In this paper, we initiate a systematic study of the theory of nonlinear second order SBVP through sample calculus approach [4] and obtain some fundamental results. We first prove an existence result for bounded nonlinearities and then develop the method of upper and lower sample solution processes by means of which the restrictive boundedness assumption can be dropped. This existence result being of theoretical value, we then proceed to extend the constructive monotone iterative technique to SBVP which yields monotone sequences that converge to extremal sample solution processes. Since the success of this technique rests heavily on the availability of stochastic maximum principle, we first prove a general result concerning random differential inequalities, a special case of which is the desired stochastic maximum principle, and then discuss the monotone technique thoroughly. Finally, the important problem of finding the error estimates between the sample solutions of SBVP and the solutions of corresponding mean BVP is considered. This necessitates proving a general stochastic comparison result for SBVP, the deterministic special case of which is it-self of very recent origin [3]. We utilize Lyapunov-like functions in obtaining this error estimate from which interesting special cases can be obtained. The present investigation is motivated by a specific problem arising in gas lubrication [2,5]. This talk is based on the joint work reported in [1,2]. Consider the stochastic boundary-value problem (SBVP for short)
where
is a set of real-valued random functions such that f(t, x( t, w), y( t, w), w) is product-measurable whenever real-valued random functions x(t, w) and y(t, w) are product-measurable; R[D, aB ] denotes a collection of all random variables defined on a complete probability space (0, .9', P) into R; for p = 0, 1; CQ, &, b,, E R[&?, R] for ZL = 0, 1, a,,, LY, 2 0, &,, /3, > 0 w.p. 1, and prime stands for the sample derivative. We shall always assume that b, E B(p) w.p. 1 for some p > 0, f~ M[Z X R X R, R[O, Iw]] and f( t, X, y, W) is sample continuous in (x, y) for each t E I. A random process x(t) = x( t, w) is said to be a sample solution process of SBVP (1) if, (i) x(t) is sample continuously differentiable; (ii) x(t) and x'(r) are product-measurable processes, which satisfy (1) w.p. 1. We shall state a basic existence theorem whose proof utilizes the stochastic version of Schauder's fixed point theorem.