An existence theorem for linear stochastic differential equations

Abstract

The purpose of this paper is to prove a general existence and uniqueness theorem (Theorem 1) for nth order linear stochastic differential equations in which derivatives are defined in the “mean-square” sense (see below for details). The results here subsume the earlier work of Edwards and Moyal [I], and the present proofs are simpler. Further, it is hoped that the present treatment exhibits the structure (and limitations) of linear stochastic differential equations in the mean-square or strong sense. This is discussed and contrasted with the “pointwise” or It6 equations in the last section. Actually, it will be seen from the present paper that the theory of linear stochastic differential equations in the strong sense merges with the theory of ordinary linear differential equations in Banach spaces, the latter spaces reducing to the scalars only when the underlying probability space contains just one point. Precise definitions of the initial value problem and of the relevant concepts will be given in the next section. The two-point boundary value problem is treated in Section 3, and a result on the sample function behavior of solutions is obtained in Section 4.