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

Conditional expected values in Markov chains are solutions to a set of associated 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 controlled global error. Three cases leading to first order partial differential equations in two variables are considered, all from finance and insurance: Option pricing in a Markov chain driven financial market; Probability distributions of cash flows generated by multi-state life insurance contracts; Reserves in life insurance when payments or intensities are path-dependent. Key-words: Continuous time Markov chain, semi-Markov models, 1st order PDE, backward differential equations, continuous differentiability, arbitrage pricing theory, life insurance, numerical solutions to PDEs. JEL classification: C63, G12, G13 Mathematics Subject Classification: 35xx, 60J25, 60J27