## 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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### References:

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