## On a partial integrodifferential equation of Seal’s type.(English)Zbl 1318.91124

Summary: In this paper we generalize a partial integro-differential equation satisfied by the finite time ruin probability in the classical Poisson risk model. The generalization also includes the bivariate distribution function of the time of and the deficit at ruin. We solve the partial integro-differential equation by Laplace transforms with the help of Lagrange’s implicit function theorem. The assumption of mixed Erlang claim sizes is then shown to result in tractable computational formulas for the finite time ruin probability as well as the bivariate distribution function of the time of and the deficit at ruin. A more general partial integro-differential equation is then briefly considered.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 35Q91 PDEs in connection with game theory, economics, social and behavioral sciences 35R09 Integro-partial differential equations 45K05 Integro-partial differential equations
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### References:

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