The proper analysis of polices under uncertainties has to deal with “hit-or-miss” type of situations by using appropriate risk measures. Formally it often requires the solution of dynamic stochastic optimization problems with discontinuous indicator functions of such events as ruin, underestimating costs and overestimating benefits. The available optimization techniques, in particular formulas for derivatives of risk functions, may not be applicable due to explicitly unknown probability distributions and essential discontinuities. The aim of this paper is to develop a solution technique by smoothing risk function over certain parameters, rather than over decision variables as in the classical distribution (generalized functions) theory. For smooth approximations we obtain explicit formulas for gradients in the form of expectations of stochastic vectors which can be viewed as a form of stochastic gradients for the original risk function. These axe used in the specific quasigradient procedure. We pay special attention to optimization of risk functions defined on trajectories of discrete time stochastic processes dependent on stopping times.
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