Smoothness of the survival probability in models with a random environment: the annuity and mixed cases

Abstract

We consider the ruin problem for an insurance company investing its whole reserve in a risky asset whose parameters depend on a Markov random environment. Using the Green's function method we prove $C^2$-smoothness of the survival probability for annuity payments under minimal assumptions-the jump distribution need only be a probability measure on the positive half-line. For two-sided jumps smoothness holds whenever, in each regime, the jump distribution has no atoms on the negative half-line or the survival probability vanishes at the origin. This condition is sharp: an isolated atom combined with a positive value at the origin makes the second derivative discontinuous, and the size of the discontinuity is computed explicitly.