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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.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H07 Stochastic calculus of variations and the Malliavin calculus
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