Optimal Capital Injections with the Risk of Ruin: A Stochastic Differential Game of Impulse Control and Stopping Approach

We consider an investment problem in which an investor performs capital injections to increase the liquidity of a firm for it to maximise profit from market operations. Each time the investor performs an injection, the investor incurs a fixed transaction cost. In addition to maximising their terminal reward, the investor seeks to minimise the risk of loss of their investment (from a possible firm ruin) by exiting the market at some point in time. We show that the problem can be reformulated in terms of a new stochastic differential game of control and stopping in which one of the players modifies a (jump-)diffusion process using impulse controls and an adversary chooses a stopping time to end the game. We show that the value of this game can be computed by solving a double obstacle problem described by a quasi-variational inequality. We then characterise the value of the game via a set of HJBI equations, considering both games with zero-sum and non-zero-sum payoff structures. Our last result demonstrates that the solution to the investment problem is recoverable from the Nash equilibrium strategies of the game.