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Malliavin calculus for two-parameter Wiener functionals. (English) Zbl 0577.60065

Let (\(\Omega\),\({\mathcal K},P)\) be the canonical stochastic basis associated to the two-parameter d-dimensional Wiener process. Consider the following system of stochastic integral equations: \[ (1)\quad X^ i_{st}=\int^{s}_{o}\int^{t}_{o}(\sum^{d}_{j=1}A^ i_ j(X_{uv})\quad dW^ j_{uv}+B^ i(X_{uv})dudv),\quad i=1,2,...,m \] where \(x\in R^ m\) and the functions \(A^ i_ j,B^ i:R^ m\to R\) have bounded derivatives of any order.
It is already known that (1) admits a continuous solution which, moreover, satisfies a certain kind of Markov property. The authors’ main result (Th.4.3) is that if the functions \(A^ i_ j\) satisfy a supplementary condition, then the law of the random vector \(X_{st}\) admits an infinitely differentiable density function with respect to Lebesgue measure for every point (s,t) not on the axes.
The condition is the following: The vector space spanned by the vector fields \((A_ j)_{1\leq j\leq d}\), \(A'_ iA_ j,1\leq i,j\leq d\), \(A'_ i(A'_ jA_ k)\), \(1\leq i,j,k\leq d\),..., \(A'_{i_ 1}(A'_{i_ 2}(...(A'_{i_{n-1}}A_{i_ n})...)\), \(1\leq i_ 1,i_ 2,...,i_ n\leq d\) at every point x is \(R^ m\).
Reviewer: G.Zbaganu

MSC:

60H20 Stochastic integral equations
60G60 Random fields
60H99 Stochastic analysis
60J65 Brownian motion
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