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Asymptotic solutions of diffusion models for risk reserves. (English) Zbl 1028.60062
Summary: We study a family of diffusion models for risk reserves which account for the investment income earned and for the inflation experienced on claim amounts. After we defined the process of the conditional probability of ruin over finite time and imposed the appropriate boundary conditions, classical results from the theory of diffusion processes turn the stochastic differential equation to a special class of initial and boundary value problems defined by a linear diffusion equation. Armed with asymptotic analysis and perturbation theory, we obtain the asymptotic solutions of the diffusion models (possibly degenerate) governing the conditional probability of ruin over a finite time in terms of interest rate.
MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
91B30 Risk theory, insurance (MSC2010)
35K20 Initial-boundary value problems for second-order parabolic equations
35B25 Singular perturbations in context of PDEs
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