Stochastic Differential and Evolution Equations

Abstract

The solution of an SDE with respect to a Wiener process is called strong (see Chap. 1, Sect. 4) if it is adapted to a Wiener flow of σ-field. In other words, a strong solution is a solution, the trajectory of which up to each moment t can be represented as a measurable mapping of the trajectory of a Wiener process also up to t. Apart from their independent significance in the theory of SDEs, strong solutions play a large role in the theory of control of diffusion processes and in filtration theory. It would seem that at present, the theory of strong solutions has more or less been constructed. Nevertheless, a number of highly important questions in it have not yet been resolved. One of the basic unsolved questions is to find natural sufficient conditions for the existence of the strong SDE solutions arising in filtration theory (the renewal problem). The Markov case (see below) has been studied fairly thoroughly. Here we do not touch upon equations in infinite-dimensional spaces (in particular, Part II of this chapter is devoted to them) or equations with jumps, equations with respect to semimartingales or semimartingales with multidimensional time. Further, we present theorems on strong solutions for SDEs with aftereffect and with random coefficients, Markov theory for a non-degenerate diffusion, diffusions with degeneracy, and counterexamples of Tsirel’son and Barlow.