Convergence of the Euler scheme for stochastic functional partial differential equations

摘要

The study of stochastic partial different equations is motivated by the fact that when one wants to model some evolution phenomena arising in physics, biology and engineering etc. However, in many applications the resulting equations have so far proved intractable to direct analytical solution. Numerical approximations, such as the Euler scheme, are therefore a vital tool in exploring model behaviour. Unfortunately, current results concerning the convergence of such schemes impose conditions on the drift and diffusion coefficients of the stochastic partial differential equation, namely the linear growth and global Lipschitz conditions, which are often not met by systems of interest. In this paper, these conditions are relaxed and prove that numerical solutions based on the Euler scheme will converge to the true solution of a broad class of stochastic partial differential equations. Several examples are studied to illustrate the results.