Preservation of probabilistic laws through Euler methods for Ornstein-Uhlenbeck process
- Schurz, Henri
2010 Mathematics Subject Classification
- 60H10 65C20 65L20 65U05
- Stochastic differential equations, Stationary Ornstein-Uhlenbeckprocess, Numerical methods, Family of implicit Euler methods, Preservation of asymptotic laws, Stability
There is a lack of appropriate replication of the asymptotical behaviour of stationary stochastic differential equations solved by numerical methods. The paper illustrates this fact with the stationary Ornstein-Uhlenbeck process and family of implicit Euler methods. For description of occuring bias, notions of asymptotical p-th mean, mean, mean square and equilibrium preservation are introduced, due to stochasticity of stationary law. Only the trapezoidal formula among these methods is optimal in the sense of replication of exact asymptotical behaviour. We also discuss the general probabilistic law of linear Euler methods. The results can be useful for implementation of stochastic-numerical algorithms (e.g. for linear-implicit methods) in several disciplines of Natural and Environmental Sciences.