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

arXiv: https://arxiv.org/abs/1805.01578 · HTML: https://mlsanity.com/paper/1805.01578/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 May 2018.

## Built on by (most cited first, 1 of 1)

- [Timing is Everything: Learning to Act Selectively with Costly Actions and Budgetary Constraints](https://mlsanity.com/paper/2205.15953/) (2022)

## Abstract (arXiv)

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.