Discretization of the Wiener-process in difference-methods for stochastic differential equations
The solution of the stochastic differential equation \(dx(t)=a(t, x(t))dt+b(t,x(t))dw(t)\), \(x(0)=x_ 0\), can be approximated by the Euler- method \(x_{n+1}=x_ n+a(t_ n,x_ n)\Delta t+b(t_ n,x_ n)\Delta w(t_ n)\), if the coefficients are uniformly Lipschitz-continuous. In numerical computations an additional approximation is necessary: the Wiener-process has to be replaced by a suitable simulation. In this paper the effect of this simulation is analysed and the error estimated in terms of the bounded-Lipschitz-distance for measures on \(R^ n\). For diffusion coefficients b, independent of x, the analysis is quite simple but problems arise if this is not the case due to multiplicative effects on the error for each time step. It would be of interest to estimate the distance between laws of the diffusion process and its computer-approximation on the space of continuous functions C[0,T] for a 'suitable' metric on the space of measures on C[0,T]. A 'suitable' metric has not yet been found but this problem could be a direction for further research.
scientific article; zbMATH DE number 3874352