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Unformatted text preview: Modified Equations for Stochastic Differential Equations * Tony Shardlow 1 † 1 School of Mathematics, Oxford Road, Manchester University M13 9PL, UK. [email protected] Abstract We describe a backward error analysis for stochastic differential equations with respect to weak convergence. Modified equations are provided for forward and backward Euler approximations to Itˆ o SDEs with additive noise, and extensions to other types of equation and approximation are discussed. 1 Introduction This paper considers the backward error analysis of stochastic differential equations (SDEs), a technique that has been of great success in interpreting numerical methods for ODEs. It is possible to fit an ODE (the so called modified equation) to a numerical method to very high order accuracy. Backward error analysis has been particularly valuable for Hamiltonian systems, where symplectic numerical methods can be approximated by a modified ODE arising from a perturbed Hamiltonian system, giving an approximate statistical mechanics for symplectic methods. See the monograph  for a review and further references. It is natural to ask whether such techniques extend to SDEs. I am unaware of any published work that has addressed this issue. We discuss modified equations for SDEs by perturbing the drift and diffusion functions by deterministic functions and looking for convergence in the weak sense of average with respect to smooth test functions. It is possible to determine a modified equation that approximates standard first order methods to second order accuracy for SDEs with additive noise. It is not possible to examine the case of SDEs with multiplicative noise, of convergence in the sense of mean square, nor is it possible to develop modified equations of higher order accuracy by working only with deterministic perturbations of the drift and diffusion coefficients. It remains to be seen whether a useful formulation of a modified equation can be introduced to describe numerical approximations of SDEs in greater generality. The paper is divided into three, each section presents the main ideas without developing any proofs. § 2 develops the modified equation for a one dimensional SDE, showing that the noise should be additive and the difficulty of dealing with higher order approximations. Modified equations are derived for the forward and backward Euler methods. In § 3, the extension to higher dimensions is discussed in relation to a Langevin equation. In § 4, we give conclusions and suggest a way of studying backward errors in the pathwise sense. 2 One dimension Consider Itˆ o SDEs on the real line (1) dX = f ( X ) dt + σ ( X ) dβ ( t ) , X (0) = Y, where f, σ : R → R are smooth functions and β ( t ) is a standard Brownian motion. Consider a numerical approximation X , X 1 , . . . parameterised by a time step Δ t that converges to the solution X ( t ) in the weak sense: for T > 0, | E φ ( X n )- E φ ( X ( n Δ t )) | = O (Δ t p ) , ≤ n Δ t ≤...
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- Fall '06