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The partial Malliavin calculus. (English) Zbl 0738.60055
Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 362-381 (1989).
[For the entire collection see Zbl 0722.00030.]
The authors give an exposition of the partial Malliavin calculus developed by Ikeda, Shigekawa, and Taniguchi, and use it to study the existence of conditional densities in the filtering problem. Their main theorem is the following: Let \((x_ t,z_ t)\) denote the solution of the Stratonovich stochastic system \[ dx_ t=X_ 0(x_ t,z_ t)dt+X_ i(x_ t,z_ t)dw^ i_ t+\tilde X_ i(x_ t,z_ t)dz^ i_ t,\;dz_ t=\ell(x_ t,z_ t)dt+d\tilde w_ t, \] where \(x_ t\in R^ n\), \(z_ t\in R^ p\) and the processes \(\{w^ i_ t, i=1,\dots,m\}\) and \(\{\tilde w^ i_ t, i=1,\dots,p\}\) are independent Brownian motions. If the vector fields \(X_ i\) and \(Y_ i\) are smooth and bounded with bounded derivatives of all orders, and satisfy the general Hörmander condition at the initial point \((x_ 0,z_ 0)\), then the law of \(x_ t\), conditioned on the history of the process \(z_ t\) up to time \(t\), has a \(C^{\infty}\) density.

60H07 Stochastic calculus of variations and the Malliavin calculus
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