A link between wave governed random motions and ruin processes.

*(English)*Zbl 1103.91045The paper links results involving wave governed random motions with problems related to the time of ruin in risk theory.

In particular, in the context of risk theory, the authors analyse problems connected to models for both positive and negative risk sums. In the first case the process \(R_t\), representing the reserve, increases linearly with rate \(c\), that is the strictly positive premium rate, during the time and decreases by jumps because of claims: \[ R_t=u+ct-S_t \] where \(R_0=u>0\), and \(S_t\) is the aggregate claims up to time \(t\).

In the second case, the reserve process \(R'_t\) decreases linearly with \(c'\), that is the strictly positive premium rate, and increases by jumps: \[ R'_t=u-c't+S_t \] where \(R_0=u>0\) and \(S'_t\) is the aggregate jumps up to \(t\).

In both cases, the first time where the reserve becomes negative, say \(T_u\) and \(T'_u\) in the case of positive and negative risk sums respectively, plays a fundamental role. In particular the finite-time ruin probabilities \(P(T_u<t)\) and \(P(T'_u<t)\) give relevant information in risk management. In this context the authors propose an algorithm for computing finite-time ruin probabilities in renewal non-Poisson risk models with exponential claims.

Applications of the results in finance close the paper.

In particular, in the context of risk theory, the authors analyse problems connected to models for both positive and negative risk sums. In the first case the process \(R_t\), representing the reserve, increases linearly with rate \(c\), that is the strictly positive premium rate, during the time and decreases by jumps because of claims: \[ R_t=u+ct-S_t \] where \(R_0=u>0\), and \(S_t\) is the aggregate claims up to time \(t\).

In the second case, the reserve process \(R'_t\) decreases linearly with \(c'\), that is the strictly positive premium rate, and increases by jumps: \[ R'_t=u-c't+S_t \] where \(R_0=u>0\) and \(S'_t\) is the aggregate jumps up to \(t\).

In both cases, the first time where the reserve becomes negative, say \(T_u\) and \(T'_u\) in the case of positive and negative risk sums respectively, plays a fundamental role. In particular the finite-time ruin probabilities \(P(T_u<t)\) and \(P(T'_u<t)\) give relevant information in risk management. In this context the authors propose an algorithm for computing finite-time ruin probabilities in renewal non-Poisson risk models with exponential claims.

Applications of the results in finance close the paper.

Reviewer: Emilia Di Lorenzo (Napoli)

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91B70 | Stochastic models in economics |

60G40 | Stopping times; optimal stopping problems; gambling theory |

35L10 | Second-order hyperbolic equations |

35R60 | PDEs with randomness, stochastic partial differential equations |

##### Keywords:

renewal models; wave-governed random motions; telegraph equation; negative risk sums; non-Poissonian ruin
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\textit{C. Mazza} and \textit{D. Rullière}, Insur. Math. Econ. 35, No. 2, 205--222 (2004; Zbl 1103.91045)

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