Max-plus Stochastic Control and Risk-sensitivity

TL;DR

This paper explores max-plus stochastic control problems, establishing their connection to risk-sensitive control, and demonstrates how the value function solves a quasivariational inequality linked to nonlinear PDEs, with applications in finance and control theory.

Contribution

It introduces a novel max-plus framework for stochastic control problems, connecting them to quasivariational inequalities and nonlinear PDEs, and provides theoretical results and practical examples.

Findings

Value function characterized as viscosity solution to QVI.

Established equivalence between QVI and nonlinear PDE with discontinuous Hamiltonian.

Demonstrated applications in finance and nonlinear H-infinity control.

Abstract

In the Maslov idempotent probability calculus, expectations of random variables are defined so as to be linear with respect to max-plus addition and scalar multiplication. This paper considers control problems in which the objective is to minimize the max-plus expectation of some max-plus additive running cost. Such problems arise naturally as limits of some types of risk sensitive stochastic control problems. The value function is a viscosity solution to a quasivariational inequality (QVI) of dynamic programming. Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous Hamiltonian is used to prove a comparison theorem for viscosity sub- and super-solutions. An example from math finance is given, and an application in nonlinear H-infinity control is sketched.