We review several problems on the quantile hedging given by [FL99].

We assume that the discounted price process of stock is given by a semimartingale on a probability space . Under self-financing, the wealth process with inital capital and a predictable strategy follows

belongs to a space of adimissible strategy given by ,

Let be a given claim.

Problem **[SP]**. Given a fixed constant , maximize success probability, i.e. find

Problem **[SR]** Given a fixed constant , maximize success ratio, i.e. find

Problem **[CP]** Given a constant , minimize the cost to beat with probability , i.e. find

Problem **[CR]** Given a constant , minimize the cost to beat with ratio , i.e. find

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