Stochastic differential equations and their applications (Q2784992)

The article provides a survey over theoretical problems arising in the area of stochastic differential equations. The material can be divided into three parts. In the first part stochastic ordinary differential equations (SODEs) in the sense of Itô, driven by Brownian motion, are treated. Definitions of weak and strong solutions of SODEs and theorems concerning the existence and uniqueness of either kind of solutions, as well as other properties of them, are given. Further, asymptotic properties, such as the existence of stationary solutions, ergodicity and stability, as treated by \textit{R. Z. Khas'minskij} [``Stochastic stability of differential equations'' (1980; Zbl 0441.60060)], are discussed. The section with the headline ``Applications'' mentions the problem of the motion of pollen grains in water. The major part of this section is devoted to the discussion of boundary value problems using diffusion type SODEs, optimal stopping, stochastic control and backward stochastic differential equations.NEWLINENEWLINENEWLINEIn the second part of the paper the formulation of SODEs with other types of driving processes is presented. Thus, SODEs with respect to processes of jump type, Lévy processes, semimartingales and nonlinear integrators are introduced, together with results concerning the existence and uniqueness of their solutions. The last part treats stochastic functional differential equations and SODEs in abstract Hilbert spaces.NEWLINENEWLINEFor the entire collection see [Zbl 0979.00017].