Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem

We consider in this paper that the reserve of an insurance company follows the classical model, in which the aggregate claim amount follows a compound Poisson process. Our goal is to minimize the ruin probability of the company assuming that the management can invest dynamically part of the reserve in an asset that has a positive fixed return. However, due to transaction costs, the sale price of the asset at the time when the company needs cash to cover claims is lower than the original price. This is a singular two-dimensional stochastic control problem which cannot be reduced to a one-dimensional problem. The associated Hamilton–Jacobi–Bellman (HJB) equation is a variational inequality involving a first order integro-differential operator and a gradient constraint. We characterize the optimal value function as the unique viscosity solution of the associated HJB equation. For exponential claim distributions, we show that the optimal value function is induced by a two-region stationary strategy (“action” and “inaction” regions) and we find an implicit formula for the free boundary between these two regions. We also study the optimal strategy for small and large initial capital and show some numerical examples.