The Strong Markov Property of Brownian Motion

2.4 The Strong Markov Property of Brownian Motion 33

34 2 Brownian Motion

setting

1fT<1gBT.!/D

BT.!/.!/ifT.!/ <1;

0 ifT.!/D 1;

then1fT<1gBTis alsoFT-measurable. To see this, we first observe that 1fT<1gBTD lim

n!1

X1 iD0

1fi2ⁿT<.iC1/2ⁿgBi2ⁿD lim

n!1

X1 iD0

1fT<.iC1/2ⁿg1fi2ⁿTgBi2ⁿ:

We then note that, for anys0,Bs1fsTgisFT-measurable, because ifAis a Borel subset ofRnot containing0 (the case where0 2 Ais treated by considering the complementary event) we have

fBs1fsTg2Ag \ fTtg D

¿ ift<s fBs2Ag \ fsTtgifts

which isFt-measurable in both cases (writefsTtg D fTtg \ fT<sg^c).

Theorem 2.20 (strong Markov property) Let T be a stopping time. We assume that P.T<1/ > 0and we set, for every t0,

B^.t^T^/D1fT<1g.BTCtBT/:

Then under the probability measure P. j T < 1/, the process .B^.t^T^//t0 is a Brownian motion independent ofFT.

Proof We first consider the case whereT < 1a.s. We fixA 2FT and0 t₁ <

<tp, and we letFbe a bounded continuous function fromR^p intoRC. We will verify that

EŒ1AF.B^.t₁^T^/; : : : ;B^.tp^T^//DP.A/EŒF.Bt₁; : : : ;Btp/: (2.1) The different assertions of the theorem then follow. First the caseA D ˝ shows that the process.B^.t^T^//t0 has the same finite-dimensional marginal distributions as B and is thus a Brownian motion (notice that the sample paths of B^.^T^/ are continuous). Then (2.1) implies that, for every choice of0 t₁ < < tp, the vector.B^.t₁^T^/; : : : ;B^.tp^T^//is independent ofFTand it follows thatB^.^T^/is independent ofFT.

Let us prove (2.1). For every integern1, and for every realt0, we writeŒtn

for the smallest real of the formk2ⁿ, withk2ZC, belonging to the intervalŒt;1/.

We also setŒ1n D 1by convention. In order to prove (2.1), we observe that we have a.s.

F.B^.t₁^T^/; : : : ;B^.tp^T^//D lim

n!1F.B^.Œt₁^Tⁿ^/; : : : ;B^.Œtp^Tⁿ^//;

2.4 The Strong Markov Property of Brownian Motion 35

hence by dominated convergence EŒ1AF.B^.t₁^T^/; : : : ;B^.tp^T^//

D lim

n!1EŒ1AF.B^.Œt₁^Tⁿ^/; : : : ;B^.Œtp^Tⁿ^// D lim

n!1

X1 kD0

EŒ1A1f.k1/2ⁿ<Tk2ⁿgF.Bk2ⁿCt₁Bk2ⁿ; : : : ;Bk2ⁿCtpBk2ⁿ/;

where to get the last equality we have decomposed the expectation according to the possible values ofŒTn. The point now is the fact that, sinceA2FT, the event

A\ f.k1/2ⁿ<Tk2ⁿg D.A\ fT k2ⁿg/\ fT .k1/2ⁿg^c isFk2ⁿ-measurable. By the simple Markov property (Proposition2.5(iii)), we have thus

EŒ1A\f.k1/2ⁿ<Tk2ⁿgF.Bk2ⁿCt₁Bk2ⁿ; : : : ;Bk2ⁿCtpBk2ⁿ/ DP.A\ f.k1/2ⁿ<T k2ⁿg/EŒF.Bt₁; : : : ;Btp/;

and we just have to sum overkto get (2.1).

Finally, whenP.T D 1/ > 0, the same arguments give, instead of (2.1), EŒ1A\fT<1gF.B^.t₁^T^/; : : : ;B^.t_p^T^//DP.A\ fT<1g/EŒF.Bt₁; : : : ;Btp/ and the desired result again follows in a straightforward way. ut

An important application of the strong Markov property is the “reflection principle” that leads to the following theorem.

Theorem 2.21 For every t> 0, set StDsup_stBs. Then, if a0and b2.1;a, we have

P.Sta;Btb/DP.Bt2ab/:

In particular, Sthas the same distribution asjBtj.

Proof We apply the strong Markov property at the stopping time Ta Dinfft0WBtDag:

36 2 Brownian Motion We already saw (Proposition2.14) thatTa < 1 a.s. Then, using the notation of Theorem2.20, we have

P.Sta;Btb/DP.Tat;Btb/DP.Tat;B^._tT^T^a^/_aba/;

sinceB^._tT^T^a^/_a DBtBTa DBtaon the eventfTa tg. WriteB⁰ DB^.^T^a^/, so that, by Theorem2.20, the processB⁰is a Brownian motion independent ofFTa hence in particular ofTa. SinceB⁰has the same distribution asB⁰, the pair.Ta;B⁰/also has the same distribution as.Ta;B⁰/(this common distribution is just the product of the law ofTawith the Wiener measure). Let

HD f.s;w/2RCC.RC;R/Wst;w.ts/bag: The preceding probability is equal to

P..Ta;B⁰/2H/DP..Ta;B⁰/2H/

DP.Tat;B^._tT^T^a^/_aba/

DP.Tat;Bt2ab/

DP.Bt2ab/

because the eventfBt 2abgis a.s. contained in fTa tg. This gives the first assertion (Fig.2.2).

- 6

Ta t

b 2a−b

Fig. 2.2 Illustration of the reflection principle: the conditional probability, knowing thatfTatg, that the graph is belowbat timetis the same as the conditional probability that the graph reflected at levelaafter timeTa(indashed lines) is above2abat timet

2.4 The Strong Markov Property of Brownian Motion 37

For the last assertion of the theorem, we observe that

P.Sta/DP.Sta;Bta/CP.Sta;Bta/D2P.Bta/DP.jBtj a/;

and the desired result follows. ut

It follows from the previous theorem that the law of the pair.St;Bt/has density g.a;b/D 2.2pab/

2t³ exp

.2ab/² 2t

1fa>0;b<ag: (2.2)

Corollary 2.22 For every a> 0, Tahas the same distribution asa²

B²₁ and has density f.t/D a

p2t³exp a²

1ft>0g:

Proof Using Theorem2.21in the second equality, we have, for everyt0, P.Tat/DP.Sta/DP.jBtj a/DP.B²t a²/DP.tB²₁a²/DP.a²

B²₁ t/:

Furthermore, since B₁ is distributed according to N.0; 1/, a straightforward

calculation gives the density ofa²=B²₁. ut

Remark From the form of the density ofTa, we immediately get thatEŒTaD 1.

We finally extend the definition of Brownian motion to the case of an arbitrary (possibly random) initial value and to any dimension.

Definition 2.23 IfZis a real random variable, a process.Xt/t0is areal Brownian motionstarted fromZ if we can writeXt D Z CBt whereBis a real Brownian motion started from0and isindependentofZ.

Definition 2.24 A random processBt D .B¹t; : : : ;B^d_t/with values in R^d is a d- dimensional Brownian motion started from 0 if its components B¹; : : : ;B^d are independentreal Brownian motions started from 0. If Z a random variable with values inR^dandXtD.Xt¹; : : : ;X_t^d/is a process with values inR^d, we say thatXis ad-dimensional Brownian motion started fromZif we can writeXtDZCBtwhere Bis ad-dimensional Brownian motion started from0and isindependentofZ.

Note that, ifXis ad-dimensional Brownian motion and the initial value ofXis random, the components ofXmay not be independent because the initial value may introduce some dependence (this does not occur if the initial value is deterministic).

In a way similar to the end of Sect.2.2, the Wiener measure in dimension d is defined as the probability measure on C.RC;R^d/ which is the law of a

38 2 Brownian Motion d-dimensional Brownian motion started from 0. The canonical construction of Sect.2.2also applies tod-dimensional Brownian motion.

Many of the preceding results can be extended to d-dimensional Brownian motion with an arbitrary starting point. In particular, the invariance properties of Proposition2.5still hold with the obvious adaptations. Furthermore, property (i) of this proposition can be extended as follows. IfX is ad-dimensional Brownian motion and˚ is an isometry ofR^d, the process.˚.Xt//t0is still ad-dimensional Brownian motion. The construction of the Wiener measure and Blumenthal’s zero- one law are easily extended, and the strong Markov property also holds: One can adapt the proof of Theorem2.20to show that, ifTis a stopping time – in the sense of the obvious extension of Definition2.18– which is finite with positive probability, then under the probability measureP. j T <1/, the processXt^.^T^/D XTCtXT, t 0, is a d-dimensional Brownian motion started from0and is independent of FT.

Exercises

In all exercises below,.Bt/t0is a real Brownian motion started from0, and St D sup₀_stBs:

Exercise 2.25 (Time inversion)

1. Show that the process.Wt/t0 defined byW₀ D 0andWt D tB₁₌tfort > 0is (indistinguishable of) a real Brownian motion started from0. (Hint: First verify thatWis a pre-Brownian motion.)

2. Infer that lim

t!1

t D0a.s.

Exercise 2.26 For every reala0, we setTaDinfft0WBtDag. Show that the process.Ta/a0has stationary independent increments, in the sense that, for every 0ab, the variableTbTais independent of the-field.Tc; 0ca/and has the same distribution asTba.

Exercise 2.27 (Brownian bridge)

We setWtDBttB₁for everyt2Œ0; 1.

1. Show that .Wt/t2Œ0;1 is a centered Gaussian process and give its covariance function.

2. Let0 < t₁ < t₂ < < tp < 1. Show that the law of.Wt₁;Wt₂; : : : ;Wtp/has density

g.x₁; : : : ;xp/Dp

2pt₁.x₁/pt₂t₁.x₂x₁/ ptptp1.xpxp1/p₁tp.xp/;

wherept.x/D ^p¹

2texp.x²=2t/. Explain why the law of.Wt₁;Wt₂; : : : ;Wtp/can be interpreted as the conditional law of.Bt₁;Bt₂; : : : ;Btp/knowing thatB₁D0.

Exercises 39 3. Verify that the two processes.Wt/t2Œ0;1and.W1t/t2Œ0;1have the same distribution (similarly as in the definition of Wiener measure, this law is a probability measure on the space of all continuous functions fromŒ0; 1intoR).

Exercise 2.28 (Local maxima of Brownian paths) Show that, a.s., the local maxima of Brownian motion are distinct: a.s., for any choice of the rational numbers p;q;r;s0such thatp<q<r<swe have

ptqsup Bt6D sup

rtsBt:

Exercise 2.29 (Non-differentiability) Using the zero-one law, show that, a.s.,

lim sup

t#0

pt D C1 ; lim inf

t#0

pt D 1:

Infer that, for everys0, the functiont7!Bthas a.s. no right derivative ats.

Exercise 2.30 (Zero set of Brownian motion) LetHWD ft2Œ0; 1WBtD0g. Using Proposition2.14 and the strong Markov property, show thatH is a.s. a compact subset ofŒ0; 1with no isolated point and zero Lebesgue measure.

Exercise 2.31 (Time reversal) We setB⁰_tDB₁B₁tfor everyt2Œ0; 1. Show that the two processes.Bt/t2Œ0;1and.B⁰_t/t2Œ0;1have the same law (as in the definition of Wiener measure, this law is a probability measure on the space of all continuous functions fromŒ0; 1intoR).

Exercise 2.32 (Arcsine law) SetTWDinfft0WBtDS₁g:

1. Show thatT < 1a.s. (one may use the result of the previous exercise) and then thatTis not a stopping time.

2. Verify that the three variablesSt,StBtandjBtjhave the same law.

3. Show thatT is distributed according to the so-called arcsine law, whose density is

g.t/D 1

t.1t/1_.0;1/.t/:

4. Show that the results of questions1. and 3. remain valid if T is replaced by LWDsupft1WBtD0g:

Exercise 2.33 (Law of the iterated logarithm) The goal of the exercise is to prove that

lim sup

t!1

p2tlog logt D1 a.s.

40 2 Brownian Motion We seth.t/Dp

2tlog logt:

1. Show that, for everyt> 0,P.St>up

t/ 2 up

2 e^u²⁼², whenu! C1.

2. Letrandcbe two real numbers such that1 <r<c²:From the behavior of the probabilitiesP.Srⁿ>c h.rⁿ¹//whenn! 1, infer that, a.s.,

lim sup

t!1

p2tlog logt 1:

3. Show that a.s. there are infinitely many values ofnsuch that BrⁿBrⁿ¹

rr1 r h.rⁿ/:

Conclude that the statement given at the beginning of the exercise holds.

4. What is the value of lim inf

t!1

p2tlog logt ?

Notes and Comments

The first rigorous mathematical construction of Brownian motion is due to Wiener [81] in 1923. We use the nonstandard terminology of “pre-Brownian motion” to emphasize the necessity of choosing an appropriate modification in order to get a random process with continuous sample paths. There are several constructions of Brownian motion from a sequence of independent Gaussian random variables that directly yield the continuity property, and a very elegant one is Lévy’s construction (see Exercise1.18), which can be found in the books [49] or [62]. Lévy’s construction avoids the use of Kolmogorov’s lemma, but the latter will have other applications in this book. We refer to Talagrand’s book [78]

for far-reaching refinements of the “chaining method” used above in the proof of Kolmogorov’s lemma. Much of what we know about linear Brownian motion comes from Lévy, see in particular [54, Chapitre VI]. Perhaps surprisingly, the strong Markov property of Brownian motion was proved only in the 1950s by Hunt [32]

(see also Dynkin [19] for a more general version obtained independently of Hunt’s work), but it had been used before by other authors, in particular by Lévy [54], without a precise justification. The reflection principle and its consequences already appeared, long before Brownian motion was rigorously constructed, in Bachelier’s thesis [2], which was a pioneering work in financial mathematics. The book [62]

by Mörters and Peres is an excellent source for various sample path properties of Brownian motion.

Chapter 3 Filtrations and Martingales

In this chapter, we provide a short introduction to the theory of continuous time random processes on a filtered probability space. On the way, we generalize several notions introduced in the previous chapter in the framework of Brownian motion, and we provide a thorough discussion of stopping times. In a second step, we develop the theory of continuous time martingales, and, in particular, we derive regularity results for sample paths of martingales. We finally discuss the optional stopping theorem for martingales and supermartingales, and we give applications to explicit calculations of distributions related to Brownian motion.

Dalam dokumen Brownian Motion, Martingales, and Stochastic Calculus (Halaman 46-54)