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<p>The generalised Ornstein–Uhlenbeck process is the solution of a non-homogeneous stochastic differential equation where the driving force and the non-homogeneity are independent Lévy processes. This model includes as particular cases numerous models of collective risk theory describing the evolution of the capital reserve of an insurance company investing its reserve in a risky asset with price dynamics given by a geometric Lévy process. We show that the exit probability of the generalised Ornstein–Uhlenbeck process on a finite time interval satisfies a partial integro–differential equation in a viscosity sense. We prove a result on the uniqueness of viscosity solutions in a boundary value problem.</p>

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Ruin problems with investments on a finite interval: PIDEs and their viscosity solutions

  • 期刊论文

  • 发表日期：

    2025年10月22日

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  • Viktor Antipov,
  • Yuri Kabanov

摘要

The generalised Ornstein–Uhlenbeck process is the solution of a non-homogeneous stochastic differential equation where the driving force and the non-homogeneity are independent Lévy processes. This model includes as particular cases numerous models of collective risk theory describing the evolution of the capital reserve of an insurance company investing its reserve in a risky asset with price dynamics given by a geometric Lévy process. We show that the exit probability of the generalised Ornstein–Uhlenbeck process on a finite time interval satisfies a partial integro–differential equation in a viscosity sense. We prove a result on the uniqueness of viscosity solutions in a boundary value problem.