We will formulate quantile hedging problem of American type. It seems no relevant solution so far in the literature

We fix a filtered probability space , let be a Brownian motion on the probability space, be the stock price given by

Suppose the wealth of the company is given by

where is the initial capital, and is an admissible strategy. We can define the super-hedging price of an American option as

where is the price of an American option at time , is the set of stopping times in . Given , our objective is to choose an admissible control so as to maximize

So we can define the value function as

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