Reduction of the integro-differential equation to an ordinary differential equation in ruin models with investment in a risky asset

Abstract

We study ruin models with the surplus invested in a risky asset and Erlang-distributed claims: non-life insurance, annuity payments and the mixed model. Assuming that the integro-differential equation for the survival probability has a classical twice continuously differentiable solution, we prove that this solution satisfies a linear ordinary differential equation. We construct explicit annihilating operators and write the coefficients of the reduced equation in closed form through binomial coefficients (for the mixed model with equal shape parameters). A converse criterion is obtained: a bounded solution of the reduced equation satisfies the integro-differential equation if and only if the residual vanishes at zero initial capital together with its derivatives of order below that of the annihilator. The exponent of the power branch at infinity is determined by the parameters of the risky asset alone and does not depend on the shape parameter. For the mixed model we classify the fundamental set of solutions: the exponentially growing modes are excluded by boundedness. The connection coefficient of the power branch is nonzero by a known power-law lower bound for the ruin probability, and in the annuity model by the reduced equation and the boundary condition at zero. Hence the ruin probability has power-law asymptotics with a positive constant for all shape parameters and all positive exponents, and a complete asymptotic expansion up to a multiplicative constant in all three models. For non-life insurance with Erlang claims of order two the expansion coefficients are computed by an explicit recurrence.