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<p>While the interarrival times of the classical Poisson process are exponentially distributed, complex systems often exhibit non-exponential patterns, motivating the use of the fractional Poisson process, in which interarrival times follow a Mittag–Leffler distribution. This paper investigates the associated risk process, describes its Cramér–Lundberg formula and establishes a relationship between the continuous premium rate and the fractional claim frequency. For a compound fractional risk process with exponential claims, we derive closed-form expressions for the finite-time ruin probability. Furthermore, for a general claim distribution, we provide ruin probability estimates that can serve as a basis for developing reinsurance strategies.</p>

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Probability of ruin within finite time and Cramér–Lundberg inequality for fractional risk processes

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月19日

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  • Nikolai Leonenko,
  • Andrey Pepelyshev,
  • Alois Pichler,
  • Enrica Pirozzi,
  • Xiangyun Meng

摘要

While the interarrival times of the classical Poisson process are exponentially distributed, complex systems often exhibit non-exponential patterns, motivating the use of the fractional Poisson process, in which interarrival times follow a Mittag–Leffler distribution. This paper investigates the associated risk process, describes its Cramér–Lundberg formula and establishes a relationship between the continuous premium rate and the fractional claim frequency. For a compound fractional risk process with exponential claims, we derive closed-form expressions for the finite-time ruin probability. Furthermore, for a general claim distribution, we provide ruin probability estimates that can serve as a basis for developing reinsurance strategies.