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An Optional Semimartingales Approach to Risk Theory

Author

Listed:
  • Mahdieh Aminian Shahrokhabadi

    (Faculty of Science, Mathematics and Statistical Sciences Department, University of Alberta, Central Academic Building, Edmonton, AB T6G 2G1, Canada
    These authors contributed equally to this work.)

  • Alexander Melnikov

    (Faculty of Science, Mathematics and Statistical Sciences Department, University of Alberta, Central Academic Building, Edmonton, AB T6G 2G1, Canada
    These authors contributed equally to this work.)

  • Andrey Pak

    (SS&C Technologies, Toronto, ON M5V 3K2, Canada
    These authors contributed equally to this work.)

Abstract

This paper aims to develop optional semimartingale methods in risk theory to allow for a larger class of risk models. Optional semimartingales are left-continuous with right-limit stochastic processes defined on a probability space where the usual conditions—completeness and right-continuity of the filtration—are not assumed. Three risk models are formulated, accounting for inflation, interest rates, and claim occurrences. The first model extends the martingale approach to calculate ruin probabilities, the second employs the Gerber–Shiu function to evaluate the expected discounted penalty from financial oscillations or jumps, and the third introduces a Gaussian risk model using counting processes to capture premium and claim cash flow jumps in insurance companies.

Suggested Citation

  • Mahdieh Aminian Shahrokhabadi & Alexander Melnikov & Andrey Pak, 2025. "An Optional Semimartingales Approach to Risk Theory," Risks, MDPI, vol. 13(4), pages 1-27, March.
    • Handle: RePEc:gam:jrisks:v:13:y:2025:i:4:p:61-:d:1617911
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    References listed on IDEAS

    as
    1. Cai, Jun & Dickson, David C. M., 2003. "Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 61-71, February.
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