A generalized defective renewal equation for the surplus process perturbed by diffusion.

*(English)*Zbl 1074.91563The authors consider the surplus process \(V(t)= u+ ct- S(t)+\sigma W(t)\), \(t\geq 0\), where \(\sigma> 0\), \(\{W(t): t\geq 0\}\) is a standard Wiener process, that is independent of the compound Poisson process \(\{S(t): t\geq 0\}\). The defective renewal equation for the expected discounted function of a penalty at the time of ruin is generalized and its asymptotic formula is proposed.

Reviewer: Alexandra Rodkina (Kingston/Jamaica)

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

##### Keywords:

Defective renewal equation; Surplus process; Diffusion process; Asymptotic formula; Associated claim size distribution; Reliability-based class
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\textit{C. C. L. Tsai} and \textit{G. E. Willmot}, Insur. Math. Econ. 30, No. 1, 51--66 (2002; Zbl 1074.91563)

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