On quasi-Monte Carlo simulation of stochastic differential equations

Abstract:

In a number of problems of mathematical physics and other fields stochastic differential equations are used to model certain phenomena. Often the solution of those problems can be obtained as a functional of the solution of some specific stochastic differential equation. Then we may use the idea of weak approximation to carry out numerical simulation. We analyze some complexity issues for a class of linear stochastic differential equations (Langevin type), which can be given by \[ dX_{t}=-\alpha X_{t}dt+\beta (t)dW_{t}, \quad X_{0}:= 0, \] where $\alpha >0$ and $\beta : [0,T]\to \mathbb {R}$. It turns out that for a class of input data which are not more than Lipschitz continuous the explicit Euler scheme gives rise to an optimal (by order) numerical method. Then we study numerical phenomena which occur when switching from (real) Monte Carlo simulation to quasi–Monte Carlo simulation, which is the case when we carry out the simulation on computers. It will easily be seen that completely uniformly distributed sequences yield good substitutes for random variates, while not all uniformly distributed (mod 1) sequences are suited. In fact we provide necessary conditions on a sequence in order to serve quasi–Monte Carlo purposes. This condition is expressed in terms of the measure of well-distributions. Numerical examples complement the theoretical analysis.