  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.
