1. Stochastic equations for Markov processes

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1 First Prev Next Go To Go Back Full Screen Close Quit 1 1. Stochastic equations for Markov processes Filtrations and the Markov property Ito equations for diffusion processes Poisson random measures Ito equations for Markov processes with jumps

2 First Prev Next Go To Go Back Full Screen Close Quit 2 Filtrations and the Markov property (Ω, F, P ) a probability space Available information is modeled by a sub-σ-algebra of F F t information available at time t {F t } is a filtration. t < s implies F t F s A stochastic process X is adapted to {F t } if X(t) is F t -measurable for each t. An E-valued stochastic process X adapted to {F t } is {F t }-Markov if E[f(X(t + r)) F t ] = E[f(X(t + r)) X(t)], t, r, f B(E) An R-valued stochastic process M adapted to {F t } is an {F t }-martingale if E[M(t + r) F t ] = M(t), t, r τ is an {F t }-stopping time if for each t, {τ t} F t. For a stopping time τ, F τ = {A F : {τ t} A F t, t }

3 First Prev Next Go To Go Back Full Screen Close Quit 3 Ito integrals W standard Browian motion (W has independent increments with W (t + s) W (t) normally distributed with mean zero and variance s.) W is compatible with a filtration {F t } if W is {F t }-adapted and W (t + s) W (t) is independent of F t for all s, t. If X is R-valued, cadlag and {F t }-adapted, then X(s )dw (s) = lim X(ti )(W (t t i+1 ) W (t t i )) max(t i+1 t i ) exists, and the integral satisfies E[( if the right side is finite. X(s )dw (s)) 2 ] = E[ X(s) 2 ds] The integral extends to all measurable and adapted processes satisfying X(s) 2 ds < W is an R m -valued standard Brownian motion if W = (W 1,..., W m ) T for W 1,..., W m independent one-dimensional standard Brownian motions.

4 First Prev Next Go To Go Back Full Screen Close Quit 4 Ito equations for diffusion processes σ : R d M d m and b : R d R d Consider X(t) = X() + σ(x(s))dw (s) + b(x(s))ds where X() is independent of the standard Brownian motion W. Why is X Markov? Suppose W is compatible with {F t }, X is {F t }-adapted, and X is unique (among {F t }-adapted solutions). Then X(t + r) = X(t) + +r t σ(x(s))dw (s) + +r t b(x(s))ds and X(t + r) = H(t, r, X(t), W t ) where W t (s) = W (t + s) W (t). Since W t is independent of F t, E[f(X(t + r)) F t ] = f(h(t, r, X(t), w))µ W (dw).

5 First Prev Next Go To Go Back Full Screen Close Quit 5 Gaussian white noise. (U, d U ) a complete, separable metric space; B(U), the Borel sets µ a (Borel) measure on U A(U) = {A B(U) : µ(a) < } W (A, t) Mean zero, Gaussian process indexed by A(U) [, ) E[W (A, t)w (B, s)] = t sµ(a B), W (ϕ, t) = ϕ(u)w (du, t) ϕ(u) = a i I Ai (u) W (ϕ, t) = i a iw (A i, t) E[W (ϕ 1, t)w (ϕ 2, s)] = t s ϕ U 1(u)ϕ 2 (u)µ(du) Define W (ϕ, t) for all ϕ L 2 (µ).

6 First Prev Next Go To Go Back Full Screen Close Quit 6 X(t) = i ξ i(t)ϕ i process in L 2 (µ) Definition of integral I W (X, t) = X(s, u)w (du ds) = U [,t] i [ E[I W (X, t) 2 ] = E i,j [ = E U ξ i(s)dw (ϕ i, s) ξ i (s)ξ j (s)ds ϕ i ϕ j dµ U ] X(s, u) 2 µ(du)ds ] The integral extends to adapted processes satisfying X(s, u) 2 µ(du)ds < a.s. U so that (I W (X, t)) 2 is a local martingale. U X(s, u) 2 µ(du)ds

7 First Prev Next Go To Go Back Full Screen Close Quit 7 ν Poisson random measures a σ-finite measure on U, (U, d U ) a complete, separable metric space. ξ a Poisson random measure on U [, ) with mean measure ν m. For A B(U) B([, )), ξ(a) has a Poisson distribution with expectation ν m(a) and ξ(a) and ξ(b) are independent if A B =. ξ(a, t) ξ(a [, t]) is a Poisson process with parameter ν(a). ξ(a, t) ξ(a [, t] ν(a)t is a martingale. ξ is {F t } compatible, if for each A B(U), ξ(a, ) is {F t } adapted and for all t, s, ξ(a (t, t + s]) is independent of F t.

8 First Prev Next Go To Go Back Full Screen Close Quit 8 Stochastic integrals for Poisson random measures X cadlag, L 1 (ν)-valued, {F t }-adapted We define I ξ (X, t) = U [,t] X(u, s )ξ(du ds) in such a way that [ ] E [ I ξ (X, t) ] E X(u, s ) ξ(du ds) U [,t] = E[ X(u, s) ]ν(du)ds U [,t] If the right side is finite, then [ ] E X(u, s )ξ(du ds) = U [,t] U [,t] E[X(u, s)]ν(du)ds

9 First Prev Next Go To Go Back Full Screen Close Quit 9 Predictable integrands The integral extends to predictable integrands satisfying X(u, s) 1ν(du)ds < U [,t] so that [ ] E[I ξ (X, t τ)] = E X(u, s)ξ(du ds) U [,t τ] for any stopping time satisfying [ ] E X(u, s) ν(du)ds < (1) U [,t τ] If (1) holds for all t, then X(u, s)ξ(du ds) is a martingale. U [,t τ] U [,t τ] a.s. X(u, s)ν(du)ds X is predictable if it is the pointwise limit of adapted, left-continuous processes.

10 First Prev Next Go To Go Back Full Screen Close Quit 1 Stochastic integrals for centered Poisson random measures X cadlag, L 2 (ν)-valued, {F t }-adapted We define I ξ(x, t) = X(u, s ) ξ(du ds) U [,t] [I ξ(x, ] in such a way that E t) 2 = E[X(u, U [,t] s)2 ]ν(du)ds if the right side is finite. Then I ξ(x, ) is a square-integrable martingale. The integral extends to predictable integrands satisfying X(u, s) 2 X(u, s) ν(du)ds < U [,t] a.s. so that I ξ(x, t τ) is a martingale for any stopping time satisfying [ ] E X(u, s) 2 X(u, s) ν(du)ds < U [,t τ]

11 First Prev Next Go To Go Back Full Screen Close Quit 11 Itô equations for Markov processes with jumps W Gaussian white noise on U [, ) with E[W (A, t)w (B, t)] = µ(a B)t ξ i Poisson random measure on U i [, ) with mean measure ν i. Let ξ 1 (A, t) = ξ 1 (A, t) ν 1 (A)t σ 2 (x) = σ 2 (x, u)µ(du) < α1(x, 2 u) α 1 (x, u) ν 1 (du) <, α 2 (x, u) 1ν 2 (du) < and X(t) = X() σ(x(s ), u)w (du ds) + U [,t] α 1 (X(s ), u) ξ 1 (du ds) U 1 [,t] α 2 (X(s ), u)ξ 2 (du ds). U 2 [,t] β(x(s ))ds

12 First Prev Next Go To Go Back Full Screen Close Quit 12 Discrete time version: Burkholder (1973). A martingale inequality Continuous time version: Lenglart, Lepingle, and Pratelli (198). Easy proof: Ichikawa (1986). Lemma 1 For < p 2 there exists a constant C p such that for any locally square integrable martingale M with Meyer process M and any stopping time τ E[sup M(s) p ] C p E[ M p/2 τ ] s τ M is the (essentially unique) predictable process such that M 2 M is a local martingale. (A left-continuous, adapted process is predictable.)

13 First Prev Next Go To Go Back Full Screen Close Quit 13 Graham s uniqueness theorem Lipschitz condition σ(x, u) σ(y, u) 2 µ(du) + β(x) β(y) + α 1 (x, u) α 1 (y, u) 2 ν 1 (du) + α 2 (x, u) α 2 (y, u) ν 2 (du) M x y

14 First Prev Next Go To Go Back Full Screen Close Quit 14 Estimate X and X solution of SDE. E[sup X(s) X(s) ] s t E[ X() X() ] + C 1 E[( +C 1 E[( +E[ +E[ U σ(x(s ), u) σ( X(s ), u) 2 µ(du)ds) 1 2 ] U 1 α 1 (X(s ), u) α 1 ( X(s ), u) 2 ν 1 (du)ds) 1 2 ] U 2 α 2 (X(s ), u) α 2 ( X(s ), u) ν 2 (du)ds] β(x(s )) β( X(s )) ds] E[ X() X() ] + D( t + t)e[sup X(s) X(s) ] s t

15 First Prev Next Go To Go Back Full Screen Close Quit 15 X has values in D and satisfies X(t) = X() + Boundary conditions σ(x(s))dw (s) + b(x(s))ds + where λ is nondecreasing and increases only when X(t) D. η(x(s))dλ(s) X has values in D and satisfies X(t) = X() + σ(x(s))dw (s) + b(x(s))ds + α(x(s ), ζ N(s )+1 )dn(s) where ζ 1, ζ 2,... are iid and independent of X() and W and N(t) is the number of times X has hit the boundary by time t.

16 First Prev Next Go To Go Back Full Screen Close Quit Martingale problems for Markov processes Levy and Watanabe characterizations Ito s formula and martingales associated with solutions of stochastic equations Generators and martingale problems for Markov processes Equivalence between stochastic equations and martingale problems

17 First Prev Next Go To Go Back Full Screen Close Quit 17 Martingale characterizations Brownian motion (Levy) W a continuous {F t }-martingale W (t) 2 t an {F t }-martingale Then W is a standard Brownian motion compatible with {F t }. Poisson process (Watanabe) N a counting process adapted to {F t } N(t) λt an {F t }-martingale Then N is a Poisson process with parameter λ compatible with {F t }

18 First Prev Next Go To Go Back Full Screen Close Quit 18 Diffusion processes σ : R d M d m and b : R d R d Consider X(t) = X() + By Itô s formula σ(x(s))dw (s) + b(x(s))ds where f(x(t)) f(x()) Af(X(s))ds = f(x(s)) T σ(x(s))dw (s) Af(x) = 1 2 aij (x) i j f(x) + b(x) f(x) with ((a ij (x) )) = σ(x)σ(x) T.

19 First Prev Next Go To Go Back Full Screen Close Quit 19 Martingale problem for A Assume a ij and b i are locally bounded, and let D(A) = Cc 2 (R d ). X is a solution of the martingale problem for A if there exists a filtration {F t } such that X is {F t }-adapted and f(x(t)) f(x()) is an {F t }-martingale for each f D(A). Af(X(s))ds Any solution of the SDE is a solution of the martingale problem.

20 First Prev Next Go To Go Back Full Screen Close Quit 2 General SDE Let Then X(t) = X() + σ(x(s ), u)w (du ds) + U [,t] + α 1 (X(s ), u) ξ 1 (du ds) U 1 [,t] + α 2 (X(s ), u)ξ 2 (du ds). U 2 [,t] β(x(s ))ds f(x(t)) f(x()) Af(X(s))ds = f(x(s)) σ(x(s), u)w (du ds) U [,t] + (f(x(s ) + α 1 (X(s ), u)) f(x(s )) ξ 1 (du ds) U 1 + (f(x(s ) + α 2 (X(s ), u)) f(x(s )) ξ 2 (du ds) U 2

21 First Prev Next Go To Go Back Full Screen Close Quit 21 Form of the generator Af(x) = 1 aij (x) i j f(x) + b(x) f(x) 2 + (f(x + α 1 (x, u)) f(x) α 1 (x, u) f(x))ν 1 (du) U 1 + (f(x + α 2 (x, u)) f(x))ν 2 (du) U 2 Let D(A) be a collection of functions for which Af is bounded. Then a solution of the SDE is a solution of the martingale problem for A.

22 First Prev Next Go To Go Back Full Screen Close Quit 22 The martingale problem for A X is a solution for the martingale problem for (A, ν ), ν P(E), if P X() 1 = ν and there exists a filtration {F t } such that f(x(t)) f(x()) is an {F t }-martingale for all f D(A). Af(X(s))ds Theorem 2 If any two solutions of the martingale problem for A satisfying P X 1 () 1 = P X 2 () 1 also satisfy P X 1 (t) 1 = P X 2 (t) 1 for all t, then the f.d.d. of a solution X are uniquely determined by P X() 1. If X is a solution of the MGP for A and Y a (t) = X(a + t), then Y a is a solution of the MGP for A. Theorem 3 If the conclusion of the above theorem holds, then any solution of the martingale problem for A is a Markov process. A is called the generator for the Markov process.

23 First Prev Next Go To Go Back Full Screen Close Quit 23 Weak solutions of SDEs X is a weak solution of the SDE if there exists a probability space on which are defined X, W, ξ 1, and ξ 2 satisfying the SDE and X has the same distribution as X. Theorem 4 Suppose that the a ij and b i are locally bounded and that for each f Cc 2 (R d ) sup f(x + α 1 (x, u)) f(x) α 1 (x, u) f(x) ν 1 (du) < x U 1 and sup f(x + α 2 (x, u)) f(x) ν 2 (du) <. x U 2 Let D(A) = C 2 c (R d ). Then any solution of the martingale problem for A is a weak solution of the SDE.

24 First Prev Next Go To Go Back Full Screen Close Quit 24 Consider the SDE X(t) = X() + Nonsingular diffusions σ(x(s))dw (s) + b(x(s))ds and assume that d = m (that is, σ is square). Let Af(x) = 1 aij (x) i j f(x) + b(x) f(x). 2 Assume that σ(x) is invertible for each x and that σ(x) 1 is locally bounded. If X is a solution of the martingale problem for A, then M(t) = X(t) is a local martingale and W (t) = b( X(s))ds σ( X(s)) 1 dm(s) is a standard Brownian motion compatible with {F X t X(t) = X() + σ( X(s))d W (s) + }. It follows that b( X(s))ds.

25 First Prev Next Go To Go Back Full Screen Close Quit 25 Natural form for the jump terms Consider the generator for a simple pure jump process Af(x) = λ(x) (f(z) f(x))µ(x, dz), R d where λ and µ(x, ) is a probability measure. There exists γ : R d [, 1] R d such that 1 f(x + γ(x, u))du = f(z)µ(x, dz). R d Let ξ be a Poisson random measure on [, ) [, 1] [, ) with Lebesgue mean measure. Then X(t) = X() + I [,λ(x(s ))] (v)γ(x(s ), u)ξ(dv du ds) [, ) [,1] [,t] is a stochastic differential equation corresponding to A. Note that this SDE is of the form above with U 2 = [, ) [, 1] and α 2 (x, u, v) = I [,λ(x)] (v)γ(x, u) and that α 2 (x, u, v) α 2 (y, u, v) dvdu λ(x) λ(y) γ(x, u)du [, ) [,1] [,1] +λ(y) γ(x, u) γ(y, u) du [,1]

26 First Prev Next Go To Go Back Full Screen Close Quit 26 More general jump terms X(t) = X() + I [,λ(x(s ),u)] (v)γ(x(s ), u)ξ(dv du ds) [, ) U [,t] where ξ is a Poisson random measure with mean measure m ν m. The generator is of the form Af(x) = λ(x, u)(f(x + γ(x, u)) f(x))ν(du). U If ξ is replaced by ξ, then Af(x) = λ(x, u)(f(x + γ(x, u)) f(x) γ(x, u) f(x))ν(du). U Note that many different choices of λ, γ, and ν will produce the same generator.

27 First Prev Next Go To Go Back Full Screen Close Quit 27 X has values in D and satisfies X(t) = X() + Reflecting diffusions σ(x(s))dw (s) + b(x(s))ds + η(x(s))dλ(s) where λ is nondecreasing and increases only when X(t) D. By Itô s formula f(x(t)) f(x()) where Af(X(s))ds Bf(X(s))dλ(s) = f(x(s)) T σ(x(s))dw (s) Af(x) = 1 2 aij (x) i j f(x) + b(x) f(x) Bf(x) = η(x) f(x) Either take D(A) = {f Cc 2 (D) : Bf(x) =, x D} or formulate a constrained martingale problem with solution (X, λ) by requiring X to take values in D, λ to be nondecreasing and increase only when X D, and f(x(t)) f(x()) to be an {F X,λ t }-martingale. Af(X(s))ds Bf(X(s))dλ(s)

28 First Prev Next Go To Go Back Full Screen Close Quit 28 X has values in D and satisfies X(t) = X() + Instantaneous jump conditions σ(x(s))dw (s) + b(x(s))ds + α(x(s ), ζ N(s )+1 )dn(s) where ζ 1, ζ 2,... are iid and independent of X() and W and N(t) is the number of times X has hit the boundary by time t. Then where and f(x(t)) f(x()) = + Af(X(s))ds f(x(s)) T σ(x(s))dw (s) (f(x(s ) + α(x(s ), ζ N(s )+1 )) Bf(X(s ))dn(s) f(x(s ) + α(x(s ), u))ν(du))dn(s) U Af(x) = 1 2 aij (x) i j f(x) + b(x) f(x) Bf(x) = (f(x + α(x, u)) f(x))ν(du). U

29 First Prev Next Go To Go Back Full Screen Close Quit Weak convergence for stochastic processes General definition of weak convergence Prohorov s theorem Skorohod representation theorem Skorohod topology Conditions for tightness in the Skorohod topology

30 First Prev Next Go To Go Back Full Screen Close Quit 3 Topological proof of convergence (S, d) metric space F n : S R F n F in some sense (e.g., x n x implies F n (x n ) F (x)) F n (x n ) = 1. Show that {x n } is compact 2. Show that any limit point of {x n } satisfies F (x) = 3. Show that the equation F (x) = has a unique solution x 4. Conclude that x n x

31 First Prev Next Go To Go Back Full Screen Close Quit 31 (S, d) X n complete, separable metric space S-valued random variable Convergence in distribution {X n } converges in distribution to X ({P Xn } converges weakly to P X ) if for each f C(S) lim n E[f(X n)] = E[f(X)]. Denote convergence in distribution by X n X. Equivalent statements {X n } converges in distribution to X if and only if or equivalently lim inf n P {X n A} P {X A}, each open A, lim sup P {X n B} P {X B}, each closed B, n

32 First Prev Next Go To Go Back Full Screen Close Quit 32 Tightness and Prohorov s theorem A sequence {X n } is tight if for each ɛ >, there exists a compact set K ɛ S such that sup P {X n / K ɛ } ɛ. n Theorem 5 Suppose that {X n } is tight. Then there exists a subsequence {n(k)} along which the sequence converges in distribution.

33 First Prev Next Go To Go Back Full Screen Close Quit 33 (E, r) D E [, ) Skorohod topology on D E [, ) complete, separable metric space space of cadlag, E-valued functions x n x D E [, ) in the Skorohod (J 1 ) topology if and only if there exist strictly increasing λ n mapping [, ) onto [, ) such that for each T >, lim sup ( λ n (t) t + r(x n λ n (t), x(t))) =. n t T The Skorohod topology is metrizable so that D E [, ) is a complete, separable metric space. Note that I [1+ 1 n, ) I [1, ) in D R [, ), but (I [1+ 1 n, ), I [1, )) does not converge in D R 2[, ).

34 First Prev Next Go To Go Back Full Screen Close Quit 34 Conditions for tightness S n (T ) collection of discrete {F n t }-stopping times q(x, y) = 1 r(x, y) Theorem 6 Suppose that for t T, a dense subset of [, ), {X n (t)} is tight. Then the following are equivalent. a) {X n } is tight in D E [, ). b) (Kurtz) For T >, there exist β > and random variables γ n (δ, T ) such that for t T, u δ, and v t δ and E[q β (X n (t + u), X n (t)) q β (X n (t), X n (t v)) F n t ] E[γ n (δ, T ) F n t ] lim δ lim δ lim sup E[γ n (δ, T )] =, n lim sup E[q β (X n (δ), X n ())] =. (2) n c) (Aldous) Condition (2) holds, and for each T >, there exists β > such that C n (δ, T ) sup sup E[ sup q β (X n (τ + u), X n (τ)) q β (X n (τ), X n (τ v))] τ S n(t ) u δ v δ τ satisfies lim δ lim sup n C n (δ, T ) =.

35 First Prev Next Go To Go Back Full Screen Close Quit 35 η 1, η 2,... iid, E[η i ] =, σ 2 = E[η 2 i ] < Then for u δ. Example X n (t) = 1 [nt] n E[(X n (t + u) X n (t)) 2 Ft Xn ] = i=1 η i [n(t + u)] [nt] σ 2 (δ + 1 n n )σ2

36 First Prev Next Go To Go Back Full Screen Close Quit 36 Uniqueness of limit Theorem 7 If {X n } is tight in D E [, ) and (X n (t 1 ),..., X n (t k )) (X(t 1 ),..., X(t k )) for t 1,..., t k T, T dense in [, ), then X n X. For the example, this condition follows from the central limit theorem.

37 First Prev Next Go To Go Back Full Screen Close Quit 37 Skorohod representation theorem Theorem 8 Suppose that X n X. Then there exists a probability space (Ω, F, P ) and random variables, Xn and X, such that X n has the same distribution as X n, X has the same distribution as X, and X n X a.s. Continuous mapping theorem Corollary 9 Let G(X) : S E and define C G = {x S : G is continuous at x}. Suppose X n X and that P {X C G } = 1. Then G(X n ) G(X).

38 First Prev Next Go To Go Back Full Screen Close Quit 38 Some mappings on D E [, ) π t : D E [, ) E π t (x) = x(t) C πt = {x D E [, ) : x(t) = x(t )} G t : D R [, ) R G t (x) = sup s t x(s) C Gt = {x D R [, ) : lim s t G s (x) = G t (x)} {x D R [, ) : x(t) = x(t )} G : D R [, ) D R [, ), G(x)(t) = G t (x), is continous H t : D E [, ) R H t (x) = sup s t r(x(s), x(s )) C Ht = {x D E [, ) : lim s t H s (x) = H t (x)} {x D E [, ) : x(t) = x(t )} H : D E [, ) D R [, ), H(x)(t) = H t (x), is continuous

39 First Prev Next Go To Go Back Full Screen Close Quit 39 Level crossing times τ c : D R [, ) [, ) τ c (x) = inf{t : x(t) > c} τ c : D R [, ) [, ) τ c (x) = inf{t : x(t) c or x(t ) c} 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

40 First Prev Next Go To Go Back Full Screen Close Quit 4 Localization Theorem 1 Suppose that for each α >, τ α n satisfies P {τ α n > α} α 1 and {X n ( τ α n )} is relatively compact. Then {X n } is relatively compact. Compactification of the state space Theorem 11 Let E E where E is compact and the topology on E is the restriction of the topology on E. Suppose that for each n, X n is a process with sample paths in D E [, ) and that X n X in D E [, ). If X has sample paths in D E [, ), then X n X in D E [, ).

41 First Prev Next Go To Go Back Full Screen Close Quit Convergence for Markov processes characterized by stochastic equations Martingale central limit theorem Convergence for stochastic integrals Convergence for SDEs driven by semimartingales Diffusion approximations for Markov chains Limit theorems involving Poisson random measures and Gaussian white noise

42 First Prev Next Go To Go Back Full Screen Close Quit 42 Martingale central limit theorem Let M n be a martingale such that lim E[sup M n (s) M n (s ) ] = n s t and in probability. Then M n cw. [M n ] t ct Vector-valued version: If for each 1 i d and for each 1 i, j d, lim E[sup Mn(s) i Mn(s ) ] i = n s t [M i n, M j n] t c ij t, then M n σw, where W is d-dimensional standard Brownian motion and σ is a symmetric d d-matrix satisfying σ 2 = c = ((c ij )).

43 Example: Products of random matrices Let A (1), A (2),... be iid random matrices with E[A (k) ] = and E[ A (k) 2 ] <. Set X = I X(k + 1) = (I + 1 A (k+1) )X(k) = (I + 1 A (k+1) ) (I + 1 A (1) ) n n n and Then X n (t) = X([nt]) M n (t) = 1 [nt] A (k). n X n (t) = X n () + k=1 dm n (s)x n (s ). M n M where M is a Brownian motion with E[M ij (t)m kl (t)] = E[A ij A kl ]t, Can we conclude X n X satisfying X(t) = X() + dm(s)x(s)? First Prev Next Go To Go Back Full Screen Close Quit 43

44 First Prev Next Go To Go Back Full Screen Close Quit 44 Example: Markov chains X k+1 = H(X k, ξ k+1 ) where ξ 1, ξ 2... are iid P {X k+1 B X, ξ 1,..., ξ k } = P {X k+1 B X k } Example: X n k+1 = X n k + σ(x n k ) 1 n ξ k+1 + b(x n k+1) 1 n Assume E[ξ k ] = and V ar(ξ k ) = 1. Define X n (t) = X[nt] n A n (t) = [nt] W n n (t) = 1 X n (t) = X n () + Can we conclude X n X satisfying X(t) = X() + + n [nt] σ(x n (s )dw n (s) b(x n (s ))da n (s) σ(x(s))dw (s) + k=1 ξ k. b(x(s))ds?

45 First Prev Next Go To Go Back Full Screen Close Quit 45 Definition: For cadlag processes X, Y, X(s )dy (s) = whenever the limit exists in probability. Stochastic integration lim X(ti )(Y (t i+1 t) Y (t i t)) max t i+1 t i Sample paths of bounded variation: If Y is a finite variation process, the stochastic integral exists (apply dominated convergence theorem) and X(s )dy (s) = X(s )α Y (ds) α Y (,t] is the signed measure with α Y (, t] = Y (t) Y ()

46 First Prev Next Go To Go Back Full Screen Close Quit 46 Existence for square integrable martingales If M is a square integrable martingale, then E[(M(t + s) M(t)) 2 F t ] = E[[M] t+s [M] t F t ] Let X be bounded, cadlag and adapted. Then For partitions {t i } and {r i } E[( X(t i )(M(t i+1 t) M(t i t)) X(r i )(M(r i+1 t) M(r i t))) 2 ] [ ] = E (X(t(s )) X(r(s ))) 2 d[m] s [ ] = E (X(t(s )) X(r(s ))) 2 α [M] (ds) (,t] t(s) = t i for s [t i, t i+1 ) r(s) = r i for s [r i, r i+1 ) Let σ c = inf{t : X(t) c} and Then for τc { X(t) t < X c σc (t) = X(c ) t σ c X(s )dy (s) = τc X c (s )dy (s)

47 First Prev Next Go To Go Back Full Screen Close Quit 47 Semimartingales Definition: A cadlag, {F t }-adapted process Y is an {F t }-semimartingale if: M V Y = M + V a local, square integrable {F t }-martingale an {F t }-adapted, finite variation process Total variation: For a finite variation process T t (Z) sup Z(ti+1 t) Z(t i t) < {t i } Quadratic variation: For cadlag semimartingale [Y ] t = lim max t i+1 t i (Y (ti+1 t) Y (t i t)) 2 Covariation: For cadlag semimartingales [Y, Z] t = lim (Y (t i+1 t) Y (t i t))(z(t i+1 t) Z(t i t))

48 First Prev Next Go To Go Back Full Screen Close Quit 48 Probability estimates for SIs Y = M + V M local square-integrable martingale V finite variation process Assume X 1. P {sup s t s X(r )dy (r) > K} P {σ t} + P { sup s t σ +P {sup s t s s X(r )dm(r) > K/2} X(r )dv (r) > K/2} P {σ t} + 16E[ σ X 2 (s )d[m] s ] K 2 + P { P {σ t} + 16E[[M] t σ] K 2 + P {T t (V ) K/2} X(s ) dt s (V ) K/2}

49 First Prev Next Go To Go Back Full Screen Close Quit 49 Good integrator condition Let S be the collection of X = ξ i I [τi,τ i+1 ), where τ 1 < < τ m are stopping times and ξ i is F τi -measurable. Then If Y is a semimartingale, then is stochastically bounded. X(s )dy (s) = ξ i (Y (t τ i+1 ) Y (t τ i )). { X(s )dy (s) : X S, X 1} Y satisfying this stochastic boundedness condition is a good integrator. Bichteler-Dellacherie: Y is a good integrator if and only if Y is a semimartingale. (See, for example, Protter (199), Theorem III.22.)

50 First Prev Next Go To Go Back Full Screen Close Quit 5 Uniformity conditions Y n = M n + A n, a semimartingale adapted to {F n t } T t (A n ), the total variation of A n on [, t] [M n ] t, the quadratic variation of M n on [, t] Uniform tightness (UT) [JMP]: H t = n=1{ t is stochastically bounded. Z(s )dy n (s) : Z S n, sup Z(s) 1} s t Uniformly controlled variations (UCV) [KP]: {T t (A n ), n = 1, 2,...} is stochastically bounded, and there exist stopping times {τn α } such that P {τn α α} α 1 and sup n E[[M n ] t τ α n ] < A sequence of semimartingales {Y n } that converges in distribution and satisfies either UT or UCV will be called good.

51 First Prev Next Go To Go Back Full Screen Close Quit 51 Basic convergence theorem Theorem. (Jakubowski, Mémin and Pagès; Kurtz & Protter) (X n, Y n ) {Ft n }-adapted in D M km Rm[, ). Y n = M n + A n an {F n t }-semimartingale Assume that {Y n } satisfies either UT or UCV. If (X n, Y n ) (X, Y ) in D M km Rm[, ) with the Skorohod topology. THEN in D M km R m Rk[, ) (X n, Y n, X n (s )dy n (s)) (X, Y, X(s )dy (s)) If (X n, Y n ) (X, Y ) in probability in the Skorohod topology on D M km Rm[, ) THEN (X n, Y n, X n (s )dy n (s)) (X, Y, X(s )dy (s)) in probability in D M km R m Rk[, ) IN PROBABILITY CANNOT BE REPLACED BY A.S.

52 First Prev Next Go To Go Back Full Screen Close Quit 52 Stochastic differential equations Suppose F : D R d[, ) D M d m[, ) is nonanticipating in the sense that where x t (s) = x(s t). U cadlag, adapted, with values in R d Y an R m -valued semimartingale. Consider X(t) = U(t) + F (x, t) = F (x t, t) F (X, s )dy (s) (3) (X, τ) is a local solution of (3) if X is adapted to a filtration {F t } such that Y is an {F t }-semimartingale, τ is an {F t }-stopping time, and X(t τ) = U(t τ) + τ F (X, s )dy (s), t. Strong local uniqueness holds if for any two local solutions (X 1, τ 1 ), (X 2, τ 2 ) with respect to a common filtration, we have X 1 ( τ 1 τ 2 ) = X 2 ( τ 1 τ 2 ).

53 First Prev Next Go To Go Back Full Screen Close Quit 53 Sequences of SDE s X n (t) = U n (t) + Structure conditions F n (X n, s )dy n (s). T 1 [, ) = {γ : γ nondecr. and maps [, ) onto [, ), γ(t + h) γ(t) h} C.a F n behaves well under time changes: If {x n } D R d[, ), {γ n } T 1 [, ), and {x n γ n } is relatively compact in D R d[, ), then {F n (x n ) γ n } is relatively compact in D M d m[, ). C.b F n converges to F : If (x n, y n ) (x, y) in D R d Rm[, ), then (x n, y n, F n (x n )) (x, y, F (x)) in D R d R m Md m[, ). Note that C.b implies continuity of F, that is, (x n, y n ) (x, y) implies (x n, y n, F (x n )) (x, y, F (x)).

54 First Prev Next Go To Go Back Full Screen Close Quit 54 Examples F (x, t) = f(x(t)), f continuous F (x, t) = f( h(x(s))ds) f, h continuous F (x, t) = h(t s, s, x(s))ds, h continuous

55 First Prev Next Go To Go Back Full Screen Close Quit 55 Convergence theorem Theorem 12 Suppose (U n, X n, Y n ) satisfies (U n, Y n ) (U, Y ) {Y n } is good X n (t) = U n (t) + {F n }, F satisfy structure conditions sup n sup x F n (x, ) <. F n (X n, s )dy n (s). Then {(U n, X n, Y n )} is relatively compact and any limit point satisfies X(t) = U(t) + F (X, s )dy (s) If, in addition, (U n, Y n ) (U, Y ) in probability and strong uniqueness holds, then (U n, X n, Y n ) (U, X, Y ) in probability.

56 First Prev Next Go To Go Back Full Screen Close Quit 56 Euler scheme Let = t < t 1 < The Euler approximation for is given by X(t) = U(t) + F (X, s )dy (s) X(t k+1 ) = X(t k ) + U(t k+1 ) U(t k ) + F ( X, t k )(Y (t k+1 ) Y (t k )). Consistency Let {t n k } satisfy max k t n k+1 tn k, and define ( ) ( ) Yn (t) Y (t n = k ) U n (t) U(t n k ), t n k t < t n k+1. Then (U n, Y n ) (U, Y ) and {Y n } is good. The Euler scheme corresponding to {t n k } satisfies X n (t) = U n (t) + F ( X n, s )dy n (s) If F satisfies the structure conditions and strong uniqueness holds, then X n X in probability. (In the diffusion case, Maruyama (1955))

57 First Prev Next Go To Go Back Full Screen Close Quit 57 Sequences of Poisson random measures ξ n Poisson random measures with mean measures nν m. h measurable For A B(U) satisfying A h2 (u)ν(du) <, define M n (A, t) = 1 h(u)(ξ n (du [, t]) ntν(du)). n M n is an orthogonal martingale random measure with [M n (A), M n (B)] t = 1 n A B h(u)2 ξ n (du [, t]) M n (A), M n (B) t = t h(u) 2 ν(du). A A B M n converges to Gaussian white noise W with E[W (A, t)w (B, s)] = t s E[W (ϕ 1, t)w (ϕ 2, s)] = t s A B h(u) 2 ν(du) ϕ 1 (u)ϕ 2 (u)h(u) 2 ν(du)

58 First Prev Next Go To Go Back Full Screen Close Quit 58 Continuous-time Markov chains X n (t) = X n () + 1 n + 1 n Assume α U 1(x, u)ν(du) =. Then X n (t) = X n () + 1 n + 1 n Can we conclude X n X satisfying X(t) = X() + + U [,t] U [,t] U [,t] U [,t] U [,t] U α 1 (X n (s ), u)ξ n (du ds) α 2 (X n (s ), u)ξ n (du ds) α 1 (X n (s ), u) ξ n (du ds) α 2 (X n (s ), u)ξ n (du ds) α 1 (X(s), u)w (du ds) α 2 (X(s ), u)ν(du)ds?

59 First Prev Next Go To Go Back Full Screen Close Quit 59 Consider Discrete-time Markov chains X n k+1 = X n k + σ(x n k, ξ k+1 ) 1 n + b(x n k, ζ k+1 ) 1 n where {(ξ k, ζ k )} is iid in U 1 U 2. µ the distribution of ξ k ν the distribution of ζ k Assume U 1 σ(x, u 1 )µ(du 1 ) = Define M n (A, t) = 1 [nt] (I A (ξ k ) µ(a)) n V n (B, t) = 1 n k=1 [nt] k=1 I B (ζ k )

60 First Prev Next Go To Go Back Full Screen Close Quit 6 Stochastic equation driven by random measure Then X n (t) X[nt] n satisfies X n (t) = X n () + σ n (X n (s), u)m n (du ds) U 1 t + b n (X n (s), u)v n (du ds) U 2 V n (A, t) tν(a) M n (A, t) M(A, t) M is Gaussian with covariance E[M(A, t)m(b, s)] = t s(µ(a B) µ(a)µ(b)) Can we conclude that X n X satisfying X(t) = X() + σ(x(s), u)m(du ds) U 1 t + b(x(s), u)ν(du)ds? U 2

61 First Prev Next Go To Go Back Full Screen Close Quit 61 H Y (ϕ, t) Good integrator condition a separable Banach space (H = L 2 (ν), L 1 (ν), L 2 (µ), etc.) a semimartingale for each ϕ H Y ( a k ϕ k, t) = k a ky (ϕ k, t) Let S be the collection of cadlag, adapted processes of the form Z(t) = m k=1 ξ k(t)ϕ k, ϕ k H. Define I Y (Z, t) = Z(u, s )Y (du ds) = U [,t] k Basic assumption: H t = {sup s t I Y (Z, s) : Z S, sup Z(s) H 1} s t is stochastically bounded. (Call Y a good H # -semimartingale.) The integral extends to all cadlag, adapted H-valued processes. ξ k (s )dy (ϕ k, s).

62 First Prev Next Go To Go Back Full Screen Close Quit 62 H Y n Convergence for H # -semimartingales a separable Banach space of functions on U an {F n t }-H # -semimartingale (for each ϕ H, Y (ϕ, ) is an {F n t }-semimartingale) {X n } cadlag, H-valued processes (X n, Y n ) (X, Y ), if (X n, Y n (ϕ 1, ),..., Y n (ϕ m, )) (X, Y (ϕ 1, ),..., Y (ϕ m, )) in D H R m[, ) for each choice of ϕ 1,..., ϕ m H.

63 First Prev Next Go To Go Back Full Screen Close Quit 63 Let Convergence for Stochastic Integrals H n,t = {sup s t I Yn (Z, s) : Z S n, sup Z(s) H 1}. s t Definition: {Y n } is uniformly tight if n H n,t is stochastically bounded for each t. Theorem 13 Protter and Kurtz (1996). Assume that {Y n } is uniformly tight. If (X n, Y n ) (X, Y ), then there is a filtration {F t }, such that Y is an {F t }-adapted, good, H # -semimartingale, X is {F t }-adapted and (X n, Y n, I Yn (X n )) (X, Y, I Y (X)). If (X n, Y n ) (X, Y ) in probability, then (X n, Y n, I Yn (X n )) (X, Y, I Y (X)) in probability. Cho (1994) for distribution-valued martingales Jakubowski (1995) for Hilbert-space-valued semimatingales.

64 First Prev Next Go To Go Back Full Screen Close Quit 64 Structure conditions X n (t) = U n (t) + Sequences of SDE s U [,t] F n (X n, s, u)y n (du ds). T 1 [, ) = {γ : γ nondecreasing and maps [, ) onto [, ), γ(t + h) γ(t) h} C.a F n behaves well under time changes: If {x n } D R d[, ), {γ n } T 1 [, ), and {x n γ n } is relatively compact in D R d[, ), then {F n (x n ) γ n } is relatively compact in D H d[, ). C.b F n converges to F : If (x n, y n ) (x, y) in D R d Rm[, ), then (x n, y n, F n (x n )) (x, y, F (x)) in D R d R m Hd[, ). Note that C.b implies continuity of F, that is, (x n, y n ) (x, y) implies (x n, y n, F (x n )) (x, y, F (x)).

65 First Prev Next Go To Go Back Full Screen Close Quit 65 SDE convergence theorem Theorem 14 Suppose that (U n, X n, Y n ) satisfies X n (t) = U n (t) + F n (X n, s, u)y n (du ds), U [,t] that (U n, Y n ) (U, Y ), and that {Y n } is uniformly tight. Assume that {F n } and F satisfy the structure condition and that sup n sup x F n (x, ) H d <. Then {(U n, X n, Y n )} is relatively compact and any limit point satisfies X(t) = U(t) + F (X, s, u)y (du ds) U [,t]

66 First Prev Next Go To Go Back Full Screen Close Quit Convergence for Markov processes characterized by martingale problems Tightness estimates based on generators Convergence of processes based on convergence of generators Averaging A measure-valued limit

67 First Prev Next Go To Go Back Full Screen Close Quit 67 Compactness conditions based on compactness of real functions Compact containment condition: For each T > and ɛ >, there exists a compact K E such that lim inf n P {X n(s) K, s T } 1 ɛ. Theorem 15 Let D be dense in C(E) in the compact uniform topology. {X n } is relatively compact (in distribution in D E [, )) iff {X n } satisfies the compact containment conditions and {f X n } is relatively compact for each f D. Note that E[(f(X n (t + u)) f(x n (t))) 2 F n t ] = E[f 2 (X n (t + u)) f 2 (X n (t)) F n t ] 2f(X n (t))e[f(x n (t + u)) f(x n (t)) F n t ]

68 First Prev Next Go To Go Back Full Screen Close Quit 68 Limits of generators X n (t) = 1 n (N b (nt) N d (nt)), where N b, N d are independent, unit Poisson processes. For f C 3 c (R) Set Af = 1 2 f. A n f(x) = n( f(x+ 1 n )+f(x 1 n ) 2 f(x)) = 1 2 f (x) + O( 1 n ). E[(f(X n (t + u)) f(x n (t))) 2 F n t ] = E[f 2 (X n (t + u)) f 2 (X n (t)) F n t ] 2f(X n (t))e[f(x n (t + u)) f(x n (t)) F n t ] u( A n f f A n f ). It follows that {X n } is relatively compact in D R [, ), or using the fact that P {sup X n (s) k} 4E[X2 n(t)] s t k 2 = 4t k 2, the compact containment condition holds and {X n } is relatively compact in D R [, ).

69 First Prev Next Go To Go Back Full Screen Close Quit 69 Limits of martingales Lemma 16 For each n = 1, 2,..., let M n and Z n be cadlag stochastic processes and let M n be a {F (Mn,Zn) t }-martingale. Suppose that (M n, Z n ) (M, Z). If for each t, {M n (t)} is uniformly integrable, then M is a {F (M,Z) t }-martingale. Recall that if a sequence of real-valued random variables {ψ n } is uniformly integrable and ψ n ψ, then E[ψ n ] E[ψ]. It follows that if X is the limit of a subsequence of {X n }, then f(x n (t)) A n f(x n (s))ds f(x(t)) Af(X(s))ds (along the subsequence) and X is a solution of the martingale problem for A.

70 First Prev Next Go To Go Back Full Screen Close Quit 7 Elementary convergence theorem E compact, E n, n = 1, 2,... complete, separable metric space. η n : E n E Y n Markov in E n with generator A n, X n = η n (Y n ) cadlag A C(E) C(E) (for simplicity, we write Af = g if (f, g) A). D(A) = {f : (f, g) A} For each (f, g) A, there exist f n D(A n ) such that sup x E n ( f n (x) f η n (x) + A n f n (x) g η n (x) ). THEN {X n } is relatively compact and any limit point is a solution of the martingale problem for A.

71 Reflecting random walk E n = {, 1 n, 2 n,...} A n f(x) = nλ(f(x + 1 n ) f(x)) + nλi {x>} (f(x 1 n ) f(x)) Let f C 3 c [, ). Then A n f(x) = λf (x) + O( 1 n ) x > A n f() = nλf () + λ 2 f () + O( 1 n ) Assume f () =, but still have discontinuity at. Let f n = f + 1 n h, f {f C 3 c [, ) : f () = }, h C 3 c [, ). Then A n f n (x) = λf (x) + 1 n λh (x) + O( 1 n ) x > A n f n () = λ 2 f () + λh () + O( 1 n ) Assume that h () = 1 2 f (). Then, noting that η n (x) = x, x E n, sup ( f n (x) f(x) + A n f n (x) Af(x) ) x E n First Prev Next Go To Go Back Full Screen Close Quit 71

72 First Prev Next Go To Go Back Full Screen Close Quit 72 Averaging Y stationary with marginal distribution ν, ergodic, and independent of W Y n (t) = Y (β n t), β n X n (t) = X() + σ(x n (s), Y n (s))dw (s) + b(x n (s), Y n (s))ds Lemma 17 Let {µ n } be measures on U [, ) satisfying µ n (U [, t]) = t. Suppose ϕ(u)µ n (du [, t]) ϕ(u)µ(du [, t]), U U for each ϕ C(U) and t, and that x n x in D E [, ). Then h(u, x n (s))µ n (du ds) h(u, x(s))µ(du ds) U [,t] U [,t] for each h C(U E) and t.

73 First Prev Next Go To Go Back Full Screen Close Quit 73 Convergence of averaged generator Assume that σ and b are bounded and continuous. Then for f Cc 2 (R), f(x n (t)) f(x n ()) Af(X n (s), Y n (s))ds = σ(x n (s), Y n (s))f (X n (s))dw (s) Af(x, y) = 1 2 σ2 (x, y)f (x) + b(x, y)f (x) is a martingale, {X n } is relatively compact, ϕ(y n (s))ds = 1 βnt ϕ(y (s))ds t ϕ(u)ν(du), β n U so any limit point of {X n } is a solution of the martingale problem for Af(x) = Af(x, u)ν(du). Khas minskii (1966), Kurtz (1992), Pinsky (1991) U

74 First Prev Next Go To Go Back Full Screen Close Quit 74 Coupled system X n (t) = X() + Y n (t) = Y () + Consequently, σ(x n (s), Y n (s))dw 1 (s) + b(x n (s), Y n (s))ds t nα(xn (s), Y n (s))dw 2 (s) + nβ(x n (s), Y n (s))ds f(x n (t)) f(x n ()) and g(y n (t)) g(y ()) Af(X n (s), Y n (s))ds nbg(x n (s), Y n (s))ds Bg(x, y) = 1 2 α2 (x, y)g (y) + β(x, y)g (y) are martingales.

75 First Prev Next Go To Go Back Full Screen Close Quit 75 Estimates Suppose that for g Cc 2 (R), Bg(x, y) is bounded, and that, taking g(y) = y 2, Bg(x, y) K 1 K 2 y 2. Then, assuming E[Y () 2 ] <, which implies E[Y n (t) 2 ] E[Y () 2 ] + (K 1 K 2 E[Y n (s) 2 ])ds E[Y n (t) 2 ] E[Y () 2 ]e K 2t + K 1 K 2 (1 e K 2t ). The sequence of measures defined by ϕ(y)γ n (dy ds) = is relatively compact. R [,t] ϕ(y n (s))ds

76 First Prev Next Go To Go Back Full Screen Close Quit 76 Convergence of the averaged process If σ and b are bounded, then {X n } is relatively compact and any limit point of {(X n, Γ n )} must satisfy: f(x(t)) f(x()) Af(X(s), y)γ(dy ds) R [,t] is a martingale for each f Cc 2 (R) and Bg(X(s), y)γ(dy ds) (4) R [,t] is a martingale for each g Cc 2 (R). But (4) is continuous and of finite variation. Therefore Bg(X(s), y)γ(dy ds) = Bg(X(s), y)γ s (dy)ds =. R [,t] R

77 First Prev Next Go To Go Back Full Screen Close Quit 77 Characterizing γ s For almost every s R Bg(X(s), y)γ s (dy) =, g C 2 c (R) But, fixing x, and setting B x g(y) = Bg(x, y) B x g(y)π(dy) =, g C 2 (R), R implies π is a stationary distribution for the diffusion with generator B x g(y) = 1 2 α2 x(y)g (y) + β x (y)g (y), α x (y) = α(x, y), β x (y) = β(x, y) If α x (y) > for all y, then the stationary distribution π x is uniquely determined. If uniqueness hold for all x, Γ(dy ds) = π X(s) (dy)ds and X is a solution of the martingale problem for Af(x) = Af(x, y)π x (dy). R

78 First Prev Next Go To Go Back Full Screen Close Quit 78 Moran models in population genetics E B σ type space generator of mutation process selection coefficient Generator with state space E n A n f(x) = n 1 B i f(x) + 2(n 2) i=1 1 i j k n (1 + 2 n σ(x i, x j ))(f(η k (x x i )) f(x)) η k (x z) = (x 1,..., x k 1, z, x k+1,... x n ) Note that if (X 1,..., X n ) is a solution of the martingale problem for A, then for any permutation σ, (X σ1,..., X σn ) is a solution of the martingale problem for A.

79 First Prev Next Go To Go Back Full Screen Close Quit 79 Conditioned martingale lemma Lemma 18 Suppose U and V are {F t }-adapted, U(t) is an {F t }-martingale, and G t F t. Then V (s)ds is a {G t }-martingale. E[U(t) G t ] E[V (s) G s ]ds

80 First Prev Next Go To Go Back Full Screen Close Quit 8 Generator for measure-valued process P n (E) = {µ P(E) : µ = 1 n n i=1 δ x i } For f B(E m ) and µ P n (E) f, µ (m) = Z(t) = 1 n n i=1 1 n (n m + 1) δ Xi (t) Symmetry and the conditioned martingale lemma imply i 1 =i m f(x i1,..., x im ). is a {F Z t }-martingale. Define F (µ) = f, µ (n) f, Z (n) (t) A n f, Z (n) (s) ds A n F (µ) = A n f, µ (n)

81 First Prev Next Go To Go Back Full Screen Close Quit 81 Convergence of the generator If f depends on m variables (m < n) A n f(x) m = B i f(x) i= (n 2) 1 + 2(n 2) m (1 + 2 n σ(x i, x j ))(f(η k (x x i )) f(x)) k=1 1 i k m 1 j k,i n m n k=1 i=m+1 1 j k,i n (1 + 2 n σ(x i, x j ))(f(η k (x x i )) f(x)) A n f, µ (n) = m B i f, µ (m) + 1 ( Φ ik f, µ (m 1) f, µ (m) ) + O( 1 2 n ) i=1 + 1 i k m m ( σ k f, µ (m+1) σf, µ (m+2) ) + O( 1 n ) k=1

82 First Prev Next Go To Go Back Full Screen Close Quit 82 Conclusions If E is compact, compact containment condition is immediate (P(E) is compact). A n F is bounded as long as B i f is bounded. Limit of uniformly integrable martingales is a martingale Z n Z if uniqueness holds for limiting martingale problem.

83 First Prev Next Go To Go Back Full Screen Close Quit The lecturer s whims Wong-Zakai corrections Processes with boundary conditions Averaging for stochastic equations

84 First Prev Next Go To Go Back Full Screen Close Quit 84 Not good (evil?) sequences Markov chains: Let {ξ k } be an irreducible finite Markov chain with stationary distribution π(x), g(x)π(x) =, and let h satisfy P h h = g (P h(x) = p(x, y)h(y)). Define V n (t) = 1 [nt] g(ξ k ). n Then V n σw, where k=1 σ 2 = x,y π(x)p(x, y) (P h(x) h(y)) 2. {V n } is not good but V n (t) = 1 [nt] (P h(ξ k 1 ) h(ξ k )) + 1 (P h(ξ [nt] ) P h(ξ )) n n k=1 = M n (t) + Z n (t) where {M n } is good sequence of martingales and Z n.

85 First Prev Next Go To Go Back Full Screen Close Quit 85 Piecewise linear interpolation of W : W n (t) = W ( [nt] [nt] ) + (t n n )n (Classical Wong-Zakai example.) W n (t) = W ( [nt] + 1 ) ( [nt] + 1 n n = M n (t) + Z n (t) ( W ( [nt] + 1 ) W ( [nt] ) n n ) ( t)n where {M n } is good (take the filtration to be F n t W ( [nt] + 1 ) W ( [nt] n n ) = F [nt]+1 ) and Z n. n Renewal processes: N(t) = max{k : k i=1 ξ i t}, {ξ i } iid, positive, E[ξ k ] = µ, V ar(ξ k ) = σ 2. N(nt) nt/µ V n (t) =. n ) Then V n αw, α = σ/µ 3/2. V n (t) = (N(nt) + 1)µ S N(nt)+1 µ n = M n (t) + Z n (t) + S N(nt)+1 nt µ n

86 First Prev Next Go To Go Back Full Screen Close Quit 86 Not so evil after all Assume V n (t) = Y n (t) + Z n (t) where {Y n } is good and Z n. In addition, assume { Z n dz n }, {[Y n, Z n ]}, and {[Z n ]} are good. X n (t) = X n () + = X n () + Integrate by parts using F (X n (t)) = F (X n ()) + F (X n (s ))dv n (s) F (X n (s ))dy n (s) + F (X n (s ))dz n (s) F (X n (s ))F (X n (s ))dv n (s) + R n (t) where R n can be estimated in terms of [V n ] = [Y n ] + 2[Y n, Z n ] + [Z n ].

87 First Prev Next Go To Go Back Full Screen Close Quit 87 Integration by parts F (X n (s ))dz n (s) = F (X n (t))z n (t) F (X n ())Z n () Z n (s )F (X n (s ))F (X n (s ))dy n (s) Z n (s )dr n (s) Z n (s )df (X n (s)) [F X n, Z n ] t F (X n (s ))F (X n (s ))Z n (s )dz n (s) F (X n (s ))F (X n (s ))d([y n, Z n ] s + [Z n ] s ) [R n, Z n ] t Theorem 19 Assume that V n = Y n +Z n where {Y n } is good, Z n, and { Z n dz n } is good. If (X n (), Y n, Z n, Z n dz n, [Y n, Z n ]) (X(), Y,, H, K), then {X n } is relatively compact and any limit point satisfies X(t) = X() + F (X(s ))dy (s) + Note: For all the examples, H(t) K(t) = ct for some c. F (X(s ))F (X(s ))d(h(s) K(s))

88 First Prev Next Go To Go Back Full Screen Close Quit 88 X has values in D and satisfies X(t) = X() + Reflecting diffusions σ(x(s))dw (s) + b(x(s))ds + η(x(s))dλ(s) where λ is nondecreasing and increases only when X(t) D. By Itô s formula where f(x(t)) f(x()) Af(X(s))ds = Bf(X(s))dλ(s) f(x(s)) T σ(x(s))dw (s) Af(x) = 1 2 aij (x) i j f(x) + b(x) f(x) Bf(x) = η(x) f(x) Either take D(A) = {f Cc 2 (D) : Bf(x) =, x D} or formulate a constrained martingale problem with solution (X, λ) by requiring X to take values in D, λ to be nondecreasing and increase only when X D, and f(x(t)) f(x()) to be an {F X,λ t }-martingale. Af(X(s))ds Bf(X(s))dλ(s)

89 First Prev Next Go To Go Back Full Screen Close Quit 89 X has values in D and satisfies X(t) = X() + Instantaneous jump conditions σ(x(s))dw (s) + b(x(s))ds + α(x(s ), ζ N(s )+1 )dn(s) where ζ 1, ζ 2,... are iid and independent of X() and W and N(t) is the number of times X has hit the boundary by time t. Then where and f(x(t)) f(x()) = + Af(X(s))ds f(x(s)) T σ(x(s))dw (s) (f(x(s ) + α(x(s ), ζ N(s )+1 )) Bf(X(s ))dn(s) f(x(s ) + α(x(s ), u))ν(du))dn(s) U Af(x) = 1 2 aij (x) i j f(x) + b(x) f(x) Bf(x) = (f(x + α(x, u)) f(x))ν(du). U

90 First Prev Next Go To Go Back Full Screen Close Quit 9 X has values in D and satisfies X n (t) = X() + λ n (t) Nn(t) n Approximation of reflecting diffusion σ(x n (s))dw (s) + b(x n (s))ds + 1 n η(x n(s ))dn n (s) Assume there exists ϕ C 2 such that ϕ, Aϕ and ϕ σ are bounded on D and inf x D η(x) ϕ(x) ɛ >. Then inf n(ϕ(x + 1 x D n η(x)) ϕ(x))λ n(t) 2 ϕ + 2 Aϕ t ϕ(x n (s)) T σ(x n (s))dw (s) and {λ n (t)} is stochastically bouned

91 First Prev Next Go To Go Back Full Screen Close Quit 91 Convergence of time changed sequence γ n (t) = inf{s : s + λ n (s) > t} t γ n (t) + λ n γ n (t) t + 1 n X n (t) X n γ n (t) X n (t) = X() + σ( X n (s))dw γ n (s) + + b( X n (s))dγ n (s) η( X n (s ))dλ n γ n (s) Check relative compactness and goodness of {(W γ n, γ n, λ n γ n )}. γ n is Lipschitz with Lipschitz constant 1, λ n γ n is finite variation, nondecreasing and bounded by t + 1 n, M n = W γ n is a martingale with [M n ] t = γ n (t). If σ and b are bounded and continuous, then (by the general theorem) {( X n, W γ n, γ n, λ n γ n )} is relatively compact and any limit point will satisfy X(t) = X() + σ( X(s))dW γ(s) + b( X(s))dγ(s) + where γ(t) + λ(t) = t. (Note that γ and λ are continuous.) η( X(s))d λ(s)

92 First Prev Next Go To Go Back Full Screen Close Quit 92 Convergence of original sequence X(t) = X() + σ( X(s))dW γ(s) + b( X(s))dγ(s) + If γ is constant on [a, b], then for a t b, X(t) D and X(t) = X(a) + a η( X(s))ds. η( X(s))d λ(s) Under appropriate conditions on η, no such interval can exist with a < b and γ must be strictly increasing. Then (at least along a subsequence) X n X X γ 1 and setting λ = λ γ 1, X(t) = X() + σ(x(s))dw (s) + b(x(s))ds + η(x(s))dλ(s)

93 Rapidly varying components W a standard Brownian motion in R X n () independent of W ζ a stochastic process with state space U, independent of W and X n () Define ζ n (t) = ζ(nt). X n (t) = X n () + Define and so that X n (t) = X n () + σ(x n (s), ζ n (s))dw (s) + M n (A, t) = V n (A, t) = I A (ζ n (s))dw (s) I A (ζ n (s))ds, U [,t] U [,t] b(x n (s), ζ n (s))ds σ(x n (s), u)m n (du ds) + b(x n (s), u)v n (du ds) First Prev Next Go To Go Back Full Screen Close Quit 93

94 First Prev Next Go To Go Back Full Screen Close Quit 94 Define Assume that Convergence of driving processes M n (ϕ, t) = V n (ϕ, t) = 1 t in probability for each ϕ C(U). ϕ(ζ(s))ds ϕ(ζ n (s))dw (s) ϕ(ζ n (s))ds U ϕ(u)ν(du) Observe that [M n (ϕ 1, ), M n (ϕ 2, )] t = ϕ 1 (ζ n (s))ϕ 2 (ζ n (s))ds t ϕ 1 (u)ϕ 2 (u)ν(du) U

95 First Prev Next Go To Go Back Full Screen Close Quit 95 Limiting equation The functional central limit theorem for martingales implies M n (ϕ, t) M(ϕ, t) where M is Gaussian with E[M(ϕ 1, t)m(ϕ 2, s)] = t s ϕ 1 (u)ϕ 2 (u)ν(du) and V n (ϕ, t) t ϕ(u)ν(du) U If {(M n, V n )} is good, then X n X satisfying X(t) = X() + U σ(x(s), u)m(du ds) + U U b(x(s), u)ν(du)ds

96 First Prev Next Go To Go Back Full Screen Close Quit 96 Recall If then and E[( U [,t] U [,t] Estimates for averaging problem M n (ϕ, t) = Z(t, u) = ϕ(ζ n (s))dw (s) m ξ k (t)ϕ k (u) k=1 Z(s, u)m n (du ds) = m k=1 ξ k (s )dm n (ϕ k, s) [ ] t Z(s, u)m n (du ds)) 2 ] = E ξ k (s)ξ l (s)ϕ k (ζ n (s))ϕ l (ζ n (s))ds k,l [ ] = E Z(s, ζ n (s)) 2 ds Assume U is locally compact, ψ is strictly positive and vanishes at, ϕ H = sup ψ(u)ϕ(u), and sup T E[ 1 1 ds] <. T ψ(ζ(s)) 2 T