for all f satisfying E[ f(x) ] <.

Transcription

1 . 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.

2 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

3 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

4 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

5 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

6 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., 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, Ichikawa, Akira (986). Some inequalities for martingales and stochastic convolutions. Stoch. Anal. Appl. 4, 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., Khas minskii, R. Z. (966b). A limit theorem for the solutions of differential equations with random right-hand sides. Theory Probab. Appl., 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, Kurtz, Thomas G. and Protter, Philip (99a). Weak limit theorems for stochastic integrals and stochastic differential quations. Ann. Probab. 9, 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

7 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, 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 Maruyama, G. (955). Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo 4, 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