TIIE DOOB-MEYER DECOMPOSITION FOR ANTICIPATING PROCESSES

In this paper we study a class of nonadapted quasimartingales defined in terms of two‐sided conditional expectations. Necessary and sufficient conditions are given for the Doob‐Meyer decomposition of such processes.


O. Introduction
In [13], Skorohod introduced a stochastic integral of non-adapted random processes with respect to a Gaussian measure with orthogonal increments. The Skorohod integral is an extension of the classical Ito integral and coincides with the adjoint of the derivative operator on the Wiener space (see [5]).
The relation between the Skorohod integral and the Malliavin calculus has been analyzed by Nualart and Zakai in [8]. More recently, a generalized or anticipating stochastic calculus based on the Skorohod integral has been developed by Nualart and Pardoux [9] (see also [12,14,15]). We also refer to [10] for an exposition of the basic ideas of this theory.
The anticipating calculus has sorne special features. One of them is that the "indefinite" Skorohod integral <loes not have the martingale property. However it possesses an orthogonality property (see Proposition 5.1 (i) in (9]) which can be formulated as On the other hand, if f : R --+ IR is a function of class C 2 , under suitable hypotheses, and with the same notations as before, it follows from the extended Ito formula that f (Xt) can be written as the sum of a process Mt satisfying (0.1), and a process of bounded variation. This gives the feeling that the property (0.1) plays the role of the martingale property in the non-adapted case.
The aim of this paper is to study a class of processes X = {Xt, t E (O, 1]} for which a generalized Doob-Meyer decomposition holds that means Xt = Mt+At, with M satisfying (0.1) and A a process of bounded variation.
In the first section we introduce the notion of S-quasimartingale. These processes are the analogue of the quasimartingales of [4] (see also [3], [6], [11]) in the non-adapted case. We also include the essential tools of the anticipating calculus which are needed in the development of our work. In the second section we give a suffi.cient condition for a S-quasimartingale to have a Doob-Meyer decomposition. Notice that, dueto the lack of adaptedness, it is not clear how to define the analogue of class D ( see [3]) in our situation.
Our sufficient condition has been inspired by the work of Brennan ([1]), where the problem of the Doob-Meyer decomposition for two-parameter quasimartingales is considered.
No ingredient of anticipating calculus is needed to prove the results of this section. In section 3, using the anticipating calculus, we give a necessary and sufficient condition ensuring the Doob-Meyer decomposition of sorne classes of S-quasimartingales, and we also prove uniqueness.  More generally, we can define the N-th derivative of F E S by Dt .. tN F = Dt 1 Dt 2 ••• DtN F, and for any p > l the space [)N,p is the completion of S with respect to the norm IIFIIN,p = IIFllu(n) + I:f=l 11 IIDk FIIP(Jk)IILP(íl).

Preliminaries
The following result will we needed in section 3.
We refer the reader to [9] (Proposition A.l) for the proof of this fact, which is an extension of a well-known result on the representation of Wiener functionals.

tingale.
Consider the simple example where I = N, the set of all natural numbers, and let X = { X n, n 2:: O} be an integrable process such that and ..ón(X);in the analogous way.
Consider the following hypothesis: (H1) The set Ix is uniformly integrable. We can state the main result of this section. respectively, in the weak topology u(L 1 , L 00 ), as n --+ oo, for any i ~ l.
Then, for any n 2 no

h=2n-nok
Hence A~ -At 2 O, because it is the weak limit of a positive sequence. The same arguments apply to {A;n, n 2 1}. We set Arn = A-;t -A;n, rn E Q.
The continuity of Q(X)¡ and Q(X)¡ entails that {Arn, n 2 1} is uniformly continuous in L 1 . In fact, assuming that r¡ < rj, we have The process {Mrn = Xrn -Arn, n 2 1} is an S-martingale. To prove this fact take r¡ < ri anda bounded random variable e which is F(r;,r;Jc-measurable. Using the same notations as before it follows that: h=2"(n)-nok Sn < tn for any n 2: 1, Sn l s and tn j t as n ----+ oo. Then by the L 1 -continuity of X. On the other hand This finishes the proof of the theorem. D

Remark.
Assume that we are dealing with the Brownian case, and that the process X in Theorem 2.1 is adapted. Then, so are A and M. Furthermore, in this case M is a martingale.

Necessary and sufficient conditions for the Doob-Meyer decomposition of an S-quasimartingale
In this section we will study the Brownian case. This is our fundamental assumption.
In the first part we prove that, if the measure induced by the process A constructed in Theorem 2.1 is absolutely continuous, then the hypotheses (H 1 ) is also necessary for the Doob-Meyer decomposition of an L 1 -continuous S-quasimartingale. In the second part we consider the set of L 1 -continuous S-quasimartingales which are S-submartingales. Vve introduce a new hypotheses (H 3 ) and prove that it is a necessary and sufficient condition for the Doob-Meyer decomposition to hold. As a by-product of our results we will obtain the uniqueness of the representation in this particular case. In the last part we will study a family of S-quasimartingales derived by transformation of Skorohod integrals.
Assume that X is a process such that X = M + A where M is an S-martingale and A an integrable process of bounded variation, with Ao = O. With the notation of the previous section we have The process A can be decomposed as the difference of two increasing integrable processes A = A(l) -A( 2 ) . Clearly ~ n(A) :::; ~n(AC 1 >)+~ n(A( 2 )). So, in arder to establish the uniform integrability of Ix we can assume without loss of generality that A is increasing.
We want to prove that for any e E L 00 we have   We can now state condition (H 3 ) as follows.
(H 3 ) The set Ix(t) = {An(X)t, n 2:: 1} is weakly uniforrnly integrable for any t E Q, where Q denotes the set of the rational dyadic nurnbers in [0,1].

Notice that (H 3 ) is weaker than (H 1 ).
The rernaining of this section is devoted to prove the following result. The proof of this theorern is based on severa! lernrnas. Hence the supremum when llell 00 :' .S k of the right hand side of (3.5) can be bounded by ¡.
By the same reason, by choosing 8 = 3 E(Ai) , the right hand side of (3. 7) can be majorized by¡.
The functional rJe can be obtained using the following argument. First consider a sequence of smooth functionals {ln, n 2:: 1} such that ln converges to e in 0 1 , 2 and a.s., when n ~ oo. Let a < b be real numbers such that a < b and l( w) E ( a, b) a.s. consider a function 1-1) E Ccf(R) such that r.p(ln) converges to e in 0 1 , 2 ' as n tends to OO. The sequence {r.p(ln), n 2:: 1} is bounded and satisfies E f 0 1 1Drr.p(ln)l 4 dr < +oo, for any n 2:: l.
(1) Assume first that X is an L 1 -continuous S-submartingale satisfying (H 3 ). The construction of the increasing process A follows by the same arguments than in the thus Mrn = Xrn -Arn, rn E Q is an S-martingale, and we continue as in the proof of  The property E f 0 In order to show that Mt is an S-martingale we will use the techniques developed in [9] for the proof of the extended Ito formula.
Let t, t' E [O, 1], t < t', and 7rn = {t = to,n < t1,n < ... < tn,n = t'} be a refining sequence of partitions of [t, t'] whose mesh tends to zero. We will write t¡ instead of ti,n, for the sake of simplicity. Let n-+oo }t (3.9) Since F is F(t,t']c-measurable, and using the duality between D and h, we have Let us now prove (3.9). Suppose that n ~ m, and for any i = 1, ... n denote by t~m) the point of the partition 1rm which is closer to tf from the left. Then we have n-1 t 1 ~ J"(X¡)(Xt,+ 1 -Xt.) 2 -¡ f"(Xs)u;dsj Finally, tends to zero in L 2 , as n tends to infinity, for each m fixed, as follows from Theorem 5.4 of [9). This completes the proof of (3.9), and the proposition is established.
• Remark. Notice that the assumption of Proposition 3.8 are not strong enough to ensure the continuity of the processes Xt and Mt (see [9]). In fact, a sufficient condition for the continuity of Xt is u E L 2 ,P with p > 4, and in this case the proof of Proposition 3.8 could be shortened.
Actually, in this case we can apply the Ito formula (see [9]) and get Mt = J; f'(Xs)usdWs with f'(X 8 )u 8 locally in L 1 , 2 .