π). We define |π₯π‘| =β
π |π₯ππ‘|, π₯+π‘ = ((π₯1π‘ )
+, . . . , (π₯ππ‘ )+) and π₯βπ‘ = ((π₯1)β, . . . , (π₯ππ‘ )
β). If π₯, π¦ are
π -dimensional vectors, we denote by π₯π¦ the scalar product π₯π¦ =β
π π₯ππ¦π.
For a trading strategy (π₯π‘, π¦π‘)π‘βπ―β, the dividends
ππ‘(π₯π‘) =πβπ=2
[(π₯ππ‘)
+π·ππ‘ β (π₯ππ‘)
βπ·π
π‘
]are received on each portfolio π₯π‘ at each session π‘ β π―β. Negative values
of ππ‘(π₯π‘) mean that the corresponding amounts should be returned rather
than received. After the dividend payments, the portfolio π₯π‘ (with the div-
idends added) is rearranged by buying and selling assets, subject to the
self-financing trading constraint, to a family of portfolios π¦π‘ = (π¦π‘,π’)π’βπ―π‘+,where π¦π‘,π’ β πΏ1
π‘ . The feasible pairs (π₯π‘, π¦π‘) form the cone
π΅π‘ =
{(π₯π‘, π¦π‘) : π₯π‘ β πΏ1
π‘ , π¦π‘,π’ β πΏ1π‘ and(
π₯π‘ ββπ’βπ―π‘+
π¦π‘,π’)+
Β· ππ‘ + ππ‘(π₯π‘) >
(π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’)β
Β· ππ‘ a.s.
}.
The above inequality represents the self-financing condition: its left-hand
side is equal to the amount of cash obtained from selling assets and receiving
dividends, and its right-hand side is the amount of cash paid for buying
assets. We denote by π¦π‘ = π¦π‘(π¦π‘) the family of portfolios (π¦π‘,π’)π’βπ―π‘+ whose
positions are given by
π¦1π‘,π’ = π¦1π‘,π’ + ππ‘,π’(π¦π‘,π’) and π¦ππ‘,π’ = π¦ππ‘,π’, π = 2, . . . , π,
where
ππ‘,π’(π¦π‘,π’) = (π¦1π‘,π’)+π π‘,π’ β (π¦1π‘,π’)βπ π‘,π’
44
is the interest paid on cash held between the trading sessions π‘ and π’.
Random Fπ‘-measurable closed cones2 ππ‘,π’ : Ξ© β 2Rπ
, π‘ β π―β, π’ β π―π‘+,specifying portfolio constraints are given. It is assumed that each cone ππ‘,π’(π)
contains all non-negative vectors in Rπ . Trading strategies are defined in
terms of the cones (π‘ β π―β)
π΅π‘ = {(π₯π‘, π¦π‘) : π¦π‘ = π¦π‘(π¦π‘), (π₯π‘, π¦π‘) β π΅π‘, π¦π‘,π’ β ππ‘,π’ a.s.}. (1.7)
Portfolios π¦π‘,π’ are obtained from π¦π‘,π’ by adding interest to the 1st (bank
account) position. Only those portfolios π¦π‘,π’ are admissible that satisfy the
constraint π¦π‘,π’ β ππ‘,π’ a.s.
Note that we assume π¦π‘ are Fπ‘-measurable, which can be explained by
that βthe portfolio for tomorrow should be obtained todayβ. Since the π-
algebras Fπ‘ are such that Fπ‘ β Fπ’ whenever π’ β π―π‘+ and are completed
by all sets of measure 0, we have πΏ1π‘ β πΏ1
π’ for π’ β π―π‘+, and so the model at
hand can be included into the framework described in Section 1.1 (where,
generally, π¦π‘,π’ β πΏ1π’).
Also note that the class of trading strategies defined by the cones π΅π‘ in
(1.7) excludes the possibility of bankruptcy, i. e. a trading strategy cannot in-
terrupt at an intermediate node of the graph. Nevertheless, the model allows
the modeling of the possibility that a trader does not fulfil the obligations
of a contract through the notion of hedging with risk.
We consider cones of risk-acceptable families of portfolios π given by
π =β¨π‘βπ―
ππ‘, where for each π‘ β π―
ππ‘ = {0} or ππ‘ ={ππ‘ β πΏ1
π‘ : π+π‘ ππ‘ β πβπ‘ ππ‘ + ππ‘(ππ‘) β π΄π‘
}(1.8)
with some given cones π΄π‘ β πΏ1π‘ (Ξ©) interpreted as cones of risk-acceptable
liquidation values (see Section 1.3). It is assumed that ππ‘ = {0} for all sink
nodes π‘ /β π―β, and for each π‘ β π― , where ππ‘ = {0}, the cone π΄π‘ contains all
non-negative integrable random variables and Eπ£ > 0 for any π£ β π΄π‘ β {0}.2A random Fπ‘-measurable closed cone in Rπ is a mapping π : Ξ© β 2R
π
such that π (π) is aclosed cone in Rπ for each π β Ξ©, and {π : π (π) β©πΎ = β } β Fπ‘ for any open set πΎ β Rπ . SeeAppendix 1 for details.
45
Let us prove an auxiliary proposition regarding the structure of the cones
ππ‘ that will be used below.
Proposition 1.3. Suppose ππ‘ = {0} is a cone of form (1.8) and ππ‘ β πΏβπ‘
is a random vector such that
(a) π1π‘ > 0,
(b) πππ‘ +π·π
π‘ 6 πππ‘/π1π‘ 6 π
π
π‘ +π·π
π‘ a.s. and πππ‘ +π·π
π‘ < πππ‘/π1π‘ < π
π
π‘ +π·π
π‘ a.s. on
the set {π : πππ‘ +π·π
π‘ < ππ
π‘ +π·π
π‘}, π = 2, . . . , π ,
(c) Eπ£π‘π1π‘ > 0 for any π£π‘ β π΄π‘ β {0}.
Then Eππ‘ππ‘ β₯ 0 for each ππ‘ β ππ‘ and Eππ‘ππ‘ > 0 for each ππ‘ β π+π‘ .
Proof. For any ππ‘ β ππ‘ = {0} we have ππ‘ππ‘/π1π‘ > π+π‘ ππ‘ β πβπ‘ ππ‘ + ππ‘(ππ‘),
consequently, ππ‘ππ‘/π1π‘ β π΄π‘, and so Eππ‘ππ‘ = E(ππ‘ππ‘/π
1π‘ )π
1π‘ > 0.
Suppose Eππ‘ππ‘ = 0. Then ππ‘ππ‘/π1π‘ = π+π‘ ππ‘ β πβπ‘ ππ‘ + ππ‘(ππ‘) = 0, which
implies that πππ‘ = 0 a.s. on the set {π : πππ‘ + π·π
π‘ < ππ
π‘ + π·π
π‘}, π = 2, . . . , π .
Then in this case (βππ‘) β ππ‘, and so Eππ‘ππ‘ > 0 for any ππ‘ β π+π‘ .
The above proposition guarantees that the cone π in the model at hand
satisfies assumption (A) with π = (ππ‘)π‘β€π ,
π1π‘ = 1, πππ‘ = (πππ‘ +π·π
π‘ + ππ
π‘ +π·π
π‘)/2 for π = 2, . . . π.
Let π be a constant such that π β€ (1 + π π‘,π’)/(1 + π π‘,π’) a.s. for all
π‘ β π―β, π’ β π―π‘+. We say that the market model at hand satisfies a margin
requirement if there exists a constant π such that 0 6 π < π and for
almost each π β Ξ© we have (cf. conditions (b) and (c) in Theorem 1.1)
ππ¦+π‘,π’ππ‘(π) > π¦βπ‘,π’ππ‘(π) for any π¦π‘,π’ β ππ‘,π’(π), π‘ β π―β, π’ β π―π‘+, (1.9)
where
π¦1π‘,π’ =(π¦1π‘,π’)
+
1 +π π‘,π’(π)β
(π¦1π‘,π’)β
1 +π π‘,π’(π), π¦ππ‘,π’ = π¦ππ‘,π’ for π = 2, . . . π.
Inequality (1.9) requires that the total short position of any portfolioπ¦π‘,π’ does not exceed on average π times the total long position, and can
be interpreted as a weak form of a margin requirement. Note that margin
46
requirements in real markets have a more complex structure, and, in partic-
ular, differentiate initial margin (the amount that should be collateralized
in order to open a position) and maintenance margin (the amount required
to be kept in collateral until the position is closed). Such additional limi-
tations typically imply (1.9) with some constant π and can be modeled by
appropriate cones ππ‘,π’.
The following theorem establishes the validity of the hedging principle for
a stock market with a margin requirement.
Theorem 1.3. The hedging principle is valid for a stock market with a
margin requirement if at least one of the following conditions is satisfied:
(i) for each π‘ β π― such that ππ‘ = {0}, the cone π΄π‘ is closed with respect to
πΏ1π‘ (Ξ©)-bounded a.s. convergence, and ππ
π‘ + π < ππ
π‘ a.s. for some π > 0
and all π‘ β π―β, π = 2, . . . , π ;
(ii) ππ‘ = {0} for all π‘ β π―β.
Proof. In order to prove the theorem we will apply Theorem 1.1 if (i)
holds and Theorem 1.2 if (ii) holds. Observe that the cones ππ‘ are closed
with respect to πΏ1-bounded a.s. convergence. The cone π is closed with
respect to πΏ1-bounded a.s. convergence if (i) holds and it is closed in πΏ1 if
(ii) holds.
Define random vectors π 1π‘ , π 2π‘,π’ β πΏβ
π‘ by
π 11π‘ = 1, π 21π‘,π’ =1
1 +π π‘,π’
, π 1ππ‘ =πππ‘ + π
π
π‘
2+π·
π
π‘, π 2ππ‘,π’ =πππ‘ + π
π
π‘
2for π > 2.
Then for any (π₯π‘, π¦π‘) β π΅π‘ we have
0 6
(π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’)+
Β· ππ‘ + ππ‘(π₯π‘)β(π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’)β
Β· ππ‘
6 π₯π‘π 1π‘ β
βπ’βπ―π‘+
π¦π‘,π’π 2π‘,π’.
From the margin requirement it follows that π¦π‘,π’π 2π‘,π’ > 0 for each π‘ β π―β,
π’ β π―π‘+, which implies that condition (a) in Theorem 1.1 and condition (aβ²)
in Theorem 1.2 are fulfilled.
47
Suppose (i) holds. To verify conditions (b) and (c) in Theorem 3.1, take
0 6 π < 1 such that π >π/π and ππ/2 > (1βπ)maxπ‘,π(ππ
π‘ +π·π
π‘) a.s. It
is straightforward to check that ππ¦+π‘,π’π 2π‘,π’ >ππ¦+π‘,π’ππ‘ and π¦
βπ‘,π’π
2π‘,π’ 6 π¦βπ‘,π’ππ‘ for
any (π₯π‘, π¦π‘) β π΅π‘, and so ππ¦+π‘,π’π 2π‘,π’ β π¦βπ‘,π’π
2π‘,π’ > 0, as follows from the margin
requirement. Thus condition (b) holds.
In order to check (c), put π₯π‘ = (π₯2π‘ , . . . , π₯ππ‘ ) and observe that π₯+π‘ π
1π‘ β
π₯βπ‘ π 1π‘ > |π₯π‘|π/2 for any (π₯π‘, π¦π‘) β π΅π‘ by virtue of the choice of π 1π‘ . Then
ππ₯+π‘ π 1π‘ β π₯βπ‘ π
1π‘ = π(π₯+π‘ π
1π‘ β π₯βπ‘ π
1π‘ )β (1βπ)π₯βπ‘ π
1π‘
> π|π₯π‘|π/2β (1βπ)π₯βπ‘ π 1π‘ > |π₯π‘|(ππ/2β (1βπ)max
π(π
π
π‘ +π·π
π‘)) > 0.
The second inequality is a consequence of the fact that if π₯1π‘ < 0 then π₯βπ‘ π 1π‘ β€
|π₯π‘|maxπ(ππ
π‘+π·π
π‘), as it follows from the inequality π₯+π‘ ππ‘βπ₯βπ‘ ππ‘+ππ‘(π₯π‘) β₯ 0,
which in turn can be obtained by combining the self-financing condition and
the margin requirement. Thus condition (c) in Theorem 3.1 holds.
Condition (d) is valid because if ππ‘ = {0} then for any ππ‘ β ππ‘ we have
ππ‘π 1π‘ > π+π‘ ππ‘ β πβπ‘ ππ‘ + ππ‘(ππ‘), which yields ππ‘π
1π‘ β π΄π‘ and Eππ‘π
1π‘ β₯ 0. Thus
condition (i) implies the hedging principle by virtue of Theorem 3.1.
If assumption (ii) is satisfied, then condition (bβ²) holds with π = π/π
(which can be proved in the same way as above), and so the hedging principle
is valid in view of Theorem 3.3.
The next theorem provides a characterisation of consistent price systems
in the model with a margin requirement. It extends the results of the papers
[11, 53], which considered finite probability spaces and markets with linear
time structure.
We denote by π *π‘,π’(π) the positive dual cone of ππ‘,π’(π), i.e. π
*π‘,π’(π) = {π β
Rπ : ππ¦ > 0 for all π¦ β ππ‘,π’(π)}.
Theorem 1.4. In the stock market model with a margin requirement, a
sequence (ππ‘)π‘βπ― , ππ‘ β πΏβπ‘ , is a consistent price system if and only if each
ππ‘ > 0 and the following conditions hold.
(i) For every π‘ β π― such that ππ‘ = {0} we have
Eπ£π1π‘ > 0 for any π£ β π΄π‘ β {0}
48
and
πππ‘ +π·π
π‘ <πππ‘π1π‘< π
π
π‘ +π·π
π‘ a.s. on the set {π : πππ‘ +π·π
π‘ < ππ
π‘ +π·π
π‘}
for all π = 2, . . . , π .
(ii) There exist Fπ‘-measurable random variables πππ‘ β [ππ
π‘, ππ
π‘], π·ππ‘ β [π·π
π‘, π·π
π‘],
π‘ β π― , π = 2, . . . , π , such that
πππ‘ = (πππ‘ +π·π
π‘)π1π‘ . (1.10)
(iii) There exist Fπ‘-measurable random variables π΅π‘,π’ β [1/(1+π π‘,π’), 1/(1+
π π‘,π’)] and random vectors ππ‘,π’ β π *π‘,π’ a.s., π‘ β π―β, π’ β π―π‘+, such that
E(π1π’ | Fπ‘) = π΅π‘,π’π1π‘ β π1π‘,π’, E(πππ’ | Fπ‘) = ππ
π‘π1π‘ β πππ‘,π’, π > 2. (1.11)
Proof. Suppose the conditions listed above are satisfied for a sequence
(ππ‘)π‘βπ― , 0 < ππ‘ β πΏβπ‘ . Put ππ‘ = (1, π2
π‘ , . . . , πππ‘ ), πβ²
π‘,π’ = (π΅π‘,π’, π2π‘ , . . . , π
ππ‘ ),
π·π‘ = (0, π·2π‘ , . . . , π·
ππ‘ ). Then for any (π₯π‘, π¦π‘) β π΅π‘ we have
E
[π₯π‘ππ‘ β
βπ’βπ―π‘+
π¦π‘,π’ππ’
]= E
[π₯π‘ππ‘ β
βπ’βπ―π‘+
π¦π‘,π’E(ππ’ | Fπ‘)
]= E
[(π₯π‘(ππ‘ +π·π‘)β
βπ’βπ―π‘+
π¦π‘,π’πβ²π‘,π’
)π1π‘
]+
βπ’βπ―π‘+
Eπ¦π‘,π’ππ‘,π’
> E
[((π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’)ππ‘ + π₯π‘π·π‘
)π1π‘
]> E
[((π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’)+
ππ‘ β(π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’)βππ‘ + ππ‘(π₯π‘)
)π1π‘
]> 0,
and so condition (1.2) involved in the definition of a consistent price system
holds. Condition (1.3) follows from Proposition 1.3. Thus, (ππ‘)π‘βπ― is a
consistent price system.
Let us prove the converse statement. Suppose (ππ‘)π‘βπ― is a consistent
price system. It can be proved by induction over the sets π― πΎ , π― πΎβ1, . . . , π― 0
(see their definition in the proof of Proposition 1.1) that each ππ‘ is strictly
positive.
Indeed, for any π‘ /β π―β any non-negative 0 = ππ‘ β πΏ1π‘ belongs to π+
π‘
49
(according to the assumption on p. 45), and so Eππ‘ππ‘ > 0 by virtue of (1.3),
which implies ππ‘ > 0 for π‘ /β π―β. Suppose ππ’ > 0 for all π’ β π― >π and
consider some π‘ β π― πβ1. Take arbitrary non-negative 0 = π₯π‘ β πΏ1π‘ and let
π¦π‘ = (π¦π‘,π’)π’βπ―π‘+ with π¦π‘,π’ = π₯π‘ for some π’ β π―π‘+ and π¦π‘,π = 0 for π = π’. Since
(π₯π‘, π¦π‘) β π΅π‘, from (1.2) we obtain Eπ₯π‘ππ‘ > Eπ₯π‘ππ’ > 0, which proves the
strict positivity of ππ‘.
For any π‘ β π― where ππ‘ = {0}, the first part of condition (i) can be
obtained by applying (1.3) to (ππ’)π’βπ β π+ with ππ’ = 0, π’ = π‘, and ππ‘ =
(π£, 0, . . . , 0) for any π£ β π΄π‘ β {0}.In order to check the second part, suppose πππ‘/π
1π‘ < ππ
π‘+π·ππ‘ on a set Ξ β Fπ‘,
P(Ξ) > 0, for some π β₯ 2. Consider ππ‘ = (π1π‘ , . . . , πππ‘ ) with π
1π‘ = β(ππ
π‘+π·ππ‘)IΞ,
πππ‘ = IΞ and πππ‘ = 0 for π = 1, π. We have ππ‘ β ππ‘ and Eππ‘ππ‘ < 0, which
contradicts (1.3). Thus πππ‘/π1π‘ > ππ
π‘ +π·ππ‘, and similarly, πππ‘/π
1π‘ 6 π
π
π‘ +π·π
π‘.
If for the set Ξβ² = {π : πππ‘+π·
ππ‘ = πππ‘/π
1π‘ < π
π
π‘+π·π
π‘}, we have P(Ξβ²) > 0, we
can define ππ‘ as above (with Ξβ² in place of Ξ) and obtain that ππ‘ β π+π‘ but
Eππ‘ππ‘ = 0, which is a contradiction. Hence πππ‘/π1π‘ > ππ
π‘ + π·ππ‘, and similarly,
πππ‘/π1π‘ < π
π
π‘ +π·π
π‘ a.s. on the set {π : πππ‘ +π·π
π‘ < ππ
π‘ +π·π
π‘}, which proves the
second part of (i).
Let us verify conditions (ii) and (iii). First we show that for almost each
π β Ξ© there exist πππ‘(π), π·
ππ‘(π), π΅π‘,π’(π), ππ‘,π’(π) satisfying (1.10)β(1.11), and
then it will be shown that they can be chosen in the Fπ‘-measurable way.
For π‘ β π―β, consider the cones defining the self-financing condition:
ππ‘(π) =
{(π₯, π¦) β Rπ(1+|π―π‘+|) :(
π₯π‘ ββπ’βπ―π‘+
π¦π‘,π’)+
Β· ππ‘(π) + ππ‘(π)(π₯π‘) >(π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’)βΒ· ππ‘(π)
}.
Property (1.2) implies the following inclusion for each π‘ β π―β:(ππ‘,βE[(ππ’)π’βπ―π‘+ | Fπ‘]
)β(ππ‘ β© (Rπ Γ ππ‘)
)*where we denote ππ‘ = {(π¦π‘,π’)π’βπ―π‘+ : π¦π‘,π’ β ππ‘,π’}. It can be verified that for
almost all π, the cone ππ‘(π), π‘ β π―β, is positively dual to the set π β²π‘(π)
50
consisting of all pairs (π,βπ) β Rπ(1+|π―π‘+|) such that
π1 = 1, ππ = ππ +π·π, π > 2,
π1π’ = π΅π’, πππ’ = ππ, π > 2, for π’ β π―π‘+,
with some π΅π’ β [1/(1 + π π‘,π’(π)), 1/(1 + π π‘,π’(π))], ππ β [ππ
π‘(π), ππ
π‘(π)], π·π β
[π·ππ‘(π), π·
π
π‘(π)]. Sinceπβ²π‘(π) is closed, we haveπ
*π‘ (π) = cone(π β²
π‘(π)), which
implies π*π‘ (π) β© ({0} Γ π *
π‘ (π)) = {0}, and so(ππ‘ β© (Rπ Γ ππ‘)
)*=π*
π‘ + {0} Γ π *π‘
by virtue of Proposition 1.7. Thus, we obtain the existence of π΅π‘,π’(π), πππ‘(π),
π·ππ‘(π), ππ‘,π’(π) satisfying (1.10), (1.11) for π‘ β π―β. In a similar way we obtain
the existence of πππ‘(π), π·
ππ‘(π) satisfying (1.10) for π‘ /β π―β.
It remains to show that π΅π‘,π’, πππ‘ , π·
ππ‘, ππ‘,π’ can be chosen Fπ‘-measurably.
For π‘ β π―β consider the closed cones
π(1)π‘ (π) =
{(π΅, π,π·, π) : ππ β [ππ
π‘(π), ππ
π‘(π)], π·π β [π·π
π‘(π), π·π
π‘(π)], π > 2,
π΅π’ β [1/(1 +π π‘,π’(π)), 1/(1 +π π‘,π’(π))], π’ β π―π‘+},
π(2)π‘ (π) =
{(π΅, π,π·, π) : πππ‘(π) = (ππ +π·π)π1π‘ (π), π > 2,
E(π1π’ | Fπ‘)(π) = π΅π’π1π‘ (π)β π1π’, π’ β π―π‘+,
E(πππ’ | Fπ‘)(π) = πππ1π‘ (π)β πππ’, π > 2, π’ β π―π‘+}
π(3)π‘ (π) =
{(π΅, π,π·, π) : ππ’ β π *
π‘,π’(π), π’ β π―π‘+}.
These cones consist of families (π΅, π,π·, π) of vectors π΅ = (π΅π’)π’βπ―π‘+ β R|π―π‘+|,
π = (ππ)π>2 β Rπβ1, π· = (π·π)π>2 β Rπβ1, π = (πππ’ | π’ β π―π‘+, π > 1} βRπ |π―π‘+| satisfying the corresponding conditions. One can see that they are
Fπ‘-measurable: π(1)π‘ is the volume bounded by Fπ‘-measurable hyperplanes,
π(2)π‘ is the intersection of Fπ‘-measurable hyperplanes, and π
(3)π‘ is positively
dual to Fπ‘-measurable closed cone {0}Γππ‘,π’ (see Remark 1.7). Consequently,
their intersection is also Fπ‘-measurable, so there exists a measurable selection
from it (Proposition 1.8), which provides the sought-for πππ‘ , π·
ππ‘, π΅π‘,π’, π
ππ‘,π’.
Condition (ii) for π‘ = π can be proved in a similar way.
51
1.6 Appendix 1: auxiliary results from functional analysis
In this appendix we provide several results from functional analysis used
in this and the next chapters.
Let (π΅,β¬, π) be a measurable space with a finite measure π. Let πΏ1 and
πΏβ be the spaces of all (equivalence classes of) integrable and essentially
bounded functions with the norms β Β· β1 and β Β· ββ, respectively. We put
Eπ₯ :=β«π₯ππ for any π₯ β πΏ1. For a cone π, we denote π+ = π β (βπ).
Proposition 1.4 (a version of the Kreps-Yan theorem). Let β and π be
closed cones in πΏ1 such that π+ = β , βπ β β, π+β©β = β and there exists
π β πΏβ satisfying Eππ₯ > 0 for all π₯ β π+. Then there exists π β πΏβ such
that Eππ₯ > 0 for all π₯ β π+ and Eππ₯ 6 0 for all π₯ β β.
Proof. If π β© (βπ) = {0} (i.e. π+ = π β {0}), the proposition follows
from a result by Rokhlin: see Theorem 1.1 in [52]. In the general case,
consider the closed cones
ππ = {π₯ β π : Eππ₯ > πβ1βπ₯β}.
Then ππ β© (βππ) = {0}, and by applying Rokhlinβs theorem to β and ππ,
we find ππ β πΏβ such that Eπππ₯ > 0 for any π₯ β ππ β {0} and Eπππ₯ 6 0 for
all π₯ β β. Without loss of generality it may be assumed that βππββ = 1.
Then the proposition holds with π =ββπ=1
2βπππ.
Remark 1.6. Note that if the space πΏ1 is separable, there always exists
π β πΏβ such that Eππ₯ > 0 for all π₯ β π+ ( = β ), and Proposition 1.4 follows
from Theorem 5 in [8]. In a non-separable πΏ1, there might exist pointed
cones π without functionals π β πΏβ, see Section 1.7.
Proposition 1.5. Let β and π be closed cones in πΏ1 such that π+ = β ,βπ β β, π+ β© β = β and there exists π β πΏβ such that Eππ₯ > 0 for all
π₯ β π+. Then for any π¦ β πΏ1 the following conditions are equivalent:
(a) π¦ β β;
(b) Eππ¦ 6 0 for any π β πΏβ such that Eππ₯ > 0 for all π₯ β π+ and Eππ₯ 6 0
for all π₯ β β.
52
Proof. Clearly (a) implies (b). To prove the converse implication suppose
π¦ β β. By separating the point π¦ from the closed convex set β, we construct
π1 β πΏβ such that Eπ1π¦ > 0 and Eπ1π₯ 6 0 for all π₯ β β. Since β β βπ,
we have Eπ1π₯ > 0 for all π₯ β π.
According to Proposition 1.4, there exists π2 β πΏβ such that Eπ2π₯ 6 0
for any π₯ β β and Eπ2π₯ > 0 for any π₯ β π+. By choosing π > 0 large
enough, we get Eππ¦ > 0 for π = ππ1 + π2. On the other hand, Eππ₯ > 0 for
all π₯ β π+ and Eππ₯ 6 0 for all π₯ β β. A contradiction.
Proposition 1.6 (Komlos theorem, [38]). Let {π₯π}πβΞ be a family of
functions in πΏ1 such that supπβΞ βπ₯πβ1 <β. Then there exists π₯ β πΏ1 and a
sequence π₯ππ such that any its subsequence π₯πππ Cesaro converges to π₯ a.s.,
i. e. πβ1(π₯ππ1 + . . .+ π₯πππ ) β π₯ a.s.
For the readerβs convenience, we provide an auxiliary (known) result used
in Theorem 1.4. For a cone π΄ β Rπ we denote by π΄* its positive dual cone,
i. e. π΄* = {π β Rπ : ππ > 0 for all π β π΄}. It is well known that π΄** = π΄ for
any closed cone π΄ β Rπ.
Proposition 1.7. Let π΄ and π΅ be closed cones in Rπ such that π΄* β©(βπ΅*) = {0}. Then (π΄ β©π΅)* = π΄* +π΅*.
Proof. It is straightforward to prove that (π΄ β©π΅)* β π΄* +π΅*. In order
to prove (π΄ β© π΅)* β π΄* + π΅*, observe that π΄ β© π΅ β (π΄* + π΅*)*. Indeed, if
π₯ β (π΄* + π΅*)* then ππ₯ > 0 for any π β π΄*, so π₯ β π΄** = π΄, and, similarly,
π₯ β π΅. Consequently, (π΄β©π΅)* β (π΄*+π΅*)** = π΄*+π΅*, where we use that
π΄* + π΅* is closed because π΄* and π΅* are closed and π΄* β© (βπ΅*) = {0}, see[50, Corollary 9.1.3].
In the rest of the appendix we provide results on random measurable sets
which were used in Section 1.5.
Let (Ξ©,F ,P) be a complete probability space andπ a complete separable
metric space endowed with the Borel π-algebra B(π). By definition, a
(closed) random set π on (Ξ©,F ,P) is a set-valued mapping π : Ξ© β 2π
from Ξ© to the family of all subsets of π such that π(π) is closed a.s.
53
A random set π is called measurable if for any open set πΊ β π
{π : π(π) β©πΊ = β } β F .
A measurable function π : Ξ© β π is called a measurable selection of a
random set π if π(π) β π(π) a.s.
The following propositions assemble properties of measurable random
sets. Their proofs can be found in [6, Ch. III] and [43, Ch. 2].
Proposition 1.8. For a set-valued mapping π : Ξ© β 2π such that π(π)
is closed and non-empty a.s. the following conditions are equivalents:
(a) π is measurable;
(b) for any π₯ β π the distance π(π₯, π(π)) is a measurable function from
(Ξ©,F ) to (R,B(R));
(c) there exists a sequence {ππ}π>0 of measurable selections of π such that
π(π) = cl{ππ(π), π > 0} a.s.;
(d) {π : π(π) β©π΅ = 0} β F for any π΅ β B(π);
(e) Graph(π) = {(π, π₯) β Ξ©Γπ : π₯ β π(π)} β F β B(π).
Proposition 1.9. Any countable intersection π(π) =βπ>0
ππ(π) of mea-
surable random sets ππ, π > 0, is also measurable.
Remark 1.7. 1) If π(π) = {π₯ β Rπ : π₯ Β· π(π) = π(π)} is a random
hyperplane in Rπ (possibly, π(π) = Rπ or π(π) = β for some π), where
π : Ξ© β Rπ is a measurable vector, and π : Ξ© β R is random variable, then
π is measurable. This follows from that the distance between any point
π₯ β Rπ and this hyperplane is a measurable function of π(π) and π(π).
2) For a random measurable set π in Rπ, which is non-empty a.s., its
positively dual cone π*(π) = {π β Rπ : π Β· π₯ > 0 for any π₯ β π(π)} is also
measurable. Indeed, let {ππ}π>0 be a sequence of measurable selections of π
such that π = cl{ππ, π > 0} a.s. Then ππ(π) = {π β Rπ : π Β· ππ(π) > 0} are
measurable random sets, and hence π* =βπ>0
ππ is also measurable.
54
1.7 Appendix 2: linear functionals in non-separable πΏ1
spaces
The aim of this appendix is to provide an example3 of a non-separable
space πΏ1 that contains a pointed non-empty cone π such that there is no
linear functional π β πΏβ strictly positive on π β {0}. This example shows
that assumption (A) imposed in Section 1.1 is essential.
Let (Ξ©,F ,P) be a probability space, where Ξ© is the space of all func-
tions π : [0, 1] β R, F is the product of a continuum of copies of the Borel
π-algebra B on R, and P is the probability measure on F such that the
(canonical) random variables ππ‘(π) := π(π‘), π‘ β [0, 1], have standard Gaus-
sian distribution (ππ‘ βΌ N (0, 1)) and are independent.
Let π be the closed convex cone in πΏ1 = πΏ1(Ξ©,F ,P) spanned on the
system of random variables ππ‘.
First we prove that the cone π is pointed, i. e. πβ© (βπ) = {0}. Indeed,suppose the contrary: there exists a non-zero random variable π β πΏ1, a
sequence ππ‘1, ππ‘2, . . . of distinct random variables ππ‘π, and two sequences
π΄π =πβ
π=1
πππ ππ‘π, π΅π =πβ
π=1
πππ ππ‘π
of non-negative linear combinations of ππ‘π such that
π΄π β π, π΅π β βπ (in πΏ1).
This implies
πΆπ :=πβ
π=1
πππ ππ‘π β 0 (in πΏ1) for πππ := πππ + πππ . (1.12)
Since ππ‘π are independent and standard Gaussian, πΆπ is a Gaussian random
variable with zero expectation and variance π£π =βπ
π=1(πππ )
2. Thus E|πΆπ| =βπ£π Β· E|π|, where π βΌ N (0, 1). According to (1.12), E|πΆπ| β 0, and
3The idea of this example was suggested by Prof. W. Schachermayer, [54].
55
consequently,
0 = limπββ
π£π = limπββ
πβπ=1
(πππ + πππ )2 > lim
πββ
πβπ=1
(πππ )2,
where the last inequality holds because πππ β₯ 0 and πππ β₯ 0. Therefore
Eπ΄2π =
πβπ=1
(πππ )2 β 0,
which implies that E|π΄π| β 0, and hence π = 0. This is a contradiction
meaning that π is a pointed cone.
Now we prove that there are no continuous linear functionals on πΏ1 which
are strictly positive on π β {0}. Suppose the contrary: such a functional π
exists. Let π be a function in πΏβ such that β¨π, π¦β© = Eππ¦, π¦ β πΏ1.
It is well-known that any F -measurable function π on Ξ© can be repre-
sented in the form
π = π(ππ‘1, ππ‘1, . . .),
where {π‘1, π‘2, . . .} is a countable subset of [0, 1] and π(Β·, Β·, . . .) is a measurable
function on the product of a countable number of copies of (R,B). Consider
any π‘ /β {π‘1, π‘2, ...}. The random variable ππ‘ is independent of ππ‘1, ππ‘1, ... and
hence independent of π. Consequently,
Eπππ‘ = Eπ Β· Eππ‘ = 0,
which is a contradiction.
56
Chapter 2
Utility maximisation in multimarket
trading
In this chapter we continue studying the general model of interconnected
markets described in the previous chapter and consider the question of choos-
ing optimal trading strategies in this model. We propose a problem where a
trader needs to manage a portfolio of assets choosing the level of consump-
tion and allocating her wealth between the assets in the way that maximises
the utility from trading. The main result of the chapter provides conditions
for the existence of supporting prices, which allow to decompose the multi-
period constrained utility maximisation problem and to reduce it to a family
of single-period unconstrained problems.
Problems of utility maximisation in multi-period trading have been widely
studied in the literature (typically for the βlinearβ time structure; see e.g.
[1, 10, 13, 25, 42]). The most closely related model to our research is the
framework of von Neumann β Gale random dynamical systems introduced
by von Neumann [68] and Gale [25, 27] for the deterministic case and later
extended to the stochastic case by Dynkin, Radner and others (see e.g. [1,
12, 13, 48, 50]). Originally, this framework addressed utility maximisation
problems in models of a growing economy, where a planner needs to choose
between consumption of commodities and using them for further production.
Our model extends results of the von Neumann β Gale framework to financial
markets. The main conceptual difference between financial and economic
models consists in the possibility of short sales, when a trader may borrow
assets from the broker. Mathematically this is expressed in that coordinates
of portfolio vectors may be negative, while in economic models commodity
vectors are typically non-negative.
57
Similarly to Chapter 1, in order to obtain the main results, we system-
atically use the idea of margin requirements on admissible portfolios, which
limit allowed leverage. As it has already been mentioned, requirements of
this type present in one form or another in all real financial markets and thus
are fully justified from the applied point of view. We prove the existence of
supporting prices in a market with a margin requirement under a condition
on the size of the margin.
The chapter is organised as follows. In Section 2.1 we describe the gen-
eral problem of utility maximisation in the multimarket model and establish
the existence of optimal trading strategies which deliver maximum values of
utility functionals. In Section 2.2 we prove the main result on the existence
of supporting prices in the model. Section 2.3 contains an application of the
general result to a specific model of an asset market with transaction costs
and portfolio constraints. In Section 2.4 we construct two examples of natu-
ral market models, where supporting prices do not exist. Section 2.5 provides
an auxiliary result from convex analysis, the Duality Theorem, which plays
the central role in the proof of the existence of supporting prices.
2.1 Optimal trading strategies in the model of
interconnected asset markets
In this chapter we use the general market model of Chapter 1, slightly
modifying it to allow consideration of utility maximisation problems.
It is assumed that a system of interconnected markets is described by
a probability space (Ξ©,F ,P), a directed acyclic graph πΊ with a finite set
of nodes π― , which are interpreted as different trading sessions, a family of
π-algebras Fπ‘, π‘ β π― , representing random factors affecting each trading
session, and a measurable space of assets (Ξ,J , π) with a finite measure π.
The meaning of these objects is the same as in Chapter 1.
For a subset π β π― we denote by πΏ1π = πΏ1
π (Ξ© Γ Ξ) the space of all
integrable functions π¦ : π ΓΞ©ΓΞ β π , and let πΏ1π‘+ = πΏ1
π―π‘+ for each π‘ β π―β.
58
To keep further notation concise, we define πΏ1π‘+ := {0} for each sink node
π‘ /β π―β (which can be interpreted as that no distribution of assets is done in
terminal trading sessions).
Trading constraints in the model are defined by some given (convex) cones
π΅π‘ β πΏ1π‘βπΏ1
π‘+, π‘ β π― . In this chapter, from the beginning, we assume that π΅π‘
are closed with respect to πΏ1-bounded a.s.-convergence, and, hence, are also
closed in πΏ1 (which follows from the fact that from any sequence converging
in πΏ1 it is possible to extract a subsequence converging with probability one).
A trading strategy π is a family of functions π = (π₯π‘, π¦π‘)π‘βπ― such that
(π₯π‘, π¦π‘) β π΅π‘ for each π‘ β π― .
For each π‘ β π―β, the function π₯π‘ represents the portfolio held before one buys
and sells assets in the session π‘ and the function π¦π‘ = (π¦π‘,π’)π’βπ―π‘+ specifies the
distribution of assets to the sessions π’ β π―π‘+. If π‘ /β π―β (i. e. π‘ is a sink
node), π₯π‘ β πΏ1π‘ specifies the terminal portfolio obtained in this session, and
π¦π‘ := 0 by the definition of πΏ1π‘+. Note that in Chapter 1 trading strategies are
indexed by π‘ β π―β, while in this chapter we also take into account terminal
pairs (π₯π‘, 0) for sink nodes π‘ /β π―β.The definition of a contract in the market is the same as in Chapter 1: it
is a family of portfolios
πΎ = (ππ‘)π‘βπ― , ππ‘ β πΏ1π‘ ,
where ππ‘ stands for the portfolio which has to be delivered β according to
this contract β in the trading session π‘.
In this chapter we will consider only exact hedging (replication) of con-
tracts, and will say that a trading strategy π = (π₯π‘, π¦π‘)π‘βπ― hedges a contract
πΎ = (ππ‘)π‘βπ― if
π₯π‘ = π¦π‘β β ππ‘ for each π‘ β π― ,
where π¦π‘β :=β
π βπ―π‘βπ¦π ,π‘ for π‘ β π―+ is the amount of assets delivered to the
trading session π‘, and π¦π‘β := 0 for a source node π‘ β π―+. In terms of the
previous chapter, the set of families of risk-acceptable portfolios is π = {0}.
59
Suppose that with each π‘ β π― , a utility function π’π‘ : πΏ1π‘ Γ πΏ1
π‘+ β R is
associated. The utility of a trading strategy π = (π₯π‘, π¦π‘)π‘βπ― is defined by
π’(π) =βπ‘βπ―
π’π‘(π₯π‘, π¦π‘).
Further, it is assumed that each function π’π‘ is upper semi-continuous
(with respect to the topology induced by the norm in πΏ1) and concave, and,
hence, π’ is also upper semi-continuous and concave.
If π‘ is not a sink node (i. e. π‘ β π―β), the utility function π’π‘ measures the
gain obtained from rearranging a portfolio π₯π‘ into a family of portfolios π¦π‘.
A simple example of such a function is
π’(1)π‘ (π₯π‘, π¦π‘) = E π
(β«Ξ
(π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’
)ππ‘ππ
)(2.1)
for a function ππ‘ = ππ‘(π, π) of asset prices and a deterministic utility function
π : R β R. Here, the utility of a pair (π₯π‘, π¦π‘) is equal to the expected π -utility
of the monetary gain from selling assets and buying new ones.
A more sophisticated model is
π’(2)π‘ (π₯π‘, π¦π‘) = Eπ
(β«Ξ
[(π₯π‘β
βπ’βπ―π‘+
π¦π‘,π’
)+
ππ‘β(π₯π‘β
βπ’βπ―π‘+
π¦π‘,π’
)βππ‘
]ππ
), (2.2)
where ππ‘ 6 ππ‘ are bid and ask price functions.
If π‘ /β π―β, the value of π’π‘(π₯π‘, π¦π‘) with π¦π‘ β‘ 0, represents the utility of a
terminal portfolio π₯π‘ and can be also defined, for example, by formulae (2.1)
or (2.2), putting π¦π‘,π’ := 0.
Remark 2.1. It is worth mentioning that the model proposed is able
to include the case when one receives a gain from just holding assets, not
necessarily consuming them. This circumstance was especially important
in von Neumann β Gale models of a growing economy, where additional
utility can be obtained from availability of goods whose consumption does
not reduce their quantity (e.g. housing, works of art, etc.). The approach
can also be used to model irrationalities in peopleβs actions, e. g. when the
valuation of additional gain or loss depends on the current wealth (such as
60
the endowment effect, see [34]).
In the financial setting, a typical example is a gain obtained from dividend
payments. The corresponding utility function can be of the form
π’(3)π‘ (π₯π‘, π¦π‘) = E π
(β«Ξ
π₯π‘π·π‘ππ
)+ E π
(β«Ξ
(π₯π‘ β
βπ’βπ―π‘+
π¦π‘,π’
)ππ‘ππ
),
where π·π‘ = π·π‘(π, π) is a function of dividend rates and π is some determin-
istic utility function.
If πΎ is a hedgeable contract (there exists at least one trading strategy
hedging it), by π(πΎ) we denote the supremum of the utilities of all trading
strategies hedging πΎ, i. e.
π(πΎ) := sup{π’(π) | π hedges πΎ}.
A trading strategy π is called an optimal trading strategy for πΎ if π’(π) =π(πΎ), i. e. π maximises the utility from trading with the contract πΎ.
Remark 2.2. Note that π is a concave function. Indeed, if πΎ = πΌπΎ1 +
(1 β πΌ)πΎ2 for hedgeable contracts πΎ1, πΎ2 and πΌ β [0, 1], then for any π > 0
there exist trading strategies π1, π2 hedging πΎ1, πΎ2 such that π(πΎπ) 6 π’(ππ)+π.
Consequently, π = πΌπ1 + (1β πΌ)π2 hedges πΎ and
π(πΎ) > π’(π) > πΌπ’(π1) + (1β πΌ)π’(π2) > πΌπ(πΎ1) + (1β πΌ)π(πΎ2)β π.
Passing to the limit πβ 0, we obtain π(πΎ) > πΌπ(πΎ1) + (1β πΌ)π(πΎ2).
The next theorem states that conditions (a), (b) of Theorem 1.1 guarantee
the existence of optimal trading strategies for hedgeable contracts.
Theorem 2.1. Suppose there exist functions π 1π‘ , π 2π‘,π’ β πΏβ
π‘ , π‘ β π―β, π’ βπ―π‘+, with values in [π , π ], where π > 0, π > 1, and a constant 0 6 π < 1
such that for all π‘ β π― , π’ β π―π‘+, (π₯π‘, π¦π‘) β π΅π‘ the following conditions are
satisfied:
(a) Eπ₯π‘π 1π‘ > Eπ¦π‘,π’π
2π‘,π’;
(b) πEπ¦+π‘,π’π 2π‘,π’ > Eπ¦βπ‘,π’π
2π‘,π’.
Then an optimal trading strategy exists for any hedgeable contract.
61
The functions π 1π‘ , π 2π‘,π’ can be interpreted as some systems of asset prices
in the same way as in Theorem 1.1.
Proof. Let πΎ be a hedgeable contract, we need to show that there exists
a trading strategy π hedging πΎ such that π’(π) = π(πΎ).
As follows from the proof of Theorem 1.1, there exists a constant πΆ
(namely, πΆ = (1 + π)/(1 β π) Β· π /π ) such that βπ¦π‘,π’β β€ πΆβπ₯π‘β for any
(π₯π‘, π¦π‘) β π΅π‘, π‘ β π―β, π’ β π―π‘+, which implies that the set of all trading
strategies hedging πΎ is bounded in πΏ1.
Take a sequence of strategies π π hedging πΎ such that π’(π π) β π(πΎ). By
using the Komlos theorem (see Proposition 1.6), we find a subsequence π ππ
Cesaro-convergent a.s. to some π (i. e. ππ := πβ1(π π1 + . . .+ π ππ) β π). Sincethe cones π΅π‘ are closed with respect to πΏ1-bounded a.s.-convergence, π is a
trading strategy and it hedges πΎ. Then we find
π’(π) > lim supπββ
π’(ππ) > lim supπββ
1
π
πβπ=1
π’(π ππ) = π(πΎ),
where in the first inequality we use that π’ is upper semicontinuous, and in
the second inequality we use that π’ is concave. Thus π is the sought-for
trading strategy.
Remark 2.3. Under the conditions of the theorem, the set of all hedge-
able contracts is a closed cone in πΏ1π― , as follows from the proof of Theorem 1.1.
2.2 Supporting prices: the definition and conditions of
existence
In this section we introduce the central notion of the chapter, a system
of supporting prices for a hedgeable contract, and provide conditions which
guarantee its existence.
Further we always assume that cones π΅π‘ and utility functions π’π‘ satisfy
62
the following natural conditions of monotonicity :
if (π₯π‘, π¦π‘) β π΅π‘, then (π₯β²π‘, π¦π‘) β π΅π‘ for any π₯β²π‘ β πΏ1
π‘ such that π₯β²π‘ > π₯π‘, (2.3)
π’(π₯β²π‘, π¦π‘) > π’(π₯π‘, π¦π‘) for any π₯π‘, π₯β²π‘ β πΏ1
π‘ , π¦π‘ β πΏ1π‘+ such that π₯β²π‘ > π₯π‘, (2.4)
where here and below all inequalities for random variables are understood to
hold with probability one.
These two properties imply that if πΎ = (ππ‘)π‘βπ― , πΎβ² = (πβ²π‘)π‘βπ― are two
hedgeable contracts and πΎβ² 6 πΎ (πβ²π‘ 6 ππ‘ for each π‘ β π― ), then π(πΎβ²) > π(πΎ),
i. e. a contract with a smaller amount of payments provides greater utility.
Indeed, if πΎ can be hedged by a trading strategy π = (π₯π‘, π¦π‘)π‘βπ― , then πΎβ² can
be hedged by π β² = (π₯β²π‘, π¦π‘)π‘βπ― with π₯β²π‘ = π₯π‘ + ππ‘ β πβ²π‘, and π’(πβ²) > π’(π).
Moreover, (2.3) implies that any non-positive contract πΎ = (ππ‘)π‘βπ― (i. e.
a contract according to which one does not make any payment, but only
receives assets) is hedgeable, e. g. by the strategy π = (π₯π‘, π¦π‘)π‘βπ― with π₯π‘ = ππ‘,
π¦π‘ = 0.
A family of functions (ππ‘)π‘βπ― , ππ‘ β πΏβπ‘ , is called a system of support-
ing prices for a hedgeable contract πΎ = (ππ‘)π‘βπ― if for any family of pairs
(π₯π‘, π¦π‘)π‘βπ― , (π₯π‘, π¦π‘) β π΅π‘, it holds that
π(πΎ) >βπ‘βπ―
[π’π‘(π₯π‘, π¦π‘) + E
( βπ’βπ―π‘+
π¦π‘,π’ππ’ β π₯π‘ππ‘ β ππ‘ππ‘
)], (2.5)
where π¦π‘,π’ππ’ := 0 for a sink node π‘.
The financial interpretation of supporting prices is that E[β
π’ π¦π‘,π’ππ’βπ₯π‘ππ‘]is equal to the expected profit from choosing the pair (π₯π‘, π¦π‘), where Eπ₯π‘ππ‘
is interpreted as the expected cost of the portfolio π₯π‘ in the session π‘ and
Eβ
π’ π¦π‘,π’ππ’ is the expected cost of the portfolios π¦π‘,π’ in the βfutureβ sessions
π’ β π―π‘+. The quantity Eππ‘ππ‘ is the expected cost of the portfolio ππ‘ delivered
according to the contract.
Observe that each ππ‘ is non-negative β otherwise for π₯π‘ = ππβπ‘ , π¦π‘ = 0 we
have (π₯π‘, π¦π‘) β π΅π‘, but the right-hand side of (2.5) exceeds the left-hand side
for a large real π > 0. Moreover, each ππ‘ is strictly positive (ππ‘(π, π) > 0 a.s.)
if π’π‘(πIΞ, 0) β +β whenever π β +β for any set Ξ β F β J of strictly
63
positive measure (otherwise take π₯π‘ = πI(ππ‘ = 0), π¦π‘ = 0 to obtain the same
contradiction).
The following theorem provides sufficient conditions for the existence of
supporting prices for non-positive contracts in the general model. A par-
ticular asset market model with transaction costs and portfolio constraints
will be studied in the next section. Note that even in natural market models
supporting prices may not exist for contracts whose portfolio positions can
be positive. An example will be provided in Section 1.5.
Theorem 2.2. Supporting prices exist for any contract πΎ 6 0 if condi-
tions (a), (b) of Theorem 2.1 and the following three conditions hold:
(c) if (π₯π‘, π¦π‘) β π΅π‘ for some π‘ β π―β, then (π₯π‘, 0) β π΅π‘ and (π¦π‘,π’, 0) β π΅π’ for
each π’ β π―π‘+;
(d) there exists a constant π΄ such that if (π₯π‘, π¦π‘), (π₯β²π‘, 0) β π΅π‘ for some π‘ β
π―β, then there exists π¦β²π‘ such that (π₯β²π‘, π¦β²π‘) β π΅π‘ and βπ¦π‘βπ¦β²π‘β 6 π΄βπ₯π‘βπ₯β²π‘β;
(e) there exists a constant π΅ such that if (π₯π‘, π¦π‘) β π΅π‘ and (π₯β²π‘, π¦β²π‘) β π΅π‘ for
some π‘ β π― , then |π’π‘(π₯π‘, π¦π‘)β π’π‘(π₯β²π‘, π¦
β²π‘)| 6 π΅(βπ₯π‘ β π₯β²π‘β+ βπ¦π‘ β π¦β²π‘β).
Condition (c) can be interpreted as a βsafetyβ requirement meaning that
it should be possible to liquidate a portfolio at any time. Condition (d) is a
continuity property of the cones π΅π‘ and condition (e) is a Lipschitz-continuity
property of the utility functions.
In order to prove the theorem, we first establish an auxiliary result show-
ing that the steepness of the function π is bounded from above at any con-
tract πΎ 6 0 (see Section 2.5 for the definitions of the steepness and a support
of a function). This will allow us to apply the Duality theorem to obtain a
support for π , which will provide a system of supporting prices.
We will use the notation β for the closed cone of all hedgeable contracts.
Lemma 2.1. If conditions (a)β(e) are satisfied, the steepness of the func-
tion π : β β R is bounded from above at any contract πΎ 6 0.
Proof. Let πΎ denote the maximal length of a directed path in the graph.
Assuming, without loss of generality, that π΄ > 1, we will show that for any
64
non-positive contract πΎ = (ππ‘)π‘βπ― , and any hedgeable contract πΎβ² = (πβ²π‘)π‘βπ―
we have π(πΎβ²)β π(πΎ) 6 π΅(2π΄)πΎ+1βπΎβ² β πΎβ.Observe that it is sufficient to consider only the case πΎβ² 6 πΎ, because
for πΎβ²β² = min(πΎ, πΎβ²) we have π(πΎβ²β²) > π(πΎβ²) according to (2.3)β(2.4) and
βπΎβ²β² β πΎβ 6 βπΎβ² β πΎβ.Let π = πΎ β πΎβ² (i. e. π = (ππ‘)π‘βπ― , where ππ‘ = ππ‘ β πβ²π‘) and π
β² = (π₯β²π‘, π¦β²π‘)π‘βπ―
be a trading strategy at which π(πΎβ²) is attained. Using assumptions (c)β(d)
we will construct a trading strategy π = (π₯π‘, π¦π‘)π‘βπ― hedging πΎ such thatβπ‘βπ―
βπ₯β²π‘ β π₯π‘β 6 (2π΄)πΎβπβ,βπ‘βπ―
βπ¦β²π‘ β π¦π‘β 6 π΄(2π΄)πΎβπβ.
Let π (π‘) denote the maximal length of a directed path emanating from a
node π‘ β π― . Define the sets π― π := {π‘ β π― : π (π‘) = πΎ β π}, π = 0, . . . , πΎ,
which form a partition of π― such that if there is a path from π‘ β π― π to
π β π― π, then π < π. Note that π― 0 contains only source nodes, and π― πΎ is
the set of all the sink nodes. Put also π― 6π =βπ6π
π― π, π― >π =βπ>π
π― π.
Let us show by induction that for each π 6 πΎ it is possible to find a
family of pairs (π₯π‘, π¦π‘) β π΅π‘, π‘ β π― 6π, such that for each π‘ β π― 6π
π₯π‘ = π¦π‘β β ππ‘ (2.6)
and βπ‘βπ― 6π
βπ₯β²π‘ β π₯π‘β 6 (2π΄)πβπ‘βπ― 6π
βππ‘β, βπ¦β²π‘ β π¦π‘β 6 π΄βπ₯β²π‘ β π₯π‘β. (2.7)
For π = 0 we can find such a family of pairs for each π‘ β π― 0 by defining
π₯π‘ = βππ‘, and using assumption (d) choosing π¦π‘ such that (π₯π‘, π¦π‘) β π΅π‘
and βπ¦β²π‘ β π¦π‘β 6 π΄βπ₯β²π‘ β π₯π‘β (observe that (π₯π‘, 0) β π΅π‘ because ππ‘ 6 0 and
monotonicity assumption (2.3) holds).
If a family (π₯π‘, π¦π‘) β π΅π‘, π‘ β π― 6π, satisfying (2.6)β(2.7), is found for some
π < πΎ, define π₯π‘ = π¦π‘β β ππ‘ for each π‘ β π― π+1, where π¦π‘β = 0 if π‘ is a source
65
node, and π¦π‘β =β
π βπ―π‘βπ¦π ,π‘ otherwise. Then
βπ‘βπ― 6π+1
βπ₯β²π‘ β π₯π‘β 6β
π‘βπ― 6π+1
(βπ¦β²π‘β β π¦π‘ββ+ βππ‘β)
6βπ‘βπ― 6π
βπ¦β²π‘ β π¦π‘β+β
π‘βπ― 6π+1
βππ‘β
6 π΄(2π΄)πβπ‘βπ― 6π
βππ‘β+β
π‘βπ― 6π+1
βππ‘β 6 (2π΄)π+1β
π‘βπ― 6π+1
βππ‘β.
Observe that (π₯π‘, 0) β π΅π‘ for π‘ β π― π+1 according to assumption (2.3) because
(π¦π‘β, 0) β π΅π‘ as follows from (c), and ππ‘ 6 0. Consequently, using (d) it is
possible to find π¦π‘, π‘ β π― π+1, such that (π₯π‘, π¦π‘) β π΅π‘, βπ¦β²π‘ β π¦π‘β 6 π΄βπ₯β²π‘ β π₯π‘β.Proceeding by induction, we find a family π = (π₯π‘, π¦π‘)π‘βπ― satisfying (2.6)β
(2.7) for π = πΎ. Then from (2.6) it follows that π is a trading strategy
hedging the contract πΎ. From (2.7) and property (e) if follows that π’(π β²) βπ’(π) 6 π΅(2π΄)πΎ+1βπβ, which proves the lemma.
Proof of Theorem 2.2. Consider any contract πΎ 6 0. According to
Proposition 2.1 (see Section 2.5), there exists a support of the function
π : β β R at πΎ, and so there exists π = (ππ‘)π‘βπ― , ππ‘ β πΏβπ‘ , such that
π(πΎβ²)β π(πΎ) 6 E[(πΎ β πΎβ²)π] for any πΎβ² β β
(π can be taken as the negative of the support).
Take any family π = (π₯π‘, π¦π‘)π‘βπ― such that (π₯π‘, π¦π‘) β π΅π‘ for each π‘ β π― .
This family is a trading strategy hedging the contract πΎβ² = (πβ²π‘)π‘βπ― with
πβ²π‘ = π¦π‘β β π₯π‘. Consequently
π(πΎ) > π(πΎβ²) + E[(πΎβ² β πΎ)π] > π’(π) + E[(πΎβ² β πΎ)π]
=βπ‘βπ―
π’π‘(π₯π‘, π¦π‘) +βπ‘βπ―
E[(πβ²π‘ β ππ‘)ππ‘]
=βπ‘βπ―
π’π‘(π₯π‘, π¦π‘) +βπ‘βπ―
E[(π¦π‘β β π₯π‘ β ππ‘)ππ‘].
66
Rearranging the terms in the second sum,
βπ‘βπ―
E[(π¦π‘β β π₯π‘ β ππ‘)ππ‘] = E
[βπ‘βπ―
βπ βπ―π‘β
π¦π ,π‘ππ‘
]β E
[βπ‘βπ―
[(π₯π‘ + ππ‘)ππ‘]
]= E
[βπ‘βπ―
βπ’βπ―π‘+
π¦π‘,π’ππ’
]β E
[βπ‘βπ―
[(π₯π‘ + ππ‘)ππ‘]
],
we obtain (2.5), which shows that (ππ‘)π‘βπ― is a system of supporting prices.
2.3 Supporting prices in an asset market with transaction
costs and portfolio constraints
In this section we prove the existence of supporting prices in a model of
an asset market proposed in Section 1.1 (and modified in a way to fit the
general model of Section 2.1).
Recall that the model considers a market of π assets represented by a
probability space (Ξ©,F ,P), a directed acyclic graph πΊ with a finite set of
nodes π― , π-algebras Fπ‘ associated with the nodes of the graph, and the finite
space of assets (Ξ,J , π), where Ξ = {1, . . . , π}, J = 2Ξ, π{π} = 1 for
each π = 1, . . . , π . It is assumed that Fπ‘ β Fπ’ for any π‘ β π―β, π’ β π―π‘+, andeach π-algebra Fπ‘ is completed by all F -measurable sets of measure 0.
Asset π = 1 is interpreted as cash (deposited with a bank account) and
assets π = 2, . . . , π as stock. For each π‘ β π― , the bid and the ask stock
prices are denoted by πππ‘ 6 π
π
π‘ and the dividend rates for long and short stock
positions are denoted by π·ππ‘ 6 π·
π
π‘. Random variables π π‘,π’ 6 π π‘,π’ stand for
the interest rates for lending and borrowing cash between the sessions π‘ β π―βand π’ β π―π‘+.
For each π‘ β π―β random Fπ‘-measurable closed cones ππ‘,π’(π), π’ β π―π‘+, aregiven and describe portfolio constraints. We define the cones π΅π‘ by formula
(1.7) for π‘ β π―β, and for sink nodes π‘ /β π―β we let
π΅π‘ ={(π₯π‘, 0) β πΏ1
π‘ β {0} : π₯+π‘ ππ‘ + ππ‘(π₯π‘) > π₯βπ‘ ππ‘ a.s.},
67
which consists of pairs (π₯π‘, 0), where the portfolio π₯π‘ has non-negative liqui-
dation value with probability one.
It is assumed that the market satisfies margin requirement (1.9),
ππ¦+π‘,π’ππ‘(π) β₯ π¦βπ‘,π’ππ‘(π) for any π¦π‘ β ππ‘,π’(π), π‘ β π―β, π’ β π―π‘+,
where π is a constant such that 0 6π < π with π β€ (1+π π‘,π’)/(1+π π‘,π’).
Theorem 2.3. 1) In the model of a stock market with a margin require-
ment, an optimal trading strategy exists for any hedgeable contract πΎ.
2) If the constant π satisfies the inequality
π 6min
{ess inf
π,π
πππ’+π·π
π’
πππ‘, ess inf
π(1 +π π‘,π’)
}max
{ess sup
π,π
πππ’+π·
ππ’
πππ‘
, ess supπ
(1 +π π‘,π’)} , π‘ β π―β, π’ β π―π‘+, (2.8)
and the utility functions π’π‘, π‘ β π― , satisfy assumption (e) of Theorem 2.2,
then a system of supporting prices exists for any contract πΎ 6 0.
Inequality (2.8) requires that the margin should be large enough (equiv-
alently, the constant π should be small enough), which protects a trader if
the price change in an unfavourable way: it will be seen in the proof that
this inequality implies the validity of condition (c) in Theorem 2.2.
Proof. In the proof of Theorem 1.3 conditions (a), (b) of Theorem 1.1,
and, hence, of Theorem 2.1, were verified for the random vectors π 1π‘ , π 2π‘,π’ β πΏβ
π‘
with
π 11π‘ = 1, π 21π‘,π’ =1
1 +π π‘,π’
, π 1ππ‘ =πππ‘ + π
π
π‘
2+π·
π
π‘, π 2ππ‘,π’ =πππ‘ + π
π
π‘
2for π > 2.
Thus, the first claim follows from Theorem 2.1.
Now we prove the second claim by verifying conditions (c)β(d) of Theo-
rem 2.2. In order to verify (c), observe that if (π₯π‘, π¦π‘) β π΅π‘ for some π‘ β π―βthen β
π’βπ―π‘+
(π¦+π‘,π’ππ‘ β π¦βπ‘,π’ππ‘) > 0
as follows from the margin requirement. Summing this inequality with the
self-financing condition and using the superlinearity (concavity and positive
68
homogeneity) of the function π₯ β¦β π₯+ππ‘ β π₯βππ‘, we obtain π₯+π‘ ππ‘ + ππ‘(π₯π‘) >
π₯βπ‘ ππ‘, i. e. (π₯π‘, 0) β π΅π‘.
Using the notation ππ‘ = (1, π2π‘ , . . . , π
ππ‘ ), ππ‘ = (1, π
2
π‘ , . . . , ππ
π‘ ) for the price
vectors, and π·π‘ = (0, π·2π‘ , . . . , π·
ππ‘ ), π·π‘ = (0, π·
2
π‘ , . . . , π·π
π‘ ) for the dividend
rates vectors, we obtain the following chain of inequalities for all π‘ β π―β,π’ β π―π‘+:
π¦+π‘,π’(ππ’ +π·π’) > π¦+π‘,π’ππ‘ Β·min
{ess inf
π,π
πππ’ +π·π
π’
πππ‘
, ess infπ
(1 +π π‘,π’)
}
>ππ¦+π‘,π’ππ‘ Β·max
{ess sup
π,π
ππ
π’ +π·π
π’
ππ
π‘
, ess supπ
(1 +π π‘,π’)
}
> π¦βπ‘,π’ππ‘ Β·max
{ess sup
π,π
ππ
π’ +π·π
π’
ππ
π‘
, ess supπ
(1 +π π‘,π’)
}> π¦βπ‘,π’(ππ’ +π·π’),
where in the second inequality we used (2.8) and in the third inequality we
used the margin requirement. This implies (π¦π‘,π’, 0) β π΅π’ and proves the
validity of condition (c).
Let us prove condition (d) for the constant π΄ defined by
π΄ = π΄ ess supπ,π‘,π’
(1 +π π‘,π’), where π΄ =
(π +π·
π
)2
Β· 1 +π
1βπ.
If |π¦π‘(π)| 6 π΄|π₯π‘(π) β π₯β²π‘(π)| for some π β Ξ©, define π¦β²π‘(π) = 0, otherwise
let π¦β²π‘(π) = π(π)π¦π‘(π), where 0 < π(π) 6 1 is such that |π¦π‘(π) β π¦β²π‘(π)| =π΄|π₯π‘(π)β π₯β²π‘(π)|. Observe that in the latter case we have
(1β π(π))|π¦+π‘ (π)| > |π₯π‘(π)β π₯β²π‘(π)| Β· π΄ π
(1 +π)π(2.9)
as is possible to obtain from the margin requirement: indeed, |π¦βπ‘,π’| 6 |π¦+π‘ | Β·ππ/π, so π΄|π₯π‘ β π₯β²π‘| = |π¦π‘ β π¦β²π‘| = (1β π)|π¦π‘| 6 (1β π)|π¦+π‘ |(1 +ππ/π) a.s.
on the set {π : |π¦π‘(π)| > π΄|π₯π‘(π)β π₯β²π‘(π)|}, which implies (2.9).
Then on the set {π : |π¦π‘(π)| 6 π΄|π₯π‘(π)β π₯β²π‘(π)|} we have
(π₯β²π‘ β π¦β²π‘)+ππ‘ β (π₯β²π‘ β π¦β²π‘)βππ‘ + ππ‘(π₯β²π‘) = π₯β²+π‘ ππ‘ β π₯β²βπ‘ ππ‘ + ππ‘(π₯
β²π‘) > 0.
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On its complement we find
(π₯β²π‘ β π¦β²π‘)+ππ‘ β (π₯β²π‘ β π¦β²π‘)βππ‘ + ππ‘(π₯β²π‘)
> (π₯π‘ β π¦π‘)+ππ‘ β (π₯π‘ β π¦π‘)βππ‘ + ππ‘(π₯π‘) + (π¦π‘ β π¦β²π‘)+ππ‘ β (π¦π‘ β π¦β²π‘)βππ‘
+ (π₯π‘ β π₯β²π‘)+ππ‘ β (π₯π‘ β π₯β²π‘)
βππ‘ + ππ‘(π₯β²π‘ β π₯π‘)
> (1β π(π))(π¦+π‘ ππ‘ β π¦βπ‘ ππ‘)β |π₯π‘ β π₯β²π‘|(π +π·) > 0.
The first inequality follows from that π₯β²π‘ β π¦β²π‘ = π₯π‘ β π¦π‘ + π¦π‘ β π¦β²π‘ + π₯β²π‘ β π₯π‘
and the functions π₯ β¦β π₯+ππ‘ + π₯βππ‘ and π₯ β¦β ππ‘(π₯) are superlinear. The
second inequality is valid because (π₯π‘β π¦π‘)+ππ‘β (π₯π‘β π¦π‘)βππ‘+ππ‘(π₯π‘) > 0 for
(π₯π‘, π¦π‘) β π΅π‘, π¦β²π‘(π) = π(π)π¦π‘(π), and (π₯π‘βπ₯β²π‘)+ππ‘β(π₯π‘βπ₯β²π‘)βππ‘ > β|π₯π‘βπ₯β²π‘|π.
In the last inequality we use that π¦+π‘ ππ‘ β π¦βπ‘ ππ‘ > (1βπ)π¦+π‘ ππ‘ according to
the margin requirement, and apply (2.9).
Consequently, using that π¦β²π‘,π’ β ππ‘,π’ a.s. as follows from the construction
of π¦β²π‘, we get (π₯β²π‘, π¦
β²π‘) β π΅π‘ and |π¦β²π‘ β π¦π‘| 6 π΄|π₯β²π‘ β π₯π‘|, so |π¦β²π‘ β π¦π‘| 6 π΄|π₯β²π‘ β π₯π‘|,
which proves the validity of (d), and completes the proof of the theorem.
2.4 Examples when supporting prices do not exist
We provide two examples, when supporting prices do not exist for certain
hedgeable contracts. In the first example we consider a market with the
margin requirement, whereπ is not small enough, and in the second example
we consider a contract with positive coordinates. These examples show that
the assumptions of Theorem 2.3 cannot be relaxed.
In the both examples we let πΊ be the linear graph with the nodes π‘ =
0, 1 (i. e. the graph 0 β 1) and the probability space (Ξ©,F , π ) consist of
Ξ© = [0, 1], the Borel π-algebra F on Ξ©, and the Lebesgue measure P. We
assume F0 is the trivial π-algebra (i. e. it consists of all sets of probability 0
and 1) and put F1 = F . In the model, two assets are traded in the market
70
with the prices ππ‘ = (π1π‘ , π
2π‘ ), π‘ = 0, 1, given by
π10 = 1/2, π1
1(π) =
β§β¨β©3, 0 6 π < 1/2,
1, 1/2 6 π 6 1,
π20 = 1/2, π2
1(π) = 1 + π.
The cones π΅π‘ are defined by
π΅0 = {(π₯, π¦) : (π₯β π¦)π0 > 0, ππ¦+π0 > π¦βπ0},
π΅1 = {(π₯, π¦) : π₯π1 > 0, π¦ = 0},
where π > 0. The utility functions are
π’0(π₯, π¦) = (π₯β π¦)π0, π’1(π₯, π¦) = E[(π₯β π¦)π1].
Example 1. Suppose π > 1/2. Consider the contract πΎ = (π0, π1) with
π0 = (β1, 0), π1 = (0, 0). Let us show that π(πΎ) = 5/2. Indeed, for any
trading strategy π = (π₯π‘, π¦π‘)π‘=0,1 hedging πΎ we have
π₯10 β π¦10 + π₯20 β π¦20 > 0, π¦10 + 2π¦20 > 0,
where the first inequality is the self-financing condition of the cone π΅0 and
the second inequality follows from that if (π¦0, 0) β‘ (π₯1, 0) β π΅1 for a constant
vector π¦0 then π¦10π11 + π¦20π
21 > 0, which is possible only if π¦10 + 2π¦20 > 0 since
π11(π) = 1, π2
1(π) = 2 for π = 1.
If π hedges πΎ, we have π₯10 = 1, π₯20 = 0, so
π¦10 + π¦20 6 1, π¦10 + 2π¦20 > 0.
Multiplying the first inequality by 2 and subtracting the second one multi-
plied by 1/2 we obtain 3π¦10/2 + π¦20 6 2. On the other hand, for any constant
vectors π₯, π¦ we have
π’0(π₯, π¦) = (π₯1 + π₯2 β π¦1 β π¦2)/2
π’1(π¦, 0) = π¦1Eπ11 + π¦2Eπ2
1 = 2π¦1 + 3π¦2/2,
so π’(π) = π’0(π₯0, π¦0) + π’1(π¦0, 0) = 1/2+ 3π¦10/2+ π¦20 6 5/2. Thus π(πΎ) 6 5/2.
71
The value 5/2 is attained at the strategy with π₯0 = (1, 0), π¦0 = π₯1 = (2,β1),
π¦1 = (0, 0), which hedges πΎ if π > 1/2 as one can verify.
Suppose there exist supporting prices π0, π1 for the contract πΎ. Then for
any (π₯π‘, π¦π‘) β π΅π‘, π‘ = 0, 1, we have
5/2 = π(πΎ) > π’0(π₯0, π¦0) + π’1(π₯1, π¦1)β (π0 + π₯0)π0 + E[(π¦0 β π₯1)π1]. (2.10)
Take the following (π₯π‘, π¦π‘):
π₯0 = (1, 0), π¦0 = (2 + π,β1β π),
π₯1 = (2 + π+ ππ,β1β π), π¦1 = (0, 0),
where π β (0, 2π β 1] is an arbitrary number and the function ππ = ππ(π)
is given by
ππ(π) = π Β· I{π > 1/(1 + π)}
It is straightforward to check that (π₯0, π¦0) β π΅0 and (π₯1, π¦1) β π΅1.
Observe that π’0(π₯0, π¦0) = 0, π’1(π₯1, π¦1) = E[(2 + π+ ππ)π11 β (1 + π)π2
1 ] >
(2 + π)Eπ11 β (1 + π)Eπ1
2 = (2 + π) Β· 2 β (1 + π) Β· (3/2) = 5/2 + π/2, and
E(π¦0 β π₯1)π1 = βπΈπππ11, so from (2.10) we have
5/2 > 5/2 + π/2β Eπππ11.
However, since π1 β πΏβ1 , there is a constant π such that |π11| 6 π a.s., so
βEπππ1 > βπEππ = βππ2/(1 + π), i.e.
5/2 > 5/2 + π/2β π Β· π2/(1 + π),
which is impossible for π > 0 small enough. The contradiction means that
supporting prices do not exist.
Example 2. Let now π > 0 be arbitrary (in particular, when π = 0,
the markets forbids short sales). Consider the contracts πΎπ = (π0, ππ1) with
π0 = (β2, 0), ππ1 = (0, 1 + ππ), where π β [0, 1/2] is an arbitrary number and
ππ is the function from Example 1.
Let us show that π(πΎπ) = 5/2 β π β Eπππ21 . Indeed, for any trading
72
strategy π = (π₯π‘, π¦π‘)π‘=0,1 hedging πΎπ we have
π¦10 + π¦20 6 2, π¦10 + 2(π¦20 β 1β π) > 0,
where the first inequality follows from the self-financing condition of the cone
π΅0, and the validity of the second inequality follows from that π11(π) = 1,
π21(π) = 2 for π = 1.
Multiplying the first inequality by 2 and subtracting the second one mul-
tiplied by 1/2 we obtain 3π¦10/2 + π¦20 6 3β π. Consequently,
π’(π) = π’0(π₯0, π¦0) + π’1(π₯1, π¦1) = π’0(π₯0, π¦0) + π’1(π¦0 + ππ1, 0)
= 1β (π¦10 + π¦20)/2 + Eπ¦10π11 + E(π¦20 β 1β ππ)π
21
= 3π¦10/2 + π¦20 β 1/2β Eπππ21
6 5/2β πβ Eπππ21 ,
so π(πΎ) 6 5/2β πβ π(π2), where π(π2) = Eπππ21 = (2π2 + 3π3/2)/(1 + π)2.
Observe that the value 5/2 β π β Eπππ21 is attained at the trading strategy
with π₯0 = (2, 0), π¦0 = (2β 2π, 2π), π₯1 = (2β 2π, 2πβ 1β ππ), π¦1 = (0, 0).
If supporting prices existed, we would have
π(πΎπ) > π’0(π₯0, π¦0) + π’1(π₯1, π¦1) + (π0 β π₯0)π0 + E(π¦0 β π₯1 + ππ1)π1
for any (π₯π‘, π¦π‘) β π΅π‘, π‘ = 0, 1.
Take π₯0 = π¦0 = (2, 0), π₯1 = (2,β1), π¦1 = (0, 0), which satisfy (π₯π‘, π¦π‘) β π΅π‘,
and give
5/2β πβπ(π2) > 5/2β Eπππ1.
As it was shown in Example 1, we have Eπππ1 = π(π2), so the above inequal-
ity is impossible for π > 0 small enough, implying that supporting prices do
not exist in the model.
73
2.5 Appendix: geometric duality
Let πΏ be a normed linear space and πΏ* its dual. Let π be a real-valued
concave function defined on a convex set π β πΏ.
The steepness of π from a point π₯ β π to a point π₯β² β π, π₯β² = π₯, is
defined by
π π₯(π₯β²) =
π(π₯β²)β π(π₯)
βπ₯β² β π₯β.
An element π β πΏ* is called a support of the function π at a point π₯ β π if
π(π₯β²)β π(π₯) 6 π Β· (π₯β² β π₯) for any π₯β² β π.
The set of all supports of π at π₯ is denoted by Ξ£π₯ (which may be empty).
For a given π₯ β π, consider the two problems:
(1) the primal problem:
find supπ₯β²βπβ{π₯}
π π₯(π₯β²);
(2) the dual problem:
find infπβΞ£π₯
βπβ.
The following theorem connects the primal and the dual problems and
plays an important role in many questions of convex analysis.
Proposition 2.1 (The Duality Theorem, [26]). The set Ξ£π₯ is not empty
if and only if π π₯(π₯β²) is bounded from above. In this case, if π₯ is not a point
where π attains its global maximum, the values of the two problems are equal.
74
Chapter 3
Detection of changepoints in asset prices
In this chapter applications of sequential statistical methods to detecting
trend changes in asset prices are considered. We propose a model where one
holds an asset and wants to sell it before a fixed time π . The price of the
asset initially increases but may start decreasing at an unknown moment of
time. The problem consists in detecting this change in the trend in order to
sell the asset for a high price.
The solution will be based on methods of changepoint (or disorder) de-
tection theory. The optimal moment of selling the asset will be represented
as the first moment of time when some statistic (the Shiryaev-Roberts statis-
tic or, equivalently, the posterior probability process), constructed from the
price sequence, exceeds a certain time-dependent threshold.
The mathematical theory of changepoint detection started developing in
1920-1950s in connection with problems of production quality control and
radiolocation (see a survey in [65]). Later it was also applied in the financial
context (see e. g. [62]). Changepoint detection methods can be divided into
the two groups: Bayesian and minimax methods. In the Bayesian setting, a
changepoint (a moment of disorder) is an unobservable random variable with
a known prior distribution, while in the minimax setting it is an unknown
parameter. In this chapter we follow the Bayesian approach.
The problem we consider was previously studied in [2, 14, 56] in the case
of continuous time when the evolution of an asset price is described by a
geometric Brownian motion, which changes its value of the drift coefficient at
an unknown moment of time (from a βfavourableβ value to an βunfavourableβ
one). The result we obtain extends the results available in the literature to
the case of discrete time. Moreover, unlike the majority of previous results,
which assume the changepoint is exponentially distributed, we do not impose
75
any assumption on its prior distribution.
The first section of the chapter describes the model of asset prices with
changepoints. In Section 3.2 we introduce the basic statistics and formulate
the main theorem, which is proved in Section 3.3. Section 3.4 contains results
of numerical simulations.
3.1 The problem of selling an asset with a changepoint
We propose a model describing the price π of an asset at moments of time
π‘ = 0, 1, . . . , π driven by a geometric Gaussian random walk with logarithmic
mean and variance (π1, π1) up to an unknown moment of time π and (π2, π2)
after π. The moment π will be interpreted as the point when the trend of the
asset changes from a favourable value to an unfavourable one, and is called
a changepoint (or a moment of disorder1).
Let π = (ππ‘)ππ‘=1 be a sequence of i.i.d. (independent and identically dis-
tributed) standard normal random variables defined on a probability space
(Ξ©,F ,P) and let the sequence π = (ππ‘)ππ‘=0 be defined by its logarithmic
increments as follows:
π0 = 1, logππ‘
ππ‘β1=
β§β¨β©π1 + π1ππ‘, π‘ < π,
π2 + π2ππ‘, π‘ > π,for π‘ = 1, 2, . . . , π,
where π1, π2 β R, π1, π2 > 0 are known numbers, π β {1, 2, . . . , π + 1} is an
unknown parameter. The equality π0 = 1 means that the prices are measured
relative to the price at time π‘ = 0 and does not reduce the generality of the
model.
We follow the Bayesian approach and assume that π is a random variable
defined on (Ξ©,F ,P), independent of the sequence ππ‘ and taking values in
the set {1, 2, . . . , π + 1} with known prior probabilities ππ‘ = P(π = π‘). The
value π1 is the probability that the changepoint appears from the beginning
1The term βdisorderβ comes from applications of the theory in production quality controlproblems, where π can represent the moment of equipment breakage (a disorder) and increase ofdefective products.
76
of the sequence ππ‘ , and ππ+1 is the probability that the changepoint does
not appear within the time horizon [0, π ]. The prior distribution function of
π is denoted by πΊ(π‘) =βπ’6π‘
ππ’.
Let ππΌ : R+ β R, πΌ 6 1, be the family of functions defined as follows2:
ππΌ(π₯) = πΌπ₯πΌ for πΌ = 0, π0(π₯) = log π₯.
The problem considered in this chapter consists in finding the moment of
time π which maximises the utility from selling the asset provided one holds
it at time π‘ = 0 and needs to sell it by π‘ = π .
Let F = (Fπ‘)ππ‘=0, Fπ‘ = π(ππ’; π’ 6 π‘), be the filtration generated by the
price sequence π. By definition, a moment π when one sells the asset should
be a stopping time of the filtration F, i. e. π should be a random variable
taking values in the set {0, 1, . . . , π} such that {π : π(π) 6 π‘} β Fπ‘ for any
π‘ = 0, . . . , π . The class of all stopping times π 6 π of F is denoted by M.
The notion of a stopping time reflects the concept that a decision to sell the
asset at time π‘ should be based only on the information obtained from the
βhistoricalβ prices π0, π1, . . . , ππ‘ and should not rely on the βfutureβ prices
ππ‘+1, ππ‘+2, which are unknown at time π‘.
Mathematically, the problem of optimal selling of the asset with respect
to the utility function ππΌ is formulated as the optimal stopping problem
ππΌ = supπβM
EππΌ(ππ ). (3.1)
Its solution consists in finding the value ππΌ and the optimal stopping time
π *πΌ, at which the supremum is attained (if such a stopping time exists).
It will be assumed that
π1 > βπΌπ21
2, π2 < βπΌπ
22
2. (3.2)
In this case the sequence (π’πΌπ‘ )ππ‘=0, π’
πΌπ‘ = ππΌ(ππ‘), is a submartingale3 when
2These functions are from capital growth theory and are known to maximize long run asymp-totic growth of wealth; see [5, 37, 40]. π0(π₯) corresponds to the full Kelly strategy and ππΌ(π₯),πΌ < 0, correspond to fractional Kelly strategies blending cash with the optimal full Kelly portfolio.
3A random sequence (ππ‘)ππ‘=0 is called a submartingale (resp., a supermartingale or amartingale)
with respect to a filtration (Fπ‘)ππ‘=0 if E(ππ‘ | Fπ‘β1) > ππ‘β1 (resp., E(ππ‘ | Fπ‘β1) 6 ππ‘β1 or
E(ππ‘ | Fπ‘β1) = ππ‘β1) for each π‘ = 1, . . . , π .
77
the logarithmic returns of π are i.i.d. N (π1, π21) random variables, and
a supermartingale when they are i.i.d. N (π2, π22) random variables. In
the former case, the value of π’π‘ increases on average as π‘ increases, so it
is profitable to hold the asset, and in the latter case the average value of
π’π‘ decreases meaning that one needs to sell the asset as soon as possible.
Consequently, the random variable π represents the moment of time when
holding the asset becomes unprofitable. Note that π is not measurable with
respect to Fπ‘ (except the trivial case when π is a constant random variable),
which can be interpreted as the unobservability of the changepoint.
It is also reasonable to assume that π1 > 0 β otherwise it is clear that
one should never stop at time π‘ = 0 (i. e. the optimal stopping time π *πΌ > 1),
and the problem can be reduced to the smaller time horizon π‘ = 1, 2, . . . , π .
Nevertheless, the result of the main theorem in the next section will be still
valid if π1 = 0.
3.2 The structure of optimal selling times
In order to formulate the main result of the chapter, introduce auxiliary
notation. Let π = (ππ‘)ππ‘=1 denote the logarithmic returns of the prices:
ππ‘ = logππ‘
ππ‘β1, π‘ = 1, . . . , π.
On the space (Ξ©,Fπ ) introduce the family of probability measures Pπ’,
π’ = 1, . . . , π+1, generated by the sequence π with the value of the parameter
π β‘ π’. Following the standard notation of the changepoint detection theory,
let4 Pβ β‘ Pπ+1, and denote by Pπ‘ = P | Fπ‘, Pπ’π‘ = Pπ’ | Fπ‘ the restrictions
of the corresponding measures to the π-algebra Fπ‘.
4Typically, Pβ denotes the measure when there is no change in the probability law of theobservable sequence on the whole time horizon (i. e. the change βoccursβ at time π‘ = β). Sincein the problem considered the time horizon is finite, this measure has the same meaning as Pπ+1.
78
Introduce the ShiryaevβRoberts statistic π = (ππ‘)ππ‘=0:
π0 = 0, ππ‘ =π‘β
π’=1
πPπ’π‘
πPβπ‘
ππ’ for π‘ = 1, . . . , π.
Using that the density πPπ’π‘ /πP
βπ‘ is given by the formula
πPπ’π‘
πPβπ‘
=
(π1π2
)π‘βπ’+1
Β· exp( π‘β
π=π’
[(ππ β π1)
2
2π21β (ππ β π2)
2
2π22
]), π’ 6 π‘,
πPπ’π‘
πPβπ‘
= 1, π’ > π‘,
it is straightforward to check that ππ‘ satisfies the following recurrent formula:
ππ‘ = (ππ‘ + ππ‘β1) Β·π1
π2exp
((ππ‘ β π1)
2
2π21β (ππ‘ β π2)
2
2π22
), π‘ = 1, . . . , π. (3.3)
Define recurrently the family of functions ππΌ(π‘, π₯) for πΌ 6 1, π‘ = π, π β1, . . . , 0, π₯ > 0 as follows. For πΌ = 0 let
π0(π, π₯) β‘ 0,
π0(π‘, π₯) = max{0, π2(π₯+ ππ‘+1) + π1(1βπΊ(π‘+ 1)) + π0(π‘, π₯)
},
where the function π0(π‘, π₯) is given by
π0(π‘, π₯) =
β«Rπ0
(π‘+ 1,
(ππ‘+1 + π₯) Β· π1
π2exp
((π§ β π1)
2
2π21β (π§ β π2)
2
2π22
))Γ 1
π1β2π
exp
(β(π§ β π1)
2
2π21
)ππ§.
For πΌ = 0 define
ππΌ(π, π₯) β‘ 0,
ππΌ(π‘, π₯) = max{0, πΌπ½π‘
[(πΎ β 1)(π₯+ ππ‘+1) + (π½ β 1)(1βπΊ(π‘+ 1))
]+ ππΌ(π‘, π₯)
},
where
π½ = exp
(πΌπ1 +
πΌ2π212
), πΎ = exp
(πΌπ2 +
πΌ2π222
),
79
and
ππΌ(π‘, π₯) =
β«RππΌ
(π‘+ 1, (ππ‘+1 + π₯) Β· π1
π2exp
((π§ β π1)
2
2π21β (π§ β π2)
2
2π22
))Γ 1
π1β2π
exp
(β(π§ β π1 β πΌπ21)
2
2π21
)ππ§.
Below it will be shown that ππΌ(π‘, π₯) + πΌ represents the maximal expected
gain one can obtain from selling the asset if the observation is started as time
π‘ with the value of the ShiryaevβRoberts statistic equal to π₯. In particular,
the value ππΌ in problem (3.1) is equal to ππΌ(0, 0) + πΌ.
The main result of the chapter consists in the following theorem about
the structure of optimal stopping times.
Theorem 3.1. The following stopping time is optimal in problem (3.1):
π *πΌ = inf{0 6 π‘ 6 π : ππ‘ > π*πΌ(π‘)}, (3.4)
where the stopping boundary π*πΌ(π‘), π‘ = 0, . . . , π , is given by
π*πΌ(π‘) = inf{π₯ > 0 : ππΌ(π‘, π₯) = 0}.
The value ππΌ of problem (3.1) is equal to ππΌ(0, 0) + πΌ.
The theorem states that the optimal stopping time is the first moment of
time when the ShiryaevβRoberts statistic exceeds the time-dependent thresh-
old π*πΌ(π‘). In order to find the function π*πΌ(π‘) numerically, one first computes
the functions ππΌ and then finds their minimal non-negative roots.
Note that it is also possible to express the optimal stopping time through
the posterior probability process π = (ππ‘)ππ‘=0 defined as the conditional prob-
ability
ππ‘ = P(π 6 π‘ | Fπ‘).
Indeed, by the generalised Bayes formula (see e. g. [39, S 7.9]) we have
ππ‘ =
π‘βπ’=1
πPπ’π‘
πPπ‘ππ’
π+1βπ’=1
πPπ’π‘
πPπ‘ππ’
, 1β ππ‘ =
π+1βπ’=π‘+1
πPπ’π‘
πPπ‘ππ’
π+1βπ’=1
πPπ’π‘
πPπ‘ππ’
=(1βπΊ(π‘))
πPβπ‘
πPπ‘π+1βπ’=1
πPπ’π‘
πPπ‘ππ’
.
80
This implies
ππ‘ =ππ‘
1β ππ‘(1βπΊ(π‘)), ππ‘ =
ππ‘
ππ‘ + 1βπΊ(π‘).
Consequently, the optimal stopping time in problem (3.1) can be equivalently
represented in the form
π *πΌ = inf{0 6 π‘ 6 π : ππ‘ > π*πΌ(π‘)}, where π*πΌ(π‘) = π*πΌ(π‘)
π*πΌ(π‘) + 1βπΊ(π‘).
This representation provides a clear interpretation of the optimal stopping
time: one should sell the asset as soon as she becomes sufficiently confident
that the disorder has already happened, which quantitatively means that the
posterior probability ππ‘ exceeds the threshold π*πΌ(π‘).
3.3 The proof of the main theorem
The proof of the theorem will be given in two steps. First, problem (3.1)
will be reduced to an optimal stopping problem for the ShiryaevβRoberts
statistic π with respect to a new probability measure on (Ξ©,Fπ ). Then we
show that ππ‘ is a Markov sequence with respect to this measure and apply
methods of Markovian optimal stopping theory to the problem.
Step 1. On the measure space (Ξ©, Fπ ) introduce the family of probability
measures QπΌ, πΌ 6 1, such that the logarithmic returns ππ‘, π‘ = 1, . . . , π , are
i.i.d π© (π1 + πΌπ21, π21) random variables under QπΌ. In particular, Q0 β‘ Pβ.
For π‘ = 1, . . . , π , the explicit formula for the density πPπ‘/πQπΌπ‘ , where QπΌ
π‘ =
QπΌ | Fπ‘, is given by the formula
πPπ‘
πQπΌπ‘
=πPπ‘
πQ0π‘
Β· πQ0π‘
πQπΌπ‘
=
(π+1βπ’=1
πPπ’π‘
πPβπ‘
ππ’
)Β· πQ
0π‘
πQπΌπ‘
=
( π‘βπ’=1
πPπ’π‘
πPβπ‘
ππ’ +π+1β
π’=π‘+1
ππ’
)Β· exp
(πΌπ1π‘+
πΌ2π212
π‘β πΌ
π‘βπ’=1
ππ
)
= (ππ‘ + 1βπΊ(π‘))π½π‘ Β· exp(βπΌ
π‘βπ’=1
ππ
).
81
We show that for any stopping time π β M it holds that
Eπ0(ππ ) = EQ0
[ πβπ‘=1
[π2ππ‘ + π1(1βπΊ(π‘))
]], (3.5)
EππΌ(ππ ) = πΌEQπΌ
[ πβπ‘=1
π½π‘β1[(π½ β π½/πΎ)ππ‘ + (π½ β 1)(1βπΊ(π‘))
]]+ πΌ for πΌ = 0,
(3.6)
where EQπΌ
denotes the expectation with respect to QπΌ. Here and below, we
defineπβ
π‘=1[. . .] = 0 if π = 0.
In order to prove (3.5), observe that
Eπ0(ππ ) = Eπβ
π‘=1
ππ‘ = EQ0
[πPπ
πQ0π
πβπ‘=1
ππ‘
]= EQ0
[(ππ + 1βπΊ(π))
πβπ‘=1
ππ‘
].
Using the βdiscrete versionβ of the integration by parts formula,
ππ‘ππ‘ =π‘β
π =1
ππ Ξππ +π‘β
π =1
ππ β1Ξππ + π0π0, (3.7)
valid for any sequences ππ‘ and ππ‘ with the notation Ξππ = ππ β ππ β1, Ξππ =
ππ β ππ β1, we obtain
Eπ0(ππ ) = EQ0
[ πβπ‘=1
(ππ‘ + 1βπΊ(π‘))ππ‘ +πβ
π‘=1
π‘β1βπ =1
ππ (ππ‘ β ππ‘β1 β ππ‘)
]. (3.8)
The sequence ππ‘ + 1 β πΊ(π‘) is a martingale with respect to the measure
Q0 since it is the sequence of the densities πPπ‘/πQ0π‘ . This implies that the
expectation of the second sum in the above formula is zero. Indeed, we have
EQ0
[ πβπ‘=1
π‘β1βπ =1
ππ (ππ‘ β ππ‘β1 β ππ‘)
]
= EQ0
[π+1βπ‘=1
π‘β1βπ =1
EQ0[ππ (ππ‘ β ππ‘β1 β ππ‘)I(π‘ 6 π) | Fπ‘β1
]]= 0,
where we use that ππ and I(π‘ 6 π) are Fπ‘β1-measurable random variables,
82
so they can be taken out of the conditional expectation, and
EQ0
(ππ‘ β ππ‘β1 β ππ‘ | Fπ‘β1) = 0, π‘ = 1, . . . , π,
as follows from that ππ‘ + 1βπΊ(π‘) is a martingale.
For the first sum in (3.8) we have
EQ0
[ πβπ‘=1
(ππ‘ + 1βπΊ(π‘))ππ‘
]
= EQ0
[π+1βπ‘=1
EQ0((ππ‘ + 1βπΊ(π‘))ππ‘I(π‘ 6 π) | Fπ‘β1
)]= EQ0
[ πβπ‘=1
[EQ0
(ππ‘ + ππ‘β1)π2 + π1(1βπΊ(π‘))]]
= EQ0
[ πβπ‘=1
[π2ππ‘ + π1(1βπΊ(π‘))]
].
In the second equality we use that I(π‘ 6 π) is an Fπ‘β1-measurable random
variable and it can be taken out of the conditional expectation, ππ‘ is indepen-
dent of Fπ‘β1, so EQ0
[(1βπΊ(π‘))ππ‘ | Fπ‘β1] = (1βπΊ(π‘))EQ0
ππ‘ = (1βπΊ(π‘))π1,and, as follows from (3.3),
EQ0
(ππ‘ππ‘ | Fπ‘β1) = (ππ‘ + ππ‘β1)π1π2
EQ0
[ππ‘ exp
((ππ‘ β π1)
2
2π21β (ππ‘ β π2)
2
2π22
)]= (ππ‘ + ππ‘β1)π2.
In the third equality we use the representation
EQ0
[ πβπ‘=1
(ππ‘ + ππ‘β1)
]= EQ0
[π+1βπ‘=1
EQ0(ππ‘ | Fπ‘β1
)I(π‘ 6 π)
]
= EQ0
[π+1βπ‘=1
EQ0(ππ‘I(π‘ 6 π) | Fπ‘β1
)]= EQ0
[ πβπ‘=1
ππ‘
].
This proves formula (3.5).
83
Let us prove (3.6). We have
EππΌ(ππ ) = πΌE exp
(πΌ
πβπ‘=1
ππ‘
)= πΌEQπΌ
[πPπ
πQπΌπ
exp
(πΌ
πβπ‘=1
ππ‘
)]= πΌEQπΌ[
(ππ + 1βπΊ(π))π½π]
= πΌEQπΌ
[ πβπ‘=1
[(ππ‘ + 1βπΊ(π‘))(π½π‘ β π½π‘β1) + π½π‘β1(ππ‘ β ππ‘β1 β ππ‘)
]]+ πΌ,
where in the third equality we use the formula for the conditional density
πPπ‘/πQπΌπ‘ , and in the last equality we use formula (3.7). Using representation
(3.3), it is possible to verify the following expression for the conditional
expectation EQπΌ
(ππ‘ | Fπ‘β1):
EQπΌ
(ππ‘ | Fπ‘β1) =πΎ
π½(ππ‘β1 + ππ‘) for π‘ = 1, . . . , π. (3.9)
To check that the right-hand side (π π») of above expression for EππΌ(ππ ) co-
incides with the right-hand side (π π») of (3.6), we show that their difference
is equal to zero:
(π π»)β (π π») = πΌEQπΌ
[ πβπ‘=1
π½π‘β1(ππ‘π½/πΎ β ππ‘β1 β ππ‘)
]
= πΌEQπΌ
[π+1βπ‘=1
π½π‘β1(ππ‘π½/πΎ β ππ‘β1 β ππ‘)I(π‘ 6 π)
]
= πΌEQπΌ
[π+1βπ‘=1
EQπΌ[π½π‘β1(ππ‘π½/πΎ β ππ‘β1 β ππ‘)I(π‘ 6 π) | Fπ‘β1
]]
= πΌEQπΌ
[π+1βπ‘=1
π½π‘β1(EQπΌ
(ππ‘ | Fπ‘β1)π½/πΎ β ππ‘β1 β ππ‘)I(π‘ 6 π)
]= 0,
where in the fourth equality we use that ππ‘β1 and I(π‘ 6 π) are Fπ‘β1-
measurable random variables, so their conditional expectations coincides
with themselves, and apply (3.9). This proves (3.6).
Step 2. For convenience of further notation, let πΉπΌ(π‘, π) denote the terms
in the sums in (3.5)β(3.6):
πΉ0(π‘, π₯) = π2π₯+ π1(1βπΊ(π‘)),
πΉπΌ(π‘, π₯) = πΌπ½π‘β1[(π½ β π½/πΎ)π₯+ (π½ β 1)(1βπΊ(π‘))].(3.10)
84
Representations (3.5)β(3.6) allow to reduce problem (3.1) to the optimal
stopping problems for the ShiryaevβRoberts statistic π
ππΌ = supπβM
EQπΌ
[πβ
π’=1
πΉπΌ(π’, ππ’)
]+ πΌ,
so that the optimal stopping times in these problems will be also optimal in
problem (3.1).
Let Mπ‘ denote the class of all stopping times π of the filtration F such
that π 6 π‘. In particular, Mπ = M.
According to the results of [64, Ch. II, S 2.15], the ShiryaevβRoberts statis-tic is a Markov sequence with respect to the filtration F under each measure
QπΌ, πΌ 6 1, since ππ‘ is a function of ππ‘β1 and ππ‘, while ππ‘ form a sequence
of independent random variables. Following the general theory of optimal
stopping of Markov sequences (see e. g. [47, Ch. I]), introduce the family of
the value functions ππΌ(π‘, π₯) for π‘ β {0, 1, . . . , π}, π₯ > 0:
ππΌ(π‘, π₯) = supπβMπβπ‘
EQπΌ
[πβ
π’=1
πΉπΌ(π‘+ π’, ππ’(π‘, π₯))
], (3.11)
where π(π‘, π₯) = (ππ’(π‘, π₯))πβπ‘π’=0 is a sequence of random variables defined by
the recurrent formula
π0(π‘, π₯) = π₯,
ππ’(π‘, π₯) = (ππ‘+π’ + ππ’β1(π‘, π₯)) Β·π1π2
exp
((ππ’ β π1)
2
2π21β (ππ’ β π2)
2
2π22
)with π1, π2, . . . being i.i.d. π© (π1 + πΌπ21, π
21) random variables with respect
to the measure QπΌ. The sums in the definition of ππΌ(π‘, π₯) are equal to zero
if π = 0, which, in particular, means that ππΌ(π, π₯) = 0 for any π₯ > 0.
The functions ππΌ(π‘, π₯) represent the maximal possible gain in the optimal
stopping problem if the observation starts at time π‘ with the value of the
ShiryaevβRoberts statistic π₯. From formulae (3.5)β(3.6), it follows that orig-
inal problem (3.1) corresponds to π‘ = 0, π₯ = 0, so the optimal stopping time
for ππΌ(0, 0) will be the optimal stopping time in (3.1), and ππΌ = ππΌ(0, 0)+π.
The well-known result of the optimal stopping theory for Markov se-
85
quences (see [47, Theorem 1.8]) states that the value functions ππΌ(π‘, π₯) satisfy
the following WaldβBellman equations for π‘ = 0, . . . , π β 1:
ππΌ(π‘, π₯) = max{0, EQπΌ
[πΉπΌ(π‘+ 1, π1(π‘, π₯)) + ππΌ(π‘+ 1, π1(π‘, π₯))]}. (3.12)
Here, 0 is the gain from instantaneous stopping in the problems at hand.
Using that ππ‘ are i.i.d N (π1+πΌπ21, π
21) random variables with respect to QπΌ
and computing the expectations EQπΌ
[. . . .] in the above equation, we obtain
that the functions ππΌ(π‘, π₯) satisfy the recurrent relations on p. 79.
From [47, Theorem 1.7] it follows that the optimal stopping time in prob-
lem (3.1) is the first moment of time when ππ‘ enters the stopping set π·πΌ:
π·πΌ = {(π‘, π₯) : ππΌ(π‘, π₯) = 0}, π *πΌ = inf{π‘ > 0 : (π‘, ππ‘) β π·πΌ}.
In order to prove representation (3.4), we show that for fixed π‘ and πΌ, the
function π₯ β¦β ππΌ(π‘, π₯) is continuous and non-increasing, and there exists π₯
such that ππΌ(π‘, π₯) = 0. The non-increasing follows from that ππ’(π‘, π₯1) >
ππ’(π‘, π₯2) whenever π₯1 > π₯2, and the coefficients π2 and πΌπ½π‘(1 β 1/πΎ) are
negative in the formulae for πΉ0(π‘, π₯) and πΉπΌ(π‘, π₯) respectively as follows from
the assumption π2 < βπΌπ22/2 (see (3.2)).
In order to prove the continuity of π₯ β¦β ππΌ(π‘, π₯), we show by induction
over π‘ = π, π β 1, . . . , 0 that for arbitrary 0 6 π₯1 6 π₯2 it holds that
ππΌ(π‘, π₯1)β ππΌ(π‘, π₯2) 6 ππΌπ‘ (π₯2 β π₯1) (3.13)
with the constants
ππΌπ‘ =
β§β¨β©|π2|(π β π‘), πΌ = 0
πΌπ½π‘(1β πΎπβπ‘), πΌ = 0.
For π‘ = π the claim is valid, because ππΌ(π, π₯) = 0 for all π₯ > 0. Suppose it
holds for some π‘ = π and consider π‘ = π β 1. From the formulae on p. 79 it
86
follows that
ππΌ(π β 1, π₯1)β ππΌ(π β 1, π₯2)
6
β§β¨β©|π2|(π₯2 β π₯1) + π0(π β 1, π₯1)β π0(π β 1, π₯2), πΌ = 0,
πΌπ½π β1(1β πΎ)(π₯2 β π₯1) + ππΌ(π β 1, π₯1)β ππΌ(π β 1, π₯2), πΌ = 0.
Further, using the induction assumption for π‘ = π , we find
ππΌ(π β 1, π₯1)β ππΌ(π β 1, π₯2) 6β«R
ππΌπ (π₯2 β π₯1)
π2β2π
exp
((π§ β π1)
2
2π21β (π§ β π2)
2
2π22
)Γ exp
((π§ β π1 β πΌπ21)
2
2π21
)ππ§ = ππΌπ (π₯2 β π₯1)
πΎ
π½.
Combining it with the previous inequality we find for πΌ = 0
ππΌ(π β 1, π₯1)β ππΌ(π β 1, π₯2) 6 (π₯2 β π₯1)(|π2|+ ππΌπ ) = ππΌπ β1(π₯2 β π₯1)
and for πΌ = 0
ππΌ(π β 1, π₯1)β ππΌ(π β 1, π₯2) 6 (π₯2 β π₯1)(πΌπ½π β1(1β πΎ) + ππΌπ πΎ/π½)
= ππΌπ β1(π₯2 β π₯1),
which proves the claim. Since 0 6 ππΌ(π‘, π₯1)β ππΌ(π‘, π₯2) because π₯ β¦β ππΌ(π‘, π₯)
is a non-increasing function, we obtain that it is continuous.
Finally by induction over π‘ = π, π β 1, . . . , 0 we prove that there exists a
root ππΌ,π‘ of the function π₯ β¦β ππΌ(π‘, π₯). For π‘ = π this is true since ππΌ(π, π₯) = 0
for all π₯ > 0. Suppose there exists a root for π‘ = π > 0. Then for π‘ = π β 1
and π₯β +β we have
EQπΌ
ππΌ(π , π1(π β 1, π₯)) 6 sup06π¦6ππΌ,π
ππΌ(π , π¦) Β·QπΌ{π1(π β 1, π₯) 6 ππΌ,π } β 0
since π¦ β¦β ππΌ(π , π¦) is a continuous function and, hence, bounded on the
segment [0, ππΌ,π ], while QπΌ{π1(π β 1, π₯) 6 ππΌ,π } β 0 for π₯ β +β as follows
from the definition of π1(π‘, π₯). On the other hand, πΉπΌ(π β 1, π₯) β ββas π₯ β +β, which follows from that, according to the assumption π2 <
βπΌπ22/2 (see (3.2)), the coefficients π2 or πΌπ½π‘β1(π½ β π½/πΎ) in front of π₯ in
formula (3.10) are negative respectively in the case πΌ = 0 or πΌ = 0. Then
87
from (3.12) we obtain the existence of the root ππΌ,π β1.
This completes the proof of the theorem.
Remark 3.1. Assumption π2 < βπΌπ22/2 was used in the proof of the
theorem to show that the functions π₯ β¦β ππΌ(π‘, π₯) have non-negative roots.
Assumption π1 > βπΌπ21/2 is not necessary for the proof, but if it does not
hold, then problem (3.1) becomes trivial: from the recurrent formula for
ππΌ(π‘, π₯) it is easy to see that ππ(π‘, π₯) = 0 for all π‘ = 0, 1, . . . , π , π₯ > 0, so the
optimal stopping time π *πΌ = 0.
Remark 3.2. Let us provide another proof of the continuity of the func-
tions π₯ β¦β ππΌ(π‘, π₯), which directly uses their definition (3.11) without relying
on the recurrent formulae.
Observe that for arbitrary 0 6 π₯1 6 π₯2 it holds that
0 6 ππΌ(π‘, π₯1)β ππΌ(π‘, π₯2) = ππΌ(π‘, π₯1)β EQπΌ
[π*πΌ(π‘,π₯2)βπ’=1
πΉπΌ(π‘+ π’, ππ’(π‘, π₯2))
]
6 EQπΌ
[π*πΌ(π‘,π₯2)βπ’=1
[πΉπΌ(π‘+ π’, ππ’(π‘, π₯1))β πΉπΌ(π‘+ π’, ππ’(π‘, π₯2))
]]
6
β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ©
πβπ‘βπ’=1
EQ0[π2(ππ’(π‘, π₯1)β ππ’(π‘, π₯2))
], πΌ = 0,
πβπ‘βπ’=1
πΌEQπΌ[π½π‘+π’(1β 1/πΎ)(ππ’(π‘, π₯1)β ππ’(π‘, π₯2))
], πΌ = 0,
where π *πΌ(π‘, π₯) = inf{π’ > 0 : (π‘ + π’, ππ’(π‘, π₯)) β π·πΌ}. From the definition of
π(π‘, π₯), using (3.9), we obtain
EQπΌ
(ππ’(π‘, π₯1)β ππ’(π‘, π₯2)) = EQπΌ[EQπΌ
(ππ’(π‘, π₯1)β ππ’(π‘, π₯2)) | Fπ’β1
]=πΎ
π½EQπΌ
(ππ’β1(π‘, π₯1)β ππ’β1(π‘, π₯2)).
By induction we find EQπΌ
(ππ’(π‘, π₯1)β ππ’(π‘, π₯2)) = (πΎ/π½)π’(π₯1 β π₯2), so
0 6 ππΌ(π‘, π₯1)β ππΌ(π‘, π₯2) 6
β§β¨β©|π2|(π β π‘)(π₯2 β π₯1), πΌ = 0,
πΌπ½π‘(1β πΎπβπ‘)(π₯2 β π₯1), πΌ = 0,
which implies that the functions π₯ β¦β ππΌ(π‘, π₯) are continuous.
88
3.4 Numerical solutions
In this section we describe a method how the functions ππΌ(π‘, π₯) and the
optimal stopping boundaries π*πΌ(π‘) can be found numerically and consider
several examples which illustrate the structure of random walks with change-
points and the structure of the stopping boundaries.
In order to find the functions ππΌ(π‘, π₯) and π*πΌ(π‘) numerically we take a
partition of R+ by points {π₯π}βπ=0, π₯π = πΞ, where Ξ > 0 is a param-
eter, and compute the values π πΌ(π‘, π₯π) approximating ππΌ(π‘, π₯π) and π*πΌ(π‘)
approximating π*πΌ(π‘) by backward induction over π‘ = π, π β 1, . . . , 0.
For π‘ = π , let π πΌ(π, π₯π) = 0 for each π > 0. Suppose π πΌ(π , π₯π), π > 0,
are found for some π > 0. In order to find π πΌ(π β 1, π₯π), π > 0, define for
any π₯ > 0
π πΌ(π , π₯) =ββπ=0
π πΌ(π , π₯π)I{π₯ β [π₯π, π₯π+1)}
and compute the values π πΌ(π β 1, π₯π), π > 0, by formulae on p. 79 with
π πΌ(π , π₯) in place of ππΌ(π , π₯) in the formulae for ππΌ(π , π₯). This can be done
in a finite number of steps, since after we find π such that π πΌ(π β 1, π) =
0 then π πΌ(π β 1, πβ²) = 0 for all πβ² > π. Proceeding by induction over
π‘ = π, π β 1, . . . , 0 we find all the values π πΌ(π‘, π₯π), π > 0. Then for each
π‘ = 0, . . . , π define π*πΌ(π‘) = π₯π0(πΌ,π‘), where π0 = π0(πΌ, π‘) is the smallest
number π0 such that π πΌ(π‘, π₯π0) = 0.
Let us show that the computational errors of the method (i. e. the differ-
ences π πΌ(π‘, π₯) β ππΌ(π‘, π₯) and π*πΌ(π‘) β π*πΌ(π‘)) can be estimated by quantities
proportional to Ξ. It will be assumed that the integral in the formulae for
ππΌ(π‘, π₯) and the elementary functions in the formulae for π πΌ(π‘, π₯) can be
computed exactly (or with negligible errors), so the computational errors
appear only due to the approximation of ππΌ(π‘, π₯) by the functions π πΌ(π‘, π₯).
Introduce the constants πΆπ‘πΌ, π‘ = 0, 1, . . . , π :
πΆπΌπ‘ =
πβπ =π‘
ππΌπ =
β§βͺβͺβ¨βͺβͺβ©|π2|
(π β π‘)(π β π‘+ 1)
2, πΌ = 0,
πΌ
(π½π+1 β π½π‘
π½ β 1β π½π+1 β πΎπ+1(π½/πΎ)π‘
πΎ β π½
), πΌ = 0.
89
Theorem 3.2. 1) For all π‘ = 0, 1, . . . , π , π₯ > 0, the estimate holds:
0 6 π πΌ(π‘, π₯)β ππΌ(π‘, π₯) 6 πΆπΌπ‘ Ξ.
2) For π‘ = 0, 1, . . . , π , the functions π*πΌ(π‘), π*πΌ(π‘) satisfy the inequalities
π*0(π‘)β
(πΆ0
π‘
|π2|+ 2
)Ξ 6 π*0(π‘) 6 π
*0(π‘),
π*πΌ(π‘)β
(πΆπΌ
π‘
πΌπ½π‘(1β πΎ)+ 2
)Ξ 6 π*πΌ(π‘) 6 π
*πΌ(π‘), πΌ = 0.
Proof. 1) The proof of the first statement is conducted by induction
over π‘ = π, π β 1, . . . , 0. For π‘ = π the estimate is true because ππΌ(π‘, π₯) =
π πΌ(π‘, π₯) = 0. Suppose it holds for π‘ = π and let us prove it for π‘ = π β 1.
According to the recurrent formulae on p. 79, 0 6 π πΌ(π β1, π₯)βππΌ(π β1, π₯)
because 0 6 π πΌ(π , π₯) β ππΌ(π , π₯) for all π₯ > 0 as follows from the inductive
assumption.
Let π(π₯) denote the largest π₯π not exceeding π₯. Then for π₯ > 0 we have
π πΌ(π β 1, π₯)β ππΌ(π β 1, π₯)
=[π πΌ(π β 1, π₯)β π πΌ(π β 1, π₯π(π₯))
]+[π πΌ(π β 1, π₯π(π₯))β ππΌ(π β 1, π₯π(π₯))
]+[ππΌ(π β 1, π₯π(π₯))β ππΌ(π β 1, π₯)
]6 0 + πΆπΌ
π Ξ+ ππΌπ β1Ξ = πΆπΌπ β1Ξ,
where we use that the difference in the second line equals zero according to
the definition of π πΌ, the difference in the fourth line is estimated from above
by ππΌπ β1(π₯β π₯π(π₯)) 6 ππΌπ β1Ξ according to (3.13), and for the third line we use
the inequality
π πΌ(π β 1, π₯π(π₯))β ππΌ(π β 1, π₯π(π₯)) 6 ππΌ(π β 1, π₯π)β ππΌ(π β 1, π₯π)
6β«R
πΆπΌπ Ξ
π1β2π
exp
(β(π§ β π1 β πΌπ21)
2
2π21
)ππ§ = πΆπΌ
π Ξ,
where the function ππΌ is defined by the same formula as ππΌ, but with π πΌ in
place of ππΌ. The inequalities obtained prove the inductive step and, conse-
quently, prove statement 1 of the theorem.
90
2) Fix πΌ 6 1. Observe that π*πΌ(π‘) 6 π*πΌ(π‘) because π πΌ(π‘, π₯) > ππΌ(π‘, π₯) for
all π‘ = 0, 1, . . . , π , π₯ > 0.
Let us prove the lower inequalities. First, suppose for some π‘ 6 π there
exists a non-negative integer number π such that
π₯π0(π‘)β1 β π₯π >
β§βͺβͺβ¨βͺβͺβ©πΆ0
π‘
|π2|Ξ, πΌ = 0,
πΆπΌπ‘
πΌπ½π‘(1β πΎ)Ξ, πΌ = 0
(3.14)
(since πΌ is fixed, it is omitted in the notation π0(πΌ, π‘)). Let π(π‘) denote the
largest such integer number. Observe that for any 0 6 π1 6 π2 < π0(π‘) it
holds that
π πΌ(π‘, π₯π1)β ππΌ(π‘, π₯π2) >
β§β¨β©|π2|(π₯π2 β π₯π1), πΌ = 0,
πΌπ½π‘(1β πΎ)(π₯π2 β π₯π1), πΌ = 0,(3.15)
as follows from the definition of π πΌ(π‘, π₯π) by the formulae on p. 79. As a
consequence,
ππΌ(π‘, π₯π(π‘)) > π πΌ(π‘, π₯π(π‘))β πΆπΌπ‘ Ξ > π πΌ(π‘, π₯π(π‘))β π πΌ(π‘, π₯π0(π‘)β1)β πΆπΌ
π‘ Ξ > 0,
where in the first inequality we use statement 1, in the second inequality we
use that π πΌ > 0, and in the last one we apply (3.14)β(3.15). This implies
π*πΌ(π‘) > π₯π(π‘). Using that π(π‘) is the largest integer number satisfying (3.14),
we see that
π₯π(π‘) >
β§βͺβͺβ¨βͺβͺβ©π₯π0(π‘)β1 β
(πΆ0
π‘
|π2|+ 1
)Ξ, πΌ = 0,
π₯π0(π‘)β1 β(
πΆπΌπ‘
πΌπ½π‘(1β πΎ)+ 1
)Ξ, πΌ = 0.
Consequently, if for some π‘ 6 π there exists a non-negative integer number
satisfying (3.14), the lower inequality in statement 2 holds for this π‘ because
π₯π0(π‘)β1 = π*πΌ(π‘)βΞ.
In the opposite case, we have π₯π0(π‘)β1 6 πΆ0π‘Ξ/|π2| if πΌ = 0 and π₯π0(π‘)β1 6
πΆπΌπ‘ Ξ/(πΌπ½
π‘(1 β πΎ)) if πΌ = 0, which also implies the validity of statement 2,
since π*πΌ(π‘) > 0.
91
Now we provide examples of the solutions found numerically for certain
values of the parameters. We let π = 100, the moment of disorder π be
uniformly distributed in the set {1, 2, . . . , 101}, and consider the values of
the parameters π1 = βπ2 = 2/π , π1 = π2 = 1/βπ . The choice of the
parameters π1, π2 of order 1/π and the choice of π1, π2 of order 1/βπ is due
to that, as follows from the invariance principle (see e. g. [35, Theorem 12.9]),
such a random walk with disorder converges weakly to a Brownian motion
with disorder on [0, 1] as π β β, which presents interest in view of the
results of the papers [2, 14, 56].
Figure 1 shows the optimal stopping boundaries for the ShiryaevβRoberts
statistic in the problem with the above values of the parameters and πΌ =
1, 0,β12 ,β1. A larger value of πΌ corresponds to a higher boundary.
Figure 2 presents the optimal stopping boundaries for the posterior prob-
ability process. Similarly, higher boundaries are obtained for larger values
of πΌ = 1, 0,β12 ,β1.
Figures 3 and 4 show the value functions π0(π‘, π₯) for the ShiryaevβRoberts
statistic and π0(π‘, π₯) = π0(π‘, π₯/(π₯ + 1 β πΊ(π‘)) for the posterior probability
process, where π‘ = 10, 20, 30, . . . , 90. Higher lines correspond to smaller
values of π‘.
On Figure 5 we provide three sample paths of the geometric Gaussian
random walk with changepoints at π = 25, 50, 75. These paths are con-
structed from the same realisation of the sequence ππ of i.i.d N (0, 1) random
variables, but with the three different values of π.
The corresponding paths of the Shiryaev-Roberts statistic and the opti-
mal stopping boundary π*0(π‘) are shown on Figure 6. The optimal stopping
times π * = 46, 57, 75 (respectively, for π = 25, 50, 75) are found as the first
moments of time the Shiryaev-Roberts statistic crosses the boundary.
92
0 20 40 60 80 100
0
5
10
15
t
bΞ±(t
)
Figure 1. Stopping boundaries π*πΌ(π‘) for the ShiryaevβRoberts statistic forπΌ = 1, 0,β1
2 ,β1 (higher lines correspond to larger values of πΌ).
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
b~Ξ±(t
)
Figure 2. Stopping boundaries π*πΌ(π‘) for the posterior probability process forπΌ = 1, 0,β1
2 ,β1 (higher lines correspond to larger values of πΌ).
93
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.1
0.2
0.3
0.4
0.5
x
V0(t
, x)
Figure 3. Value functions π0(π‘, π₯) for the ShiryaevβRoberts statistic forπ‘ = 10, 20, . . . , 90 (higher lines correspond to smaller values of π‘).
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
x
V~0(t
, x)
Figure 4. Value functions π0(π‘, π₯) for the posterior probability process forπ‘ = 10, 20, . . . , 90 (higher lines correspond to smaller values of π‘).
94
0 20 40 60 80 100
0
1
2
3
4
5
6
7
t
St
Figure 5. Sample paths of the random walk with changepoints π = 25, 50, 75.
0 20 40 60 80 100
0.0
0.5
1.0
1.5
2.0
2.5
3.0
t
Οt
Figure 6. Paths of the ShiryaevβRoberts statistic with π = 25, 50, 75.
95
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