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

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  • Ragnar Norberg

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. Copyright Springer-Verlag Berlin/Heidelberg 2005

Suggested Citation

  • Ragnar Norberg, 2005. "Anomalous PDEs in Markov chains: Domains of validity and numerical solutions," Finance and Stochastics, Springer, vol. 9(4), pages 519-537, October.
    • Handle: RePEc:spr:finsto:v:9:y:2005:i:4:p:519-537
    DOI: 10.1007/s00780-005-0157-8
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    Cited by:

    1. Christiansen, Marcus C. & Djehiche, Boualem, 2020. "Nonlinear reserving and multiple contract modifications in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 187-195.
    2. Djehiche, Boualem & Löfdahl, Björn, 2016. "Nonlinear reserving in life insurance: Aggregation and mean-field approximation," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 1-13.
    3. Jacek Jakubowski & Mariusz Niewk{e}g{l}owski, 2013. "Risk-minimization and hedging claims on a jump-diffusion market model, Feynman-Kac Theorem and PIDE," Papers 1305.4132, arXiv.org, revised Jul 2013.

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