A Comparison Theorem for Solutions of Stochastic Differential Equations and Its Applications

A new kind of comparison theorem in which an SDE is compared with two deterministic ODEs is established by means of the generalized sample solutions of SDEs. Using this theorem, we can compare solutions of two SDEs with different diffusion coefficients and obtain some asymptotic estimations for the paths of diffusion processes. In the investigation of solutions of stochastic differential equations (SDEs), comparison theorems are very powerful tools as in the case for deterministic ones. But so far most of these theorems have dealt with those SDEs with the same diffusion coefficient (cf. [1, 3—7, 9, 10]) except in [8] where a very special case involving two different diffusion coefficients has been discussed (cf. Example 2). In this paper, we use the method of generalized sample solutions of SDEs (cf. [5, 11]) to establish a new kind of comparison theorem in which an SDE is compared with two deterministic ODEs. The conditions imposed here are weaker than those in [5 and 6] and the proof is much simpler. On the other hand, a discontinuous right-hand side is allowed. So it seems more appropriate to the stochastic optimal control problems. We begin with a lemma. LEMMA. Assume that two functions f{t,x) and f(t,x) are defined on some domain G in R2 satisfying the Carathéodory conditions, that is, they are measurable in t, continuous in x and dominated by a locally integrable function m(t) in the domain G. Let (ín,xo) an(^ (*o,¿o) be two points in G such that xq < xq, x{t) be any solution to the initial value problem (1) xt = f{t,xt), x{t0) = x0 and x(t) be the maximal solution to the problem tn\ ■ tit \ 14. \ (2) Xt=f{t,Xt), X{t0) = X0.