optimal investment under dynamic risk constraints and partial information

This article was downloaded by: [Seton Hall University]On: 30 November 2014, At: 12:27Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Quantitative FinancePublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/rquf20

Optimal investment under dynamic risk constraintsand partial informationWolfgang Putschögl a & Jörn Sass ba UniCredit Bank Austria AG, Risk Integration , Risk Architecture & Risk Methodologies,Julius Tandler Platz 3, 1090 Vienna, Austriab Department of Mathematics , University of Kaiserslautern , PO Box 3049, 67653Kaiserslautern, GermanyPublished online: 04 Jan 2010.

To cite this article: Wolfgang Putschögl & Jörn Sass (2011) Optimal investment under dynamic risk constraints and partialinformation, Quantitative Finance, 11:10, 1547-1564, DOI: 10.1080/14697680903193413

To link to this article: http://dx.doi.org/10.1080/14697680903193413

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/rquf20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/14697680903193413

http://dx.doi.org/10.1080/14697680903193413

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Quantitative Finance, Vol. 11, No. 10, October 2011, 1547–1564

Optimal investment under dynamic risk

constraints and partial information

WOLFGANG PUTSCHOGLy and JORN SASS*z

yUniCredit Bank Austria AG, Risk Integration, Risk Architecture & RiskMethodologies, Julius Tandler Platz 3, 1090 Vienna, AustriazDepartment of Mathematics, University of Kaiserslautern,

PO Box 3049, 67653 Kaiserslautern, Germany

(Received 3 April 2008; in final form 6 July 2009)

We consider an investor who wants to maximize expected utility of terminal wealth. Stockreturns are modelled by a stochastic differential equation with non-constant coefficients.If the drift of the stock returns depends on some process independent of the driving Brownianmotion, it may not be adapted to the filtration generated by the stock prices. In such a modelwith partial information, due to the non-constant drift, the position in the stocks variesbetween extreme long and short positions making these strategies very risky when tradingon a daily basis. To reduce the corresponding shortfall risk, motivated by Cuoco, He andIssaenko [Operations Research, 2008, 56, pp. 358–368.] we impose a class of risk constraints onthe strategy, computed on a short horizon, and then find the optimal policy in this class.This leads to much more stable strategies that can be computed for both classical drift models,a mean reverting Ornstein–Uhlenbeck process and a continuous-time Markov chain withfinitely many states. The risk constraints also reduce the influence of certain parametersthat may be difficult to estimate. We provide a sensitivity analysis for the trading strategy withrespect to the model parameters in the constrained and unconstrained case. The results areapplied to historical stock prices.

Keywords: Portfolio optimization; Utility maximization; Risk constraints; Limited expectedshortfall; Hidden Markov model; Partial information

1. Introduction

We formulate a financial market model which consists

of a bank account with stochastic interest rates and n

stocks whose returns satisfy a Stochastic Differential

Equation (SDE) with a stochastic drift process. The

investor’s objective is to maximize the expected utility

of terminal wealth over a finite time horizon.If the drift process is not adapted to the filtration of the

driving Brownian motion and the investor can only use

the information he gets from observing the stock

prices, this leads to a model with partial information.

To compute strategies explicitly under this realistic

assumption, we have to specify a model for the drift

of the stocks, typically with some linear Gaussian

Dynamics (GD) (see for example Lakner 1998, Pham

and Quenez 2001, Brendle 2006, Putschogl and Sass 2008)

or as a continuous-time Markov chain with finitely many

states. The filter for the first model is called a Kalman

filter. The latter model was proposed in Elliott and Rishel

(1994) and we refer to it as the Hidden Markov Model

(HMM); the corresponding filter is called the HMM

filter. It satisfies a lot of stylized facts observed in stock

markets (cf. Ryden et al. 1998 for related regime switching

models). Efficient algorithms for estimating the param-

eters of this model are available (cf. Elliott 1993, James

et al. 1996, Hahn et al. 2007a). It has also been used

in the context of portfolio optimization (e.g. in Sass and

*Corresponding author. Email: [email protected]

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online � 2011 Taylor & Francis

http://www.tandfonline.comhttp://dx.doi.org/10.1080/14697680903193413

Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

Haussmann 2004, Martinez et al. 2005, Rieder andBauerle 2005, Putschogl and Sass 2008). The filters canbe described as the solution of one Stochastic DifferentialEquation (SDE) in the HMM case or by one SDE andan ordinary differential equation for the second momentin the Kalman case. For more models which allowfor finite-dimensional filters we refer to Sekine (2006).We also allow in our stock models for non-constantvolatility (cf. Hobson and Rogers 1998, Haussmann andSass 2004, Hahn et al. 2007b).

Various risk measures have been applied recentlyto measure and control the risk of portfolios. Usuallythe risk constraint is static, i.e. a risk measure likeValue-at-Risk (VaR) or some tail-expectation-basedrisk constraint like Limited-Expected-Loss (LEL) orLimited-Expected-Shortfall (LES) has to hold at theterminal trading time (see, for example, Duffie andPan 1997, Basak and Shapiro 2001, Gundel and Weber2008, Gabih et al. 2009). In Cuoco et al. (2007), a riskconstraint is applied dynamically for a short horizonwhile the investor strives to solve the optimizationproblem with a longer time horizon. An optimal con-sumption and investment problem with a dynamic VaRconstraint on the strategy is studied in Yiu (2004). In bothpapers dynamic programming techniques are applied andthen numerical methods are used to solve the resultingHamilton–Jacobi–Bellman equation. Their findings indi-cate that dynamic risk constraints are a suitable methodfor reducing the risk of portfolios. In Pirvu (2007), themodel of Yiu (2004) is generalized. For risk managementunder partial information we also refer to Runggaldierand Zaccaria (2000). Also in practice (motivated by BaselCommittee proposals) it is common to reevaluate riskconstraints frequently (e.g. on a daily or weekly basis)with a short time horizon (cf. Jorion 2000). Motivated byCuoco et al. (2007), we impose a slightly different classof risk constraints on the strategy, computed on a shorthorizon. An additional motivation for using dynamicrisk constraints is that they can be specified in such a waythat they limit the risk caused by trading at discrete times.Due to the non-constant drift in the models with partialinformation, portfolio optimizing strategies might haveextreme long and short positions. This is no problemwhen trading continuously, since these strategies adaptto the continuous price movements. But if we tradeonly, say, daily, this can already lead to severe losses,e.g. having a long position of 400% leads to bankruptywhen stock prices fall by more than 25%, as we observefor the market data in section 9.3. Also extreme shortpositions pose a problem. In section 5 we will see thatdynamic risk constraints can limit these positions andthus put a bound on this discretization error. Moreprecisely, these dynamic risk constraints lead to convexconstraints on the strategy. These may depend on time,e.g. for non-constant volatility models. A verificationresult for finding an optimal strategy can then beprovided analogously to the classical theory for convexconstraints using implicitly the separation principle forfiltering which states that we can do filtering first andoptimization afterwards (cf. Gennotte 1986). In special

cases also the existence of a solution can be guaranteed.For logarithmic utility an analytic solution can bederived in a fairly general market model under partialinformation.

Further, the risk constraints reduce the influenceof certain parameters which may be difficult to estimate.We investigate the impact of inaccuracy that a parameter,e.g. the volatility of the stocks, has on the strategy andcompare the results of the constrained case with thoseof the unconstrained case. Finally, the results are appliedto historical stock prices. The results under dynamicrisk constraints indicate that they are a suitable remedyfor reducing the risk and improving the performanceof trading strategies.

The paper is organized as follows. In section 2 weintroduce the basic model and the risk neutral measurewhich we need for filtering and optimization. In section 3we introduce the optimization problem of maximizingexpected utility from terminal wealth. We show howto use time-dependent convex constraints in section 4.In section 5 we apply risk constraints dynamically on thestrategy and show how to derive optimal strategiesunder those constraints using the results of section 4.We introduce Gaussian dynamics for the drift in section 6and a hidden Markov model in section 7. We investigatethe impact of inaccuracy that a parameter has on thestrategy in section 8, where we also compare the resultsof the constrained case with those of the unconstrainedcase. Finally, we provide a numerical example in section 9where we illustrate the strategies in the constrained caseand also apply them to real stock data.

Notation. The symbol > will denote transposition. Fora vector v, Diag(v) is the diagonal matrix with diagonal v.For a matrix M, diag(M) is the vector consisting of thediagonal of the matrix M. We use the symbol 1n for then-dimensional vector whose entries all equal 1. The symbol1n�d denotes the n� d-dimensional matrix whose entries allequal 1. The symbol Idn denotes the n-dimensional identitymatrix. Moreover, FX ¼ ðFX

t Þt2 0,T½ � stands for the filtra-tion of augmented �-algebras generated by the F -adaptedprocess X¼ (Xt)t2[0,T]. We write x� for the negative part ofx: x�¼max {�x, 0}, and xþ for the positive part of x:xþ¼max {x, 0}. We denote the kth component of a vectora by ak. The kth row and column of a matrix A are denotedby (A)k . and (A). k, respectively.

2. The basic model

In this section we outline the basic market model.We have to start with general conditions that allow usto change from the original measure to the risk neutralmeasure. Filtering and optimization will then be doneunder the risk neutral measure.

Let (�,A, P) be a complete probability space, T40 thefixed finite terminal trading time, and F ¼ (F t)t2[0,T ]

a filtration in A satisfying the usual conditions, i.e. F isright-continuous and contains all P-null sets. We caninvest in a money market with constant interest rate r

1548 W. Putschogl and J. Sass

Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

and n risky securities (stocks). The correspondingdiscount factors �¼ (�t)t2[0,T] read �t¼ exp (�rt). Theprice process S¼ (St)t2[0,T], St ¼ ðS

ð1Þt , . . . ,S

ðnÞt Þ>, of the

stocks evolves as

dSt ¼ DiagðStÞð�t dtþ �t dWtÞ, S0 ¼ s0,

where W¼ (Wt)t2[0,T] is an n-dimensional standardBrownian motion with respect to P. The return processR¼ (Rt)t2[0,T] associated with the stocks is defined bydRt¼Diag(St)

�1 dSt, i.e.

dRt ¼ �t dtþ �t dWt:

We assume that the Rn-valued drift process �¼ (�t)t2[0,T]

is progressively measurable with respect to the filtrationF and the R

n�n-valued volatility matrices (�t)t2[0,T] areprogressively measurable w.r.t. FS and �t is non-singularfor all t2 [0,T ].

Definition 2.1: We define the market price of risk�¼ (�t)t2[0,T] by �t ¼ �

�1t ð�t � r 1nÞ and the density process

Z¼ (Zt)t2[0,T] by

Zt ¼ exp

�

Z t

0

�>s dWs�1

2

Z t

0

k�sk2 ds

!, t 2 ½0,T �: ð1Þ

Assumption 2.2: SupposeZ T

0

�k�tk þ k�tk

2�dt51 and

Z T

0

k�tk2 dt51, ðP� a.s.Þ

and that Z is a martingale with respect to the filtration Fand the probability measure P. Further we demandfor �t ¼ ð�

ð1Þt , . . . ,�ðnÞt Þ

Ehk�tk

i51, t 2 ½0,T � and E

Xni¼1

�ðiÞ0

� �4" #51,

i ¼ 1, . . . , n:

Next, we introduce the risk neutral probability measure~P by d ~P ¼ ZT dP, where Z is defined as in equation (1).We denote by ~E the expectation operator under ~P.Girsanov’s theorem guarantees that d ~Wt ¼ dWt þ �t dt isa ~P-Brownian motion with respect to the filtration F .Thus, also the excess return process ~R ¼ ð ~RtÞt2½0,T �,

d ~Rt ¼ dRt � r1n dt ¼ ð�t � rÞdtþ �t dWt ¼ �t d ~Wt,

is a martingale under ~P; and the price process has under ~Pdynamics

dSt ¼ DiagðStÞðr1n dtþ �td ~WtÞ:

We model the volatility as

�t ¼ �ð�tÞ

in terms of the m-dimensional factor process �¼ (�t)t2[0,T]with dynamics

d�t ¼ �ð�tÞdtþ �ð�tÞd ~Wt, ð2Þ

where � and � are Rm-valued. Further, we demand that �,

� and � as well as x}Diag(x)�(x) satisfy the usualLipschitz and linear growth conditions. We cite the nextlemma from proposition 2.1 of Hahn et al. (2007b) butprovide a more detailed proof here.

Lemma 2.3: We have F S ¼ FR ¼ F~W ¼ F

~R.

Proof: Due to the Lipschitz and linear growth condi-tions, the system consisting of d ~Rt ¼ �ð�tÞd ~Wt and ofequation (2) has a unique strong solution ð ~R, �Þ.In particular, F

~Rt � F

~Wt for all t2 [0,T ]. To show the

other inclusion, note that t} �t(!)¼ �(�t(!)) is continuousand hence for At ¼ �t�

>t

Aijt ¼ lim

h&0

1

h

Z t

0

Aijs ds�

Z t�h

0

Aijs ds

� �

¼ limh&0

1

h½ ~R

i, ~R

j�t � ½

~Ri, ~R

j�t�h

� �,

where [X,Y ] denotes the quadratic covariation processof X and Y. In particular, At is F

~Rt -measurable. Choosing

a fixed algebraic scheme to compute the root �t of At, wecan assume – without l.o.g. – that �t is F

~Rt -measurable.

Thus, d ~Wt ¼ ð�tÞ�1 d ~Rt shows that F

~Wt � F

~Rt for all

t2 [0,T ]. Therefore, F~W ¼ F

~R. Using that xi�(x) iscontinuous and

Aijt ¼

limh&01h ½S

i,Sj �t � ½Si,Sj �t�h

� �SitS

jt

,

similar arguments as above imply F S ¼ F~W. Constant r

implies FR ¼ F~R. œ

Remark 1: The filtration FS is the augmented naturalfiltration of ~W and ð ~W,F SÞ is a ~P-Brownian motion.So every claim in L2ð ~P,FS

TÞ can be hedged by classicalmartingale representation results. Thus, the model iscomplete with respect to FS.

Example 2.4: For one stock (n¼ 1) a class of volatilitymodels which satisfy our model assumptions wasintroduced by Hobson and Rogers (1998). As the factorprocesses introduced in equation (2) we use the offsetfunctions �( j ) of order j,

�ð j Þt :¼

Z 10

e�uð ~Rt � ~Rt�uÞjdu, t 2 ½0,T �, j ¼ 1, . . . ,m:

Contrary to Hobson and Rogers (1998), we use in thedefinition of the offset functions the excess return of thestock instead of the discounted log-prices. The offsetfunction of order j can be written recursively as

d�ð j Þt ¼ j�ð j�1Þt d ~Rt þj ð j� 1Þ

2�ð j�2Þt d ½ ~R

�t� �ð j Þt dt

¼

j ð j� 1Þ

2�ð j�2Þt �2t � �

ð j Þt

!dtþ j�ð j�1Þt �t d ~Wt

for j¼ 1, . . . ,m, where �(�1) :¼ 0 and �(0) :¼ 1. Here, [�]denotes the quadratic variation. In the special case m¼ 1we have

d�ð1Þt ¼ �t d ~Wt � �ð1Þt dt:

Optimal investment under dynamic risk constraints 1549

Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

In Hobson and Rogers (1998) the model

�t ¼ �ð�ð1Þt Þ :¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ "

��ð1Þt

�2r^ �

is considered, which we call the HR-model. The minimumand maximum volatility are given by �40 and �4 �,respectively, and "� 0 scales the influence of the offsetfunction. For more details and further examples we referto Hahn et al. (2007b).

3. Optimization

We consider the case of partial information, i.e. we canonly observe the interest rates and stock prices. Therefore,only the events of FS are observable and the portfolio hasto be adapted to FS.

Definition 3.1: A trading strategy ¼ (t)t2[0,T] is ann-dimensional FS-adapted, measurable process satisfyingZ T

0

�jp>s �sj þ k�

>s psk

2�ds51 a.s.

In the definition above, t denotes the wealth invested inthe stocks at time t. We denote the corresponding fractionof wealth invested in the stocks at time t by �pt ¼ pt=Xp

t ,t2 [0,T ]. For initial capital x040 the wealth processXp ¼ ðXp

t Þt2 0,T½ � corresponding to the self-financingtrading strategy is well defined and satisfies

dX pt ¼ p>t ð�t dtþ �t dWtÞ þ ðX

pt � 1>n ptÞr dt, X p

0 ¼ x0:

ð3Þ

Definition 3.2: A trading strategy is called admissiblefor initial capital x040 if X p

t � 0 a.s. for all t2 [0,T ].We denote the class of admissible trading strategies forinitial capital x0 by A(x0).

Definition 3.3: A utility function U :¼ [0,1)!R[ {�1} is strictly increasing, strictly concave, twicecontinuously differentiable, and satisfies limx!1

U 0(x)¼ 0 and limx#0U0(x)¼1. Further, I denotes the

inverse function of U 0.

Assumption 3.4: We demand that I satisfies

Ið yÞ � cð1þ y�aÞ and jI 0ð yÞj � cð1þ y�bÞ ð4Þ

for all y2 (0,1) and for some positive constants a, b, c.

Well known examples for utility functions are thelogarithmic utility function U(x)¼ log(x) and the powerutility function U(x)¼ x�/� for �51, � 6¼ 0. For a givenutility function we can now formulate the followingoptimization problem.

Optimization Problem 3.5: We consider the problem ofmaximizing the expected utility from terminal wealth,i.e. for given U

maximize EhUðX p

TÞ

iover p 2 Aðx0Þ

under the condition E½U�ðXpTÞ�51.

Under partial information with a stochastic drift thisproblem has first been addressed in Lakner (1995, 1998).His results show that the optimal terminal wealth can beexpressed in terms of the conditional density

t ¼ E ZtjFSt

�,

which is the filter for the martingale density. It will beconvenient to denote by ~ ¼ �t t, t2 [0,T ], the discountedconditional density. In the subsequent sections we willsee that the computation of the corresponding optimalstrategy is based on the filter �t ¼ E½�tjF

St � for the drift�t.

Definition 3.6: We introduce the function X : (0,1)}(0,1] by

Xð yÞ ¼ Eh

~ TIð y ~ TÞi: ð5Þ

Theorem 3.7: Suppose that X ( y)51 for everyy2 (0,1). Then there exists a unique number y 2 (0,1)such that X ( y)¼ x0. The optimal terminal wealth reads

XT ¼ Ið y ~ TÞ: ð6Þ

If Ið y ~ TÞ 2 ID1,1 then the unique optimal trading strategyfor optimization problem 3.5 is given by

pt ¼ ��1t �Tð�

>t Þ�1 ~E

hDtX

T

��F St

i¼ ��1t �Tð�

>t Þ�1 ~E

hI0ð y ~ TÞ y

Dt ~ T

��FSt

i:

For a drift process with linear Gaussian dynamics orgiven as a continuous timeMarkov chain explicit solutionsare provided in Lakner (1998) and Sass and Haussmann(2004), respectively. Here D denotes the Malliavinderivative; for a definition of the space ID1,1 and anintroduction to Malliavin calculus we refer to Ocone andKaratzas (1991).

Proof: Lemma 6.5 and theorem 6.6 in Lakner (1995)yield the first statement and equation (6), respectively.The only difference is that we have to look at the dis-counted density ~ , since we consider also a non-zerointerest rate r. That E½U�ðXp

TÞ�51 follows as inKaratzas and Shreve (1998, theorem 3.7.6). Usingmartingale representation arguments, Lakner (1995,theorem 6.6) further shows that the optimal investmentstrategy is uniquely given by

�TXT ¼ x0 þ

Z T

0

�tðt Þ>�t d ~Wt:

On the other hand, for Ið y ~ TÞ 2 ID1,1 Clark’s formula

�TXT ¼ x0 þ

Z T

0

~E½DtXT j F

St �> d ~Wt

holds, see Ocone and Karatzas (1991, proposition 2.1).By comparison we get the representation for . œ

4. Time-dependent convex constraints

Under full information, convex constraints on thestrategy have been examined in detail in Cvitanic and


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

Karatzas (1992), Cvitanic (1997) and under partial

information in Martinez et al. (2005) and Sass (2007).

The latter only consider time-independent constraints and

we cannot use the first results directly, since the model

with partial information does not satisfy their assump-

tions (Brownian filtration). But following Sass (2007)

we can apply filtering techniques and transform our

market model to a model under full information w.r.t.

FS, which is a Brownian filtration – see remark 1. Then

we can adapt the theory of time-dependent constraints

as outlined in Cvitanic and Karatzas (1992, section 16) to

our model.Using the filter

�t ¼ Eh�tjF

St

i,

we can reformulate the model with respect to the

innovation process V¼ (Vt)t2[0,T ] defined by

Vt ¼Wtþ

Z t

0

��1s ð�s� �sÞds¼

Z t

0

��1s dRs �

Z t

0

��1s �s ds:

ð7Þ

We assume FV¼F

S. For FSF

V we would need instead

of the standard martingale representation a representa-

tion for FS-martingales with respect to FV (cf. Sass 2007,

remark 3.4). We can now write our model under full

information with respect to FS as

dRt ¼ �t dtþ �t dVt, t 2 ½0,T �:

The whole theory from time-independent constraints

carries over to the case of time-dependent constraints

with some minor modifications (cf. Cvitanic and Karatzas

1992, section 16). Next, we will impose constraints on �t,the fraction of wealth invested in the stocks at time t.

We define the random set-valued process K0,T¼ (Kt)t2[0,T],

where Kt represents the constraints on portfolio propor-

tions at time t and is given for !2� by a non-empty

closed convex set Kt(!)�Rn that contains 0.

Definition 4.1: A trading strategy is called K0,T-

admissible for initial capital x040 if Xpt � 0 a.s. and

�pt 2 Kt for all t2 [0,T ]. We denote the class of K0,T-

admissible trading strategies for initial capital x0 by

A0,T(x0).

The constrained optimization problem then reads as

follows.

Optimization Problem 4.2: We consider the problem of

maximizing expected utility from terminal wealth, i.e. for

given U and x040

maximize EhUðXp

TÞ

iover p 2 A0,Tðx0Þ

under the condition E½U�ðXpTÞ�51.

For each t we define the support function �t:R

n}R[ {þ1} of �Kt by

�tð yÞ ¼ �tð yjKtÞ ¼ supx2Kt

ð�x>yÞ, y 2 Rn:

For each (t,!) the mapping y} �t(yjKt(!)) is a lowersemi-continuous, convex function on its effective domain~Kt ¼ fy 2 R

n: �tð yÞ51g 6¼ ;. Then, ~Kt is a convex cone,called the barrier cone of �Kt.

Example 4.3:

(i) The unconstrained case corresponds to Kt¼Rn;

then ~Kt ¼ 0f g and �t� 0 on ~Kt for all t2 [0,T ].(ii) Typical time-dependent constraints might be given

by Kt ¼ fx 2 Rn: lðiÞt � xi � u

ðiÞt , i ¼ 1, . . . , ng,

where lðiÞt , u

ðiÞt take values in (�1, 0] and [0,1),

respectively. Then ~Kt ¼ Rn and �tð yÞ ¼Pn

i¼1ðuðiÞt ð y

ðiÞÞ�� lðiÞt ð y

ðiÞÞþÞ.

(iii) Time-dependent constraints might also be givenby closed convex sets Kt�B"t(0) for some "t40,where B"t(0) denotes the ball with centre 0 andradius "t. If Kt is also bounded, then ~Kt � R

n.

We introduce dual processes � : [0,T ]��!R, �t 2 ~Kt,which are FS

t -progressively measurable processes andsatisfy

E

"Z T

0

�k�tk

2 þ �tð�tÞ�dt

#51:

We denote the set of dual processes by H and the subsetof uniformly bounded dual processes by Hb.

Assumption 4.4: For all �t2H, (�t(�t))t2[0,T] is alsoFS

t -progressively measurable.

For example 4.3 (iii), whether assumption 4.4 holdsdepends on the boundary of Kt. Example 4.3 (ii) is aspecial case of example 4.3 (iii) for which assumption 4.4is satisfied if l

ðiÞt , u

ðiÞt are F S

t -progressively measurableand square-integrable (cf. Cvitanic and Karatzas 1992,section 16).

For each dual process �2H we introduce a new interestrate process r�t ¼ rþ �tð�tÞ and the corresponding newdiscount factor is given by ��t ¼ �t expð�

R t0 �sð�sÞdsÞ.

Further, we consider a new drift process ��t ¼ �tþ

�t þ �tð�tÞ1n. Then the new market price of risk reads��t ¼ �

�1t ð�

�t � r�t1nÞ ¼ �

�1t ð�t � r 1n þ �tÞ and the new

density process � ¼ ð �t Þt2½0,T � is given by

�t ¼ exp

�

Z t

0

ð��s Þ> dVs �

1

2

Z t

0

k��sk2 ds

!, t 2 ½0,T �:

Moreover, we introduce the notation ~ �t ¼ ��t �t . If

� is amartingale under P and

R t0 k�

�sk

2 ds51 a.s. for allt2 [0,T ], then P� defined by dP� ¼ �T dP would be aprobability measure and d ~W

�

t ¼ dVt þ ��t dt a Brownian

motion under P�. Note that � would be a martingale,if �2Hb. Thus, we consider a new market with bondprices S0,� and stock prices Si,� for i¼ 1, . . . , n and�¼ (�1, . . . , �n) given by

S0,�t ¼ S

ð0Þt exp

�Z t

0

�sð�sÞds

�,

Si,�t ¼ S

ðiÞt exp

�Z t

0

��sð�sÞ þ �

is

�ds

�:


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

The wealth process X,� then satisfies

��tXp,�t ¼ x0 þ

Z t

0

��sXp,�s ð�

ps Þ>�s d ~W

�

s , Xp,�0 ¼ x0:

Further, we introduce the functions X� as the analogue

of equation (5) where ~ is substituted by ~ �. Given a dual

process � we strive to solve the optimization problem

in the new market under no constraints in the class of

admissible trading strategies for initial capital x0 which

is denoted by A�0,Tðx0Þ.

Assumption 4.5: Let K0,T be such that ~Kt � ~K, �t(� jKt(!))is continuous on ~K for all t2 [0,T ], !2�, and that all

constant, ~K-valued processes � belong to H.

Weaker conditions can be formulated for the case

where 0 =2Kt or ~Kt is not constant (cf. Cvitanic and

Karatzas 1992, assumption 16.2). For constraint sets as in

example 4.3, assumption 4.5 is satisfied (cf. Cvitanic and

Karatzas 1992, section 16). We can now formulate

a proposition like that of Cvitanic (1997, lemma 11.6)

that allows us to compute optimal strategies.

Proposition 4.6: Suppose x040 and E½U�ðXpTÞ�51 for

all 2A0,T(x0). A trading strategy 2A0,T(x0) is optimal

for optimization problem 4.2 if, for some y40, � 2H with

X�51 for all y40,

X pT ¼ Ið y ~ TÞ, X

� ð yÞ ¼ x0,

where ~ T ¼ ~ �

T . Further, � and � have to satisfy the

complementary slackness condition

�tð�t Þ þ ð�

pt Þ>�t ¼ 0, t 2 ½0,T �:

The proof works analogously to the proofs in Cvitanic

and Karatzas (1992) (cf. Cvitanic 1997) and also the direct

proof in Sass (2007, proposition 3.2). In the proof it is

shown that y, � as given in the proposition solve the dual

problem

~Vð yÞ ¼ inf�2H

Eh

~Uð y ~ �TÞi, ð8Þ

where U is the convex dual function of U

~Uð yÞ ¼ supx40

nUðxÞ � xy

o, y4 0:

These proofs are carried out for time-independent

constraints Kt�K; for the differences resulting from

time-dependent constraints, we refer to Cvitanic and

Karatzas (1992, section 16).

Remark 2: Cvitanic and Karatzas (1992, theorem 13.1)

provides conditions that guarantee the existence of an

optimal solution under full information. If FV¼F

S holds

then the same result holds under partial information as in

the case of full information. For power utility U(x)¼x�/�with �2 (0, 1) the conditions of Cvitanic and Karatzas

(1992, theorem 13.1) are satisfied if assumption 4.4 and

4.5 hold. For logarithmic utility, we give in what follows

an explicit representation of the optimal solution under

time-dependent constraints.

5. Dynamic risk constraints

In this section we want to apply risk constraints on the

wealth dynamically. Some motivation for dynamic risk

constraints has been given in the introduction. In Cuoco

et al. (2007) the risk constraint is applied dynamically

under the assumption that the strategy is unchanged for

a short time, i.e. the portfolio manager trades continu-

ously to maintain a constant proportion of his wealth

invested in the risky assets for this short time horizon.

In Yiu (2004) the constraint on the strategy is applied

dynamically for a short time horizon assuming that the

wealth invested in the risky assets remains constant

during this period. Our approach works also for these

approximations. However, we make a slightly different

approximation which is of special interest in particular

when optimizing under partial information. When apply-

ing the optimal strategies to market data we can only

trade at discrete times, e.g. daily, and we observe that very

big short and long positions may occur when using a

non-constant drift model. For daily trading and utility

functions with low risk aversity this may result in

bankruptcies. Therefore, instead of assuming strategy or

wealth to be constant, we use the assumption that the

number of shares remains constant between the discrete

trading times for computing the risk. Then the risk

constraints allow us to measure and control the risk

caused by discretization.We can write the dynamics of the wealth process as

dXpt ¼ N>t DiagðStÞð�t dtþ �t dWtÞ þ ð1� 1>n �

pt ÞX

pt r dt,

where Nt :¼ XptDiagðStÞ

�1�pt represents the number of

stocks in the portfolio.

Limited Expected Loss (LEL) constraint. The wealthprocess satisfies under risk neutral dynamics

dXpt ¼ ð�

pt Þ>Xp

t �t d~Wt þ Xp

t r dt:

If we cannot trade in (t, tþDt), then the difference

DXpt ¼ Xp

tþDt � Xpt reads

DXpt ¼ Xp

t

�1� ð�pt Þ

>1n

�expðrDtÞ þ Xp

t ð�pt Þ>

� exp

"Z tþDt

t

r�1

2diagð�s�

>s Þ

� �ds

þ

Z tþDt

t

�s d ~Ws

#� Xp

t

¼ Xpt exp

�rDt

�� Xp

t þ exp�rDt

�ð�pt Þ

>Xpt

�

"exp

��1

2

Z tþDt

t

diagð�s�>s Þds

þ

Z tþDt

t

�s d ~Ws

�� 1n

#,


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

where we write exp(a)¼ (exp(a1), . . . , exp(an))> for

a¼ (a1, . . . , an)>. Next, we impose the relative LEL

constraint

~EhðDXp

t Þ�jFS

t

i� "t, ð9Þ

with "t ¼ LXpt . Then, any loss in the time interval

[t, tþDt) could be hedged with a fraction L of the

portfolio value at time t, when we assume that – even if we

restrict ourselves to trading at discrete times – we can still

trade at these times with options available in the

continuous-time market. Note that for hedging the

loss we then only need European call and put options.

We define

KLELt :¼

n�pt 2 R

n�� ~EhðDXp

t Þ�jFS

t

i� "t

o:

For n¼ 1 we obtain KLELt ¼ ½�lt, �

ut �, where we can find �ut

and �lt numerically as the maximum and minimum of �for which inequality (9) is satisfied (cf. section 9.1).

Remark 3: For constant volatility and interest rates,

�ut and �lt are time-independent and we can then use the

results from Sass (2007) directly to solve optimization

problem 4.2. Figure 1 illustrates the bounds on the

strategy for various values of L and Dt. Figures 3 and 4

show the bounds for fixed values of L and for �¼ 0.2 and

r¼ 0.02. Figure 5 shows KLEL for n¼ 2 where we used

L¼ 0.01, r¼ 0.02 and � ¼�

0:3 �0:15�0:15 0:3

�.

Limited Expected Shortfall (LES) constraint. Further,we introduce the relative LES constraint as an extension

to the LEL constraint

~EhðDXp

t þ qtÞ�jFS

t

i� "t, ð10Þ

with "t ¼ L1Xpt and qt ¼ L2X

pt . For L2¼ 0 we obtain the

LEL constraint. Then, any loss greater than a fraction L2

of the portfolio value in the time interval [t, tþDt) couldbe hedged with a fraction L1 of the portfolio value at

time t. Again we need for hedging only European call and

put options. We define

KLESt :¼

n�p 2 R

n�� ~EhðDXp

t þ qtÞ�jFS

t

i� "t

o:

Analogously to the LEL constraint we obtain for n¼ 1 the

interval KLESt ¼ ½�lt, �

ut �. Figure 2 shows the bounds on �pt

for various values ofL1 andL2 and for �¼ 0.2 and r¼ 0.02.

Figure 6 shows KLES for n¼ 2 where we used L1¼ 0.01,

L2¼ 0.05, r¼ 0.02 and � ¼�

0:3 �0:15�0:15 0:3

�.

The next lemma guarantees that also in the multi-

dimensional case we are again in the setting of convex

constraints.

Lemma 5.1: KLELt and KLES

t are convex.

Proof: It suffices to prove the lemma for KLESt . Suppose

�t, �0t 2 KLES

t and Xt ¼ X0t 4 0. We want to show for

2 (0, 1) that ��t ¼ �t þ ð1� Þ�0t 2 KLES

t , i.e.

~EhðDX ��t

t þ qtÞ�jFS

t

i� "t,

0 1 2 3 4 50

2

4

6

8

Upp

er b

ound

onη

π

Δ t (in days)

L = 2%

L =1.2%

L = 0.4%

Figure 3. Upper bound on � (LEL).

0

5

0

1

2−10

0

10

L (in % of Wealth)Δ t (in days)

Bou

nds

onη

π

Figure 1. Bounds on � (LEL).

0

5

10

0

1

2−10

0

10

L1 L2

Bou

ndso

nη

π

Figure 2. Bounds on � (LES).


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

where we use the notation

DX ��tt ¼ Xt

�expðrDtÞ � 1þ expðrDtÞð ��tÞ

>:

�

exp

��1

2

Z tþDt

t

diagð�s�>s Þds

þ

Z tþDt

t

�s d ~Ws

�� 1n

��

¼ DX�tt þ ð1� ÞDX�0tt :

Due to the convexity of x} x� and the linearity of ~E we

obtain

~EhðDX ��t þ qtÞ

�jFS

t

i¼ ~E

hðDX�tt þ ð1� ÞDX

�0tt þ qt þ ð1� ÞqtÞ

�jFS

t

i� ~E

hðDX�t þ qtÞ

�jFS

t

iþ ð1� Þ ~E

hðDX�t þ qtÞ

�jFS

t

i� "t þ ð1� Þ"t ¼ "t: h

Reduction of risk. In the introduction we motivated theconstraints above as a remedy to reduce the discretization

error which comes from investing at discrete trading times

only while the market evolves continuously. In this

section we illustrate how these constraints are related

to certain convex risk measures of the terminal wealth.

This analysis shows that by imposing the dynamic risk

constraints motivated from discrete trading, we indeed

control the risk associated with our portfolio. We show

this by an example for the LES constraint and interest

rate r¼ 0.To this end, suppose that we have split [0,T ] in N

intervals [tk, tkþ1] of length Dt¼N�1T. By Xk we denote

the wealth at tk obtained by trading at discrete times

t0, . . . , tk�1. So XN is the terminal wealth when we apply

the continuous-time optimal strategy at these discrete

times only. Then Xkþ1¼XkþDXk, where DXk is defined

as for DX pt for t¼ tk (preceding equation 9). Therefore,

XN ¼ x0 þXN�1k¼0

DXk: ð11Þ

We impose the risk constraints (10) dynamically. Since

for the choices "t ¼ L1Xpt , qt ¼ L2X

pt all terms are

proportional to X pt , (10) does not depend on Xp

t .

Therefore, DXk satisfies the LES constraint

~E DXk þ L2Xkð Þ��FS

tk

h i� L1Xk, k ¼ 0, . . . ,N� 1:

ð12Þ

By adding L2

PN�1k¼0 Xtk � q for q40 on both sides of (11),

taking the negative part, using that x} x� is subadditive

and taking expectation w.r.t. ~P, we get from the LES

constraints

~E XN � qþ L2

XN�1k¼0

Xk

!�" #

¼ ~E x0 � qþXN�1k¼0

DXk þ L2Xkð Þ

!�" #

� ðx0 � qÞ� þXN�1k¼0

~E DXk þ L2Xkð Þ�

½ �

� ðx0 � qÞ� þXN�1k¼0

L1~E½Xk�:

Suppose now that XN is integrable under ~P. Then

Xk ¼ ~E½XN j FSk �, in particular ~E½Xk� ¼ x0. So we have

shown

�ðXNÞ :¼ ~E XN � qþ L2

XN�1k¼0

~E½XN j FStk�

!�" #

� ð ~E½XN� � qÞ� þNL1~E½XN�: ð13Þ

We define the acceptance set

R ¼ fX 2 L1ð ~PÞ : �ðX Þ � ð ~E½X � � qÞ� þNL1~E½X �g:

−2 −1 0 1 2

−2

−1

0

1

2

η 1

η2

Figure 5. KLEL for n¼ 2.

0 1 2 3 4 5−8

−6

−4

−2

0

Low

er b

ound

on

ηπ

Δ t (in days)

L = 2%

L= 1.2%

L = 0. 4%

Figure 4. Lower bound on � (LEL).


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

Then, since (i) inf{m2R :m2R}¼ 051, (ii) Y�X,

X2R implies Y2R and since (iii) R is convex, we can

define a convex risk measure by

�ðX Þ :¼ inffm 2 R : mþ X 2 Rg, X 2 L2ð ~PÞ,

see Foellmer and Schied (2004, section 4.1) for corre-

sponding results, definitions and motivation. So, by

imposing dynamic risk constraints we also put a bound

on a convex risk measure of the terminal wealth. The

limiting cases of inequality (13) have a good interpreta-

tion. For example, for L1¼ �N�1, �40, and L2¼ 0,

inequality (13) corresponds to a bound on the expected

shortfall

�ðXNÞ ¼ ~E½ðXN � qÞ�� ðx0 � qÞ� þ �x0:

This is a risk constraint used in the literature quite often

(cf. the references provided in the introduction). Another

extreme choice would be the portfolio insurer problem.

Say we want to be sure that we get back at least a fraction

�2 (0, 1) of our initial capital, i.e. we require

PðXN � �x0Þ ¼ 1: ð14Þ

Then, by defining L2 as (1�L2)N¼� and choosing L1¼ 0

we get from our dynamic LES constraints (12) at each

time tk that

Xkþ1 � ð1� L2ÞXk ¼ DXk þ L2Xk � 0 ~P� a.s.,

k ¼ 0, . . . ,N� 1: ð15Þ

Since ~P is equivalent to P, this also holds under P and thus

we get by iteration

XN � ð1� L2ÞXN�1 � ð1� L2Þ2XN�2

� � � � � ð1� L2ÞNx0 ¼ �x0:

Therefore, the dynamic risk constraints in constraints (12)

imply equality (14).

Remark 4: Note that the static risk constraint obtained

in (13) from the dynamic constraints only illustrates the

relation to convex risk measures on the terminal wealth,

since the bounds might be very poor. Instead, we get a

sharp bound for the following constraint: by comparing

(12) and (15) we see that we control the maximum shortfall

in one period,

max ~E ðXkþ1 � ð1� L2ÞXkÞ�

½ � : k ¼ 0, . . . ,N� 1n o

� L1x0,

which has a straightforward interpretation. But a defini-

tion corresponding to that for � above would not lead to

a convex risk measure since � would not be monotone.

One can construct examples where a wealth process Y

with Yk�Xk ( ~P� a.s.) might not be preferred to X. The

reason is that Y might have a much higher variance

and thus ~PðYk � ð1� L2ÞYk�1 5Xk � ð1� L2ÞXk�1Þ4 0

can occur. Note that the stronger dependency on the

variance is good in our case since it yields narrower

bounds on the possible long and short positions, and

the discretization error is mainly due to extreme

positions which cannot be adjusted when trading only in

discrete time.

Other constraints. To apply a Value-at-Risk constraintwe proceed as follows. Under the original measure, DXp

t

is given by

DXpt ¼ Xp

t exp�rDt

�� Xp

t þ ð�pt Þ>Xp

t

�

�exp

� Z tþDt

t

�s �

1

2diagð�s�

>s Þ

�ds

þ

Z tþDt

t

�s dWs

�� expðrDtÞ

�:

We impose the relative VaR constraint on the loss ðDXpt Þ�,

P�ðDXp

t Þ�� LXp

t jFSt ,�t ¼ �t

�� , ð16Þ

for some prespecified probability �. Since the probability

is computed under the original measure P we need the

(unknown) value of the drift. As we optimize under

partial information and cannot observe the drift directly

we consider for the drift �t the filter for the drift �t.

At time zero we have no information and we can

assume a stationary distribution for the drift. Under full

information we would not need to make such an

approximation, under partial information this allows us

to keep the model tractable.In the one-dimensional case n¼ 1 the risk constraint

equation (16) results in an upper bound �ut and lower

bound �lt on the strategy �pt and we can find numerically

the maximum and minimum of �pt for which equation (16)

is satisfied. Note that for n41 the set KVaR may not be

convex. For L¼ 0.1, �¼ 0.05, r¼ 0.02 and

B ¼2 0 0:5 �10

2 �10 0:5 0

� �,

Q ¼

�20:16 0:08 20 0:08

20 �40:08 20 0:08

20 0:08 �20:16 0:08

20 0:08 20 �40:08

0BBB@

1CCCA,

� ¼0:04 0:05

0:05 0:05

� �,

Figure 7 illustrates the set KVaR at time t¼ 0 where we

consider a stationary distribution for the drift. The

construction of the example is based on a Markov chain

with extreme states and a correlation structure which –

without further noise – does not allow to have very

long positions in both stocks simultaneously. Due to the

volatility matrix with low volatility level and high

correlation these features carry over to the stock model.

Therefore, if we are very long in one stock, the risk can

only be reduced by choosing a short position for the other

stock, which can be seen very well in Figure 7. At time

t40 when we also have information about the filter

for the drift we can find a modified example for which

KVaR is not convex either.


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

Analogously to the previous case it is possible to apply

a large class of other risk constraints on the strategy like aConditional Value-at-Risk (CVaR) constraint. The CVaR

at a given confidence level is the expected loss given thatthe loss is greater than or equal to the VaR (see, for

example, Rockafellar and Uryasev 2000, 2002). Note thatthe (relative) LEL/LES and the (C)VaR constraints do

not result in a substantially different risk behaviour.This is due to the assumption that we cannot trade in the

time interval for which the constraint is applied. A similarresult would hold if the strategy (i.e. the fraction of wealth

invested in the risky assets) were assumed to be constantin the interval in which the risk constraint is applied.

For a static risk constraint, i.e. if the constraint has to

hold at the terminal time only and we can tradedynamically, the strategies under the LEL and VaRconstraint exhibit quite different risk profiles (cf. Basakand Shapiro 2001).

Optimization. Under dynamic risk constraints we haveto solve optimization problem 4.2 for specific constraintsets Kt, e.g. Kt ¼ KLES

t , t2 [0,T ]. Using proposition 4.6we can show in the one-dimensional case, whereKt ¼ ½�

lt, �

ut �, �

lt 5 05 �ut , the following corollary.

Corollary 5.2: Suppose n¼ 1 and that the boundaries �l

and �u are FS-progressively measurable and squareintegrable. Then for U(x)¼ log(x) the optimal riskyfraction for the constrained problem is given by

�ct :¼ �pt ¼

�ut if �ot 4 �ut ,

�ot if �ot 2h�lt, �

ut

i,

�lt if �ot 5 �lt,

8>><>>:

where

�ot :¼ �pt ¼1

�2tð�t � rÞ:

is the optimal risky fraction without constraints.

So we cut off the strategy obtained under no con-straints if it exceeds �u or falls below �l. For completeness,we provide a direct proof based on proposition 4.6.

Proof: For these constraints, assumptions 4.4 and 4.5are satisfied (cf. the references given after those assump-tions). For U¼ log, we have X �( y)¼ y�1 for all dualprocesses �, hence y ¼ x�10 . The dual process correspond-ing to the strategy in the corollary is

�t ¼

�2t ð�ut � �

ot Þ, if �ot 4 �ut ,

0, if �0t 2 ½�lt, �

ut �,

�2t ð�lt � �

ot Þ, if �ot 5 �lt,

8><>: t 2 ½0,T �,

and one can verify directly that �ct is optimal in theauxiliary market given by �, since the market price ofrisk is ��t ¼ �

�1t ð�t � r� �t Þ ¼ �t�

ct . As in the uncon-

strained Merton problem the corresponding terminalwealth then has the same form as in proposition 4.6.Finally, we can compute

�tð�t Þ ¼ sup

�2½�lt,�ut �

ð��t Þ ¼ ��ct�t ,

so the complementary slackness condition holds. h

Also in the multidimensional case, under assumptions4.4 and 4.5 the problem can be solved up to a pointwiseminimization which characterizes the dual process(cf. Cvitanic 1997, example 12.1). For example, in casessuch as example 4.3 (iii), we have ~K ¼ R

n and thusRockafellar (1970, theorem 10.1) implies that � iscontinuous in y, so assumption 4.5 holds. If �t is, forexample, also continuous in t, this would yield assump-tion 4.4. To get constraint sets and support functions ofsuch a form is mainly due to conditions on the volatility

−5 0 5

−6

−4

−2

0

2

4

6

η1

η2

Figure 6. KLES for n¼ 2.

−20 −10 0 10 20

−20

−10

0

10

20

η 1

η2

Figure 7. KVaR for n¼ 2.


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

matrices �t, e.g. some boundedness from above and away

from 0 would be needed.The strategies corresponding to logarithmic utility

under no constraints maximize the average rate of returns

resulting in risky strategies. The dynamic risk constraints

result in higher risk aversion. In the unconstrained

case it is possible to use Malliavin calculus to obtain

rather explicit representations of the strategies for general

utility. If it is possible under constraints to find the

optimal dual process � and to show that ��TIð y� ~ �TÞ lies

in the domain ID1,1 of the Malliavin differential operator,

then we can use Clark’s formula (cf. Ocone 1984) to

obtain

�pt ¼��t

��1ð�>t Þ

�1E�hDt

��T Ið y

� ~ �TÞ�� FS

t

i,

where E� denotes expectation under P�, and y� is such that

the budget constraint is satisfied. Clearly, this is a difficult

task. One approach would be to formulate the dual

problem (8) as a Markovian stochastic control problem

for controls �. This can be done in terms of the

innovations process (7), if we can compute the filter by

solving a finite number of stochastic differential equations

(see Sass 2007, p. 233, for the HMM). If this problem

can be solved in a first step, one might be able to derive

conditions on � which guarantee ��TIð y� ~ �TÞ 2 ID1,1. This

is subject for future research.

6. Gaussian dynamics for the drift

In this section we model the drift as in Lakner (1998) as

the solution of the SDE

d�t ¼ �ð �� tÞ dtþ �d �Wt , ð17Þ

where �W is an n-dimensional Brownian motion with

respect to (F , P), independent of W under P, and �,�2R

n�n, �� 2 Rn. We assume that � is non-singular and

that �0 follows an n-dimensional normal distribution with

known mean vector �0 and covariance matrix �0.We are in the situation of Kalman filtering with signal

� and observation R, and �t ¼ E½�t j FSt � ¼ E½�t j F

Rt �

and �t ¼ E½ð�t � �tÞð�t � �tÞ>j F S

t � are the unique

FS-measurable solutions of

d�t ¼

h�� tð�t�

>t Þ�1��t þ � ��

idtþ �tð�t�

>t Þ�1 dRt,

ð18Þ

_�t ¼ ��tð�t�>t Þ�1�t � ��t � �t�

> þ ��>, ð19Þ

with initial condition ð�0, �0Þ and � is conditionally

Gaussian (cf. Liptser and Shiryayev 1978, theorem 12.7).

Proposition 6.1: For the GD we have F S ¼ FR ¼

F~W ¼ FV.

Proof: Lemma 2.3 ensures FS ¼ F~W ¼ F

~R and equa-

tion (7) yields FV 7FS. The Ricatti equation (19) has

a continuous unique solution. Further we can write

equation (18) in terms of the innovation process V and the

system

d�t ¼

h�� tð�t�

>t Þ�1þ �tð�t�

>t Þ�1��t þ � ��

idt

þ �tð�t�>t Þ�1�t dVt,

dRt ¼ �t dtþ �t dVt,

has a strong solution since it satisfies the usual Lipschitz

and linear growth conditions (cf. Liptser and Shiryayev

1978, p. 29, Note 3). œ

Hence, we can use the results in section 4 that guarantee

the existence of a solution for certain utility functions and

constraint sets. The conditional density can be repre-

sented in terms of the conditional mean �. The next

theorem corresponds to Lakner (1998, theorem 3.1).

Theorem 6.2: The process �1 ¼ ð �1t Þt2½0,T � satisfies

the SDE

d �1t ¼ �1t ð�t � r1nÞ

>ð�>t Þ

�1 d ~Wt ,

and we have the representation

t ¼ exp

�

Z t

0

ð�s � r1nÞ>ð�>s Þ

�1 d ~Ws

þ1

2

Z t

0

k��1s ð�s � r1nÞk2ds

!:

Assumption 6.3: Suppose that k��1t k, t 2 ½0,T �, is

uniformly bounded by a constant c140. For the constantsa, b of assumption 3.4 let

trð�0ÞþTk�k2 �1

Tc21c2maxt2½0,T� ke��tk2,

where c2�max{360, (8a� 3)2� 1, (16bþ 1)2� 1}.

Note that if � is a positive symmetric matrix then

maxt�T ke��tk2� n (cf. Lakner 1998, p. 84). Depending

on the utility function, this assumption ensures that the

variance of the drift � is small compared to the variance

of the return R. The following lemma states that this is

sufficient for the assumptions of sections 2 and 3 to hold.

Lemma 6.4: For Z as in equation (1), where the drift � isdefined as in equation (17), assumption 2.2 and the

conditions of theorem 3.7 are satisfied.

Proof: The verification of the first part of assumption

2.2 can be done as in the proof of Lakner (1998, lemma

4.1). The proof of the second part follows from the

finiteness of the corresponding moments of the Gaussian

distribution. With some modifications based on the non-

constant volatility and the dependence of assumption 6.3

on the utility function via the constants a and b, it can be

shown similarly as in Lakner (1998, lemma 4.1) alongthe lines of lemmas A.1 and A.2 in Lakner (1998), that

Zs2Lq(P) for q� 5, Z�1s 2 LqðPÞ for q� 4, and

s 2 L5ð ~PÞ, �1s 2 Lpð ~PÞ for p � maxf4, 2a, 4ðb� 1Þg,

s 2 ½0,T �:


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

Following the arguments in Lakner (1998) or Putschogland Sass (2008) this implies Ið y ~ TÞ 2 ID1,1 for y40.As in Putschogl and Sass (2008) it can also be shown that~E½supt2½0,T�

�at �51, which guarantees X (y)51 for all

y40. œ

Remark 5: A special case of GD is the Bayesian case,where we assume that the drift �t � �0 ¼ ð�

ð1Þ0 , . . . ,�ðnÞ0 Þ is

an (unobservable) F 0-measurable Gaussian randomvariable with known mean vector �0 and covariancematrix �0. Then we can even solve the multidimensionalfiltering equation explicitly (cf. Liptser and Shiryayev1978, theorem 12.8), and the solution reads

�t ¼

�1n�n þ �0

Z t

0

ð�s�>s Þ�1 ds

��1

�

��0 þ �0

Z t

0

ð�s�>s Þ�1 dRs

�,

�t ¼

�1n�n þ �0

Z t

0

ð�s�>s Þ�1 ds

��1�0:

7. A hidden Markov model for the drift

In this section we model the drift process � of the returnas a continuous-time Markov chain given by �t¼BYt,where B2R

n�d is the state matrix and Y is a continuous-time Markov chain with state space the standard unitvectors {e1, . . . , ed} in R

d. The state process Y is furthercharacterized by its rate matrix Q2R

d�d, whereQkl ¼ limt!0

1t PðYt ¼ eljY0 ¼ ekÞ, k 6¼ l, is the jump rate

or transition rate from ek to el. Moreover,k ¼ �Qkk ¼

Pdl¼1,l6¼k Qkl is the rate of leaving ek.

Therefore, the waiting time for the next jump isexponentially distributed with parameter k, and Qkl/kis the probability that the chain jumps to el when leavingek for l 6¼ k.

For filtering we need the risk neutral probabilitymeasure ~P introduced in section 2. Let us introduce themartingale density process Z as in equation (1) with�t¼�Yt where �t ¼ �

�1t ðB� r1n�dÞ. Hence, the process Z

satisfies

dZt ¼ �Ztð�sYsÞ>dWs:

Then (�tYt)t2[0,T] is uniformly bounded, Z¼ (Zt)t2[0,T] isa martingale under ~P and assumption 2.2 is satisfied.We are in the situation of HMM filtering with signal Yand observation R. We denote the normalized filter Yt ¼

E½YtjFSt �. Besides the conditional density t ¼ E½ZtjF

St �

we need the unnormalized filter Et ¼ ~E½Z�1T YtjFSt �.

Theorem 7.1 : The unnormalized filter satisfies

Et ¼ E Y0½ � þ

Z t

0

Q>Es dsþ

Z t

0

DiagðEsÞ�>s d ~Ws, t 2 0,T½ �:

The normalized filter is given by Yt ¼ tEt, where �1t ¼ 1>d Et.

This is Haussmann and Sass (2004, theorem 4.2), whichextends Elliott (1993, theorem 4) to non-trivial volatility.

We cite the next proposition from Sass (2007,

proposition 4.5).

Proposition 7.2: For the HMM we have F S ¼ FR ¼

F~W ¼ FV.

Hence, we can use the results in section 4 which

guarantee existence of a solution for certain utility

functions and constraint sets. Next, we cite the following

corollary about the conditional density from

Haussmann and Sass (2004, corollary 4.3) and the

subsequent lemma from Haussmann and Sass (2004,

lemma 2.5, proposition 5.1).

Corollary 7.3: The processes and �1 ¼ ð �1t Þt2½0,T � are

continuous FS-martingales with respect to P and ~P,

respectively. Moreover,

�1t ¼~E Z�1t jF

St

�and

�1t ¼ 1þ

Z t

0

ð�sEsÞ> d ~Ws, t 2 0,T½ �:

Lemma 7.4: For all q� 1 and t2 [0,T ] we have t2Lq(P)

and �1t 2 Lqð ~PÞ.

Due to the boundedness of �, for uniformly bounded

��1t a result like lemma 6.4 holds for the HMM without

any further assumptions.

8. Sensitivity analysis

In this section we investigate the impact of inaccuracy

that a parameter �, e.g. �¼ i, has on the strategy � and

compare the results for the unconstrained case with

several constrained cases.There are several possibilities to compute the sensitiv-

ity. The finite difference method or resampling method

consists in computing E ½�pt�on a fine grid for � and using

a forward or central difference method to get an estimator

of the gradient of E ½�pt�(cf. Glasserman and Yao 1992,

L’Ecuyer and Perron 1994). This method yields biased

estimators and the computations may be time-consuming.

If � is not smooth, Malliavin calculus may be applied

(under some technical conditions) and the sensitivity may

be written as an expression where the derivative of �

does not appear (cf. Gobet and Munos 2005). If the

interchange of differentiation and expectation is justified

we may put the derivative inside the expectation and thus

get an unbiased estimator. This method has been

proposed in Kushner and Yang (1992) and is called the

pathwise method. We provide a sensitivity analysis for the

HMM using the pathwise method. In particular, we want

to compute ðq=q&ÞE ½�pt�:

In the following we compute the sensitivity for the

HMM and logarithmic utility w.r.t. various parameters.

For the GD an analysis can be performed analogously.

Since we want to analyse the HMM we use constant


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

volatility; for ease of notation we set the interest rates

equal zero. In the unconstrained case we consider

qq&

Eh�ot

i¼

qq&

Ehð��>Þ�1�t

i:

Using the filter for the HMM we obtain

qq&

Ehð��>Þ�1BYt

i¼

qq&

Ehð��>Þ�1B tEt

i

¼qq&

~Ehð��>Þ�1BEt

i:

The pathwise derivative of E with respect to � under ~P

which we denote by _E reads

_Et ¼

Z t

0

ð _Q>Es þQ> _EsÞds

þ

Z t

0

�Diagð _EsÞ�

> þDiagðEsÞ _�>�d ~Ws:

The SDE for ðE, _E Þ> satisfies the Lipschitz and linear

growth conditions that ensure the existence of a strong

solution and ~E½j1>d_Etj

2�51 for t2 [0,T ]. Thus, we can

further compute (cf. Gobet and Munos 2005)

qq&

~E ð��>Þ�1BEt �

¼ ~Eqq&ð��>Þ�1B� �

Et þ ð��>Þ�1B _Et

�:

ð20Þ

In the one-dimensional constrained case we consider

below the derivative fails to exist at �ot ¼ �ut and �ot ¼ �

lt.

In our examples this event has probability 0, hence

qq&

Eh�ct

i¼ ~E

"qq&

��2B

�Et þ �

�2B _Et

� �� 1 �ct2ð�

lt,�

ut Þf g

#:

ð21Þ

In what follows, we present a sensitivity analysis for the

parameters involved in computing the strategy; the

parameters used for the calculations are provided in

table 1. For the constrained case we consider the LEL risk

constraint. The results should always be considered with

respect to the absolute value of the parameters, e.g. for �we obtain rather large values – meaning that �t wouldchange by these values if � were to increase by one,

however � taking typically small values is unlikely to be

misspecified by a value close to one. On the other hand,

the jump rates i can take large values and are difficult toestimate (cf. Hahn et al. 2007a). Hence, the sensitivitieswith respect to different parameters cannot be compareddirectly.

Analysis for E0. We choose a state of the normalizedfilter as starting value for the unnormalized filter, i.e.for d¼ 2 we have Et0¼ (x, 1�x)> for x2 [0, 1]. Thesensitivities equation (20) and equation (21) for �¼Et0are illustrated in figure 8. We see that the influence of thestart value Et0 is high at the beginning but decreasesquickly.

Analysis for r. Depending on the filter, the tradingstrategy may be positive or negative, i.e. we may be longor short in the risky assets. Since the trading strategy isbasically inversely proportional to the square of thevolatility it is more reasonable to consider the effect of thevolatility on the absolute value of the strategy insteadof the strategy itself, i.e. we consider the sensitivity of j�ct j.In general, the derivative fails to exist at j�ct j ¼ 0, but thisevent has probability 0. Hence, j�ct j is almost surelydifferentiable with respect to �. The sensitivities equation(20) and equation (21) for �¼ � are illustrated in figure 9.The volatility affects the strategy considerably.

Analysis for Q. For d¼ 2 the rate matrix has the entries�Q11¼Q12¼ 1 and Q21¼�Q22¼ 2. The sensitivitiesequation (20) and equation (21) for �¼ 1 and �¼ 2 areillustrated in figures 10 and 11, respectively. The resultsindicate that the impact of the rate matrix on the strategyis not very high.

Analysis for B. For d¼ 2 we have B¼ (b1,�b2). Thesensitivities equation (20) and equation (21) for �¼ b1and �¼ b2 are illustrated in figures 12 and 13, respectively.The strategy is very sensitive with respect to the states

10 20 30 40 50 60−2

0

2

4

6

8

10

Sens

itivi

ty

Time(in days)

Unconstrained

L =0.5%

L =1%

L =2%

Figure 8. Sensitivity w.r.t. Et0.

Table 1. Parameters for the numerical example.

Time horizon, T 0.25 (three months)Dt 0.004 (daily values)Initial wealth, X p

0 1Parameters for the drift:

State vector, B (0.5, �0.4)>

Rate matrix, Q�15 1515 �15

� �Initial state for thefilter E, E0

(0.5, 0.5)>

(stationary distribution of Y )Volatility, � 0.2


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

of the drift. The constraints help to reduce the influence

of the state vector.For a short discussion on the relevance for portfolio

optimization, please refer to section 9.2.

9. Numerical example

9.1. Implementation

To find the bounds �lt and �ut for one of the constraints

equations (9), (10) and (16) we can proceed as follows.For non-constant volatility we approximate

expf� 12

R tþDtt diagð�s�

>s Þ dsg by expf� 1

2 diagð�t�>t ÞDtg and

expfR tþDtt �s d ~Wsg by expf�tD ~Wtg where D ~Wt ¼

~WtþDt � ~Wt. If the constraints are satisfied for �ot thenthe constraints are not binding and �ct ¼ �

ot (cf. corollary

5.2). Let us consider the case in which the constraintsare not satisfied. The constraints are certainly satisfiedfor �t¼ 0. Hence, we can use a bisection algorithm withstart-interval ½�ot , 0� for �

ot 5 0 or ½0, �ot � for �

ot 4 0 to

obtain �lt and �ut , respectively. The strategy is then given

by �ct ¼ �lt or �ct ¼ �

ut , respectively. To compute the

expected values in equations (9), (10) and (16) we canuse Monte Carlo methods. We only need to do thenecessary simulations once, then it is possible to computethe expected value for different values of �t very easily andthe bisection algorithm works quite efficiently.

10 20 30 40 50 60−0.4

−0.3

−0.2

−0.1

0

Sens

itivi

ty

Time (in days)

Unconstrained

L = 0.5%

L = 1%

L = 2%

Figure 10. Sensitivity w.r.t. 1.

10 20 30 40 50 60−14

−12

−10

−8

−6

−4

−2

Sens

itivi

ty

Time (in days)

Unconstrained

L = 0 .5%

L = 1%

L = 2%

Figure 13. Sensitivity w.r.t. b2.

10 20 30 40 50 60−40

−30

−20

−10

0

Sens

itivi

ty

Time (in days)

Unconstrained

L = 0.5%

L = 1%

L = 2%

Figure 9. Sensitivity w.r.t. �.

10 20 30 40 50 600

0.1

0.2

0.3

0.4

Sens

itivi

ty

Time (in days)

Unconstrained

L = 0.5%

L = 1%

L = 2%

Figure 11. Sensitivity w.r.t. 2.

10 20 30 40 50 600

5

10

15

Sens

itivi

ty

Time (in days)

Unconstrained

L = 0 .5%

L = 1%

L = 2%

Figure 12. Sensitivity w.r.t. b1.


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

For the unnormalized filter E we compute robustversions which we denote by ED (cf. Sass and Haussmann2004). By robust we mean that the differential equationsdefine versions that depend continuously on the observa-tion path (cf. Clark 1978, James et al. 1996). Subsequentlywe consider an equidistant discretization (ti)i¼ 0, . . . ,N ofthe interval [0,T ], where ti¼ iDt and Dt¼T/N. Then ED

is given by

EDti¼ �ti ½Idd þ DtQ>

�EDti�1:

Here, we use the notation

�ti ¼ Diagð ð1Þti , . . . , ðd Þti Þ,

ðkÞti ¼ exp�bðkÞti

�>D ~Wti �

1

2kbðkÞti k

2Dt

� �,

where D ~Wti ¼~Wti �

~Wti�1 and bðkÞti ¼ �

�1tiBEk.

9.2. Simulated data

We illustrate the strategy using simulated data forlogarithmic utility. We consider the HMM for the driftand the LEL risk with L¼ 0.01. The setup of this exampleis given in table 1 except that we use the HR-model withparameters ¼ 7, �¼ 0.2, "¼ 70, �0¼ 0. Figures 14–19illustrate the strategy.

For heavy market movements, i.e. for sharp increasesor pronounced drops in stock prices, also the offsetfunction increases or decreases (figure 17). Big absolutevalues of the offset function result in high volatility(figure 16). Since in volatile markets investing in a riskyasset becomes riskier, the bounds on the strategy, whichrepresents the fraction of wealth invested in the riskyasset, become narrower as can be seen from comparingfigure 16 with the bounds �l and �u in figure 19. Forlogarithmic utility the optimal strategy without con-straints is essentially proportional to the filter � ¼ BYplotted in figure 18. Under dynamic risk constraints theoptimal strategy for logarithmic utility then correspondsto the strategy under no constraints capped or floored

at the bounds resulting from the dynamic risk constraints

as can be seen in figure 19.Summarizing, in times of heavy market movements the

volatility increases and the bounds on the strategy become

0 0.2 0.4 0.6 0.8 170

80

90

100

110

120

S

Time

Figure 14. Stock S.

0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

1.4

1.6

Xπ

Time

Figure 15. Wealth X.

0 0.2 0.4 0.6 0.8 1−0.2

−0.1

0

0.1

ξ(1)

Time

Figure 17. Offset �(1).

0 0.2 0.4 0.6 0.8 1

0.2

0.25

0.3

0.35

σ

Time

Figure 16. Volatility �.


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

more narrow, thus avoiding extreme positions whichmight even lead to bankruptcy if we can only trade on adiscrete time scale. Therefore the strategies becomemore robust. During big changes in the markets also theparameters describing the markets might change. Andfor constant coefficients it is known that the impact ofparameter misspecification can be much worse thaninfrequent trading (cf. Rogers 2001). The effect on thestrategy through parameter misspecification is reducedsignificantly by dynamic risk constraints as is indicatedby the sensitivity analysis in the previous section. Overallthis makes the strategy more robust or ‘less risky’. Therelation to rigorous risk measures was discussed insection 5.

9.3. Historical data

For the numerical example we consider logarithmic utilityin the one-dimensional case and the LEL risk constraint.We assume constant volatility and constant interest rates.We consider now 20 stocks of the Dow Jones IndustrialIndex and use daily prices (adjusted for dividends andsplits) for 30 years, 1972–2001, each year consisting ofapproximately 252 trading days where a stock was chosenonly if it had such a long history. For the interest rates weconsider the corresponding historic fed rates. For eachstock we use parameter estimates for the HMM basedon a Markov chain Monte Carlo (MCMC) method

(cf. Hahn et al. 2007a). Also the parameters for the GDare obtained from a multiple-block MCMC samplerbased on time discretization similar to the samplerdescribed in Hahn et al. (2007a). The parameter estimatesare based on five years with starting years 1972, 1973, . . . ,1996. For the interest rate we use the average fed rateof the fifth year. Based on these estimates we computein the subsequent year the optimal strategy. Hence, weperform 500 experiments whose outcomes we average.As a starting value for the filter when computing thestrategy we use the stationary distribution in the firstyear and then the last value of the filter that was obtainedby the optimization in the preceding year. We start withinitial capital X0¼ 1. We compare the strategy based onHMM, GD and Bayes with the Merton strategy, i.e. thestrategy resulting from the assumption of a constant drift,and with the buy and hold (b&h) strategy. We computethe average of the utility which is �1 if we go bankruptat least once. In this case we also list the average utilitywhere we only average over those cases in which we don’tgo bankrupt. The results are presented in table 2, wherewe used the LEL risk constraint and computed the mean,median, and the standard deviation. We face twoproblems when applying results of continuous-timeoptimization to market data: model and discretizationerrors. The 13 bankruptcies for the HMM mainly fall on

Table 2. Numerical results (logarithmic utility).

UðXTÞ Mean Median St.dev. Bankrupt

Unconstrained:

b&h 0.1188 0.1195 0.2297 0Merton �1 (0.0331) 0.0826 0.4745 2GD �1 (�1.2600) �1.5175 1.0849 79Bayes �1 (0.0204) 0.0824 0.5024 2HMM �1 (�0.0026) 0.0277 0.9228 13

LEL risk constraint (L¼0.5%):Merton 0.1003 0.0988 0.1591 0GD 0.0252 0.0294 0.1767 0Bayes 0.1002 0.0988 0.1595 0HMM 0.1285 0.1242 0.2004 0

LEL risk constraint (L¼1.0%):

Merton 0.0927 0.0967 0.2871 0GD �0.0531 �0.0474 0.3274 0Bayes 0.0921 0.0962 0.2894 0HMM 0.1513 0.1387 0.3626 0

LES risk constraint (L1¼1.0%, L2¼5%):






0 0.5 1

−0.4

−0.2

0

0.2

0.4

Time

Drift

Filter

Figure 18. Drift � and Filter BY.

0 0.5 1

−2

−1

0

1

2

Time

ηη

u

ηl

Figure 19. Strategy �.


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

Black Monday 1987, where single stocks had losses upto 30%. Owing to the non-constant drift, very big shortand long positions may occur for the HMM (cf. Sass andHaussmann 2004). For GD the positions are even moreextreme, since the drift process is unbounded. For dailytrading without risk constraints and risky utility functionsthis may result in bankruptcies; and also the Merton andBayes strategies go bankrupt twice. The latter is due tothe fact that the parameters for the Merton strategy areestimated over a relatively short period and can resultin some cases also in quite long positions which led tobankruptcy in 1987. The HMM is less prone to parametermisspecification, since changes in the drift parameter areincluded in the model, and with more states it could alsoexplain big price movements. But it is still a model withcontinuous asset prices and exhibits extreme positionswhich make the strategy less robust when trading atdiscrete times only.

The results under dynamic risk constraints suggest thatthey are a suitable remedy for the discretization error.We can improve the performance for all models comparedto the case of no constraints and the strategies don’t gobankrupt anymore. In particular the HMM strategy withrisk constraints clearly outperforms all other strategies.

10. Conclusion

In this paper we show how dynamic risk constraints onthe strategy can be applied to make portfolio optimiza-tion more robust. We derive explicit trading strategieswith risk constraints under partial information. Further,we analyse the dependence of the resulting strategies onthe parameters involved in the HMM. The results underdynamic risk constraints indicate that they are a suitableremedy for reducing the risk and improving the perfor-mance of trading strategies.

Acknowledgements

The authors thank the Austrian Science Fund FWF,Project P17947-N12, the German Research FoundationDFG, Heisenberg Programme, for financial support, andtwo anonymous referees for their stimulating commentsand suggestions. The opinions expressed in this paper arethose of the authors and do not necessarily reflect theviews of their employers.

References

Basak, S. and Shapiro, A., Value-at-risk-based risk manage-ment: optimal policies and asset prices. Rev. Finan. Stud.,2001, 14, 371–405.

Brendle, S., Portfolio selection under incomplete information.Stochast. Process. Appls., 2006, 116, 701–723.

Clark, J., The design of robust approximations to the stochasticdifferential equations of nonlinear filtering. In Proceedings ofthe 2nd NATO Advanced Study Institute Conference on

Communication Systems and Random Process Theory,Darlington, UK, 8–20 August 1977, pp. 721–734, 1978(Sijthoff &Noordhoff: Alphen aan den Rijn, TheNetherlands).

Cuoco, D., He, H. and Issaenko, S., Optimal dynamic tradingstrategies with risk limits. Oper. Res., 2008, 56, 358–368.

Cvitanic, J., 1997, Optimal trading under constraints. InProceedings of the Conference on Financial Mathematics,Bressanone, Italy, 8–13 July 1996, edited by B. Biais andW.J. Runggaldier. Lecture Notes in Mathematics Vol. 1656,pp. 123–190, 1997 (Springer-Verlag: Berlin).

Cvitanic, J. and Karatzas, I., Convex duality in constrainedportfolio optimization. Ann. Appl. Probab., 1992, 2, 767–818.

Duffie, D. and Pan, J., An overview of value at risk.J. Derivatives, 1997, 4, 7–49.

Elliott, R., New finite-dimensional filters and smoothers fornoisily observed Markov chains. IEEE Trans. Inform. Theor.,1993, 39, 265–271.

Elliott, R. and Rishel, R., Estimating the implicit interest rate ofa risky asset. Stochast. Process. Appls., 1994, 49, 199–206.

Follmer, H. and Schied, A., Stochastic Finance: an Introductionin Discrete Time, 2nd ed., 2004 (Walter de Gruyter: Berlin).

Gabih, A., Sass, J. and Wunderlich, R., Utility maximizationunder bounded expected loss. Stochast. Models, 2009, 25,375–409.

Gennotte, G., Optimal portfolio choice under incompleteinformation. J. Finan., 1986, 41, 733–746.

Glasserman, P. and Yao, D., Some guidelines and guaranteesfor common random numbers. Mgmt Sci., 1992, 38, 884–908.

Gobet, E. andMunos, R., Sensitivity analysis using Ito-Malliavincalculus andmartingales, and application to stochastic optimalcontrol. SIAM J. Control Optim., 2005, 43, 1676–1713.

Gundel, A. and Weber, S., Utility maximization under ashortfall risk constraint. J. Math. Econ., 2008, 44, 1126–1151.

Hahn, M., Fruhwirth-Schnatter, S. and Sass, J., Markov chainMonte Carlo methods for parameter estimation in multi-dimensional continuous time Markov switching models.RICAM-Report No. 2007-09, 2007a.

Hahn, M., Putschogl, W. and Sass, J., Portfolio optimizationwith non-constant volatility and partial information. Brazil. J.Probab. Statist., 2007b, 21, 27–61.

Haussmann, U. and Sass, J., Optimal terminal wealth underpartial information for HMM stock returns. In Mathematicsof Finance, Contemporary Mathematics, Vol. 351,pp. 171–185, 2004 (Amer. Math. Soc.: Providence, RI).

Hobson, D. and Rogers, L., Complete models with stochasticvolatility. Math. Finan., 1998, 8, 27–48.

James, M., Krishnamurthy, V. and Le Gland, F., Time discret-ization of continuous-time filters and smoothers for HMMparameter estimation. IEEE Trans. Inf. Theor., 1996, 42,593–605.

Jorion, P., Value at Risk: the New Benchmark for ManagingFinancial Risk, 2nd ed., 2000 (McGraw-Hill: New York).

Karatzas, I. and Shreve, S., Methods of Mathematical Finance,1998 (Springer-Verlag: New York).

Kushner, H. and Yang, J., A Monte Carlo method forsensitivity analysis and parametric optimization of nonlinearstochastic systems: the ergodic case. SIAM J. Control Optim.,1992, 30, 440–464.

Lakner, P., Utility maximization with partial information.Stochast. Process. Appls., 1995, 56, 247–273.

Lakner, P., Optimal trading strategy for an investor: the caseof partial information. Stochast. Process. Appls., 1998, 76,77–97.

L’Ecuyer, P. and Perron, G., On the convergence rates ofIPA and FDC derivative estimators. Oper. Res., 1994, 42,643–656.

Liptser, R. and Shiryayev, A., Statistics of Random Processes II:Applications, 1978 (Springer-Verlag: New York).

Martinez, M., Rubenthaler, S. and Thanre, E., Misspecifiedfiltering theory applied to optimal allocation problems infinance. NCCR-FINRISK Working Paper Series 278, 2005.


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

Ocone, D., Malliavin’s calculus and stochastic integral repre-sentations of functionals of diffusion processes. Stochastics,1984, 12, 161–185.

Ocone, D. and Karatzas, I., A generalized Clark representationformula, with application to optimal portfolios. Stochast.Stochast. Rep., 1991, 34, 187–220.

Pham, H. and Quenez, M.-C., Optimal portfolio in partiallyobserved stochastic volatility models. Ann. Appl. Probab.,2001, 11, 210–238.

Pirvu, T., Portfolio optimization under the value-at-riskconstraint. Quant. Finan., 2007, 7, 125–136.

Putschogl, W. and Sass, J., Optimal consumption and invest-ment under partial information. Decis. Econ. Finan., 2008, 31,137–170.

Rieder, U. and Bauerle, N., Portfolio optimization withunobservable Markov-modulated drift process. J. Appl.Probab., 2005, 42, 362–378.

Rockafellar, R.T., Convex Analysis, 1970 (Princeton UniversityPress: Princeton, NJ).

Rockafellar, R. and Uryasev, S., Optimization of conditionalvalue-at-risk. J. Risk, 2000, 2, 21–41.

Rockafellar, R. and Uryasev, S., Conditional value-at-risk forgeneral loss distributions. J. Banking Finan., 2002, 26,1443–1471.

Rogers, L.C.G., The relaxed investor and parameter uncer-tainty. Finan. Stochast., 2001, 5, 131–154.

Runggaldier, W. and Zaccaria, A., A stochastic controlapproach to risk management under restricted information.Math. Finan., 2000, 10, 277–288.

Ryden, T., Terasvirta, T. and Asbrink, S., Stylized facts of dailyreturn series and the hidden Markov Model. J. Appl.Econom., 1998, 13, 217–244.

Sass, J., Utility maximization with convex constraints andpartial information. Acta Applic. Math., 2007, 97, 221–238.

Sass, J. and Haussmann, U., Optimizing the terminal wealthunder partial information: the drift process as a continuoustime Markov chain. Finan. Stoch., 2004, 8, 553–577.

Sekine, J., Risk-sensitive portfolio optimization underpartial information with non-Gaussian initial prior, 2006(preprint).

Yiu, K., Optimal portfolios under a value-at-risk constraint.J. Econ. Dynam. Contr., 2004, 28, 1317–1334.


Dow

nloa

ded

by [

Seto

n H

all U

nive

rsity

] at

12:

27 3

0 N

ovem

ber

2014

optimal investment under dynamic risk constraints and partial information

Documents