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.
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Mathematics Subject Classification:
35B65, 45D05, 60J25, 60J27
JEL Classification:
C63, G12, G13
This work was partly supported by the Mathematical Finance Network under the Danish Social Science Research Council, Grant No. 9800335.
Insightful comments from an anonymous referee helped improve the paper.
Manuscript received: October 2002; final version received: February 2005
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Norberg, R. Anomalous PDEs in Markov chains: Domains of validity and numerical solutions. Finance Stochast. 9, 519–537 (2005). https://doi.org/10.1007/s00780-005-0157-8
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DOI: https://doi.org/10.1007/s00780-005-0157-8