Stochastic boundary value problems (Q796898)

Two theorems are stated on the existence of a sample solution process of the stochastic boundary value problem \[ x''(t,\omega)=f(t,x(t,\omega),x'(t,\omega),\omega) \] \[ B_ 0(\omega)\times(0,\omega)=b_ 0(\omega),\quad B_ 1(\omega)\times(1,\omega)=b_ 1(\omega). \] A theorem is given which establishes the existence of two sequences: one that converges uniformly and monotonically with probability 1 to the sample minimal solution process and one to the sample maximal solution process. An upper bound is given for the difference between the sample solution process of the stochastic boundary value problem and the solution of the corresponding mean boundary value problem.