A Lagrangian scheme for numerical evaluation of the noncausal stochastic integral

摘要

We are concerned with a noncausal approach to the numerical evaluation of the stochastic integral f dW t with respect to Brownian motion. Viewed as a special case of the numerical solution (in strong sense) of the SDE, it may be believed that the precision level of such an approximation scheme that uses only a finite number of increments kW = W(tk+1) - W(tk) of Brownian motion, would not exceed the order O 1 n where n is the number of steps for discretization. We present in this note a simple but not trivial example showing that this belief is not correct. The discussion is developed on the basis of the noncausal theory of stochastic calculus introduced by the author.