Anticipating stochastic differential equations: Regularity of the law (Q677474)

Continuing to work on the way initiated by \textit{T. Masuda} [J. Math. Kyoto Univ. 32, No. 3, 527-531 (1992; Zbl 0768.60054)] and \textit{M. E. Caballero}, \textit{B. Fernández} and \textit{D. Nualart} [Stochastics Stochastics Rep. 52, No. 3/4, 303-322 (1995; Zbl 0864.60040)] the authors study sufficient conditions for the existence and regularity of densities for the probability law of the solution to a multi-dimensional stochastic Stratonovich differential equation of diffusion type with smooth diffusion and drift functions \(A_j\), \(1\leq j\leq 4\), and \(A_0\), resp., and an anticipating initial condition \(X_0\) satisfying some assumptions on its Malliavin derivative. Two different types of conditions are developed: First a restricted Hörmander condition on the diffusion functions \(A_1,\dots,A_k\), which has to be satisfied on the whole Euclidean space, and secondly an unrestricted Hörmander condition which is similar to the classical one (one takes the Lie algebra generated by \(A_j\), \([A_0,A_j]\), \(1\leq j\leq 4)\), except that the Lie brackets have to be calculated now at the random point \(X_0\). Finally, a highly degenerate case is studied.