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Existence of densities of solutions of stochastic differential equations by Malliavin calculus. (English) Zbl 1198.60025
In the present paper, the \(d\)-dimensional stochastic equations: \[ \begin{gathered} dX(t)= \sum^r_{j=1} \sigma_j(t, X(t))\,dB^j(t)+ b(t, X(t))\,dt,\\ X(0)= x_0\in\mathbb{R}^d\end{gathered}\tag{1} \] are considered. The coefficients of this equations are not Lipschitz continuous and have their densities or not.
Let \((\Omega,{\mathfrak I},P)\) be a probability space which is an orthogonal product measure space of an abstract Wiener space \((B,H,\mu)\) and another probability space \((\Omega',{\mathfrak I}',v)\).
The \(V_h(B\times\Omega')\) is the total set of random variables \(F\) on \((\Omega,{\mathfrak I},P)\) such than there exists a random variable \(\widehat F\) on \((\Omega,{\mathfrak I},P)\) that \(F=\widehat F\) a.s. and \(\widehat F(x+ th,\omega')\) is a function of bounded variation on any finite interval with respect to \(t\) for all \(x\) and \(\omega'\).
The author gives a criterion that a random variable belongs to the class \(V_h\) and studies relation between the solution of stochastic differential equation and the class \(V_h\). It shows that in the special case the solution of (1) have densities.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
Full Text: DOI
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