The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations.

*(English)*Zbl 0851.60054Summary: We consider the numerical approximation to strong solutions of stochastic differential equations (SDE’s) using a fixed time step and given only the increments of the Brownian path over each time step. Using the approach generalised by Ben Arous, Castell and Hu, of approximating the solution to an SDE over small time by the solution to a time inhomogeneous ordinary differential equation (ODE), we obtain ODE’s which, as the number of time steps increases, yield an asymptotically efficient sequence of approximations to the solution of an SDE, where the concept of asymptotic efficiency is that of Clark and Newton. We distinguish between the two cases of an SDE driven by a one-dimensional Brownian path or satisfying the commutativity condition on the one hand and an SDE driven by a multidimensional Brownian path and with a noncommutative Lie algebra on the other hand. When the ODE’s presented are solved numerically, the property of asymptotic efficiency is presented as long as the solution is accurate enough. The methods of this paper represent an alternative and easily generalisable way of looking at the approximation of strong solutions to SDE’s.

##### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

65F05 | Direct numerical methods for linear systems and matrix inversion |