## A remark on the chaos expansion of a diffusion. (Une remarque sur le développement en chaos d’une diffusion.)(French)Zbl 0754.60054

Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 165-168 (1989).
[For the entire collection see Zbl 0722.00030.]
The author gives a proof of a theorem of Meyer and Leandre, exposed at Oberwolfach (“Stochastic Analysis”), stating that, if $$(W^ i)$$ is a $$d$$-dimensional Brownian motion, if $$X^ j_ t$$ is an $$R^ m$$-valued solution of $$dX^ j_ t=\sum A^ j_ i(X_ s)dW^ i_ s+A^ j_ 0(X_ s)ds$$, where $$A_ i$$ are $$C^ \infty$$-functions, with all their derivatives uniformly bounded, if $$h$$ is a $$C^ \infty$$-function having, together with all its derivatives, a polynomial growth and if $$t$$ is fixed, then the projection of $$h(X_ t)$$ on the $$n$$-th chaos of $$W$$ is represented as a (multiple) stochastic integral (of order $$n$$) of a continuous kernel. The proof uses the fact that $$h(X_ t)\in D_ \infty$$ (in usual notations of the Malliavin calculus) and relies on two lemmas. The first states that, for every $$p>1$$, the $$n$$-th derivative $$D^{j_ 1,\dots,j_ n}_{t_ 1,\dots,t_ n}(X^ i_ t)$$ is $$L^ p$$-bounded, uniformly in $$t_ 1,\dots,t_ n$$, $$t\leq c$$, and gives a relation which appears as a stochastic integral equation in that derivative, with the value in $$\max t_ i$$ expressed in terms of the $$(n-1)$$-derivatives of $$A^ i_{j_ k}(X_{t_ k})$$. The second lemma gives the possibility of applying the Kolmogorov criterion for sample path continuity in $$t_ 1,\dots,t_ n$$ to the $$n$$-derivative above.

### MSC:

 60H07 Stochastic calculus of variations and the Malliavin calculus 60J60 Diffusion processes 60H20 Stochastic integral equations 60H05 Stochastic integrals

Zbl 0722.00030
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