1 Brownian Local Time

Transcription

1 1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) = 1 with probability one. For µ the Lebesgue measure. What about the time the Wiener process spends near zero? Define 1 L t (x) = lim ɛ 4ɛ µ({ s t : W s x ɛ}). Paul Léve first proved that L exists and is finite and non-trivial (not identically equal to zero). He first referred to it as le mesure du voisinage or the measure of the vicinity (of the Wiener process). We start first with the occupation time of the Wiener process. Let F t be a filtration on (Ω, F, P ). F t may be the (usual augmentation of) σ-algebra generated by a k-dimensional Wiener process W t up to time t: F t = σ({w s : s t}). We also denote {P } as the probability measure corresponding to the Wiener process with W =. This can be generalized for W = z easily. For any B R, Γ t (B) = χ {Ws B} ds = measure{ s t : W s B} Then Γ(B) as a process of t, is adapted and continuous with respect to F t. We can write the above equation using L t, Γ t (B) = 2L t (x, ω) dx (1) In terms of the Radon-Nikodyn derivative B L t = 1 dγ t 2 dx or the rate of change of the occupation measure with respect to the Lebesgue measure. Definition of L t Let L = {L t (x, ω); (t, x) [, ) R, ω Ω} be a random field with values in [, ), s.t. for all (t, x) the RV L t (x) is F t -measurable. Suppose that there exists Ω F such that P (Ω ) = 1 and for each ω Ω (t, x) L t (x) is continuous and equation (1) holds, then L is called Brownian Local Time (BLT). 1.1 Intuition of BLT Formally we want L t (a, ω) = δ(w s a) ds

2 So let s find a way to use Itô s rule to derive an equation for L. Want to find a φ(x) such that dφ(w t ) = φ (W t )dw t φ (W t )(dw t ) 2 and φ = δ. Using the idea of distributional derivatives define { x < a φ(x) = x x a Then, distributionally, φ (x) = for x < a and φ (x) = 1 x a. Then or We write dφ(w t ) = H(W t a) dw t δ(w t a)(dw t ) 2 dφ(w t ) = H(W t a) dw t δ(w t a)dt L t (a) = (W t a) + (W a) + χ {Ws a} dw s (2) Now you can see why we chose to define 2L t before because of the factor of 1/2 that arises from the Itô formula. This discussion can be made rigorous using a regularization technique, see Proposition (3.6.8 of [2]) for full proof. The existence of Brownian Local Time is proven in the Trotter Existence Theorem [2]. Important Remark: The process (W t a) + is a continuous, non-negative submartingale (because it is a convex function of a continuous submartingale Prop of [2]). Thus it admits a Doob decomposition, (W t a) + = (W a) + + M t (a) + A t (a), t <, where M t (a) is a martingale, and A t (a) is a continuous, increasing process. Thus we know that L t is increasing. Also, by symmetry, we can take equation (2) and reflect it about the line y=a. Then adding it to equation (2) we get 2L t (a) = W t a W a sgn(w s a) dw s ; t <. where sgn(x) = 1 x < and 1 for x. However, it doesn t matter how we define sgn. The above equation gives a definition for L t and W t in terms of each other. 2 Connection of Local Time to The Running Maximum of BM Lemma: (The Skorohod equation (1961)) Let z be a given number and y( ) = {y(t); t < } a continuous function with y() =. Then there exists a unique continuous function k( ) = {k(t); t < }, such that 2

3 1. x(t) = z + y(t) + k(t), is a positive continuous function for all t < 2. k() =, k( ) is nondecreasing, and 3. k( ) is constant on the sets {t ; x(t) } which implies χ {x(s)>} dk(s) =. k(t) = max[, max{ (z + y(s))}], t < s t Proof. With the definition of k, (i),(ii) are done. For the flat off part (iii) we need to show χ {x(s)>ɛ} dk(s) = for all ɛ >. Let (t 1, t 2 ) {s ; x(s) > ɛ} and (z + y(s)) = k(s) x(s) k(t 2 ) ɛ; t 1 s t 2 because x(s) > ɛ. However, because k is non-decreasing, k(t 2 ) = max[k(t 1 ), max t 1 s t 2 { (z + y(s))}] max[k(t 1 ), k(t 2 ) ɛ] Thus k(t 2 ) = k(t 1 ). For uniqueness, let k and k be continuous with i iii and x and x be the corresponding solutions. Suppose there exists T > with x(t ) > x(t ). Then define τ = max{ t T ; x(t) x(t) = }. But k is flat on {u ; x(u) > }, thus k(τ) = k(t ) and therefore < x(t ) x(t ) = k(t ) k(t ) k(τ) k(τ) = x(τ) x(τ) because y is the same for each x. This is a contradiction because x(τ) x(τ) =. Thus x(t ) x(t ) and k k. Now repeat for k k. Back to 2L t (a) = W t a W a or for a = and W =, First note that W t = sgn(w s ) dw s +2L t () }{{} B t = E[ B t 2 ] = B t sgn(w s a) dw s ; t <. sgn(w s ) 2 ds = t. (3) Thus B t is a Brownian Motion. Also, by the Skorohod equation above, L t is nondecreasing and. χ {R\{}} dl t () = 3

4 At the same time, note that where y = B, z = and k = 2L. or k(t) = max[, max{ (z + y(s))}] s t 2L t = max[, max s t {(B t)}] in the sense of distribution. Theorem(P. Lévy 1948). The pairs of processes {(M W t W t, M W t ); t < } and {( W t, 2L t (); t < } where M W t = max s t W s, have the same laws under P. Proof. Because of uniqueness of the Skorohod equations, k = 2L t and k = M B t. Also, W t = M W t B t for P almost every ω Ω. Because we have shown that B t is a Brownian motion [Eq. (3)], both B and W are Brownian motions starting at the origin under P. 3 Existence and Uniqueness for SDE with one discontinuity 3.1 Weak Solution Consider the one dimensional SDE dx t = b(x t ) dt + σ(x t ) dw t, (4) where b and σ are bounded measurable functions. Definition: A weak solution of equation (4) is a triple (X, W ), (Ω, F, P ), {F t }, where (Ω, F, P ) is a probability space, and {F t } is a filtration satisfying the usual conditions. X = {X t, F t ; t < } is a continuous, adapted R valued process. and W = {W t, F t ; t < } is a 1-D Brownian motion. For every t <, P ( [ b(x s ) + σ 2 (X s ) ds < ) = 1 The integral version of equation (4) holds almost surely for t <. Ex: Tanaka Consider, X t = sgn(x t ) dw t, t < (5) Note that if (X, W ), (Ω, F, P ), {F t } is a weak solution then ( X, W ), (Ω, F, P ), {F t } is also a weak solution. First note that X t is a 1-D Brownian motion (i.e. X t = t), and thus given X = {X t, Ft X ; t < }, where Ft X is the augmented filtration under P, then W t = sgn(x s ) dx s, 4

5 is a BM adapted to {Ft X }. It can be shown that (X, W ), (Ω, F, P ), {Ft X } is a weak solution to equation (5) where {Ft X } (see Corollary [2]). Assume that X t is a strong solution with Ft X Ft W for all t. Then from equation (5), X is a BM and from the Tanaka Formula, W t = sgn(x s ) dx s = X t 2L X t (). Consequently Ft W F X t, and thus Ft X F X t by assumption, for all t. This is a contradiction; the information generated by X can not contain information about the original process. 3.2 Existence and Uniqueness Proof Consider SDE (4) where σ(x) = D 1 for X > and D for X, b is some piecewise continuous function with possibly one jump at X =, and X =. I.e. { b1 (X dx t = t ) dt + D 1 dw t X t > b (X t ) dt + D dw t X t. A system similar to the one above is studied in [5, 6] and referred to as sliding dynamics. First intuition would be to build a solution of X, then at the random time that X t hits X =, use the definitions of σ and b to build the solution again. However, we know that given W = and any time interval t [, ɛ] ɛ >, W t will hit zero infinitely often in that time interval. We would expect X to do the same. By Theorem [2], σ > ɛ for some ɛ > implies there exists a weak solution. We will use this fact to prove the existence and uniqueness of a strong solution. To do so we use a theorem that, given a weak solution and pathwise uniqueness, the weak solution is actually a strong one. First we define pathwise uniqueness as: Definition: Suppose that whenever (X, W ), (Ω, F, P ), {F t } and ( X, W ), (Ω, F, P ), { F t } are weak solutions to SDE (4), with a common BM W on a common probability space, and P (X = X ) = 1, then pathwise uniqueness hold for equation (4) if P (X t = X t, t < ) = 1. Then a theorem by Yamada and Watanabe imply there exists a unique solution of SDE (4) with a given initial value on any filtered probability space carrying a BM W t. Moreover, this solution is a strong one. Theorem 1 (Yamada and Watanabe (1971) Corr 3.23 [2]). Weak existence and pathwise uniqueness imply strong existence. Before proving pathwise uniquness, we must construct a probability space and measure which we can measure two weak solutions of SDE (4). We will not discuss this step here, but refer to Theorem [1], Theorem [4], for examples of how to do so. Theorem 2 (Le Gall 1985, [3]). Suppose that there exists ɛ > such that σ(t, x) > ɛ, and there exists a strictly increasing function f : R R such that (σ(t, x) σ(t, y)) 2 f(x) f(y) for all (t, x, y). Then pathwise uniqueness holds for SDE (4). 5

6 Note: For the example of σ(x) = D 1 for X > and D for X, we consider the function { (D1 D f(x) = ) 2 + X, X X, X <. Proof. We begin by assuming there exists two solutions X 1 and X 2 on the same probability space (Ω, F, P ), with the same Wiener process W t, and initial condition X 1 = X 2 = X. Then for Y = X 1 X 2, we have χ {Ys>} Y s d Y s = (σ(xs 1 ) σ(xs 2 )) 2 χ Xs 1 Xs 2 {X 1 s Xs 2 } ds (f(xs 1 ) f(xs 2 )) 2 χ Xs 1 Xs 2 {X 1 s Xs 2 } ds < Taking C 1 approximations of f, then the integral above is finite (still needs work, what is the assumption needed for f? And using C 1 approximations in what norm?). Similarly, by the occupation time formula, χ {Ys>} Y s d Y s = 1 y LY t (y) dy, and since it is finite, and becuase 1/y dy = for an ɛ-neighborhood on the positive n + ɛ () side of (n + ɛ ()), then L X1 X 2 t () = for all t, almost surely. We now show that min{x 1, X 2 } = X 1 X 2 and max{x 1, X 2 } = X 1 X 2 are solutions to SDE (4). Note that By the Tanaka formula X 1 t X 2 t = X 1 X 2 = 1 2 (X1 + X 2 ) 1 2 X1 X 2. However, L X1 X 2 t () = a.s. for all t. Thus sgn(x 1 t X 2 t ) d(x 1 t X 2 t ) + 2L X1 X 2 t (). d(x 1 X 2 ) t = 1 2 [b(x1 t ) + b(x 1 t )] ds (σ(x1 t ) + σ(x 1 t )) dw t 1 2 [sgn(x1 t X 2 t )[b(x 1 t ) b(x 2 t )] dt + (σ(x 1 t ) σ(x 2 t )) dw t ]. You can check above, that when X 1 > X 2, only the X 2 terms remain. Thus d(x 1 X 2 ) = b((x 1 X 2 ) t ) dt + σ((x 1 X 2 ) t ) dw t, and X 1 X 2 is a solution to SDE (4). Similarly for X 1 X 2, therefore, by weak uniqueness X 1 X 2 = X 1 X 2 and thus X 1 = X 2 which gives pathwise uniqueness. 6

7 References [1] Nobuyuki Ikeda and Shinzo Watanabe. Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, [2] Ioannis Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, [3] J.-F. Le Gall. One-dimensional stochastic differential equations involving the local times of the unknown process. In Stochastic analysis and applications (Swansea, 1983), volume 195 of Lecture Notes in Math., pages Springer, Berlin, [4] Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, [5] D. J. W. Simpson and R. Kuske. Stochastically Perturbed Sliding Motion in Piecewise- Smooth Systems. ArXiv e-prints, April 212. [6] D. J. W. Simpson and R. Kuske. The Positive Occupation Time of Brownian Motion with Two-Valued Drift and Asymptotic Dynamics of Sliding Motion with Noise. ArXiv e-prints, April 212. [7] G. George Yin and Chao Zhu. Hybrid switching diffusions, volume 63 of Stochastic Modelling and Applied Probability. Springer, New York, 21. Properties and applications. 7