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In the simplest case, we obtain a general solution to a problem of minimizingan integral of a nondecreasing right continuous stochastic process from zero tosome nonnegative random variable tau, under the constraints that for somenonnegative random variable T, tau is between zero and T a.s. and the expectedvalue of tau is some alpha. The nondecreasing process and T are allowed to bedependent. In fact a more general setup involving sigma-finite measures, ratherthan just probability measures is considered and some consequences for familiesof stochastic processes are given as special cases. Various applications areprovided.

Minimizing a stochastic convex function subject to stochastic constraints and some applications