an:05655696
Zbl 1198.60025
Kusuoka, Seiichiro
Existence of densities of solutions of stochastic differential equations by Malliavin calculus
EN
J. Funct. Anal. 258, No. 3, 758-784 (2010).
00256298
2010
j
60H10 60H07
stochastic differential equation; Malliavin calculus; absolute continuity; existence of densities; existence of fundamental solutions
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.
Maria Stolarczyk (Katowice)