-V THE?" LIBRARY Michigan State University This is to certify that the dissertation entitled Weak Convergence of Distribution-Valued Semimartingales and Associated SDE's presented by Ravi Chari has been accepted towards fulfillment ofthe requirements for Ph.D. degree in Statistics ff Major professor Dr. V. Mandrekar Date M MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 ——‘\ fi—wr )V1£SI_} RETURNING MATERIALS: Place in book drop to LIBRARJES remove this checkout from Jun-zyll-L. your record. FINES will be charged if book-is returned after the date stamped below. WEAK CONVERGENCE 0F DISTRIBUTION-VALUED SEMIMARTINGALES AND ASSOCIATED SDE'S by Ravi T. Chari A THESIS Submitted to Michigan State University in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 1984 Weak Convergence of Distribution-Valued Semimartingales and Associated SDE's ABSTRACT by Ravi T. Chari We study in this work the weak convergence of solutions of certain linear d) . These extend stochastic differential equations taking values in S'CR the works of Holley and Stroock (1978) and Kallianpur and Nolpert (1984). We start by proving the existence and uniqueness of solutions of certain linear SDE's with S'CRd)-valued square integrable martingales as the driv- ing term. The weak convergence result entails the functional central limit theorem for semimartingales taking values in S'(Rd) . This is proved in a more general context of semimartingales taking values in the dual of a Frechét nuclear space. An example from Neurophysiology is worked out. i” ACKNOWLEDGEMENTS I would like to thank my advisor Professor V. Mandrekar, for his help during the preparation of this thesis and Professor H. L. Koul, for his care- ful reading of the thesis. I would like to thank Professors Sheldon Axler and R. Erickson for serving on my committee. Finally,thanks to Carol Case for her fast typing under time pressure. .1 TABLE OF CONTENTS Chapter Page I NOTATIONS AND PRELIMINARIES ............... 3 II FUNCTIONAL CENTRAL LIMIT THEOREM FOR SEMIMARTINGALES TAKING VALUES IN THE DUAL E', OF A FRECHET NUCLEAR SPACE 14 2.1 Auxiliary Results .................... 14 2.2 The Increasing Process and the Functional CLT ...... 20 2.3 CLT and Functional CLT for E'-Valued Triangular Arrays. . 26 2.4 Example ......................... 27 III LINEAR S'(Rd)-VALUED SDE'S AND WEAK CONVERGENCE TO AN ORNSTREIN-UHLENBECK (O—U) PROCESS ............ 30 3.1 Existence and Uniqueness of Solutions of Certain Linear S'(Rd)-Valued 305's ................... 32 3.2 Weak Convergence to an O-U Process ............ 37 3.3 Example ......................... 41 NOTATION ................................ 42 REFERENCES ............................... 43 1““ INTRODUCTION Our aim in this work is to study the weak convergence of solutions of certain linear stochastic differential equations (SDE) taking values in S'CRd) --the space of tempered distributions in IRd . We first prove the existence and uniqueness (Theorem 3.1.1) for solutions of such linear SDE's where the driving term is a s'CRd)-valued square integrable martingale (for a definition see Chapter I). Existence and uniqueness in particular cases of such solutions have been studied previously in [7], [13] among others. Holley and Stroock . T __ . [7] have studied the existence and uniqueness in law when the martingale driv- ing term is a Wiener process in these spaces. Kallianpur and Wolpert [13] have studied this problem in the case where the martingale is a centered Poisson process. Next, we take up the weak convergence of the solutions of such SDE's to an Ornstein-Uhlenbeck (O-U) process in the space D([O¢»), S'CRd)) (D(S'(Rd)) for short). Similar results have occurred in E8] and [13]. Holley and Stroock [8] have studied the weak convergence result in the case where the martingale term has an increasing process which admits a density with respect to Lebesgue measure. Kallianpur and Wolpert [13] have studied the case where the martingale term--a centered Poisson process, converges to a Wiener process in S'CRd). Their techniques are particular to the properties of Poisson process and cannot be generalized to our case. The basic approach to our problem is that, if the martingale term in the linear SDE converges weakly to a Wiener process in D([O, w); S'GRd)), then the corresponding solutions of such SDE's converges weakly to an O-U process. ‘ The results of [13], in case of S'(Rd) valued processes, follow from our work. tam 'x-Lm‘ ‘ ‘ aim: ‘-r,-m' WA"A;.;1£_W 2.3-Iii V." In Chapter II, we take up the question of-when a martingale and in gen- eral a semi-martingale taking values in S'(Rd) converges weakly to a Wiener process. Since .S'CRd) is the dual of a Frechét nuclear space, we study the problem in the context of martingales and semi-martingales taking values in the dual of Frechét nuclear spaces. In the case of real valued square integrable martingales, functional central limit theorems involve the increasing processes associated with these martingales [15]. We prove the existence and uniqueness of a positive operator-valued analog of such increasing processes corresponding to square integrable martingales in these spaces. We give sufficient conditions, analogous to [15], for the weak convergence of a semi-martingale to a Wiener process to take place in a metrisable subspace of D([O, 0°), E') . Here E' is the dual of a Frechét nuclear space. As a consequence of these results we prove the central limit theorem and the functional central limit theorem for E'-valued triangular arrays. In addition to the ideas in [15], the main technique in our work, is the result on tightness for probability measures on D(E') developed by Mitoma in [18]. This result, along with necessary prerequisites and notations are discussed in the next section. CHAPTER 1 PRELIMINARIES AND NOTATION 1.1 Frechét Nuclear Spaces. In the following a vector space is a vector space over R . A nonnegative function p:E +-R+ on a vector space EC is called a seminorm if it satisfies the following conditions: alpU+yF5MO+pwl waeE m mnd=lwa) v xen,v er A seminorm p on a vector space E is called Hilbertian if p(x + y)2 + p(x - y)2 = 2p(X)2 + 2 pm2 If p is a Hilbertian seminorm then it is easily checked that p(x.y) = é-{p(x + y)2 - p(x - y)2} defines a symmetric bilinear form on E and p(x, x) = p(x)2 . p(x, y) is called the inner product corresponding to p(x) . For a Hilbertian seminorm p , let Np = {x e E: p(x) = 0} Then Ep/Np is a pre-Hilbert space, where ED is the space E with the topology given by p . Thus Ep , the completion of Ep/Np , is a Hilbert space. We shall say p is a separable Hilbertian seminorm if Ep is a separable Hilbert space. 00 o J=1 be an ortho- Let (H, <-, -> ) be separable Hilbert space and let {hj} normal basis (ONB) in H . Let L(H) denote the set of bounded linear operators from H into H . An operator T in L(H) is said to be positive if .3‘3 V x e H . Every positive operator T e L(H) is known to have a unique positive square root S e L(H) (cf. [20] Theorem 12.33) i.e., a positive operator S such that T = S2 . We write 5 = T>2 . Let 01(H) = {T e L(H) : T is positive and E < hj Thj > < o=>} . i=1 01 is called the class of positive, nuclear (or trace class) operators. For T e 01(H) , define ”TH1 = .2 . U “1 is independent of the ONB {h.}. 3:13 J J An operator T in L(H) is said to be a Hilbert-Schmidt operator iff T*T 6 01(H) , where T* denotes the adjoint of T . A topological vector space (TA/S) E , is said to be countably Hilbertian, if there exists a countable family {pk}:_1 of separable Hilbertian seminorms determining the topology of E . We can and do assume that these seminorms are a directed family. This is with- out loss of generality since the norms {H ”k}:=1 defined by k 1 U - 11.. = (.2 pzt-nz K = 1, 2, 3:1 are such that n ”1, 5_ H ”2 f_... and the topologies generated by these countable seminorms is the same as those generated by {pk}:=1 . A neighborhood basis of 0 in this topology is given by sets of the form Uk = {f E E : “fuk < 1} k = 1, 2, ... Clearly such a topology is locally convex. Moreover E is metrisable, and the metric is translation invariant. Indeed set, oo _.“f-g”. d(f. 9) = X 2 J ———l =1 1+Hf-9Hj Then it is easily checked that the topology generated by d is the topology generated by the norms {n “k’ k.: 1} A complete, locally convex, metric space, whose metric is translation invariant is known as a Frechét space. Thus a complete, countably Hilbertian TVS E is a Frechét space. Define Ek (= E” ”k) as before. Then I _“ El: E23... 3E It can be shown that E is complete if and only if E = 3 Ep . P=1 As the norms ” ”k are increasing in k , the identity mapping 1mn : En + Em definesa bounded linear operator for m < n . A complete, countably Hilbertian TVS E , is said to be a Frechét nuclear space if to every m.: 1 there exists an ".2 m such that the map * T = ' l i : + ' . mn mn En En TS a nuclear operator In such a case TI/2 : En + En will be a Hilbert-Schmidt operator. For a TVS F, let F' denote its topological dual, i.e., the set of all continuous linear functionals on F . For a Frechét nuclear space E , with Ek, k=1, 2, ... as before, we have E1CEZC°" and E =kl.=J1Ek For x in E' and f in E we will denote the duality between them by (f, x) . Thus, by definition the map f3 E -> (f, x) 612 is continuous. If x 6 E' , then define ”XH-k = sup [(f, x)| p ”f”k=1 We have 11x10: no-1: : lelLZ : A strong neighborhood of O in E' is given by sets of the form V(A) = {x e E' : sup [(f, x)| < I} f e A where A is an arbitrary bounded set in E . (Recall that a set A in E is bounded iff ufflk < c for all f in A , for some K': 1 , c > 0) . The topology defined on E' with the strong neighborhoods as a neigh- borhood basis at O is called the strong dual topology. We shall always consider the strong dual topology on E' . d) of The classical example of a Frechét nuclear space is the space 3(R d rapidly decreasing functions on ‘R [23]. To describe this space, we first set up some standard notation. d d We will denote a xeR by (x1,...,xd) . Amultiindexa on R is d-tupheof nonnegative integers (a1....,qd) . By la] we will mean a1 + a2 + ... + ad . .‘al Da will denote °' for any multiindex a . C11 dd 8x1 ...axd The space SCRd) can now be described as the space of all real, infinitely differentiable functions f , satisfying [le = sup sup d (1 + |x|2)N [(Daf)(x)| < m IalfN x 612 2 2 1 for N = O, I, 2, ... where |xl2 = x + ... + xd The collection of norms I define a locally convex topology which W makes SCRd) into a Frechét space (cf. [20]). In our work it is more convenient to work with an equivalent collection of norms { , n > O} which generate the same topolgoy on SCRd) . These 1 n ._ I have previously occurred in other works, such as in [7], [10] or [21]. Given k 3_O define the Hermite polynomials hk : R.» R by 2 2 L -L - hk(x) = 012 2k k1) 2(-1>k exp(§-)(%;)k(expt—§—)) Then {hk’ k30} form an ONB in L2(R) . For a multiindex a on le define d _ by ha(x) - ha1(x1)... - ha (xa) . C]. 2< d) . h on]? 0. Clearly {ha } form an ONB in L IR Hill2 ={éd lf(x)l2dx and = x f(x)g(x)dx Rd denote the norm and inner product on LZCRd) . th . 2 d . . The k norm ” “k in L CR.) 15 defined as follows (1,1,1) mi = Z (2|a|+d)k <16, ha>2 for f in L2(Rd) la|=0 Define the space (Ek, <-, ->k) by _ 2 d . , - (1.1.2) Ek-{feL(R) . |1rukk = g (2|a|+d) f, g e Ek . Here E0 = L2(Rd), Ek is a Hilbert space, and 3(Rd) = ‘3 E Further, It can be checked from equation(1.1.1)that the identity map a o , u * o . o Tn’n+d+1 En 15 such that l . E E TS : E n,n+d+1 1n,n+d+1 n+d+1 T n+d+1 T n+d+1 nuclear. ‘The dual, S'(Rd) is known as the space of tempered distributions. We define H ”-k by 1xn-k = §<2Ial+d)‘p (ha. x) 2 for x 6 S'(Rd) I and henceforth we shall denote (ha, x) by x(ha) V a . and E'k = {x eS'(Rd) : nxu_k < oo} k = o, 1, Then E; is a Hilbert space with inner product _k = X xa(ha)y(ha)(2lal + d)’p X. y e El; (1 (Efi <-, ->_k) is the dual of (Ek, <-, ~>k) . 3(Rd) c.... c E1 c: Eoc Eic.... cS'(Rd) If x e E5 and f 6 Ep , the duality takes the form (f, x) = Z X(ha) and “X“..k= f E3(1Rd)~ {O} 1&1ng 1.2 E'-valued processes and semi-martingales Throughout, we will be dealing with "weak" semi-martingales taking values in the dual of a Frechét nuclear space. Before defining it we need the following. Let (Q,F,P) be a complete probability space and (Ft, tip) be a non-decreasing family of sub-o-algebras of F satisfying the usual conditions [12], i.e., Ft+ = Ft and F0 contains all P-null sets of F . The filtration (Ft’ tip) will be denoted by f_. Let E bea Frechétnuclear space and E' its dual. Let B(E') denote the cylinder o-algebra on E‘, i.e., the smallest o-algebra on E' which makes all the maps x9 E' + (f, x) 6R measurable for f in E . A measurable map X : (O, F) + (E', B(E')) will be called a E'-valued random variable. Thus X is a E'-valued random variable if for each f in E . (f, X) is a real random variable. 1 A collection X = {X(t), tip} of E'-valued random variables on (O, F) is called an E'-valued stochastic process. An E'-valued process X on (O, F, f, P) is said to be flffadapted if (f, X(t)) is Ft adapted for each f in E and for all t.: 0 . Let Y = (V(t), t:O) and X = (X(t), tZO) be two E'-valued processes on (O, F, f; P) . Y is said to be a modification of X if V t 3_O P[X(t) = Y(t)] = 1 . An E' or Ek-valued process X (X(t), tip) is said to be a cadlag process if in the topology of E' or E; the process X has left limits at each t‘: O and is right continuous. Recall that a real valued process X (= X(t), tip) is said to be a semi—martingale if X admits a decomposition of the form X = M + A where M = (M(t), E39) is a cadlag local martingale [12] and A = (A(t); tip) is a cadlag adapted process of finite variation on compacts. ' We will extend this definition for E'-valued processes. 1.2.1 An E'-valued process X = (X(t), t30) is said to be a semi-martingale (martingale) on (O, F. E, P) if a) X is ffadapted and b) the real process t |+ (f, X(t)) is a semi-martingale (martingale) for each f in E . In the theory of real semi-martingales two processes play an important role-~the increasing process associated with a square integrable martingale and the dual predictable projection of the integer-valued random measure as- sociated with the jumps of a semi-martingale. We describe these briefly. For more on this subject see [12] (Chapters II and III). 10 Let X = (X(t), E29) be a real martingale on (O, F, f, P) satisfying EX2(t) < m for each t.: O . Then the Doob-Meyer decomposition shows that there exists an increasing predictable process, denoted by . Satisfying t Fr X2(t) - is an ffmartingale. is unique up to P-equivalence and is t called the increasing process associated with X . Let X = (X(t); tzO) be a real semi-martingale on (n, F, F, P) . By definition, it is a cadlag adapted process. Define the integer-valued random measure associated with the jumps of X by N((O, t], A) = 2 [AX(S) e A] 0 . Next we discuss an example of an E-valued martingale without jumps with the explicit form for <(f, X)> . 1.3 The Wiener Martingale One of the most important examples of an E-valued martingale is the Wiener martingale, the analog of Brownian motion on these spaces. Let Q : E x E +1R be a continuous, positive definite bilinear form. 11 An adapted E‘-valued process W = (W(t), tip) on (O, F, f, P) is said to be an [- P Wiener martingale if a) W(O) 5 0 a.s. P b) the map t 1+ W(t) is continuous. c) {(f, W(t)) t.: O , f e E} is a centered Gaussian system with covariance function E(f, W(t))(g. W(s)) = (US) Q(f. 9) V f, g e E d) W(t) - W(s) is independent of FS V s < t . Observe that these conditions imply that W is an E'-valued .5- P martingale satisfying E(f, W(t))2 < m for all f in E and t.: 0 . By (a). (c), (d) it follows that the increasing process associated with W is given by tQ(f;f) i.e., t l+ (f, W(t))2 - tQ(f,f) is a martingale for each f in E If Qflfif).: CHfHk V f E E , for some k‘: 1 and some c > O , then by the Bochner-Minlos theorem ([6] Theorem 3.1) there exists a m > k such that we can find a version of W taking values in E% . In such a case we will call W an E$-Wiener martingale. Typically, the kinds of processes which we shall be considering will not be continuous E'-valued process, but those which are cadlag in E' . We describe these spaces now, along with the result of Mitoma [18] on tightness of probability measures on such spaces. 1.4 The spaces D(E'), D(Eé) . Recall that we only consider the strong dual topology on E' . With this topology let D(E') = D(R+; E') denote the space of all cadlag mappings from R+ to E' . Let D(Eé) = DCR+; E6) denote the space of all cadlag mappings from 1R+ to E6 . The topology on these spaces is given by the Lindvall metrics 12 described in [19]. B(D(E')) is the smallest o-algebra on D(E') which makes all the maps x(-) ems') 1+ ((12 x(t1)) . (f2. x(t2)) .....(fk. xukm 6R ,r in E; k_>_1. B(D(E‘SH k measurable for 0‘: t1 < t2 < ... O , p > 0 and T > 0 there exists a 6 > 0 such that whenever ufflk §_ 6 , then Pn{x(-) e D(E') : sup |(f, x(t))| > e}.: p for all n.: 1 th Let D = D(R+; R) . Define the map II(f) : D(E') + D by N(flm f in E . 1.4.1 Theorem ([18]): Let {Pn} Dill be a sequence of probability measures on (D(E') , B(D(E')) . a) If {Pnio n(1=)‘1}n>1 tight in D(E') . is tight in D for every f in E , then {Pn} is b) If {Pn} n': 1 is uniformly k-continuous and {Pn} n.: 1 is tight in D(Ev) , then {Pn}113_1 is tight in D(Eé) for some p.: k . f1,...,fm c) Let H be the mapping that carries t1,...,tm the point x of D(E') to the point ((f1 X(tl),...,(f 13 If {Pn} is tight in D(E') and for any finite number of elements f1,... in E and time points t1,...,tm we have f1’000,f -1 f1’.l.,f P <1H( m) = Q m where n t t t1,0.l’tm 1,000, m fl’. .,fm m Q is a probability measure on 2R , then there exists a unique t . t 1’ ’ m probability measure P on D(E') such that Pn converges weakly to P . If moreover {Pn} is uniformly k-continuous, then Pn = P on D(EA) for some 9.3 k . CHAPTER II FUNCTIONAL CLT FOR SEMIMARTINGALES TAKING VALUES IN THE DUAL OF A FRECHET NUCLEAR SPACE Recall that, corresponding to every locally square integrable martingale, M , 2-A isa there exists a predictable increasing process A , such that M local martingale. This is the famous Doob-Meyer decomposition of the sub- martingale M2 . We generalize this result to the case where M is a E'-valued locally square integrable martingale, E being a nuclear Frechét space. We then go on to prove the functional central limit theorem for E'-valued semi-martin- gales. Although very useful, this turns out to be rather straightforward to prove, given Theorem 1.4.1 and the results of [15]. In fact, the only point where one encounters some difficulty is, in proving the convergence of the finite dimensional distributions. This is handled by a variation of the results of [15] and is stated in Proposition 2.1.6 below. As a consequence of this theorem, we prove the central limit theorem and the invariance principle for E'-valued triangular arrays. An example which hints at the applicability of the functional central limit theorem, follows. We start by proving some auxilliary results needed for our main theorems. 2.1 Auxilliary results. The proposition below is a regularization theorem due essentially to Ito [10]. Our proof follows closely, Theorem 3.1 in [10]. 2.1.1 Proposition. Let Y(= Y(f), f in E ) be a family of real random variables such that (l) Y(clf + czg) = c1Y(f) + c2Y(g) a.s. where the exceptional set may depend c1. c2, f, 9 (ii) EY(f)2_: CHI”: for some c > o , p-: 1 l4 15 then Y has a version in EA for some q > p and there exists a constant K such that Envui, : a Proof: The map Y : Epl+ L2(Q, F, P) is clearly a bounded linear operator. Let {hi} i.: 1 be a countable dense set in E . Let {e?} i.: 1 be for each fixed n a complete orthonormal system in En , obtained from {hi} by the Schmidt orthogonalization. Since E is nuclear there exists a q > p such that Z ue‘luz = z < co . Let {e9} j> 1 be the ONB in E' dual to {eq} . a J p 1 — q 1 q = Set Y(ej) Xj . Then E( X?) < c e9 2 = ct g, _ guy, 52 Hence ZX§(w) < m for all m in 1 , with P(Ql) = 1 . Let X(w) = . Z X.(w)eq for w in O , j J J 0 otherwise Then X(w) 6 EA for each w and X(w) is measurable in F/B(Eé) in w . X is thus an EA variable and E(llxufq) = e<1x§1 _<_ e. As in [10] it is easy to show that X(f) = Y(f) a.s. for all f in E . Q.E.D. The following proposition helps us in formulating the analog of an increasing process associated with square integrable E'-valued martingales. The pro- position is easily proved by mimicking the proof given in [25] for continuous Hilbert space valued martingales. So we shall only present a sketch of the proof here. 2.1.2 Proposition. Let (H, flol', <-, ->) be a real, separable Hilbert space. Let (M, f, P) be a H-valued process which is cadlag in H , with 16 E amuuz < m for each t.: 0 . Assume that (M, E, P) is a martingale in the sense of definition 1.2.1. Then there exists a unique increasing process A(t) with values in the positive operators 01(H) c:L(H) , right continuous for the norm H ”1 of 01(H) , such that for each y, z in H , the real process t (+ is a predictable process of finite variation and t 1+ <2. M(t)> - is an .E martingale. Proof: Since is a real, locally square integrable martingale for each y in H , it follows that there exists a unique (upto P-equivalence) increasing, predictable process A(t, y) (a unique, upto P-equivalence, 2 - Mn 1) ( <2, M(-)> - A(-, y, 2)) are .E- martingales. If '{en} is an ONB predictable process of finite variation A(t, y, 2)) such that for H , the argument in [25] shows that under the condition that E"M(t)[[2 < w , the series ct(y) = n2mg, en> A(t, e 2 - o.(y) is an '5 martingale. Thus o.(y) and A(-, y) are , e ) is convergent and in fact n m P-equivalent. Similarly A(-, y, z) is P-equivalent to the process t 1+ 2 <2 em>A(t, e em) and hence there exists a process .A(-) n n’ A(-, y, z) = with E(Trace A(t)) < w Since A(-, y, z) is predictable so is the process t |+ . Since A(o, y) is increasing, we have, for each y in H and h > o .3 . Thus A(t+h) - A(t) is a positive operator. Finally, for h > 0 , we have 17 “A(t+h) - A(t IA )1 A(t+h) - AM In”) I! ll g IA % for all t , t+h 3 T Since t l+ A(t,y) is right continuous, it follows by dominated con- vergence theorem that the map t |+ A(t) is right continuous for the norm of L(H) and 01(H) Q.E.D. We will have occassion to use the continuous mapping theorem (Theorem 5.1 of [2]) for mappings from D(E') to E' . However, since D(E') is not, in general, a metric space, a proof is needed for the theorem. 2.1.3 Proposition: Let Qn’ Q be probability measures on (D(E'), B(D(E'))) and let h be a continuous map from D(E') to E' . If Qn = Q then th'l a Qh'l. Proof. As in [24] Qn a Q iff for any closed set G lim sup Qn(G) 3 0(6) n Applying this result to the closed set G = h'1(F) we have the desired result. Q.E.D. 2.1.4 Remark: This result can be extended to maps h , which are discontinuous, provided the set Dh , of discontinuities of h , has Q measure zero. How- ever, unlike the case where E = Rd , it is not always true that Oh is measurable! The following is an easy but technical lemma, needed in the proof of our main theorem. 18 2.1.5 Lemma: Let {Y3 (8)}:=1 be a sequence of random variables for each a > 0 and j = 1,...., k . Suppose Y2(e) 3'0 as n +~w for j = 1,...,k . Then there exists a sequence en , decreasing to zero such that Y2(en) E 0 as n +'m for j = 1,...,k . Proof. This is the analog of Lemma 3 of [15]. However the proof given there has a minor error. n = n n Set mj(e) E[|Yj(e)I/(1 + IYj(e)|J Clearly, Yg(e)-E o as n +'w iff m3(e)‘+ o as n +-m for j = 1,...,k . So let n1 = min {n0 : max m0(1) §_1 V n.: no} lffififi J . . 1 1 and for l > 2 n. = min In > n. : max mQ(w) < 7- V n > n } — l o 1-1 lijile—l -o '{ni} is obviously an increasing sequence. Let -1. En - i if ni < n.: n1.+1 Then an decreases to zero. Further, for j = 1,...,k n _ n l_ . mj(€n) — mj(i) if ni.: n.: ni+1 1 . §_ 7- if "i < n §_ni+1 = En + o as n + w Q.E.D. Finally, we need a proposition, which is based on the argument in [15] to show convergence of the finite dimensional distributions of martingales. We recast their technique there so that it fits in our context. 2.1.6 Proposition: Let (M", f?) be a sequence of real valued, locally square integrable martingales satisfying (a) sup IAMn(t)[ < dn for all (n,T) in N x R+ tt K 02(t) for all t in [0, T] where t 1+ 02(t) is a positive, non-decreasing function on [0, m) . Then (C) E[exp 'iMn(T)] + exp [-%-02(T)] Proof: We will give an outline of the proof, and refer the reader to [15] for the details. First, consider the case where (d) T i a for all n where 02(T) + 1.: a . To prove (c) it suffices to show that (e) E{exp (iMn(T) + %-02(T)}-+ 1 as n + w . Set A"(t) = '%-t + 4} f (e1X - 1 - ix)N”(ds, dx) lx|_<_dn _ -1 nc n - 2-t + b (t) where Nn(ds, dx) is the compensator of the random measure Nn(ds, dx) corresponding to the jumps of the process Mn and Mnc denotes the continuous martingale part of Mn . Define 2%) = exp (iMn(t))/1E(t, A") where lE(t, A") ‘= exp(An(t)) 11 (1+ b”(s))e'bn(5) o+bfin5%eQn T which under (b) is equivalent to showing that . 2A [T (91x - 1 - ix + %-)Nn(ds, dx) P>-0 o led “n 2 d T ix . X en n n But If f (e - 1 - lX + —-)N (ds, dx)| < +-O . Assumption (d) can be disposed of, by a standard argument using stop- ping times as in [15]. Q.E.D. 2.2 The increasing process and the functional CLT. 2 < m We first show that for every E'-valued martingale satisfying E(f, M(t)) for each f in E and all t positive, we can associate an increasing process of the type described in Proposition 2.1.2. 2.2.1 Theorem: Let (M(o), E, P) be an E'-valued martingale satisfying E(f, M(t))2 < w for all (t, f) in IR+ E . Then for any T > 0 , there exists a unique, increasing positive—operator valued process A(t) of the type described in Proposition 2.1.2, taking values in E5 for some p(= pT) such that t |+ (f, M(tAT))(g, M(tAT) - _p is an E_ martingale for all f, g in E . 2 Erggf: Let T > 0 be arbitrary and define X(f) = sup (f, M(t)) and 035T VH})=EXH). 21 By Ooob's inequality VT(f)-: 4E(f M(T))2 < w for all f in E . The map f |+ VT(f) is lower semi-continuous since if fn + f in E then by Fatou's lemma lim inf VT(fn)-: E lim inf X(fn) n+0!) 11-1-00 .3 E X(f) =vT(f) Moreover {f : VT(f) < m} = E Since E is complete it follows by the Baire Category theorem (cf. Problem C, Chapter 3, 9 [14]) that there exists a p_: 1 , C, k > 0 such that VT(f)-: k for all f in E satisfying ufflp.: c Thus v C f < k for all f in E TUTTI—I; ’- But VT(a f) = a‘2 VT(f) for a in IR . Hence VT”) i €451”!ng By Propositon 2.1.1 M(-) has Eé- regularization for some q > p and EuM(t)u§q.g L for some constant L , for all O.§ t.fi T . Proposition 2.1.2 now gives the desired result. Q.E.D. The use of the Baire category theorem in this context is suggested by [17] 2.2.2 Remark: We call A(t) the increasing process associated with M . In the main theorem, stated below, the process A(t, e, f) will play a crucial role. We desribe it in the next paragraph. Let (X(o), f3 be an E'-valued semi-martingale. Then, for any 6 > 0 , X(t, f) has a decomposition of the form (cf. [15] equation 45). 22 (f, X(t)) = [ 2 f x N(f,{s},dx)] + [Bc(t,f) + Z I N(f,{s},dx)] 01 0 € 0 , and all f in E . Assume Xn(0) E 0 a.s. Let (W(-),F) be a Wiener martingale, with Q(-,-) as its covariance operator. Suppose for any (t,e,f,g) in iR+ x (0.1] x E x E , we have (A) ft x fi”(f.ds.dx) 3 o 0 le>e (B) sup (3"(s,e.f)| 3 o Ot E t Q(f9 g) n L - I then X -+ W in D(E ) . 2.2.5 Corollary: Let (Mn, F", P”) be a sequence of E'-valued martingales 2 < m for all (n, t, f) satisfying M"(0) s 0 a.s. for all n and E(f, M"(t)) in N x R + xE . Suppose for each (t, T, e, f, g) in '12+ x R+ x (O, 1] x E x E the fellowing is true. 23 . P (2.2.6) It I x2 Nn(f,ds,dx) + o 0 |x|>e P (Law 6 flumzp+ mu.m “roitil n where A" is the increasing process associated with Mn and the pn is as in'Theorem 2.2.1 then L M” + w in D(E') Proof of Theorem 2: By Theorem 1.4.1, it suffices to show that (a) (f M"(-)) is tight in D for each f in E and (b) for any 0‘: t1 < t2 < ... < tk = T any f fk in E and any k.Z 1 , we have L 1,... k ((f,. x"(tp)....,(tk. x“(tk>) ((f,. w(tp).....(fk. w(tk)) in R Since ((f, X"(-)), f?) is a real semi-martingale, Theorem 1 of [15] assures us that (f, X"(-)) is tight in D for each f in E . In order to prove (b), write, for any 8 > 0 [(fl, Xn(t1)),...,(fk, x"(tk))1 =[an(t1,e,f1),...,an(T,e,fk)] + [Bn(t1,e,f1),...,Bn(T,e,fk)] + [Yn(t1,e,f1),...,yn(T,e,fk)] + [An(t1,e,f1),...,An(T,e,fk)] By equation(46L(47)and(50)0f [15], the first three terms on the right converge in law, in IRk, to the zero vector. Thus, to prove (b) it suffices to show that, for a choice of a (possibly depending on n ) Y"(e) = [An(t1,e,f1)....An(T,e,fk)] E Y = [(f1 w(t1),...(rk W(Tk)] Since t (+-t:Q(f,g) is continuous for all f, g in E , (C) implies that P sup |>t - to(f.g) 1+ 0 th Applying Lemma 2.1.5, we can get a sequence en decreasing to zero such that for all f1,...fk in E P ') An('s€ :f')>t ' t Q(f.ia fj)| +0 Tl sup | t izl jZI a1 aj t-1AtjAt and by (C) ew = u"(t, c, f) + B"(t, e, f) . Observe that B"(-, e, f) is a square integrable martingale of finite variation with t= “lo I x 2Nn(f,ds,dx). 0IX|>8 Thus by(2.2.6) t 3 o . Also = <(f M"(-))(g M“(-)> -pn t t + + <8" ( .s.f)a“(-.e.g)>t 25 It follows by(2.2.6)and the Kunita-Watanabe inequality that the last three terms converge to zero in probability showing that (C) holds. Q.E.D. We now study the weak convergence of semi-martingales taking values in the Hilbert space ER , to a EA Wiener martingale. Under an additional condition on the semi-martingales Xn(-) , it is shown that conditions (A), (sup B) and (C) of Theorem 2.2.4 suffice to prove weak convergence'in the Polish space D(Eé) . 2.2.8 Theorem: Let (Xn(-), f?) be a sequence of E'-valued semi-martingales satisfying X"(0) E O a.s. and PE sup [(fi X (t))| > e] f-CT e Oith for some k.: 1 , CT > O and for all f in E, e > O, T > O . Let (W(o),f) be a Eé-Wiener'martingale, with covariance operator Q . If Xn L satisfies conditions (A), (sup B) and (C) of Theorem 2, then Xn + W in D(Eé) for some q > k . Proof: By Theorem 1.4.1 the Xn's are uniformly p-continuous. Hence conditions L (A), (sup B) and (C) suffice to show that Xn + W in D(Eé) for some q > k . Q.E.D. 2.2.9 Corollary: Let (Mn(-),fh) be a sequence of locally square integrable E'-valued martingales satisfying E(f,Mn(t))2 i Ct “fnk for some k > O , C > O t and for all t > O , f in E , n.: 1 . Let (W, E) be an Eé- Wiener martin- gale. If M” satisfies(2.2.6)and(2.2.7)of Corollary 2.2.5, then M" 5 w in D(Eé) for some q > k . Proof: By Ooob's inequality 4C Pt sup |(f.M”(t))l > e1:%E(f.M”(T))2:—%((fnk Ofith e e 26 Thus Mn is uniformly kecontinuous and the result follows by Theorem 2.2.8. Q.E.D. We now prove a few results as applications of Theorems 2 and 3. 2.3 CLT and functional CLT for E'-valued triangular arrays. 2.3.1 Theorem: Let {Xnk’ k=1,...kn, n=1, 2,...) be a triangular array of E'-valued random variables defined on a probability space (O, F, P) such that for each f in E, {(fxnk), k=1,...,kn} is an i.i.d. sequence for each n . Let (W(-),£) O i t _<_1) be a E'-Wiener martingale with Q as its covariance operator. Suppose for every f, g in E , we have knP[I(f,Xn1)l > e] + O for every 5 > O anE[(f, Xn1)[l(f: Xn1)| _<_ 83' ‘* O for every E: > 0 kn Cov ((f,Xn1)[l(f,Xn1)| 5_ei . (9.xn1)[|(g,xnl)| §_e]) + Q(f. 9) Then, kn L Sn = X an T N(la ') in E' i=1 and the process Xn given by [kntJ x"(t) = Z an 0 5_t 5.1 i=1 . . n L . satisfies X + W in D([O 1]; E') Proof: Let F2 = c{Xn(S, -) , s §_t} . f? (= F1: 0.: t.§ 1) Then (Xn , _f") is a E'-valued semi-martingale with a canonical decomposition, in which B"°(t, f) e o , (f,X"(t))c e o for all t in [0 11. f in E . 27 ~n - F q .- w N (f: (0 t], A) ' ekntJPL(f an) E A.) By Theorem 2.2.4, it follows now, that Xn E W in D([O I]; E') . The map h : D([O 1]; E') + E' given by h(x) = x(1) is obviously a continuous map. Thus by Proposition 2.1.3, x”(1) = 5n k W(I) in E' 2.4 Example. Let (O, F, f, P) be a complete probability space with a filtration satisfying the usual conditions. In what follows we will use the notation and results developed in [9] (Chapters 1 and 3) . Let s t+ V(s) be 61 ffadapted process on (O, F, P) taking values in R.xiRd , which is quasi-left continuous (cf. definition 3.1 in [9].) Let N(ds, da, dx) denote the counting measure associated with p i.e. N(t, A x B) = N((O t] x A x B) = Z IAxB(v(s)) Sft for A 6 802) and B 6 B(Rd) . Let N(ds, da, dx) denote the compensator of N . A We shall make the following assumption on N . t A (2.4.1) For every t > O . E f f f a2 f2(X) N(ds, da, dx) < CHI“ 0 it iri " k for all f e SCRd) , where H ”k is as defined in 1.1.1. C - a positive constant, for some k.i 1 . Then (see page 63 of [9]) the real valued process defined by X(t, f) = f a f(x) [N(ds,da,dx) - N(ds,da,dx] (0 t]xRde is a ‘57P square integrable martingale with EX2(t, r) = E f a2 f (o t]xRde 2(x) N(ds,da,dx)_: cufuk . 28 By Proposition 2.1.1, we can find a Eq-version of the stochastic process X , denoted, X again for a q.: k such that for all t': O X(t, f) = (f X(t)) . In [13].the interpretation for N(ds, da, dx) is that N(t, AxB) is the number of voltage pulses of sizes a e A CR , arriving at sites x e B cRd at times 5 < t . Let (9”, F", F", P") be a sequence of complete probability spaces with filtrations .fn satisfying the usual conditions. Let Nn(ds, da, dx) be a sequence of point processes on (9", F", fr, P") of the type described earlier, with their compensators N"(ds, da, dx) satisfying assumption 2.4.1. Construct the Eé-valued processes Xn as before by t (f x"(t))= f f a f(x) (Nn - Nn)(ds, da, dx) . 0 Rde (2.4.2) Theorem. Suppose that for all (t, f, g) 6 (IR x 3(1Rd) x S(IRd)) we have A P (2.4.3) It 1 la f(x)]Nn(ds, da, dx) + o 0 Rde t 2 “n P 2.4.4 f f a f(x) g(x) N (ds, da, dx) + t 0 12de with = as in 1.1.2. 0 Then Xn = W where W is a Wiener martingale in 3(Rd) . Remark; Observe that the above conditions are in terms of N" and not in terms of the compensator of the jump measure corresponding to Xn , as are the conditions of Theorem 2.2.4. Proof: Observe that by 2.4.4 the sequence {(f X”) oil} is tight in D and hence by Theorem 1.4.1, Xn is tight in D(S'CRd)) Note that A(f,Xn(s)) = f a f(x) Nn({s}, da, dx) Thus sup lu(f,Xn(s))| < sup Ila f(x)) Nn({s}, da, dx) EET _'S O 0 |x|>e T n P and f f lxl p (f, ds, dx) + O 0 lx|>1 Thus once again by Lenglart's inequality we have (2.4.6) IT I (x) vn(f, ds, dx) 5 o lel>1 From the proof of Corollary 2 in [15], we have Bn(t. e, f) = -&; f x vn(f, ds, dx) |x|>1 where an is as in(2.2.3). Equation(2.4.6)now implies (sup B) of Theorem 2.2.4, finishing off the proof. Q.E.D. CHAPTER III DISTRIBUTION VALUED SDE'S AND THEIR WEAK CONVERGENCE We will prove here the existence and uniqueness (upto a modification) of a cadlag adapted process, taking values in S'CRd)--the space of tempered distributions on IRd . These processes could be thought of as generalizations of linear stochastic differential equations. We then take up the weak conver- gence of such processes to a generalized Ornstein-Uhlenbeck (O-U) process. The analogy to keep in mind is the following. Let (D, F, f, P) be a probability with a filtration f_ satisfying the usual conditions. Let b be a constant dxd matrix and B(-) a d-dimensional ffp Brownian motion. Then there exists a unique (upto P equivalence) Rd-valued cadlag adapted process X(-) called the O-U process satisfying (3.0.1) X(t) = X(O) + if b-X(s)ds + B(t) a.s. P tb °° tk k In fact if we define e = Z —7- b then k=O ' the solution is given by (3.0.2) X(t) = ebt. X(O) + ebt- rot e‘bs 03(5) as can be easily seen by Ito's formula. Observe also that if we define a semi-group of operators T from IRd t to IRd by T(t)x = ebtx for x Ele, then its infinitesimal generator A , is simply the matrix b . Thus equation 3.0.1 in terms of this A would read (3.0.3) X(t) = X(O) + ft () A -X(s)ds + B(t) Our first aim is to generalize this result to processes, X(v) , taking values in S'GRd) with more general semi-groups Tt and the Brownian term replaced by S'CRd)-valued square integrable martingales, i.e., equations of the form (3.0.4) (f X(t)) = (f X(O)) + r; (Af X(u-))dur + (f M(t)) a.s. P forall f in 3(Rd), v tzo. 3O 31 In the case when the martingale M(-) is a Wiener martingale in S'ORd) , the existence and uniqueness in law of a process satisfying(3.0.4)was proved using a martingale problem approach in [7]. They called such a process a generalized O-U process. When M(-) is a generalized, centered Poisson process, and the semi—groups T satisfy certain assumptions, then the existence and uniqueness of t process of the form 3.0.4 was proved in [13], for general nuclear spaces. In case of S'CRd)-valued processes, their assumptions on Tt are not necessary as we shall show. At a later stage we shall explain how each of these methods is con- nected to our method. If Mn's are a sequence of locally square integrable S'CRd)-valued martingales converging weakly to a Wiener martingale, then one expects the corresponding solutions Xn(o) to converge weakly to a generalized 0+U process. We prove this. It must be pointed out that in [8] Holley and Stroock have also studied similar limit theorems. However their methods are different from ours. Roughly speaking, they study the weak convergence of processes of the form (4), where the real square integrable martingales (f, Mn(-)) have associated increasing processes which admit a density with respect to Lebesgue measure, for every f in SCRd) . In [13], weak convergence of such processes is studied where Mn's are centered Poisson processes. Again, their methods are particular to the properties of Poisson processes, and do not carry over to the general case. The generalized 0-U process turns out to be a useful limiting case in the study of infinite particle systems ([8]) . More recently, these processes have played an important role in the development of the Malliavin calculus ([22]). Processes of the form(3.0.4)have been used to model neuronal behavior in the nervous system ([13]. Throughout we shall make the following assumptions. We will call the space SCRd) 2( d by E and its dual by E' . Thus E0 is the space L IR ) . 32 Let A : E + E be a bounded linear operator admitting a dissipative extension A' (i.e., §_0 for all x e E Assume there is a strongly O)' continuous semi-group of bounded linear operators {Tt, t 3_0} on E into itself such that t x(T f) - x(f) = 4i x(ATSf)ds v t > 0 x in E' and f in E . As mentioned in [7], it is easily checked that tA tA th = e f a.e. for f in E (where e is the semi-group of self-adjoint contractions on E , generated by A') and Ath = TtAf . 0 Let (O, F, F, P) be a complete probability space, with a filtration satisfying the usual conditions. Let (M(t), f, P) be a Eé-valued locally square integrable martingale for some p.: 1 and satisfying moreover M(O) E O , E”M(t)ng < c < w for all t-: 0 , and some c > 0 . Let N be a EB-valued random variable satisfying EHNHEp < c . 3.1.1 Theorem. There exists a unique (upto a modification) adapted Eé-valued cadlag process X(-) satisfying (3.1.2) (- X(t)) = (- N) + (f(A- X(u-))du + (-,M(t)) a.s. P v t.: 0 2 with E{[X(t)”_p 3.2c V t.: 0 Proof: Let 0n = card {a : lal §_n} and ba = . Consider the system B B of On real-valued stochastic differential equations given by _ t X3(t) - N(ha) + 4) (:B% n . Since the coefficient of the drift term-- I b x8 is clearly Lipschitz in x , independent of t and I b B XQ(u-) is F measurable, we have by the theorem of Doléans-Dade [4], that the systems of equations has a unique cadlag adapted solution. 33 Let n > m . Consider t 0 X:(t) - X2(t) = {l g baB(Xg(u_) ' Xg(u-))du 1f la] f'm N(ha) + {f g baa xg(u-)du + M(t)(ha) if m < Ial §_n and the function F e 020R) given by F(x) = x2C(a) where C(a) = (2|a| + d)"p Then by Ito's formula([12]). we have for (0| 5 m . E(xg(t) - x§(t)) = 2C(e) 1} (xg(s-) - xm(s-)) ) b For m < Idl.: n we get F(xg(t) - x:(t))= C(a)N2(ha) + 20(0) (fx3(s-) g has X2(s-)ds + 2C(a) 4} X2(s-)dM(s)(ha) + C(a) t + C(u) z [Xn(s)2 - Xn2(s-) - 2Xn(s-)AXn(s)] 0t + Z AM2(s)(ha)] . This equals st + sEtAM (s)(ha)) . Thus if we define Xn(t) = Z X2(t)ha, then 0: (We) - Wolf, = §F - X3”) = E C(q) N2(h ) + 2 ft ds (el=m+1 “ + g C( ) ft Xn(s )dM(s)(h ) + 2 g C( )[M( )(h )] OI. " 0L ' . la|=m+1 0 a a Iql=m+1 q t By the dissipativity of A , we have on taking expectations n EHXn(t) - Xm(t)u§ .i E Z C(a)N2(h ) + E E C(a)M(t)2(h ) . p Ial=m+1 a We have used here the fact([12])that M2(c)(ha) - [M(-)(ha)]t is a ffP martingale 34 As Euanp = E g C(a)N2(ha) < C and E”M(t)”§p < C s we have lim sup E fT “x“(t) - xm(t)u§ dt = 0 v T > 0 . (new n>m 0 p Thus by the Riesz-Fischer theorem there exists an adapted process X(-) in E5 such that . T ~ n 2 _ Tim E ‘b u.X(t) - x (t) ”_p dt - 0 n-)oo Set X(t) = N + {f(A X(u-))du + M(t) Clearly, X is cadlag, adapted. Observe that Xn(t) converges in norm to Y(t) and hence converges weakly to X(t) in H = L2((O, F, P); E6) in the sense that for every f in ED we have E(f, x"(t)) + E(f, X(t)) We will now show that Xn(t) also converges weakly to X(t) in H . Since weak limits in Hilbert spaces are unique, this will imply that X(t) = X(t) a.s. P for each t.: 0 and thus X(t) = N + ft 0 (A, X(u-))du + M(t) a.s. P - Observe that N converges weakly to N and M(t) converges weakly to M(t) in H 2 Thus to show that Xn(t) converges weakly to X(t), it suffices to show that E A; (Af, Xn(u-))du converges to E A; ( But Xn(°) converges in norm to X(-) in the Hilbert space K = L Af X(u-))du . 2([0 T] x O (t E6) = {f : [O T]'+ E5 such that Q; E W(t)”?p dt < co} . Therefore Xn converges weakly to X(-) in K . This by definition implies that E {f (Af,Xn(u-))du converges to E {f (Af,X(u-))du for any t': O . This finishes the proof of the existence. 35 To prove the uniqueness, let X(-) , Y(-) be two cadlag adapted EB-valued processes satisfying (1). We will show that X(t) = Y(t) a.s. P V t.Z 0 . To do this it suffices to show that X(t)(ha) = Y(t)(ha) , since the action of X(t) or Y(t) on E6 is given by (i, X(t)) = Z X(t)(h ). Cl G. For any n.: 1 , define the operator Pn : EO + E by Pnf Z ha lalgn Observe that for t_: 0 and lal f'n . X(t)(hal - Y(t)(ha) = r: (Aha,X(u-) - Y(u-))du . Set 2 t = Z x t - Y t h h () (4)511“) (ma), then PnZ(t) E E0 V ”.i 1 . By applying Ito's formula to the function D xb Z x: for xeR”, wehave = Z [(X(t) - Y(t))(ha)]2 = 2 4; du By the dissipativity of A , and the above equation, we have “Pn Z(t)([2 : O a.s. P . Since n was arbitrary this gives us the uniqueness. Lastly, since EHXn(t) - X(t)”§p + 0 and Ean(t)”§p.: 20 we get W(t)“?p < 20 v t > 0 . Q.E.D. 3.1.3 Remark: We have not shown here that E sup ((X(t)”2 < w for any T > 0 t for a bounded linear operator B from E0 to E:, then by Ito's formula (3.1.4) F((f.X(t)) - 13(Ai,x(u) -1111].ng X(u )) - ;g(4f,x(s))ds)du is a _F_-P martingale for F 6 C502). Conversely if(3.1.4) is f—P martingale for every F 6 C302), then by choosing F(x) = x and F(x) = x2 , we have + _ t t l (f, X(t)) )0 t (e [(f,X(t)) - 10H”, X(u))du] (Af, X(u))du = Y(t, f) 2 - 1(1an are both continuous local martingales. By Lévy's characterization of Brownian motion, it is seen that Y(-, f) is a Brownian motion with covariance operator Q defined earlier. Holley and Stroock make (3.1.4) their defining condition for the O-U process and then proceed to show the existence and uniqueness of a measure P , satisfying (3.1.4). They explicitly find the conditional distribution of (f, X(t)) given F5 for s < t . Their method however, will fail if we replace the Wiener martingale by a general locally square integrable martingale, for then the Markovian structure of the process X(o) is destroyed. To see how the method of Kallianpur and Wolpert [13] works assume that the matrix of A(= ((baB))) is diagonal with respect to the orthonormal basis (ha) i.e., = bus = 0 if (3 # B . By Theorem 3.1.1 we get the 37 existence of a unique EE-process X(o) satisfying (ha. X(t)) = We) + (few X(u-)du + M(t)(n,) a.s. P By Ito's formula we get tb (3.1.5) (ha, X(t)) = e O‘O‘N(ha) + if e(t'5)bao: d(M(s)(ha) Thus X(-) can be explicitly solved in terms of M, N . In [13], instead of assuming that the matrix of A is diagonal,they impose conditions on A' which ensure that -A' has countably many eigen-vectors f1, f2, f3,... and corre- sponding eigenvalues 0.: 11.: 12 form {(1 + Xj)'2r1< m for some r .3 ... satisfying a growth condition of the 2( d) is constructed >0. AnONBfor L R 1 using these fj's , which diagonalize the matrix of A'. The growth condition on the eigenvalues enables them to construct a nuclear space E c:L2(Rd) . If a process is now defined by(3.1.5),they show that it has Eé-version for some q.: 1 and solves (3.1.2). Clearly such a program can be carried out for more general L2 2(12“) spaces than L and this is exactly what they do. But as our theorem shows, in the case of LZCRd) their assumptions are not necessary. 3.2 Weak convergence to an O-U process. We turn next to the study of weak convergence of E'-valued processes of the form (1). Suppose Mn(-) is a sequence of Eé-valued locally square integrable 2 11.p Suppose also that Nn is a sequence of Eé-valued random variables satisfying martingales satisfying EHMn(t) < C say, on a filtration (an, F", En, P"). Ennnufp < C . Let X"(-) be the unique Eé-valued cadlag adapted process satisfying (.,x”(t)) = (-.~“) + 1 It seems intuitively clear that such equations should be robust in the sense that if Mn converges weakly to a Wiener martingale in D(EB) , then the corresponding solutions should converge weakly, too. This is what we will set 38 out to do now. We shall put conditions on Mn that ensure its weak convergence to W and prove under these conditions the weak convergence of Xn to an O-U process. We start by making the setting of our problem more precise. Let (0", F", f?, P") (O, F, f, P be a sequence of filtrations n31 ’ N) satisfying the usual conditions. We shall say that a continuous E'-valued process X(o) is a f_- PN O-U process starting at N in E' and characteristic A if PN [X(O) = N] = 1 and V F 6 C200?) we have I 2 (3.2.1) F((f. X(t)) - rot (Af, X(u))du) - 11ng F"((f, X(u)) - )0“ (Af, X(s))ds)du is f-- PN martingale. Let Nn be a sequence of EB-valued random variables satisfying EHNnngp.: C for some p.: 1 , C > 0 . Let (Mn, 5?, P") be a sequence of locally square integrable E5 martingales satisfying M”(0) a 0 v n.: 1 and Eunnufp E c . Let Nn(f, ds, dx) denote the dual predictable projection of the integer valued measure corresponding to the jumps of (f M"(-)) . Let X"(-) denote the unique, cadlag, adapted, EB-valued process satisfying (3.2.2) (f, x"(t)) = (f, N") + ft (Af, x”(u-))du + (f, M"(t)) a.s. Pn o with Eux”(t)n§p : 2c . 3.2.3 Theorem. Suppose for all (8, f, t) in (0 1] x E x R+ we have . P (3.2.4) ii I x2 N“(r, ds, dx) + 0 as n e e IX1>€ P (3.2.5) <(f M") (g Mn)> + t as n +,m V g in E t 39 (3.2.6) Nn(f) = N(f) in IR , for a deterministic N in E . Then Xn = X in D(Eé) for a q 3.p where X is an O—U process starting at N with characteristic A . Proof; Our first task will be to show that the Xn's are uniformly PO-Contin- uous for a 90.3 1 i.e. V e > 0 , V P > 0 , T > 0 6 > 0 , such that PnEsup ((f, x”(t))| > e] < 0 if ”fflp < 6 for all n.: 1 . th " 0 T' To this end, observe that 1(i. X"(t))12 _<_ 31(i.N")12 + 3t (51(Af.X"(u-)12du + 1(i.11"(t))12 Thus E sup [(f x"(t))12.: 3E(f N“)2 + 3T 4; E(Af X”(u-))2du + 3E sup (f M"(t))2 tET CET 5 3“““6 + 3T (J EHX“ SUCh that ”Afflpgf'Kanp for all f in E . So we get 0 2 E sup (i.x”(t))2 3 “in; (3c + 6CKT + 3C) 0 th proving that Xn's are uniformly Q)-continuous. According to Theorem 1.4.1, if Xn's are uniformly gj-continuous, and tight in D(E') , then they are tight in D(Eé) for some q 3_Q). To show that Xn's are tight in D(E') it suffices by Theorem 1.4.1 to show that for any f in E , (f. X"(-)) is tight in D . To do this it suffices to show that Nn(f) is tight in R , Y"(o, f) = 4; (Af,Xn(u-))du is tight in o and (i,M"(-)) is tight in o . By(3.2.6), Nn(f) is tight in IR . By 3.2.5 and Theorem 1 of 1151, 40 (r, N”(-)) is tight in 0 . We are left to show that {Yn(-, f)} is tight in D . nil Let T2 be the family of all stopping times with respect to the natural 0 field of Y" and which are bounded by 2 . Let 0 < 6 < 1 and note that for T 6 T2 we have E(Y”(T + 3 f) Y"(T f))2 - E Tia " 2 T+5 " 2 3 - 3 - (Afgx (U'))C1U : 6 f E(Af,x (U')) dU T T 2 §_6 . zc - (Atup Thus if Tn is a sequence of stopping times, with Tn in T2 and O < on < 1 is any sequence of numbers with on + 0 we have n n 2 E(Y (Tn + on,f)_- Y (Tn,f)) + 0 So by the theorem of Aldous 111. {v"(-,f)} is tight in 0 for each f in E . n31 Consequently X"(-) is tight in D(E') . To prove the uniqueness of the limit point, notice that Nn converges weakly to N in E' , so we can, without loss of generality assume that X"(0) e 0 V n‘: 1 and show convergence to an 0-U process with characteristic A, starting at 0 . By conditions 3.2.4 and 3.2.5 and Corollary 2.2.9, M" =1W in D(Eé) , where W is ii Eé-Wiener martingale on (O, F, F, P) . By Theorem 3.1.1 there exists a unique cadlag adapted E' process X such that q (f, X(t)) = n; (Af, X(u))du + (r, W(t)) a.s. P n So we have shown that if X k converges weakly then (x""(-) - 10°01» x"'<(u-))du) (= N“(-)) .. (x<-) - [0 (A-. X(u))du) Ito's formula now shows that every weak subsequential limit satisfies 2 F((f, X(t)) - 4} (Af, X(u))du) - 1§u_, 4; F"((f, X(s)) - (f(Ai;X(u))du)as is an f}P martingale. By Theorem 1.2.3 of [7] we get Xn a X in D(Eé) . Q.E.D. 41 3.3 Example. Consider the set up of example 2.4. We shall use the notation of that example. Thus xn defined by (f, x"(t)) -- rot f d a f(x)(Nn - N")(ds, da, dx) Ex]? is a martingale for each f in SCRd) . By Theorem 3.1.1 there exists a unique process Yn in E4 such that (f, v"(t) ) = (f (Af. v"(u-) )du + (f, x"(t) ) for any dissipative operator A satisfying the conditions of Theorem 3.1.1. By Example 2.4 and Theorem 3.2.3 Yn converges in law to an O-U process starting at 0 in D(Eé) . 42 NOTATION Frechét Nuclear space Hilbertian semi-norms determining the topology of E , k = 1, 2,... . completion of E with respect to H “k topological dual of Ek norm and inner product on EQ topological dual of E , with the strong dual topology generic elements of E the smallest o-algebra on E' that makes all the maps x 9 E')+ x(f) measurable, for f in E the Borel o-algebra of EL right continuous with left limits the space of all cadlag maps from ZR+ to a topological space Z . Also, 0 = D(R+, R) the smallest o-algebra on D(E') that makes all the maps z 3 D(E') it (2(t1, f1),... able for 0.: t ,z(tk,fk)) 6 le , measur— < t ...