This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called BrennanSchwartz algorithm for the valuation of American put options. (1950). 90A09, 60G40, 60J60, 65K10, 65M 10.
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