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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.

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