Time-Consistent Risk Measures for Continuous-Time Markov Chains

TL;DR

This paper introduces a novel framework for time-consistent risk evaluation in continuous-time Markov systems, utilizing dual representations, semi-derivatives, and differential equations to generalize classical Markov process analysis.

Contribution

It develops a new approach to risk measurement in continuous-time Markov systems, including a differential equation characterization and discrete approximation methods.

Findings

Derived a system of differential equations for risk evaluation.

Introduced semi-derivatives of risk multikernels.

Constructed convergent discrete-time risk approximations.

Abstract

We develop an approach to time-consistent risk evaluation of continuous-time processes in Markov systems. Our analysis is based on dual representation of coherent risk measures, differentiability concepts for multivalued mappings, and a refined concept of time consistency. We prove that the risk measures are defined by a family of risk evaluation functionals (transition risk mappings), which depend on state, time, and the transition function. Their dual representations are risk multikernels of the Markov system. We introduce the concept of a semi-derivative of a risk multikernel and use it to generalize the concept of a generator of a Markov process. Using these semi-derivatives, we derive a system of ordinary differential equations that the risk evaluation must satisfy, which generalize the classical backward Kolmogorov equations for Markov processes. Additionally, we construct convergent discrete-time approximations to the continuous-time risk measures.