×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bouleau, N.; Hirsch, F., Dirichlet forms and analysis on Wiener space, (1991), Walter de Gruyter Berlin/New York · Zbl 0748.60046
[2] Hirsch, F., Propriété d’absolue continuité pour LES équations différentielles stochastiques dépendant du passé, J. funct. anal., 76, 193-216, (1988) · Zbl 0638.60068
[3] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1989), North-Holland/Kodansha · Zbl 0684.60040
[4] H. Kaneko, S. Nakao, A note on approximation for stochastic differential equations, in: Séminaire de Probabilités XXII, 1988, pp. 155-162 · Zbl 0647.60072
[5] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1991), Springer-Verlag Berlin/Heidelberg · Zbl 0734.60060
[6] S. Kusuoka, D. Stroock, Application of the Malliavin calculus, part I, in: Proceedings of the Taniguchi Intern. Symp. on Stochastic Analysis, Kyoto and Katata, 1982, pp. 271-306
[7] Nualart, D., The Malliavin calculus and related topics, (2006), Springer-Verlag Berlin/Heidelberg · Zbl 1099.60003
[8] Revuz, D.; Yor, M., Continuous martingales and Brownian motion, (1999), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0917.60006
[9] Saks, S., Theory of the integral, (1937), Hafner New York, Warszawa-Lwow
[10] Shigekawa, I., Stochastic analysis, (2004), American Mathematical Society
[11] Williams, D., Probability with martingales, (1991), Cambridge University · Zbl 0722.60001
[12] Yamada, T.; Watanabe, S., On the uniqueness of solutions of stochastic differential equations, J. math. Kyoto univ., 11, 1, 155-167, (1971) · Zbl 0236.60037
[13] Ziemer, P., Weakly differentiable functions, (1989), Springer-Verlag New York/Tokyo · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.