Ruin problems with assets and liabilities of diffusion type

Ruin and related problems are studied for a risk business with compounding assets when the cash flow and the cumulative interest rate are diffusion processes with coefficients depending on the time and on the current cash balance. Differential equations are obtained for the probabilities of ruin at a given date, in finite time, and in infinite time. Some previously known explicit formulas related to Brownian motion come out as special cases. Relationships between crossing probabilities and transition probabilities are investigated and, in particular, existing results on the probability distribution of the running maximum of a Brownian motion and on the relationship between the probability of ruin and on the probability distribution of the discounted total payments are generalized. Proofs rest on a martingale technique.