PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS

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1 MATHEMATICAL TRIPOS Part III Thursday, 5 June, 214 1:3 pm to 4:3 pm PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

2 1 2 Let X be a continuous local martingale with X =, such that E( X p/2 t ) < for all t and p 2. (a) By applying Itô s formula to X t p or otherwise, show that if X is uniformly bounded then E( X t p ) p(p 1) ( ) E sup X s p 2 X t 2 s t for all p 2. Conclude that there is a constant C p, depending only on p, such that ( ) E sup X s p C p E( X p/2 t ). s t Show that inequality ( ) remains valid even if X is unbounded. [You may use without proof Doob s inequality: for any continuous martingale M we have ( ) E sup M s p p p s t (p 1) pe( M t p ) ( ) for all p > 1.] (b) Show that Y t = X 4 t 6X2 t X t +3 X 2 t defines a martingale. Assuming that E(X 2 t) = t, Cov(X 2 t, X t ) = for all t, show that E(X 4 t) 3t 2 ; furthermore, show that if E(X 4 t) = 3t 2 for all t, then X is a Brownian motion.

3 2 3 (a) Let (M t ) t be a continuous local martingale with M = with respect to a filtration (F t ) t satisfying the usual conditions. State and prove the Dambis Dubins Schwarz theorem in terms of M under the extra assumption that M is strictly increasing. (b) Let X and Y be independent Brownian motions, and let R t = Xt 2 +Y2 t. Show that there exists a Brownian motion W and an increasing adapted process A such that R 2 t = 2W A(t) +2t. Now let Z t = Y s dx s X s dy s. Show that there is a Brownian motion B which is independent of W and such that Z t = B A(t). 3 Letg n (t) = 2cos[(n 1 2 )πt]andh n(t) = 2sin[(n 1 2 )πt], andletw beabrownian motion. In this question you may use without proof the fact that the collections (g n ) n 1 and (h n ) n 1 are both orthonormal bases of L 2 [,1]. (a) Letξ n = 1 g n(t)dw t. Showthatthesequence(ξ n ) n 1 areindependentn(,1) random variables. (b) Show that ξ n = (n 1 2 )π 1 h n(t)w t dt and hence conclude that 1 (c) Compute the Laplace transform W 2 t dt = n=1 E(e λ 1 W2 t dt ) 1 (n 1 2 )2 π 2ξ2 n. in terms of λ. [You might find it useful to note that ( ) x 2 1+ (n 1 = coshx 2 )2 π 2 for all x R.] n=1 [TURN OVER

4 4 4 Let Z be a positive continuous uniformly integrable martingale with Z = 1, defined on a probability space (Ω,F,P). Define an equivalent measure Q P on (Ω,F) by the density dq dp = Z. Let X be a continuous P-local martingale with X =, and let Y t = X t logz,x t. Show that the process Y is a Q-local martingale. [If you use Girsanov s theorem, you must prove it.] Now supposew is Brownian motion defined on the same probability space (Ω,F,P) and suppose that W generates the filtration. Show that there exists a predictable process α such that and such that the process α 2 sds < almost surely for all t Ŵ t = W t α s ds is a Q-Brownian motion. [You may appeal to any standard integral representation results if clearly stated.] 5 LetM beacontinuous, non-negative local martingale suchthatm = 1and M t almost surely as t. (a) If M is strictly positive, show that M t = e Xt X t/2 for a continuous local martingale X such that X = almost surely. (b) For each a > 1, let T a = inf{t : M t > a}. Show that P(T a < ) = P(supM t > a) = 1/a. t [Hint: Compute the expected value of M t Ta = a1 {Ta t} +M t 1 {Ta>t}.] (c) Let W be a Brownian motion. Find the density functions of the following random variables. 1. sup t τ( b) W t where τ( b) = inf{t : W t < b} and b >. 2. sup t (W t λt) for λ >.

5 5 6 Consider the stochastic differential equation dx t = b(x t )dt+σ(x t )dw t, ( ) where W is a Brownian motion and the functions b and σ are bounded and smooth. Assume that for every square-integrable ξ independent of W, there exists a unique strong solution X such that X = ξ and sup t E(X 2 t) <. Let f : R R be a smooth function and let u : [, ) R R be a bounded and smooth solution of the PDE with boundary condition (a) Show that u(t,x) = E[f(X t ) X = x] u t = b(x) u x σ(x)2 2 u x 2, u(,x) = f(x) for all x R. Suppose b/σ 2 is locally integrable and let p(x) = C ( x ) σ(x) 2 exp 2b(s) σ(s) 2ds where C > is chosen so that p(x)dx = 1. Assume x2 p(x)dx <. (b) Briefly show that u(t, x)p(x)dx = f(x)p(x)dx for all t. You may integrate by parts and apply Fubini s theorem without justification. Now suppose there is a constant k > such that 2(x y)[b(x) b(y)]+[σ(x) σ(y)] 2 k(x y) 2. (c) Let Y be another strong solution of ( ). Show that E[(X t Y t ) 2 ] E[(X Y ) 2 ]e kt. [Hint: You may use this version of Gronwall s lemma: If h is locally integrable and h(t) h(s) k then h(t) h()e kt for all t.] (d) Show that u(t,x) for all x R. s h(u)du for all s t, f(y)p(y)dy as t. [TURN OVER

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