We present the main results on the pricing and hedging of American derivatives by extending to continuous time the ideas introduced in the discrete-market setting in Section 2.5. Even in the simplest case of the Black-Scholes market model, the hedging and pricing problems for American options need very refined mathematical tools. In the complete-market setting, Bensoussan  and Karatzas ,  developed a probabilistic approach based upon the notion of Snell envelope in continuous time and upon the Doob-Meyer decomposition. The problem was also studied by Jaillet, Lamberton and Lapeyre  who employed variational techniques, and by Oksendal and Reikvam , Gatarek and Świech  in the framework of the theory of viscosity solutions. American options for models with jumps were studied among others by Zhang , Mulinacci , Pham , Levendorskii , Ekström , Ivanov , Lamberton and Mikou , Bayraktar and Xing .
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