Stochastic Flows and Jump-Diffusions

Recent development of stochastic differential equations on Euclidean space driven by a Wiener process and a Poisson random measure is discussed in this book. Solutions of equations are jump-diffusion processes, including diffusion processes as special cases. It is shown that the solution of an equation defines a stochastic flow of diffeomorphisms. Natures of jump-diffusions such as Kolmogorov equations and dual (backward) processes are studied through stochastic flows. Then the existence of the smooth density of transition functions of non-degenerate jump-diffusion is examined. It is shown that the density function is the fundamental solution of the heat equation associated with the generator of a diffusion (partial differential operator) and that of a jump-diffusion (integro-differential operator). Further, the short time estimate of the density function is derived. The basic tool for the study is Malliavin calculus, which is known as a powerful tool for the study of diffusion processes on the Wiener space. In this book, Malliavin calculus on Wiener-Poisson space is studied as an attempt to dock Malliavin's work on the Wiener space and Picard's work on the space of Poisson random measure. Using this calculus, a criterion is obtained for which the law of a given Wiener-Poisson functional (random variable) should have a smooth density. The criterion works well for jump-diffusions. Further, the diffeomorphic property of stochastic flows is used for proving that the density function is the fundamental solution. At the end of the monograph, the density problems are discussed for jump-diffusions on non-Euclidean space. Included are jump-diffusions killed outside of a sub-domain of Euclidean space and those defined on manifolds.