Optional Stopping Theorems

58 3 Filtrations and Martingales

˝ D f1; 1g, with the probability measurePdefined byP.f1g/DP.f1g/D 1=2. Let"be the random variable ".!/ D !, and let the process.Xt/t0 be defined byXt D 0if0 t 1, andXt D "ift> 1. Then it is easy to verify thatX is a martingale with respect to its canonical filtration.Ft^X/(which is complete since there are no nonempty negligible sets!). On the other hand, no modification ofXcan be right-continuous attD1. This does not contradict the theorem since the filtration is not right-continuous (F₁^X_C 6DF₁^X).

(ii) Similarly, to show that the right-continuity of the mapping t ! EŒXt is needed, we can just takeXtD f.t/, wheref is any nonincreasing deterministic function. If f is not right-continuous, no modification of X can have right- continuous sample paths.

3.4 Optional Stopping Theorems 59 and we get thatX12L¹. The right-continuity of sample paths (which we have not yet used) allows us to remove the restrictiont2Din the limit (3.4). ut Under the assumptions of Theorem3.19, the convergence ofXttowardsX1may not hold in L¹. The next result gives, in the case of a martingale, necessary and sufficient conditions for the convergence to also hold inL¹.

Definition 3.20 A martingale.Xt/t0 is said to beclosed if there exists a random variableZ2L¹such that, for everyt0,

XtDEŒZjFt:

Theorem 3.21 Let X be a martingale with right-continuous sample paths. Then the following properties are equivalent:

(i) X is closed;

(ii) the collection.Xt/t0is uniformly integrable;

(iii) Xtconverges a.s. and in L¹as t! 1.

Moreover, if these properties hold, we have XtDEŒX1jFtfor every t0, where X12L¹is the a.s. limit of Xtas t! 1.

Proof The fact that (i))(ii) is easy: IfZ2L¹, the collection of all random variables EŒZ j G, whenG varies over sub--fields of F, is uniformly integrable. If (ii) holds, in particular the collection .Xt/t0 is bounded in L¹ and Proposition3.19 implies thatXtconverges a.s. toX1. By uniform integrability, the latter convergence also holds inL¹. Finally, if (iii) holds, for everys0, we can pass to the limitt! 1in the equalityXsDEŒXtjFs(using the fact that the conditional expectation is continuous for theL¹-norm), and we getXsDEŒX1jFs. ut We will now use the optional stopping theorems for discrete martingales and supermartingales in order to establish similar results in the continuous time setting.

Let.Xt/t0be a martingale or a supermartingale with right-continuous sample paths, and such thatXtconverges a.s. ast! 1to a random variable denoted byX1. Then, for every stopping timeT, we writeXT for the random variable

XT.!/D1fT.!/<1gXT.!/.!/C1fT.!/D1gX1.!/:

Compare with Theorem3.7, where the random variableXT was only defined on the subsetfT < 1g of˝. With this definition, the random variableXT is still FT-measurable: Use Theorem3.7and the easily verified fact that 1fTD1gX1 is FT-measurable.

Theorem 3.22 (Optional stopping theorem for martingales) Let.Xt/t0 be a uniformly integrable martingale with right-continuous sample paths. Let S and T be two stopping times with ST. Then XSand XTare in L¹and

XS DEŒXT jFS:

60 3 Filtrations and Martingales

In particular, for every stopping time S, we have XSDEŒX1jFS; and

EŒXSDEŒX1DEŒX₀: Proof Set, for every integern0,

Tn D X1 kD0

kC1

2ⁿ 1fk2ⁿ<T.kC1/2ⁿgC 1 1fTD1g

and similarly

Sn D X1 kD0

kC1

2ⁿ 1fk2ⁿ<S.kC1/2ⁿgC 1 1fSD1g:

By Proposition3.8,.Tn/and.Sn/are two sequences of stopping times that decrease respectively toTand toS. Moreover, we haveSnTnfor everyn0.

Now observe that, for every fixedn,2ⁿSn and2ⁿTn are stopping times of the discrete filtration Hk^.n/ WD Fk=2ⁿ, and Y_k^.n/ WD Xk=2ⁿ is a discrete martingale with respect to this filtration. From the optional stopping theorem for uniformly integrable discrete martingales (see AppendixA2) we get thatY₂^.nn^/SnandY₂^.nn^/Tnare in L¹, and

XSnDY₂^.nnS^/nDEŒY₂^.nn^/Tn jH₂^.nnS^/nDEŒXTn jFSn (here we need to verify thatH₂^.nnS^/_n DFSn, but this is straightforward).

LetA2FS. SinceFS FSn, we haveA2FSn and thus EŒ1AXSnDEŒ1AXTn: By the right-continuity of sample paths, we get a.s.

XSD lim

n!1XSn; XTD lim

n!1XTn:

These limits also hold inL¹. Indeed, thanks again to the optional stopping theorem for uniformly integrable discrete martingales, we haveXSn DEŒX1jFSnfor every n, and thus the sequence.XSn/is uniformly integrable (and the same holds for the sequence.XTn/).

3.4 Optional Stopping Theorems 61 TheL¹-convergence implies thatXS andXT belong toL¹, and also allows us to pass to the limitn! 1in the equalityEŒ1AXSnDEŒ1AXTnin order to get

EŒ1AXSDEŒ1AXT:

Since this holds for everyA2 FS, and since the variableXSisFS-measurable (by the remarks before the theorem), we conclude that

XS DEŒXT jFS;

which completes the proof. ut

We now give two corollaries of Theorem3.22.

Corollary 3.23 Let.Xt/t0 be a martingale with right-continuous sample paths, and let ST be two bounded stopping times. Then XSand XT are in L¹and

XS DEŒXT jFS :

Proof Leta 0such thatS T a. We apply Theorem3.22to the martingale

.Xt^a/t0which is closed byXa. ut

The second corollary shows that a martingale (resp. a uniformly integrable martingale) stopped at an arbitrary stopping time remains a martingale (resp. a uniformly integrable martingale). This result will play an important role in the next chapters.

Corollary 3.24 Let.Xt/t0 be a martingale with right-continuous sample paths, and let T be a stopping time.

(i) The process.Xt^T/t0is still a martingale.

(ii) Suppose in addition that the martingale.Xt/t0 is uniformly integrable. Then the process .Xt^T/t0 is also a uniformly integrable martingale, and more precisely we have for every t0,

Xt^TDEŒXTjFt: (3.5)

Proof We start with the proof of (ii). Note thatt^T is a stopping time by property (f) of stopping times. By Theorem3.22,Xt^T andXT are inL¹, and we also know thatXt^TisFt^T-measurable, henceFt-measurable sinceFt^T Ft. So in order to get (3.5), it is enough to prove that, for everyA2Ft,

EŒ1AXTDEŒ1AXt^T: Let us fixA2Ft. First, we have trivially

EŒ1_A\fTtgXTDEŒ1_A\fTtgXt^T: (3.6)

62 3 Filtrations and Martingales

On the other hand, by Theorem3.22, we have Xt^T DEŒXT jFt^T;

and we notice that we have bothA\ fT>tg 2FtandA\ fT >tg 2FT(the latter as a straightforward consequence of the definition ofFT), so thatA\ fT > tg 2 Ft\FTDFt^T. Using the preceding display, we obtain

EŒ1A\fT>tgXTDEŒ1A\fT>tgXt^T: By adding this equality to (3.6), we get the desired result.

To prove (i), we just need to apply (ii) to the (uniformly integrable) martingale

.Xt^a/a0, for any choice ofa0. ut

Applications Above all, the optional stopping theorem is a powerful tool for explicit calculations of probability distributions. Let us give a few important and typical examples of such applications (several other examples can be found in the exercises of this and the following chapters). LetBbe a real Brownian motion started from0. We know thatBis a martingale with continuous sample paths with respect to its canonical filtration. For every reala, setTaDinfft0WBtDag. Recall that Ta<1a.s.

(a) Law of the exit point from an interval. For everya< 0 <b, we have P.Ta <Tb/D b

ba ; P.Tb<Ta/D a ba:

To get this result, consider the stopping time T D Ta ^Tb and the stopped martingaleMt D Bt^T (this is a martingale by Corollary3.24). Then,jMjis bounded above byb_ jaj, and the martingaleMis thus uniformly integrable.

We can apply Theorem3.22and we get

0DEŒM₀DEŒMTDb P.Tb<Ta/Ca P.Ta<Tb/:

Since we also haveP.Tb <Ta/CP.Ta<Tb/D1, the desired result follows. In fact the proof shows that the result remains valid if we replace Brownian motion by a martingale with continuous sample paths and initial value0, provided we know that this process exits.a;b/a.s.

(b) First moment of exit times. For everya > 0, consider the stopping timeUa D infft0W jBtj Dag. Then

EŒUaDa²:

To verify this, consider the martingale Mt D B²_t t. By Corollary 3.24, Mt^Ua is still a martingale, and therefore EŒMt^Ua D EŒM₀ D 0, giving EŒ.Bt^Ua/² D EŒt^Ua. Then, on one hand,EŒt^Uaconverges toEŒUaas

3.4 Optional Stopping Theorems 63 t! 1by monotone convergence, on the other hand,EŒ.Bt^Ua/²converges to EŒ.BUa/²Da²ast! 1, by dominated convergence (note that.Bt^Ua/²a²).

The stated result follows. We may observe that we haveEŒUa <1in contrast with the propertyEŒTaD 1, which was noticed in Chap.2.

(c) Laplace transform of hitting times. We now fixa> 0and our goal is to compute the Laplace transform ofTa. For every2R, we can consider the exponential martingale

N_tDexp.Bt² 2 t/:

Suppose first that > 0. By Corollary3.24, the stopped processN_{t^T a} is still a martingale, and we immediately see that this martingale is bounded above by e^a, hence uniformly integrable. By applying the last assertion of Theorem3.22 to this martingale and to the stopping timeSDTa(or toSD 1) we get

e^aEŒe^22TaDEŒNTaDEŒN₀D1 : Replacingbyp

2, we conclude that, for every > 0, EŒe^TaDe^a

p2: (3.7)

(This formula could also be deduced from the knowledge of the density ofTa, see Corollary2.22.) As an instructive example, one may try to reproduce the preceding line of reasoning, using now the martingaleN_tfor < 0: one gets an absurd result, which can be explained by the fact that the stopped martingale N_t^Ta isnot uniformly integrablewhen < 0. When applying Theorem3.22, it is crucial to always verify the uniform integrability of the martingale. In most cases, this is done by verifying that the (stopped) martingale is bounded.

(d) Laplace transform of exit times from an interval.With the notation of (b), we have for everya> 0and every > 0,

EŒexp.Ua/D 1 cosh.ap

2/:

To see this, first note thatUaandBU_aare independent since, using the symmetry property of Brownian motion,

EŒ1fB_UaDag exp.Ua/DEŒ1fB_UaDag exp.Ua/D 1

2EŒexp.Ua/:

Then the claimed formula is proved by the same method as in (c), writing EŒN_Ua D EŒN₀ D 1and noting that the application of the optional stopping

64 3 Filtrations and Martingales

theorem is justified by the fact thatN_{t^U a} is bounded above by e^a. See also Exercise3.27for a more general formula.

We end this chapter with the optional stopping theorem for nonnegative super- martingales. This result will be useful in later applications to Markov processes.

We first note that, if.Zt/t0is a nonnegative supermartingale with right-continuous sample paths,.Zt/t0 is automatically bounded inL¹ sinceEŒZt EŒZ₀, and by Theorem3.19,Zt converges a.s. to a random variableZ1 2 L¹ as t ! 1. As explained before Theorem3.22, we can thus make sense ofZTfor any (finite or not) stopping timeT.

Theorem 3.25 Let.Zt/t0be a nonnegative supermartingale with right-continuous sample paths. Let U and V be two stopping times such that U V. Then, ZUand ZV are in L¹, and

ZUEŒZV jFU:

Remark This implies thatEŒZUEŒZV, and sinceZUDZV DZ1on the event fUD 1g, it also follows that

EŒ1fU<1gZUEŒ1fU<1gZVEŒ1fV<1gZV:

Proof In the first step of the proof, we make the extra assumption thatUandVare bounded and we verify that we then haveEŒZU EŒZV. Letp 1be an integer such thatUpandVp. For every integern0, set

UnD

pX2ⁿ1 kD0

kC1

2ⁿ 1fk2ⁿ<U.kC1/2ⁿg; VnD

pX2ⁿ1 kD0

kC1

2ⁿ 1fk2ⁿ<V.kC1/2ⁿg

in such a way (by Proposition3.8) that.Un/and.Vn/are two sequences of bounded stopping times that decrease respectively toUandV, and additionally we haveUn Vnfor everyn 0. The right-continuity of sample paths ensures thatZUn !ZU

andZVn !ZVa.s. asn! 1. Then, by the optional stopping theorem for discrete supermartingales in the case of bounded stopping times (see AppendixA2), with respect to the filtration.Fk=2^nC1/k0, we have for everyn0,

ZUnC1EŒZUn jFUnC1:

Setting Yn D ZUn and Hn D FUn, for every integer n 0, we get that the sequence.Yn/n0 is a backward supermartingale with respect to the filtration .Hn/n0. Since, for everyn 0,EŒZUn EŒZ0(by another application of the discrete optional stopping theorem), the sequence.Yn/n0 is bounded in L¹, and by the convergence theorem for backward supermartingales (see AppendixA2), it converges inL¹. Hence the convergence ofZUn toZUalso holds inL¹and similarly

Exercises 65 the convergence ofZVntoZVholds inL¹. SinceUn Vn, by yet another application of the discrete optional stopping theorem, we have EŒZUn EŒZVn. Using the L¹-convergence ofZUn andZVn we can pass to the limitn ! 1and obtain that EŒZUEŒZVas claimed.

Let us prove the statement of the theorem (no longer assuming thatUandVare bounded). By the first step of the proof applied to the stopping times0andU^p, we haveEŒZU^pEŒZ₀for everyp 1, and Fatou’s lemma givesEŒZUEŒZ₀ <

1and similarlyEŒZV < 1. FixA 2 FU FV and recall our notationU^A for the stopping time defined byU^A.!/DU.!/if!2AandU^A.!/D 1otherwise (cf. property (d) of stopping times). By the first part of the proof, we have, for every p1,

EŒZ_U^A_^pEŒZ_V^A_^p:

By writing each of these two expectations as a sum of expectations over the setsA^c, A\ fUpgandA\ fU>pg, and noting thatU>pimpliesV>p, we get

EŒZU1A\fUpgEŒZV1A\fUpg: Lettingp! 1then gives

EŒZU1A\fU<1gEŒZV1A\fU<1g:

On the other hand, the equalityEŒZU1A\fUD1g D EŒZV1A\fUD1gis trivial and by adding it to the preceding display, we obtain

EŒZU1AEŒZV1ADEŒEŒZV jFU1A:

Since this holds for everyA 2 FU andZU isFU-measurable, the desired result

follows. ut

Exercises

In the following exercises, processes are defined on a probability space.˝;F;P/

equipped with a complete filtration.Ft/t2Œ0;1. Exercise 3.26

1. LetM be a martingale with continuous sample paths such thatM₀ D x2 RC. We assume thatMt 0for everyt 0, and thatMt ! 0whent ! 1, a.s.

Show that, for everyy>x, P

sup

t0Mty D x

y:

66 3 Filtrations and Martingales

2. Give the law of

sup

tT₀Bt

whenBis a Brownian motion started fromx> 0andT₀Dinfft0WBtD0g.

3. Assume now thatBis a Brownian motion started from0, and let > 0. Using an appropriate exponential martingale, show that

supt0.Btt/

is exponentially distributed with parameter2.

Exercise 3.27 Let B be an .Ft/-Brownian motion started from 0. Recall the notationTx D infft 0 W Bt D xg, for everyx 2 R. We fix two real numbers aandbwitha< 0 <b, and we set

T DTa^Tb : 1. Show that, for every > 0,

EŒexp.T/D cosh.^bCa₂ p 2/

cosh.^ba₂ p 2/: (Hint: One may consider a martingale of the form

MtDexp

p2.Bt˛/t Cexp

p

2.Bt˛/t

with a suitable choice of˛.)

2. Show similarly that, for every > 0,

EŒexp.T/1fTDTagD sinh.bp 2/

sinh..ba/p 2/: 3. Recover the formula forP.Ta<Tb/from question (2).

Exercise 3.28 Let.Bt/t0 be an.Ft/-Brownian motion started from0. Leta > 0 and

aDinfft0WBttag:

1. Show thatais a stopping time and thata<1a.s.

Exercises 67

2. Using an appropriate exponential martingale, show that, for every0, EŒexp.a/Dexp.a.p

1C21//:

The fact that this formula remains valid for2 Œ¹₂; 0Œcan be obtained via an argument of analytic continuation.

3. Let 2 RandMt D exp.Bt ₂²t/. Show that the stopped martingaleM_a^t

is closed if and only if 1. (Hint: This martingale is closed if and only if EŒM_aD1.)

Exercise 3.29 Let .Yt/t0 be a uniformly integrable martingale with continuous sample paths, such thatY₀ D 0. We setY1 D limt!1Yt. Letp 1 be a fixed real number. We say that Property (P) holds for the martingaleY if there exists a constantCsuch that, for every stopping timeT, we have

EŒjY₁YTj^pjFTC:

1. Show that Property (P) holds forYifY1is bounded.

2. LetBbe an.Ft/-Brownian motion started from0. Show that Property (P) holds for the martingaleYt D Bt^1. (Hint: One may observe that the random variable sup_t1jBtjis inL^p.)

3. Show that Property (P) holds forY, with the constantC, if and only if, for any stopping timeT,

EŒjYTY1j^pC PŒT <1:

(Hint: It may be useful to consider the stopping timesT^Adefined forA2FT in property (d) of stopping times.)

4. We assume that Property (P) holds for Y with the constant C. Let S be a stopping time and letY^S be the stopped martingale defined byY_t^S D Yt^S (see Corollary3.24). Show that Property (P) holds for Y^S with the same constant C. One may start by observing that, if S and T are stopping times, one has Y_T^S DYS^TDY_S^T DEŒYTjFS.

5. We assume in this question and the next one that Property (P) holds forY with the constantCD1. Leta> 0, and let.Rn/n0be the sequence of stopping times defined by induction by

R₀ D0 ; RnC1DinfftRnW jYtYRnj ag .inf¿D 1/:

Show that, for every integern0,

a^pP.RnC1<1/P.Rn<1/:

68 3 Filtrations and Martingales

6. Infer that, for everyx> 0, P

sup

t0Yt>x

2^p2^px=2:

Notes and Comments

This chapter gives a brief presentation of the so-called general theory of processes.

We limited ourselves to the notions that are needed in the remaining part of this book, but the interested reader can consult the treatise of Dellacherie and Meyer [13,14] for more about this subject. Most of the martingale theory presented in Sections 3 and 4 goes back to Doob [15]. A comprehensive study of the theory of continuous time martingales can be found in [14]. The applications of the optional stopping theorem to Brownian motion are very classical. For other applications in the same vein, see in particular the book [70] by Revuz and Yor. Exercise3.29is taken from the theory of BMO martingales, see e.g. [18, Chapter 7].

Chapter 4

Continuous Semimartingales

Continuous semimartingales provide the general class of processes with continuous sample paths for which we will develop the theory of stochastic integration in the next chapter. By definition, a continuous semimartingale is the sum of a continuous local martingale and a (continuous) finite variation process. In the present chapter, we study separately these two classes of processes. We start with some preliminaries about deterministic functions with finite variation, before considering the corresponding random processes. We then define (continuous) local martingales and we construct the quadratic variation of a local martingale, which will play a fundamental role in the construction of stochastic integrals. We explain how properties of a local martingale are related to those of its quadratic variation.

Finally, we introduce continuous semimartingales and their quadratic variation processes.