Statistics & Probability Letters 16 (1993) 391-395 North-Holland

8 April 1993

Worthy martingales and integrators

Ely Merzbach Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

Moshe Zakai * Faculty of Electrical Engineering, Technion - Israel Institute of Technology, Haifa, Israel

Received May 1990 Revised November 1991

Abstract: The class of worthy martingales parameterized by time and a general state space was introduced by Walsh (Lecture Notes in Math. No. 1180, 1986). It serves as a class of L*-stochastic integrators which seem natural for modelling the driving force in stochastic partial differential equations. Here we extend this class and show that this extension yields the most general class of L*-martingale integrators.

AM8 1980 Subject Classification: 60648, 60657, 60HOS.

Keywords: Worthy martingale; integrator; stochastic integral; bimeasure; predictable o-field.

1. Introduction and notation

In the study of stochastic partial differential equations, it is often necessary to integrate over the time parameter R, and a general state space. For this purpose, Walsh (1986) introduced the concept of worthy martingale which is, very roughly, a process parameterized by a time parameter and a general state space parameter. It is a martingale in the time direction, an L2-measure in the state space direction, and there exists a random measure on the product space which dominates the covariation process associated with the worthy martingale.

In this note we extend the class of worthy martingales, replacing the domination assumption by the weaker assumption that the covariation process induces an L2-multimeasure. We present an example which shows that the extended class is a strict extension of the class of worthy martingales. We characterize our class of processes as being the class of L2-martingale integrators for predictable integrands of two variables: time and points of the state space. Notice that the fundamental difference between the setup of stochastic multiparameter integration theory (as, for example, in Cairoli and Walsh, 1975, or Merzbach and Zakai, 1990) and the setup of this note (as well as Walsh, 1986) is that in our context there is only one direction of evolution (in time) and therefore the predictability is defined only with respect to the time direction.

Correspondence to: Ely Merzbach, Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel. e-mail: merzbach@bimacs.bitnet.

* Work partly supported by the Fund for Promotion of Research at Technion.

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For the notation we follow essentially Walsh (1986). The underlying probability space (0, 9, P) is equipped with a filtration {Ftl, t > 0) which satisfies the usual assumptions. The state space will be a Lusin space (G, G) (i.e. a measurable space homeomorphic to a Bore1 subset of line. This, for example, includes all Euclidean spaces), where G is the Bore1 a-field on G. The predictable a-field 9 on 0 x G x R, is the a-field generated by the ‘rectangles’ of the form F XA x (s, t], where F E FS, s < t and A E G. A function defined on n X G X R, will be defined to be predictable if it is g-measurable. A predictable function is said to be elementary if it is of the form X. 1, . Ica,bl where X is a bounded S$ measurable random variable and A E G. A finite sum of elementary predictable functions is called a predictable simple function.

Let (G, G) and (F, E> be two measurable spaces. A bimeasure is a function p from the pairs A, B where A E G and B E [F to the real numbers such that for every A E G, P(A, . > is a measure on IF and for every B E [F, p(., B) is a measure on G. A bimeasure is said to be finite if sup I P(A, B) I < w where the supremum is over all the pairs A, B. Note that if p(A, B) is a positive measure on G X F and I P(A, B)I <p(A, B) then p can be extended to a measure on G X [F.

2. Integrators

Definition 1. A process (M,(A), t E R,, A E G} is said to be a quasi-worthy martingale if: (i) M,(A) = 0 for all A E G.

(ii) For every A E ‘I$ (M,(A), S,, t 2 0) is a martingale. (iii) For every t > 0, M,( .) is a sigma-finite L’(O)-valued measure on G. (iv) Let

Q(A, B, [(VI) = (M.(A), M.(B)>,

denote the finite covariation process associated with the martingales M,(A) and M,(B). Then, for every sequence of simple predictable functions fJx, t) converging to zero for every x and t and such that I f,<x, t)l Q 1, it holds that

n~m ,’ ,,x,.fnC~~ t>fn( y, t)Q(dx, dy, dt) = 0 lim // (*)

in L1-norm, for any T < 03 and for any G’ in G such that Q(A, B, [0, T]) < co for every A, B c G’.

Remarks. 1. The name ‘quasi-worthy martingale’ is provisional, and as will be shown in the theorem below, a better name is ‘L2-martingale integrator’.

2. Walsh (1986) introduced the notion of worthy martingales. It is a process which satisfies (i), (ii) and (iii), but instead of (iv) it is assumed that there exists a random a-finite predictable measure on G x G x B(R+) which dominates Q. We refer the reader to Walsh (1986) for details on the definition and properties of worthy martingales which include orthogonal measure martingales and martingale measures with nuclear covariance. It is easily verified that every worthy martingale is a quasi-worthy martingale, but conversely, as we shall see by an example, our class is strictly larger than the class of worthy martingales.

3. Condition (iv) implies that E/T/o,, f(x, t)g(y, t>Q(dn, dy, dt) can be extended to become a bounded bilinear functional on 9 x9 and this bilinear functional can be represented by a bimeasure on 9 X9 (Merzbach and Zakai, 1986).

We proceed now to define the stochastic integral with respect to quasi-worthy martingales. The construction follows essentially Walsh (1986). Let f = X. IA * Zca,bI be an elementary predictable function.

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Define

(**I

and extend by linearity to simple predictable functions. Let M be a fixed quasi-worthy martingale and Q(. , . , . ) the covariation process associated with it.

For simple predictable functions f(x, t>, g(x, t) set

(f* da=E/ f(x, s>g( y, s)Q(dx, dy, ds). GxGXR,

Since Q is a covariation function <f, fje 2 0, therefore (‘, . > defines a norm on a class of predictable functions as follows:

Let PQ define the completion of the simple predictable functions with respect to the seminorm

11 f 11 Q = (f, f)t?” as follows. Let 3?’ denote the class of bounded predictable functions f such that there exists a sequence of simple predictable functions (f,J converging pointwise to f. Then 2 is a vector space, contains the constants and is closed under uniform and increasing convergence. Since the class of simple predictable functions is a subclass of z, closed under products, then by the monotone class theorem, 3 contains all the bounded predictable functions, hence every bounded predictable function is the pointwise limit of a sequence of simple predictable functions. Let f(x, t> > 0 be any bounded predictable function then there exists a sequence of simple predictable functions f,,(x, t> such that for almost all w, 0 <f,(x, t) +f(x, t) for all x, t. Let m(k) > n(k), m(k), n(k) + a,

1 fm(k) - fnckj I + 0 as k + CO

hence, by (iv), 11 fmckj -fnckj II Q --f 0, therefore II fmckj II Q converges to some limit. Therefore f(x, t) is in the Q-completion of the simple predictable functions denoted Pap.

Turning now to the definition of the stochastic integral, note that it follows from ( * * > that for simple predictable f(x, t),

E(f+W’))2 = IIf. b(x) . $o,&) II:!

and the extension of ( * * > to bounded predictable integrands follows directly from this isometry.

Definition 2. A process {X,(A), t a 0, A E G} is said to be an L*-martingale integrator if it is a martingale in the time direction and if the stochastic integral f + X,, as defined above, can be extended linearly for all bounded predictable processes and satisfies the following dominated convergence property: Let I fJw, x, t) I G c, n = 1, 2,. . . , f, and f predictable, and as n + ~0, fJw, x, t> - f(o, x, t) for all x, t and almost all w, then

f;X,(A) -f-X,(A) in L2(fl, KI, p)

for every A E G for which X,(A) is a.s. finite.

This definition is the standard extension of the usual definition of integrator. See Merzbach and Zakai (1986) for examples and references for different properties of multi parameter integrators. Clearly, it is possible to extend the class of integrators further - by considering semimartingales instead of martingales or by considering Lp spaces with p < 2, but only L2-martingale integrators will be considered here.

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We can state now the main result of this note:

Theorem. The process (M,(A), t E R,, A E G} is an L*-martingale integrator if and only if it is a quasi-worthy martingale. Moreover, for all bounded f ~9~ the process If. M,(A), t E R,, A E G) is again an L*-martingale integrator and the following hold:

(a) (f-M(A), f.M.(B)),,=~~‘/,~~f(X, s)f(y, s)Q(dx, dy, ds).

(b) E[(f*M,JA))*] =Ilf. n,OJOj(t). w>ll;-

Proof. If M,(A) is a quasi-worthy martingale, then exactly as in Walsh (19861, (a) and (b) hold for simple predictable functions and for such a function f, the process f. M,(A) satisfies properties (i>, (ii) and (iii> of Definition 1. Remark 3 from above ensures that Q is a bimeasure on 9 X9. If Q is finite, the following Lemma 2.1 of Merzbach and Zakai (1986) (or by a monotone class argument), we obtain that the expression in (a) is still a bimeasure, and then f * M,(A) satisfies property (iv). The extension for the case where Q is sigma-finite is obvious. Therefore f. M,(A) is a quasi-worthy martingale. Conversely, if M,(A) is an L2-martingale integrator, then Q(A, B, [O, tl) exists as a finite variation process in the time direction, and is positive definite. By the integrator property, it follows that (iv) is satisfied. 0

3. An L2-martingale integrator which is not a worthy martingale

Let g(dh, dh’) be a positive definite bimeasure on [O, 11 X [O, 11 which is not a measure. For an example of such a bimeasure cf. p. 19 of Chang and Rao (1986). Since for any smooth f(e), 0 6 LO, 11,

f(e) de

it follows that g([O, A], [0, A’]) is a positive definite L* kernel on [O, 11 X [O, 11 (cf. Blei, 1989; Chang and Rao, 1986; Nualart and Zakai, 1990; for the existence of bimeasure integrals). Therefore we may write (cf., for example, Riesz and SZ-Nagy, 19.55, p. 243):

g([O, Al, [O, A’]) = iA;4i(A)&(A’)

with Ai>0 and +&.I, i = 1, 2 ,..., orthonormal on [0, 11. Let y(t), i = 1, 2,. . . , be independent Brownian motions. Set

n=l

Note that for this M, Q< A, B, dt) = g(A, B) dt is non random and condition (iv) follows ,directly from the Grothendieck inequality (cf. e.g., Theorem 1.1 of Gilbert, Ito and Schreiber, 1985; or Blei, 1989). Consequently M is an L* martingale integrator, however in this case Q cannot be dominated by a measure since g(dA, dh’) as chosen above cannot be extended to become a measure on the product space.

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Acknowledgement

The constructive criticism of an anonymous referee is gratefully acknowledged.

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