Anomalous PDEs in Markov chains: Domains of validity and numerical solutions

Abstract

Conditional expected values in Markov chains are solutions to a set of backward differential equations, which may be ordinary or partial depending on the number of relevant state variables. This paper investigates the validity of these differential equations by locating the points of non-smoothness of the state-wise conditional expected values, and it presents a numerical method for computation of such expected values with a controlled global error. Two cases leading to first order partial differential equations in two variables are considered, both from finance and insurance: option pricing in a Markov chain driven financial market, and probability distributions of discounted cash flows generated by multi-state life insurance contracts