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Malliavin calculus for two-parameter Wiener functionals. (English) Zbl 0595.60065

Consider the system of m stochastic equations in the plane: \[ X^ i_ z=x^ i+\int_{[0,z]}(A^ i_ j(X_ r)dW^ j_ r+B^ i(X_ r)dr),\quad i=1,2,...,m \] where \(x\in R^ m\) and \(W=(W_ 1,...,W_ d)\) is the d-dimensional two-parameter Wiener process. Suppose now that the functions \(A^ i_ j\), \(B^ i\) have all the derivatives bounded and that the vector space spanned by the vector fields \(A_ 1,...,A_ d\), \(A_ i^{\nabla}A_ j\), \(1\leq i,j\leq d\), \(A_ i^{\nabla}(A_ j^{\nabla}A_ k)\), \(1\leq i,j,k\leq d,...\), \(A^{\nabla}_{i_ 1}(A^{\nabla}_{i_ 2}(...(A^{\nabla}_{i_{n-1}}A_{i_ n})\)...), \(1\leq i_ 1,i_ 2,...,i_ n\leq d,..\). has full rank at the point x, where \(A_ i^{\nabla}A_ j\) is the covariant derivative of \(A_ j\) in the direction of \(A_ i.\)
Then, for any z not on the axes, the law of the random vector \(X_ z\) admits an infinitely differentiable density. That is the main result of the paper. In proving the statement a variant of Malliavin calculus is used (Lemmas 4.1 and 4.2).
Reviewer: Gh.Zbaganú

MSC:

60H20 Stochastic integral equations
60G50 Sums of independent random variables; random walks
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