From rough-path estimates to multi-level monte carlo

Abstract. Discrete approximations to solutions of stochastic differential equations are well-known to converge with “strong ” rate 1/2. Such rates have played a key-role in Giles ’ multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the compu-tational effort necessary for the evaluation of diffusion functionals. In the present article similar results are established for large classes of rough differential equations driven by Gaussian processes (including fractional Brownian motion with H> 1/4 as special case). We consider implementable schemes for large classes of stochastic differential equations (SDEs) (1) dYt = V0 (Yt) dt+ d∑ i=1 Vi (Yt) dX i t (ω) driven by multidimensional Gaussian signals, say X = Xt (ω) ∈ Rd. The interpretation of these equations is in Lyons ’ rough path sense [LQ02, LCL07, FV10b]. This requires smoothness/boundedness conditions on the vector fields V0 and V ≡ (V1,..., Vd); for the sake of this introduction, the reader may assume bounded vector fields with bounded derivatives of all order (but we will be more specific later). This also requires a “natural ” lift of X (·, ω) to a (random) rough path X · (ω) = 1