The case where $b$ and $a$ does not depend on $Y$ and $Z$

Several methods were considered.

Pure Monte Carlo Methods. 

Pure Monte-Carlo methods are based on a discrete time approximation of the forward-backward equation. Once discretized the forward process can be simulated. These simulations are then used to compute the conditional expectations involved in the backward discrete time approximation of $(Y,Z)$. Two different methods can be used to compute these conditional expectations.

a. The Longstaff-Schwartz (Carriere) approach consists in approximating the conditional expectations by regressions on a given basis of functions. This provides a very powerful, and, easy to implement, algorithm for which convergence has been shown to hold in the case of the American option pricing problem. However, no rate of convergence is given and the choice of the basis is a quite difficult problem. It has been used successfully in up to 20 factors models. See [C] , [LS], [CLP].

b. The Malliavin approach consists in re-writing the conditional expectations as the ratio of two unconditional expectations that can be estimated by standard Monte-Carlo methods. Upper bounds for the rate of convergence are proved. Contrary to the previous approach, this algorithm also provides good approximations for the greeks, i.e. the gradient of the associated value function (which is related to the $Z$, see above). So far, the existing algorithm has a too important complexity, which explains why it has only been tested in small dimensions (up to 5). Some numerical improvements have been proposed, reducing the complexity from $N^{2}$ to $N\ln (N)^{d}$, where $N$ is the number of simulated paths. This work is still in progress, the aim being to develop an algorithm such that the major part of the work is done before a reward function is specified, so as to reduce as much as possible the effective computation time once a particular payoff is defined. See [BET], [BT].

Grid approximations. 

The Quantization approach consists in approximating the original forward process by a discrete time process which evolves on some finite grid. The grid is constructed so has to provide the best $L^{p}$ approximation. It was first applied to the pricing of American options in dimension up to $10$. The construction of the optimal grid (and the computation of the associated transition probabilities) is very time consuming, but it can be done once for all. Once the grid, which is independent of the payoff function, is constructed, it provides a very quick algorithm for pricing American options on different payoffs, whenever they are written on the same assets. As in the Malliavin and Longstaff-Schwartz approach, the use of a good control variable is required. Extensions to non-linear filtering, optimal control and Asian-type options have also been studied. See [BP1], [BP2], [PP1], [PP2].

Dual formulation. 

This algorithm is based on a dual formulation for problem (1.1.1):

$$v(0,S_{0})=\inf E\left[ \sup_{t\le T}(e^{-rt}g(S_{t})-M_{t})\right]$$

where the $\inf$ is taken over a well suited set of martingales $M$. This algorithm consists in providing a upper bound for $v$ by choosing some martingale $M^{*}$ and computing

$$E\left[ \sup_{t\le T}(e^{-rt}g(S_{t})-M_{t}^{*})\right]$$

In cases where a good martingale $\hat M$ can be found, typically when

$$E[e^{-r(T-t)}g(S_{T})\vert S_{t}]\;=:\;\hat M_{t}$$

is known as a function of $S$, this provides a quite sharp upper bound for the price of the American option. Numerical experiments, up to dimension $15$, have been performed. See [R].

Cubature on Wiener spaces. 

Cubature formula on Wiener spaces have been developed in [VL] in order to construct probability measures with finite support which approximate the Wiener measure in the sense that the expectation of iterated Stratonovich integrals under the approximating measure and the Wiener measure are close. In a sense, this approach is similar to that of the quantization since it allows to reduce to a finite dimensional setting. So far, it has been used to develop high order numerical schemes for high dimensional SDE's and semi-elliptic PDE's.

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