WIAS Preprint No. 112, (1994)

A note on pathwise approximation of stationary Ornstein-Uhlenbeck processes with diagonalizable drift


  • Schurz, Henri

2010 Mathematics Subject Classification

  • 65U05 60H10 58G32 34F05


  • Stochastic differential equations, additive noise, implicit Euler methods


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 with real-diagonalizable drift and the familiy of implicit Euler methods. For the description of the occuring bias the notions of asymptotical p-th mean, mean, variance and equilibrium preservation are introduced. The main result can be useful for the implementation of numerical algorithms requiring more precise long-term runs, such as in discrete parametric estimation or in numerical computation of top Lyapunov exponents.

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