Download - Martingales et al.gugushvilis/Slides1.pdf · Martingales et al. Shota Gugushvili Basic concepts Stopping times and martingale transforms Filtration De nition Let F = (F n) n 0, where

Martingales etal.

ShotaGugushvili

Basic concepts

Stoppingtimes andmartingaletransforms

Martingales et al.

Shota Gugushvili

Leiden University

Amsterdam, 30 October 2013

Martingales etal.

ShotaGugushvili

Basic concepts


Outline

1 Basic concepts

2 Stopping times and martingale transforms

Martingales etal.

ShotaGugushvili

Basic concepts


Stochastic process

The probability space (Ω,F ,P) is fixed throughout and allthe quantities introduced are defined on this space.

Definition

A stochastic process X = (Xt)t∈T is a collection of randomvariables Xt indexed by a set T . A discrete time stochasticprocess is the one for which T is finite or countable.

The name ‘time’ (for t) refers to the fact that stochasticprocesses are used to model random dynamic phenomenaevolving over the course of time, e.g. asset prices.

A discrete time stochastic process is nothing else but asequence of random variables.

We concentrate on the case T = 0, 1, 2, . . . and writeX = (Xn)n≥0 for the process.

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Basic concepts


Filtration

Definition

Let F = (Fn)n≥0, where each Fn is a σ-algebra satisfyingFn ⊂ F , and moreover Fn ⊂ Fn+1, for all n ≥ 0. Thesequence F is called the filtration.

Definition

If X is a stochastic process, then the filtration FX := (FXn )n≥0

generated by X is defined by FXn := σ(X0, . . . ,Xn). FX is also

called the natural filtration associated with X .

Filtration can be thought of as a flow of information,where Fn represents all the information available up totime n.

Some notation: F∞ := σ(∪∞n=0Fn). Obviously, Fn ⊂ F∞.

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Adapted process

Definition

Let a filtration F be given. A process Y is called F-adapted (oradapted to F, or just adapted), if for all n the random variableYn is Fn-measurable: Yn ∈ Fn.

Obviously, X is adapted to FX .

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Martingale

Definition

A stochastic process M = (Mn)n≥0 is called a martingale (orF-martingale), if it is adapted to a filtration F, ifMn ∈ L1(Ω,F ,P) for all n ≥ 0 and if

E [Mn+1|Fn] = Mn a.s. (1)

Equation (1) is called the martingale property of M.

The equality (1) should be read in the sense that Mn is aversion of the conditional expectation E [Mn+1|Fn].

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Basic concepts


Equivalent definition

Definition

A stochastic process M = (Mn)n≥0 is called a martingale (orF-martingale), if it is adapted to a filtration F, ifMn ∈ L1(Ω,F ,P) for all n ≥ 0 and if

E [Mm|Fn] = Mn a.s.

for all m ≥ n + 1.

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Examples

Example

Let X be a process consisting of independent random variablesXn, with n ≥ 1 and assume that Xn ∈ L1(Ω,F ,P) for all n.Put X0 = 0 and Sn =

∑nk=0 Xk =

∑nk=1 Xk . Take F = FX . X

is a martingale iff E [Xn] = 0 for all n.

Example

Let X be a process consisting of independent random variablesXn, with n ≥ 1 and assume that Xn ∈ L1(Ω,F ,P) for all n.Put X0 = 1 and Pn =

∏nk=0 Xk =

∏nk=1 Xk . Take F = FX . P

is a martingale iff EXn = 1 for all n.

Example

Let E [|Y |] <∞ and F be a filtration. The process M withMn = E [Y |Fn], n ≥ 0 is a martingale.

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Submartingale and supermartingale

Definition

A stochastic process X = (Xn)n≥0 is called a submartingale (orF-submartingale), if it is adapted to a filtration F, ifXn ∈ L1(Ω,F ,P) for all n ≥ 0 and if

E [Xn+1|Fn] ≥ Xn a.s. (2)

A stochastic process X = (Xn)n≥0 is called a supermartingale(or F-supermartingale), if −X is a submartingale.

Inequality (2), valid for all n ≥ 0 is called thesubmartingale property of X .

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Expectations

A martingale has a constant expectation (E [Xn] = const.for all n), a submartingale has an increasing expectation(E [Xm] ≥ E [Xn] for m ≥ n) and the supermartingale hasa decreasing expectation (E [Xm] ≤ E [Xn] for m ≥ n).

One can also talk informally about no trend, increasingtrend and decreasing trend for these three processes.

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Basic concepts


Examples

Example

Let X be a process consisting of independent random variablesXn, with n ≥ 1 and assume that Xn ∈ L1(Ω,F ,P) for all n.Put X0 = 0 and Sn =

∑nk=0 Xk =

∑nk=1 Xk . Take F = FX . X

is a submartingale if E [Xn] ≥ 0 and a supermartingale ifE [Xn] ≤ 0 for all n.

Example

Let X be a process consisting of positive independent randomvariables Xn, with n ≥ 1 and assume that Xn ∈ L1(Ω,F ,P) forall n. Put X0 = 1 and Pn =

∏nk=0 Xk =

∏nk=1 Xk . Take

F = FX . P is a submartingale if EXn ≥ 1 and asupermartingale if EXn ≤ 1 for all n.

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Predictable process

Definition

Given a filtration F, a process Y = (Yn)n≥1 is calledF-predictable (or just predictable) if Yn ∈ Fn−1, n ≥ 1. Aconvenient additional convention is to set Y0 = 0.

Useful interpreation is this: a predictable process Y is a(trading or gambling) strategy. It tells you what youraction at time n is going to be, given that you use yourinformation available at time n − 1.

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Basic concepts


Discrete time stochastic integral

Definition

Let Y and X be two stochastic processes. A stochastic processS = Y · X defined by

S0 = 0, Sn =n∑

i=1

Yi∆Xi , n ≥ 1

is called a discrete time stochastic integral.

The name derives from the analogy to∫ t

0YtdXt ,

a stochastic integral (whatever that is).

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Basic concepts


Martingale transform

Theorem

Let X be an adapted process and Y a predictable process.Assume that the Xn are in L1(Ω,F ,P) as well as the Yn∆Xn.Let S = Y · X . The following results hold.

(i) If X is martingale, so is S .

(ii) If X is a submartingale (supermartingale) and if Y isnonnegative, also S is a submartingale (supermartingale).

In case (i) one calls S a martingale transform.

I have never seen anybody to use the term sub- orsupermartingale transform though (case (ii)).

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Basic concepts


Stopping time

Definition

Let F be a filtration. A mapping T : Ω→ 0, 1, 2, . . . ∪ ∞is called a stopping time if for all n ∈ 0, 1, 2, . . . it holds thatT = n ∈ Fn.

T is a stopping time iff T ≤ n ∈ Fn is true for alln ∈ 0, 1, 2, . . .. Alternatively, iff T > n ∈ Fn is truefor all n ∈ 0, 1, 2, . . ..The event T =∞ can be written as (∪∞n=0T = n)c .Since T = n ∈ Fn ⊂ F∞, we have T =∞ ∈ F∞.Hence the requirement T = n ∈ Fn extends to n =∞.

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Example

Example

Let F be a filtration and X an adapted process. Let B ∈ B bea Borel set in R and let T = infn ≥ 0 : Xn ∈ B. Then T is astopping time.

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Stopped process

Definition

If X is an adapted process and T a stopping time, we definethe stopped process XT by XT

n (ω) := XT (ω)∧n(ω), n ≥ 0.

Tn(ω) := T (ω) ∧ n is a stopping time.

XT0 = X0 and XT

n (ω) = Xn(ω) for n ≤ T (ω). The latterexplains the name stopped process.

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Basic concepts


Stopped processes and martingales et al.

Theorem

If X is an adapted process and T a stopping time, then XT isadapted too. Moreover, if X is a supermartingale, so is XT andthen EXT

n ≤ EX0. If X is a martingale, then XT is amartingale too and EXT

n = EX0.

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Doubling strategy

Martingales got their name from a gambling strategyconsisting in doubling the bet until the first win and thenimmediately stopping to play.

This also gives an example of a martingale S (and astopping time T ), such that E [Sn] 6= E [ST ].

Brief terminological remark: there exists a certain relationof martingales to harmonic functions. Sub- andsupermartingales are related to subharmonic andsuperharmonic functions, hence their names.

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Expectations

Theorem

Let X be supermartingale and T an a.s. finite stopping time.Then EXT ≤ EX0 under either of the assumptions

(i) X is a.s. bounded from below by random variableY ∈ L1(Ω,F ,P), or

(ii) T is bounded, i.e. there exists N <∞ such thatP(T ≤ N) = 1 or

(iii) The process ∆X is bounded by a constant C andET <∞.

If X is a martingale, then EXT = EX0 under (ii) and (iii) andalso under the assumption (iv) that X is bounded.

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Example

Example (Wald’s identity)

Let ξ1, ξ2 . . . be i.i.d. with E [|ξ1|] <∞. Then

E [ξ1 + . . . ξn] = nE [ξ1].

Wald’s identity generalises this to the case when n is replacedwith a stopping time T :

Let T ≥ 1 (with E [T ] <∞) be an Fξ = (Fξn)n≥0 stoppingtime with

Fξn = σ(ξ1, . . . , ξn).

ThenE [ξ1 + . . .+ ξT ] = E [ξ1]E [T ],

provided |ξ1| ≤ C (a weaker condition E [|ξ1|] <∞ suffices,but anyway).

Download - Martingales et al.gugushvilis/Slides1.pdf · Martingales et al. Shota Gugushvili Basic concepts Stopping times and martingale transforms Filtration De nition Let F = (F n) n 0, where

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