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. AMS subject classifications

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We study a recursive procedure, based on the Euler scheme with decreasing step, for the computation of the invariant distribution of a Brownian diffusion process satisfying weak stability conditions. We are able to extend some of the results of [5] to diffusions for which the drift size at infinity is very small.

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