Peak Time-Windowed Risk Estimation of Stochastic Processes

TL;DR

This paper introduces a novel method to estimate upper bounds on extreme time-windowed risks in stochastic processes using infinite-dimensional linear programming and semidefinite relaxations, with convergence guarantees.

Contribution

It formulates the risk estimation as an infinite-dimensional linear program and develops a tractable semidefinite hierarchy for accurate approximation.

Findings

Method successfully estimates risk bounds for example processes.

Semidefinite relaxations converge to true risk values.

Applicable to various time-windowed risk metrics like mean and shortfall.

Abstract

This paper develops a method to upper-bound extreme-values of time-windowed risks for stochastic processes. Examples of such risks include the maximum average or 90% quantile of the current along a transmission line in any 5-minute window. This work casts the time-windowed risk analysis problem as an infinite-dimensional linear program in occupation measures. In particular, we employ the coherent risk measures of the mean and the expected shortfall (conditional value at risk) to define the maximal time-windowed risk along trajectories. The infinite-dimensional linear program must then be truncated into finite-dimensional optimization problems, such as by using the moment-sum of squares hierarchy of semidefinite programs. The infinite-dimensional linear program will have the same optimal value as the original nonconvex risk estimation task under compactness and regularity assumptions, and the sequence of semidefinite programs will converge to the true value under additional properties of algebraic characterization. The scheme is demonstrated for risk analysis of example stochastic processes.