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This article focuses on the mathematical problem of existence and uniquenessof BSDE with a random terminal time which is a general random variable but nota stopping time, as it has been usually the case in the previous literature ofBSDE with random terminal time. The main motivation of this work is a financialor actuarial problem of hedging of defaultable contingent claims or lifeinsurance contracts, for which the terminal time is a default time or a deathtime, which are not stopping times. We have to use progressive enlargement ofthe Brownian filtration, and to solve the obtained BSDE under this enlargedfiltration. This work gives a solution to the mathematical problem and provesthe existence and uniqueness of solutions of such BSDE under certain generalconditions. This approach is applied to the financial problem of hedging ofdefaultable contingent claims, and an expression of the hedging strategy isgiven for a defaultable contingent claim or a life insurance contract.

Hedging of Defaultable Contingent Claims using BSDE with uncertain time horizon