Lecture 12. F o s, (1.1) F t := s>t

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1 Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let F be the Borel σ-algebra on C generated by the topology of uniform convergence on compact sets. On the measurable space (C, F), we can impose a one-parameter family of probability measures (P x ) x R (with expectations (E x ) x R ), which are the laws of a Brownian motion starting with B 0 = x almost surely. A natural filtration on C is to let F o t = σ(ω s : ω C, 0 s t), where ω s : C R is the coordinate map at time s. However it turns out to be more convenient to define a right-continuous filtration F t := s>t F o s, (1.1) which allows an infinitesimal peek into the future. Note that F t is right-continuous since F t = s>t F s. One important advantage of the right-continuity is that a time of the form τ = inf{t 0 : B t O} for an open set O R is a stopping time w.r.t. F t, but not w.r.t. Ft o. We shall see that for the Wiener measure (i.e., the measure for a standard Brownian motion), F t and Ft o differ only by sets of measure zero. Instead of using the filtration F t, quite often people also use the so-called augmented filtration for Brownian motion, which is obtained by completing Ft o with sets of measure 0 under the Wiener measure. We now show that Brownian motion satisfies the Markov property, which intuitively means that conditional on (B t ) 0 t s, the law of (B s+t ) t 0 is the same as that of a Brownian motion starting at B s at time 0. A slight complication arises because with respect to the filtration F t, the above statement should really be modified to say that we condition on (B t ) 0 t s +. However, the theorem below will imply that this makes no difference. Theorem 1.1 Markov property If s 0 and f : C R is bounded and measurable, then for all x R, E x f((bs+t ) t 0 ) F s = EBs f(( Bt ) t 0 ) a.s., (1.2) where B is a Brownian motion starting at x with E x denoting its expectation, and given B s, B is an independent Brownian motion starting at B s at time 0. Proof. We sketch the proof. To verify (1.2), it suffices to show that E x 1A f((b s+t ) t 0 ) = E x 1A E Bs f(( Bt ) t 0 ) A F s. (1.3) For ω C, we first fix f to be of the form f((ω t ) t 0 ) = 1 n f i (ω ti ), (1.4)

2 where 0 < t 1 < < t n, and f i : R R are bounded and measurable. Fix h (0, t 1 ). Then for A Fs+h o of the form A = {ω C : ω si A i, 1 i m}, where 0 < s 1 < < s m s + h and A i are Borel sets, we can easily verify that E x 1A f((b s+t ) t 0 ) n = E x 1 A f i (B s+ti ) = E x 1 n A E Bs+h f i ( B ti h). (1.5) By the π-λ theorem, (1.5) holds for all A Fs+h o F s. Given any A F s, we can let h 0 in (1.5). By writing E Bs and E Bs+h explicitly as integration w.r.t. Gaussian densities, and using the fact that a.s. lim h 0 B s+h = B s, we can then apply dominated convergence theorem to deduce that n lim E Bs+h f i ( B n ti h) = E Bs f i ( B ti ). h 0 Therefore (1.2) holds for all f of the form (1.4). We can then use the monotone class theorem to deuce (1.2) for all bounded measurable f. One consequence of the Markov property of (B t ) t 0 w.r.t. the filtration F t is that Theorem 1.2 For any bounded measurable function f : C R and for all s 0 and x R, E x f((b t ) t 0 ) F s = E x f((b t ) t 0 ) F o s. (1.6) Proof. By the monotone class theorem, it suffices to consider f of the form in (1.4), and we can separate the indices t i into t i s and t i > s and write f((b t ) t 0 ) = g 1 ((B t ) 0 t s )g 2 ((B t ) t>s ). Then E x f F s = g 1 ( (Bt ) 0 t s ) Ex g 2 F s. By the Markov property (1.2), E x g 2 F s = E Bs g 2 (( B t ) t 0 ) F o s. Since F o s F s, this implies that E x g 2 F s = E x g 2 F o s, and hence E x f F s = E x f F o s. Setting f = 1 A for A F s in Theorem 1.2 shows that F s and Fs o are equivalent up to sets of measure zero. Furthermore, if we set s = 0 and let f = 1 A for A F 0, then we obtain 1 A = P x (A) so that P x (A) {0, 1} for all A F 0. This is known as Blumenthal s 0-1 law. Theorem 1.3 Blumenthal s 0-1 law For any x R, P x (A) {0, 1} for all A F 0. The σ-field F 0 is called the germ field, which is trivial by Blumenthal s 0-1 law. We remark that Blumenthal s 0-1 law is valid for more general Markov processes under suitable continuity assumptions. Here is an interesting application for Brownian motion. Theorem 1.4 Let τ = inf{t > 0 : B t > 0}. Then P 0 (τ = 0) = 1. Proof. Note that {τ = 0} F 0. Therefore by Blumenthal s 0-1 law, P 0 (τ = 0) {0, 1}. On the other hand, Therefore we must have P 0 (τ = 0) = 1. P 0 (τ = 0) = lim t 0 P 0 (τ t) lim t 0 P 0 (B t > 0) = 1 2. Theorem 1.4 shows that almost surely, there is an infinite sequence of times t n 0 with B tn > 0, and by symmetry, there also exists an infinite sequence of times s n 0 with B sn < 0. Thus by the a.s. continuity of Brownian sample paths, (B t ) t 0 crosses level 0 infinitely often in any neighborhood of t = 0. This is consistent with what we know from the law of the iterated logarithm for Brownian motion near t = 0. 2

3 2 Brownian motion: the strong Markov property The strong Markov property for Brownian motion basically says that conditioned on (B s ) 0 s τ for any stopping time τ, (B τ+t ) t 0 is distributed as a Brownian motion starting from B τ at t = 0. First let us define stopping times. Definition 2.1 Stopping times A random variable τ : C(0, ), R) 0, is called a stopping time if for all t 0, {ω C : τ(ω) t} F t. The corresponding stopped σ-field F τ is defined as {A F : A {τ t} F t t 0}. Note that because of the right continuity of (F t ) t 0, {τ t} F t for all t 0 if and only if {τ < t} F t for all t 0. Example 2.2 Examples of stopping times 1 If O R is an open set, then τ O = inf{t 0 : B t O} is a stopping time. 2 If K R is a closed set, then τ K = inf{t 0 : B t K} is a stopping time. 3 If T n is a sequence of stopping times and T n T (resp. T n T ), then T is a stopping time. 4 If S and T are stopping times, then S T and S T are stopping times. We leave the verification of these assertions as an exercise. We now state formally the strong Markov property for Brownian motion. Theorem 2.3 Strong Markov property Let f s (ω) : 0, ) C R be jointly measurable in s 0, ) and ω C. If τ is a stopping time, then for all x R, E x fτ ((B τ+t ) t 0 ) F τ = EBτ fτ (( B t ) t 0 ) a.s., (2.7) where B is a Brownian motion starting at x, and given B τ, B is an independent Brownian motion starting at B τ. The strong Markov property can be proved by first proving the statement for stopping times which take values in a countable discrete set, which follows from the Markov property. General stopping times can then be approximated by discrete stopping times. The only thing that we need to carry out this approximation is the continuity of the Brownian motion transition kernel in both space and time, which is known as the Feller property for Markov processes. See 1, Sec. 7.3 for a detailed proof. We now apply the strong Markov property to deduce some interesting properties for Brownian motion. Theorem 2.4 Zeroes of Brownian motion Let (B t ) t 0 be a standard Brownian motion with B 0 = x R. Let Z = {t 0 : B t = 0} be the zero set of B. Then almost surely, Z is a perfect set, i.e., Z is closed and every point in Z is an accumulation point. Furthermore, almost surely Z has Lebesgue measure 0. 3

4 Proof. Since B t is almost surely continuous in t, Z is a closed set. If with positive probability, Z contains an isolated point t ω so that B s 0 for all s (t ω ɛ ω, t ω + ɛ ω ) for some ɛ ω depending on B, then we can find a, δ Q with a > δ such that with positive probability, B has a unique zero t ω (a δ, a + δ). Let τ := inf{t a δ : B t = 0}, which is a stopping time. Since P x (B a δ = 0) = 0, we have P x (τ = a δ) = 0. Therefore by assumption, with positive probability τ = t ω, and it is an isolated zero of B in (a δ, a + δ). However, this is not possible by the strong Markov property, which implies that conditional on (B s ) 0 s τ, (B τ+s ) s 0 is distributed as a Brownian motion B starting at 0, and by Blumenthal s 0-1 law, t = 0 is a.s. an accumulation point of the zeros of ( B t ) t 0. The fact that Z almost surely has Lebesgue measure zero follows from Fubini s theorem, since T T E 1 {Bt=0}dt = P(B t = 0)dt = 0 for any T > Theorem 2.5 Reflection principle Let (B t ) t 0 be a standard Brownian motion with B 0 = 0. Let M t := sup 0 s t B s be the running maximum of B t. Then for any a > 0 and t > 0, P(M t a) = 2P(B t a) = P( B t a). (2.8) Note that with B 0 = 0 and a > 0, {M t a} = {τ a t}, where τ a := inf{t 0 : B t = a}. Proof. Since τ a is a stopping time, so is τ a t, and hence by the strong Markov property, E1 {Bt a} F t τa = 1 {τa t}p a ( B t τa a) = {τ a t}, where B is an independent Brownian motion starting from a. Taking expectation on both sides then yields (2.8). 3 Brownian motion as a martingale Most of the results for discrete time martingales, such as Doob s inequality, upcrossing inequality, martingale convergence theorems, optional stopping theorem, have analogues for continuous time martingales, provided we assume that the continuous time martingale has sample paths which are right continuous with left hand limits, known as càdlàg paths by its abbreviation from French. In particular, the optional stopping theorem states the following. Theorem 3.1 Optional stopping theorem Let (X t ) t 0 be a continuous time martingale with cádlág sample paths adapted to a right-continuous filtration (F t ) t 0, i.e., for any 0 s t <, EX t F s = X s a.s. Then for any two stopping times 0 σ τ, if (X t τ ) t 0 is uniformly integrable, then EX τ F σ = X σ a.s. (3.9) In particular, if τ is a bounded stopping time, then (X t τ ) t 0 is uniformly integrable. We collect here some most common functionals of Brownian motion which are martingales. Theorem 3.2 Martingales for Brownian motion The following functionals of (B t ) t 0 are martingales w.r.t. (F t ) t 0 : B t, B 2 t t, B 3 t 3tB t, e θbt θ2 t/2. 4

5 Remark. Theorem 3.2 can be easily verified using the independent Gaussian increment properties of (B t ) t 0. In fact, for any function f(t, x) with t f x f 0, f(t τ L, B t τl ) is a martingale for any L > 0, where τ L := inf{s 0 : B s L}. This includes the case when f(t, x) = f(x) is a harmonic function. The statement can be proved by Ito s formula. Remark. Applying the optional stopping theorem to the martingale B t, we can derive the exit probabilities of B t from the end points of an interval a, b. Using B 2 t t, we can compute the expected exit time of B t from a, b. 4 Infinitesimal generator of a Brownian motion For a discrete time Markov chain X with state space S and transition matrix Π, the operator Π I plays an important role. For any bounded measurable f : S R, f(x n ) f(x 0 ) n 1 i=0 (Π I)f(X i) is in fact a martingale. Furthermore, if (Π I)f = 0, in which case we call f a harmonic function for the Markov chain with transition matrix Π, f(x n ) is a martingale. We now introduce the continuous time analogue of Π I for Brownian motion, called the infinitesimal generator of Brownian motion. If (X t ) t 0 is a continuous time Markov process with state space S which is a complete separable metric space, then the semigroup associated with X is defined as the family of operators (S t ) t 0 acting on the class of bounded continuous functions f C b (S, R) with (S t f)(x) = E x f(x t ) for all x S. By the Markov property, it is easy to check that S t S s f = S s S t f = S t+s f, which together with the lack of inverse accounts for the name semigroup. The infinitesimal generator is then defined via S h f f Lf = lim. (4.10) h 0 h The class of f C b (S, R) for which the limit Lf C b (S, R) exists and the convergence takes place in C b (S, R) with sup norm is called the domain of the generator L. The generator L together with its domain uniquely determine the Markov process. Although one could also consider the adjoint semigroup (S t ) t 0 which acts on probability measures on S, functions are easier to handle because measures can in general be singular. Let us check that 1 2, where denotes the Laplacian, is the infinitesimal generator for Brownian motion, and the class of twice continuously differentiable functions with compact support, Cc 2, belongs to the domain of 1 2. Let f C2 c. Then by Taylor expansion, (S h f)(x) f(x) = Ef(x+B h ) f(x) = E f (x)b h + 1 x+bh 2 f (x)bh 2 + x In particular, (S h f)(x) f(x) 1 2 f x+b h (x)h E x y x y x (f (z) f (x))dzdy. f (z) f (x) dzdy E φ( B h )Bh 2, where φ(r) = sup x R, y x r f (y) f (x) is bounded with φ(r) 0 as r 0 by the assumption that f Cc 2. Since Eφ( B h )Bh 2 lim = lim Eφ( h B 1 )B 2 h 0 h h 0 1 = 0, it follows that h 1 (S h f f) converges in sup-norm to 1 2 f as h 0. The same argument applies to higher-dimensional Brownian motions. Ito s formula is also based on such Taylor expansions. Similar to the discrete time setting, if f is bounded and f = 0, then f(b t ) is a martingale. 5

6 5 Transience and recurrence of a Brownian motion in R d One-dimensional Brownian motion is clearly recurrent in the sense that it visits every point in R infinitely often, as can be seen from the law of the iterated logarithm. A d-dimensional Brownian motion is simply an R d -valued process B t := (B (1) t,, B (d) t ), where the coordinates are independent one-dimensional Brownian motions. It turns out that in d 3, B t is transient in the sense that B t a.s.; and in d = 2, B t a.s. returns to each open set infinitely often as t, but does not return to any fixed deterministic point. The proof rests on finding a suitable martingale. Recall that the fundamental solution for the Laplacian is log x d = 2, φ(x) = 1 x d 2 d 3. In particular, φ(x) = 0 for all x 0. Then for any 0 < r < R <, φ(b t τr τ R ) is actually a bounded positive martingale, where τ a = inf{s 0 : B s = a}. It is not difficult to show that if B 0 = x with x (r, R), then τ r τ R < almost surely. Therefore by the optional stopping theorem, φ(x) = P x (τ r < τ R )φ(r) + P x (τ R < τ r )φ(r), which implies that P x (τ r < τ R ) = φ(r) φ(x) φ(r) φ(r) = log R log x log R log r d = 2, R 2 d x 2 d R 2 d r 2 d d 3. (5.11) For d = 2, as we let R, we find that P x (τ r < ) = 1. Therefore by the strong Markov property, B t must visit the ball {z R 2 : z r} infinitely often a.s. as t. Since r > 0 can be arbitrary, B t must visit each open set in R 2 infinitely often a.s. If we fix R and let r 0, then we find that P x (τ R < τ 0 ) = 1. Since R can be arbitrary, this implies that B t almost surely never visits the origin, or any other deterministic point. For d = 3, fixing r and letting R gives P x (τ r < ) = rd 2 < 1. Therefore B x d 2 t is transient. It is easy to see that there is zero probability that B t stays confined in a finite ball for all time. Therefore for R > B 0, τ R < almost surely. By the strong Markov property, P BτR τ r < rd 2, which tends to 0 as R. Therefore, almost surely, B R d 2 t > r for all t sufficiently large. Since r > 0 is also arbitrary, this implies that B t as t. In fact, B t for a d-dimensional Brownian motion is a process on 0, ) called the d- dimensional Bessel process, which can be constructed from a 1-dimensional Brownian motion by adding a location-dependent drift to. References 1 R. Durrett, Probability: Theory and Examples, 2nd edition, Duxbury Press, Belmont, California,