Mean-Variance Hedging on uncertain time horizon in a market with a jump

In this work, we study the problem of mean-variance hedging with a random horizon T ^ tau , where T is a deterministic constant and is a jump time of the underlying asset price process. We rst formulate this problem as a stochastic control problem and relate it to a system of BSDEs with jumps. We then provide a veri cation theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from ltration enlargement theory.