Hi,many problems in quantitative finance may be formulated as a stochastic control problem (Merton problem), or an optimal stopping problem (American or Bermudan options for example). In the simplest case, when your variables are brownians etc, you derive a HJB equation with that or that set of boundary conditions. You may also get a free boundary problem, with constraints on the function or on its gradient, which is linked to variational inequalities. Often numerical methods are the only way of solving the problem.In general case of n variables you have a n-dimensional pde. In d< 3 there are at least two complementary numerical methods of solving the pde: Monte Carlo, which follows from probabilistic interpretation of the solution of your PDE, and finite difference. When d<3, in order to test Monte Carlo one may compare the results of the two. In high dimensional case there is only one possibility, namely - Monte Carlo.

QuoteOriginally posted by: rezaas far as I've seen PDE/Trees are better adapted to American contracts, the American MC's exist but are so-so ...on the other hand, MC's are better adapted to path-dependent contracts, again path-dependent trees/ PDEs exist but are harder to implement ...People are not aware of the cautiousness needed in MC with path-dependency because it is too simple. Try to check in a 1-dim MC the convergence rate for a barrier option (up & out call for instance) or more simply E[max(t<T) W_t]. You'll be amazed of its bad performance. Then you realize that you will need really dirty probabilistic tricks to get back to good convergence.... no free lunch here !