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Markets with asymmetrical information are generally studied from a wealth optimization point of view. We focus here on a hedging problem for a financial agent who has an additional information on the market. We extend the results given for hedging strategies with fixed terminal time to the case of a random terminal time. In particular, we provide tools to understand the behavior of American option hedging by an insider. To achieve this aim, we prove the existence and uniqueness of backward stochastic differential equations solutions, when terminal time is random, under an initially enlarged filtration. [ABSTRACT FROM AUTHOR]