Risk and Extended Expected Utility Functions: Optimization Approaches

Ermoliev, Y.M. & Norkin, V.I. (2003). Risk and Extended Expected Utility Functions: Optimization Approaches. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-03-033

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Abstract

The proper analysis of policies under uncertainties has to deal with "hit-or-miss" type situations by using approximate risk functions, which can also be viewed as so-called extended expected utility functions. Formally this 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 the risk function over certain parameters, rather than over decision variables as in the classical distribution (generalized functions) theory. For smooth approximations we obtain gradients in the form of expectations of stochastic vectors which can be viewed as a form of stochastic gradients for the original risk function. We pay special attention to optimization of risk functions defined on trajectories of discrete time stochastic processes with stopping times, which is critically important for analyzing regional vulnerability against catastrophes.

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