A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion

Abstract

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Lévy area terms.

Approximation of stationary solutions to SDEs driven by multiplicative fractional noise

“…When H > 1/2 this discretization procedure is based on an Euler type scheme, but the case H < 1/2 involves the introduction of someLévy area correction terms of Milstein type (see [8]) or products of increments of B H if one desires to deal with an implementable numerical scheme (cf. [9]). This new setting has tremendous effects on the proof of Propositions 4 and 5.…”

Numerical Schemes for Rough Parabolic Equations

“…Thanks to the bounds exhibited in [9] (and recalled in Proposition 2.1), we can immediately apply Theorems 1.1 and 1.2 to the fBm situation so as to retrieve almost sure convergence results. Let us focus for instance on the rough case:…”

“…there exists a function C = C β,η : (R + ) 2 → R + bounded on bounded sets such that if y is the mild solution of (9) in B γ ′ with initial condition ψ and y M,N is the path generated by the Milstein scheme (13), one has…”

“…By keeping track of the constants exhibited at each step of the reasoning, one soon realizes that the almost sure estimate (8) cannot be turned into an L 1 estimate, i.e., the random variable C ψ,B is not integrable (see for instance the intermediate bound (62) for the path Y M,N ). Note that such average estimates remain an open problem as soon as the Hurst index is strictly smaller than 1/2, even for rough standard systems (see [9]). Remark 1.8.…”

“…To be more specific, the following bound has been proved in [9], and it allows us to derive Corollary 1.3 from Theorem 1.2.…”