Radoniﬁcation of cylindrical semimartingales on

In this work we prove that a cylindrical semimartingale on a Hilbert space becomes a semimartingale with values in the aforementioned Hilbert space if we compose it with three Hilbert-Schmidt operators


Introduction
This paper represents a portion of a joint research project which is uncompleted due to the untimely death of the first author.The result (cf.Theorem 3.2) derived here was first shown in the thesis of the second author in the framework of nuclear spaces and was extended to Banach spaces by Laurent Schwartz (cf.[4]).In our project we had chosen, for the pedagogical reasons, to present it in the simple setting of Hilbert spaces, where everything can be done by straightforward calculations. 2

Notations and Preliminaries
Let (n, 7, P) be a probability space equipped with a right continuous, increasing filtration of sub-sigma algebras of .F, denoted by E (o,1~ ~.We suppose that ~'o contains all the P-negligeable sets and smallest sigma algebra containing all the elements of this filtration is :F. We shall denote by S0 the space of real-valued semimartingales defined relative to the filtration mentioned above, we suppose that they are equipped with the metric topology defined by where ~ denotes the set of predictable, simple processes.Recall that under this topology 6ĩ s a (non-locally convex) Fréchet space (cf.[2], [3]).Let H a separable Hilbert space any continuous, linear mapping from H into ?~will be called a cylindrical semimartingale.We will say that ~ is radonified to an H-valued semimartingale with a linear operator u on H if there exists an H-valued semimartingale, say K such that = (K, h) for any h H, where the equality is to be understood with respect to the equivalence classes of 5~. 3
Proof: We have, using the last lemma where denotes the Gauss measure on R" whose density is given by ( 2 Proof: Since e > 0 is arbitrary, we see that P sup +°° =1, , P{ ( ( u ( e j ) ) 2 t + } = 1 , hence the sum ((u(ej))tej converges in H almost surely uniformly with respect to t E [0,1].D Let us denote by Y2 the H-valued process defined by Yz = 03A3 DYs 1{~0394Ys~>1}, at with the convention Yo_ = 0. Since Y is a cadlag process, YZ is of finite variation (it has finite number of jumps on each bounded interval).Let us denote by Yl the process Yt' = Y -Y2t .
Then Y1 has uniformly bounded jumps.Theorem 3.1 Suppose that X : H -~ S° is a linear, continuous mapping.Then the pro- cess Yl constructed above defines a continuous, linear mapping from H into the space of special semimartingales.Furthermore, if we denote by M(~) and A(~) respectively, the local martingale and predictable, finite variation parts of (yl, ~), ~ E H, then ~ ~--> M(~) and ~ ,-~ ~4(~) are linear, continuous maps from H into S°.
is predictable for any ~ E H, the left hand side of the last equality is a predictable process of finite variation and it is also a local martingale, hence both sides are constant, since all the processes are zero at t = 0 (this follows from the construction of Yl and the definition of special semimartingales), this constant is equal to zero, and this proves the linearity of the maps 03C6 ~ A(03C6) and 03C6 ~ M(03C6).
Let us now show the continuity of these mappings: let ~ E H, by definition of a local martingale, there exists a sequence of stopping times (Sk : k e ltl) increasing to one such that, for any k E I N , { M ( 0 3 C 6 ) t Ŝk ; t E [0,1]} is a uniformly integrable martingale and t E ~0,1~ } is of integrable, total variation.Since A(~) is predictable we have By construction ~0394Y1t~ ~ 1 for any t > 0, hence | 0 3 9 4 A ( 0 3 C 6 ) t | ~ 0 3 C 6 ãlmost surely and |0394M(03C6)t| 2~03C6ã lmost surely.Suppose now that -a 0 in H and (M(~k); k E IN) converges to some m in SO. (A(~k); k E IN) converges then to some a E S°.From the above majorations, we know that the processes (A(~~); k E IN) and (M(~k); k E N) have uniformly bounded jumps.It is well-known that (cf.[2] and [3]) the set of local martingales and the predictable processes of finite variation having uniformly bounded jumps are closed in S'°.Therefore m is a local martingale and a is a process of finite variation, moreover, we have m +a == 0, , consequently m = a = 0, and the closed graph theorem implies the continuity of M and A.

D
From the Corollary 3.1, if u H --~ H is a Hilbert-Schmidt operator, then M o ul and A o ui are two cadlag processes, in fact since M o ui has uniformly bounded jumps, it is an H-valued local martingale.In fact we have Corollary 3.2 Suppose that M is a cylindrical local martingale with uniformly bounded jumps.Then, for any Hilbert-Schmidt operator u on H, Mou is an H-valued local martingale with bounded jumps.
We proceed with the following result which is of independent interest: Lemma 3.3 Let us denote by V° the space of real-valued cadlag processes of almost surely finite variation endowed with the metric topology defined by Suppose that B : H -V° is a linear, continuous map and v is a Hilbert-Schmidt operator on H. Then the set : 10|dB(v(03C6))s| = +~ for some 03C6 ~ H} is a negligeable set.
Proof: Let A be a finite partition of [0,1] of order n ~ IN.Let us denote by B0394 the linear, continuous map from H into where In = 1Rn with 11-topology and denotes the space of the equivalence classes of l1n-valued random variables under the topology of convergence in probability.
We have a similar result to that of Lemma 3.1: for any f > 0, there exists a b > 0 such for any § E H, where the sup with respect A is taken on a sequence of increasing partitions of [0,1] in such a 'way that it can be obtained as a monotone limit and (.1.)denotes the scalar product in To see 3.1, we proceed as in the proof of Lemma 3.1, namely, suppose that E > 0 is given.Choose 61 > 0 such that expi03B1| e/2 if |03B1| 61.Let also 62 = inf (~1, a ( 1 + E -1)j.Since the map ~ H d~(B(~), o) is continuous, there is some 6 > 0 such that for any 6, we have f(jB(~), 0) From the Chebytchev inequality, Now, denoting by the total variation of B(~), we have for any § with hence, combining the inequalities 3.2 and 3.5 and noting that all of them are controlled with we can pass to the limit on the partitions and we obtain the inequality 3.1.As in the Lemma 3.2, we have where (ej;j E IN) is a complete, orthonormal basis of H. Using the Gaussian trick as in the proof of Lemma 3.2, the last term above can be majorated by where sup0394 means, as explained before, that we take the supremum with respect to an increasing sequence of partitions.Since E is arbitrary, it follows that P sup /' |dB(v(03C6))s| +~} = 1. .a We can now announce the main result: Theorem 3.2 Suppose that X : H --~ S° is a linear, continuous mapping and that us, i = 1, 2, 3 are three Hilbert-Schmidt operators on H. Then there exists an H-valued semimartin- where At is a cylindrical local martingale and ~4 is a cylindrical predictable process of finite variation and both have uniformly bounded jumps.As we have seen in the Corollary 3.2, M o u2 gives an H-valued local martingale.Lemma 3.3 implies that almost surely the process ~4 o U2 is of finite weak variation.Let us denote by V[O,l] the space of finite measures on [0,1] endowed with the total variation norm.Then from the Lemma 3.3, we see that there exists a negligeable subset N of H such that for any 03C9 Nc the map 03C6 ~ A o u2(03C9,03C6) is a linear, continuous mapping from JEf into V[O,l]. .Therefore, there exists some /?(~) ~ 0 such that |Ã o u2(03C9, 03C6)|1 ~ 03C1(03C9)~03C6~, for any 03C6 H and 03C9 Nc.Let us denote the process A o u: by L. Using an auxiliary Gaussian measure on H with covariance operator ~3 o t~, it is easy to see that ~ ~u3~203C1(03C9), since the above majoration is uniform in A, we have obviously sup 0 3 A 3 Lt i + 1 0 Lti 0 u3~ +00 , , ~ A almost surely and this completes the proof of the theorem.
a Remark Let us note that, if X is a cylindrical local martingale, its radonification k o ui o us o U3 is not necessarily an ~-valued local martingale but it is a semimartingale, this phenomena is due to the "unbounded jumps" of ~.