STOCHASTIC DIFFERENTIAL-EQUATIONS FOR RUIN PROBABILITIES

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite ti me. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump o r between the jumps. The key point is to define a process of condition al ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes toge ther with the change-of-variable formula or the martingale representat ion theorem for point processes, we obtain differential equations for evaluating the probability of ruin. Numerical illustrations are given by solving a partial differential equation numerically to obtain the p robability of ruin over a finite time horizon.