. Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. An (H, H)-valued random variable X is independent of D if and only if P ({X Γ} D) = P {X Γ}P (D) for all Γ H and D D. Prove that if X is independent of D, then for all f satisfying E[ f(x) ] <. E[f(X) D] = E[f(X)], 2. Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. Let (H i, H i ), i =, 2, be measurable spaces, and suppose that X is an H -valued random variable and Y is an H 2 -valued random variable define on (Ω, F, P ). Suppose that X is independent of D and Y is D-measurable. Let µ X denote the distribution of X. Let ψ : H H 2 R be bounded and H H 2 -measurable, and define ϕ(y) = H ψ(x, y)µ X (dx). Show that E[ψ(X, Y ) D] = ϕ(y ). 3. Let {F t } be a filtration. τ is an {F t }-stopping time if {τ t} F t for all t, and the information available at the random time τ is F τ = {A F : A {τ t} F t, t }. (a) Show that τ is F τ -measurable. (b) Suppose P {τ = a} >. Show that for an integrable random variable Z E[Z F τ ]I {τ=a} = E[Z F t ]I {τ=a}. (c) Let τ be a discrete stopping time with range {t, t 2,...}. Show that E[Z F τ ] = E[Z F tk ]I {τ=tk }. k= 4. Let τ τ 2 be {F t }-stopping times, and for k =, 2,..., let ξ k be F τk - measurable. Define X(t) = ξ k I [τk,τ k+ )(t). Show that X is {F t }-adapted. k= 5. Let ξ be a Poisson process with mean measure ν m, compatible with {F t }. Let Z be cadlag with values in L 2 (ν) and adapted to {F t }, and define X(t) = Z(u, s ) ξ(du ds). U [,t] Let f C 2 (R). Represent f(x(t)) f(x()) as an integral involving ξ. (In other words, apply Itô s formula to f(x(t)) and express the result interms of ξ.) 6. Let Y be cadlag and suppose T t (Y ) < for all t >. Describe [Y ] t.
7. Let d = and Af(x) = 2 a(x)f (x) + b(x)f (x). Assume that a(x) > for each x and that /a(x) is locally bounded. If X is a solution of the martingale problem for A, then is a local martingale. Show that is a standard Brownian motion. M(t) = X(t) W (t) = b( X(s))ds dm(s) a( X(s)) 8. The generator for a process with independent increments can be written as 2 σ2 f (x) + bf (x) + (f(x + u) f(x) I { u } uf (x))ν(du), where ν satisfies u2 ν(du) <. Show how to represent the process in terms of a standard Brownian motion W and a Poisson random measure ξ on (, ) [, ) with mean measure ν m. 9. For i =,..., m, let X i be a solution of the martingale problem for A i. Suppose that X,..., X m are independent. Show that X = (X,..., X m ) is a solution of the martingale problem for ( m m ) m {( f i, f i i= i= k= A k f k f k ) : f k D(A k )}.. Let Y in E be a solution of the martingale problem for A, and for β : E [, ), let X satisfy ( ) X(t) = Y β(x(s))ds. Show that X is a solution of the martingale problem for βa.. Let λ : R d [, ) be measurable, and let µ(x, dz) be a transition function on R d. There exists γ : R d [, ] R d such that f(x + γ(x, u))du = f(z)µ(x, dz). R d Let ξ be a Poisson random measure on [, ) [, ] [, ) with Lebesgue mean measure. Show that X(t) = X() + I [,λ(x(s ))] (v)γ(x(s ), u)ξ(dv du ds) [, ) [,] [,t] 2
is a stochastic differential equation corresponding to A given by Af(x) = λ(x) (f(z) f(x))µ(x, dz). R d 2. Suppose X has values in D and satisfies X(t) = X()+ σ(x(s))dw (s)+ b(x(s))ds+ α(x(s))dw λ(s)+ η(x(s))dλ(s) where λ is nondecreasing and increases only when X(t) D and W is a standard Brownian motion independent of W. (If n(x) is the inward normal vector at x D, then we require η(x) n(x) > and n(x) T α(x) =.) Derive the martingale problem satisfied by X. 3. Show that I [+ n, ) I [, ) in D R [, ) but that (I [+ n, ), I [, )) does not converge in D R 2[, ). (It does converge in D R [, ) D R [, ).) 4. For each of the following mappings, verify the stated continuity properties. (E, r) is a complete, separable metric space; D E [, ) is the space of cadlag E-valued functions with the Skorohod topology; C F denotes the set of continuity points of a mapping F. (a) π t : D E [, ) E is defined by π t (x) = x(t). Then C πt = {x D E [, ) : x(t) = x(t )} (b) G t : D R [, ) R is defined by G t (x) = sup s t x(s). Then C Gt = {x D R [, ) : lim s t G s (x) = G t (x)} {x D R [, ) : x(t) = x(t )} (c) G : D R [, ) D R [, ) is defined by G(x)(t) = G t (x). Then G is continous. (d) H t : D E [, ) R is defined by H t (x) = sup s t r(x(s), x(s )). Then C Ht = {x D E [, ) : lim s t H s (x) = H t (x)} {x D E [, ) : x(t) = x(t )} (e) H : D E [, ) D R [, ) is defined by H(x)(t) = H t (x). Then H is continuous. (f) τ c : D R [, ) [, ) is defined by τ c (x) = inf{t : x(t) > c}, and τ c : D R [, ) [, ) is defined by τ c (x) = inf{t : x(t) c or x(t ) c}. Then G τc = G τ c = {x : τ c (x) = τ c (x)}. Note that τ c (x) τ c (x) and that x n x implies τ c (x) lim inf n τ c (x n ) lim sup τ c (x n ) τ c (x). n 3
5. Suppose X(t) = X() + σ(x(s))dw (s) + where σ and b are bounded. Estimate E[(X(t + h) X(t)) 2 Ft X ]. 6. Let Y be a semimartingale, and define b(x(s))ds Y n (t) = Y ( k n ) Show that {Y n } is a good sequence. k n t < k + n. 7. Let ξ be a Poisson random measure on U [, ) with mean measure ν m, and let U n U n+ U with U = n U n. Define ξ n (ϕ, t) = ϕ(u)ξ(du ds) ϕ L (ν) U n and ξ n (ϕ, t) = ϕ(u) ξ(du ds). U n Show that {ξ n } is uniformly tight for H = L (ν) and that { ξ n } is uniformly tight for H = L 2 (ν). 8. Prove the conditioned martingale lemma. 9. Let N be a unit Poisson process and let W n (t) = nt ( ) N(s) ds n Show that there exist martingales M n such that W n = M n + V n and V n, but T t (V n ). Apply the martingale central limit theorem to show that W n W where W is standard Brownian motion. 2. Let W n be as in Problem 9. Let σ have a bounded, continuous derivative, and let X n (t) = σ(x n (s))dw n (s). Show that X n X for some X and identify the stochastic differential equation satisfied by X. Hint: Write X n (t) = σ(x n (s ))dm n (s) + σ(x n (s ))dv n (s). () Integrate the second term on the right of () by parts, and show that the sequence of equations that results, does satisfies the conditions of the SDE convergence theorem. 4
Central limit theorem for Markov chains. (Problems 2-28.) Let ξ, ξ,... be an irreducible Markov chain on a finite state space {,..., d}, let P = ((p ij )) denote its transition matrix, and let π be its stationary distribution. For any function h on the state space, let πh denote i π ih(i). 2. Show that is a martingale. n f(ξ n ) (P f(ξ k ) f(ξ k )) k= 22. Show that for any function h, there exists a solution to the equation P g = h πh, that is, to the system p ij g(j) g(i) = h(i) πh. 23. The ergodic theorem for Markov chains states that j lim n n n h(ξ k ) = πh. k= Use the martingale central limit theorem to prove convergence in distribution for W n (t) = [nt] (h(ξ k ) πh). n 24. Use the martingale central limit theorem to prove the analogue of Problem 23 for a continuous time finite Markov chain {ξ(t), t }. In particular, use the multidimensional theorem to prove convergence for the vector-valued process U n = (Un,..., Un) d defined by Un(t) k = nt (I {ξ(s)=k} π k )ds n 25. Explore extensions of Problems 23 and 24 to infinite state spaces. k= Limit theorems for stochastic differential equations driven by Markov chains 26. Show that W n defined in Problem 23 and U n defined in Problem 24 are not good sequences of semimartingales. (The easiest approach is probably to show that the conclusion is not valid.) 27. Show that W n and U n can be written as M n + Z n where {M n } is a good sequence and Z n. 5
28. (Random evolutions) Let ξ be as in Problem 24, and let X n satisfy Ẋ n (t) = nf (X n (s), ξ(ns)). Suppose i F (x, i)π i =. Write X n as a stochastic differential equations driven by U n, give conditions under which X n converges in distribution to a limit X, and identify the limit. References Burkholder, D. L. (973). Distribution function inequalities for martingales. Ann. Probab., 9-42. Cho, Nahnsook (995). Weak convergence of stochastic integrals driven by martingale measure. Stochastic Process. Appl. (to appear). Ethier, Stewart N. and Kurtz, Thomas G. (986). Markov Processes: Characterization and Convergence. Wiley, New York. Graham, Carl (992). McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets. Stochastic Process. Appl. 4, 69-82. Ichikawa, Akira (986). Some inequalities for martingales and stochastic convolutions. Stoch. Anal. Appl. 4, 329-339. Jakubowski, Adam (995). (preprint) Continuity of the Ito stochastic integral in Hilbert spaces. Jakubowski, A. Mémin, J. and Pages, G. (989). Convergence in loi des suites d intégrales stochastique sur l espace D de Skorohod. Probab. Theory Related Fields 8, -37. Khas minskii, R. Z. (966a). On stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl., 2-228. Khas minskii, R. Z. (966b). A limit theorem for the solutions of differential equations with random right-hand sides. Theory Probab. Appl., 39-46. Kurtz, Thomas G. (992). Averaging for martingale problems and stochastic approximation. Applied Stochastic Analysis. Proceedings of the US-French Workshop. Lect. Notes. Control. Inf. Sci. 77, 86-29. Kurtz, Thomas G. and Protter, Philip (99a). Weak limit theorems for stochastic integrals and stochastic differential quations. Ann. Probab. 9, 35-7. Kurtz, Thomas G. and Protter, Philip (99b). Wong-Zakai corrections, random evolutions, and simulation schemes for sde s. Stochastic Analysis: Liber Amicorum for Moshe Zakai. Academic Press, San Diego. 33-346. 6
Kurtz, Thomas G. and Protter, Philip (996). Weak convergence of stochastic integrals and differential equations. II. Infinite-dimensional case. Probabilistic models for nonlinear partial differential equations, Lecture Notes in Math., 627, 97-285. Lenglart, E., Lepingle, D. and Pratelli, M. (98). Presentation unifiee de certaines inegalites des martingales. Seminares de probabilités XIV. Lect. Notes in Math., Springer, Berlin. 26-6. Maruyama, G. (955). Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo 4, 48-9. Pinsky, Mark A. (99). Lectures on random evolution. World Scientific Publishing Co., Inc., River Edge, NJ. Protter, Philip (99). Stochastic Integration and Differential Equations. Springer-Verlag, New York. 7