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| We investigate an investment-consumption problem under the threat of a market crash, where the interest rate of the bond is stochastic. Inspired by the recent work of Desmettre et al. (2013), we model the market crash as an uncertain event $(\tau,l)$, where $\tau$ denotes the crash time and $l$ the crash size. While the stock price is driven by a linear SDE at normal times $t\tau$, it loses a fraction $l$ of its value at the crash time $\tau$. The investor wants to maximize the expected discounted utility of consumption over an infinite time horizon in the worst-case scenario. We solve the problem by applying the following three ideas. First, we determine the optimal post-crash strategy by solving a classical stochastic control problem. Then, we reformulate the worst-case problem into a `controller-vs-stopper' game in order to obtain the optimal pre-crash strategy. Finally, we apply a martingale approach and characterize the worst-case optimal investment-consumption strategy. |
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