From Rough Path Estimates to Multilevel Monte Carlo

Abstract

Abstract. New classes of stochastic differential equations can now be studied using rough path theory (e.g. Lyons et al. [LCL07] or ). In this paper we investigate, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense. Our focus lies on numerical implementations, and more specifically on the saving possible via multilevel methods. Our analysis relies on a subtle combination of pathwise estimates, Gaussian concentration, and multilevel ideas. Numerical examples are given which both illustrate and confirm our findings.

Symplectic Runge–Kutta methods for Hamiltonian systems driven by Gaussian rough paths

“…Moreover, note that the estimate in Theorem 5.2 is valid for the case when ρ ∈ [1, 3/2), which is similar to the simplified step-2 Euler scheme in [6]. To construct a numerical scheme for ρ ∈ [1,2) and fill the gap between the convergence rates of two parts in (5.3), we use a higher stage symplectic Runge-Kutta method with local order τ ≥ 4 when applied to classical ordinary differential equations. If γ > max{2ρ, τ − 1}, then the estimate in (5.4) for the global error will hold for any 0 < η < τ 2ρ − 1.…”

“…Figure 2 shows the evolution of domains in the phase plane for one sample path. The initial domain is a square with four corners at (1, 1), (2, 1), (2, 2) and (1,2). Images at t = 0.4, 1.6, 8, are presented under the exact mapping and the three numerical methods.…”

“…Conservation of quadratic invariant. For the initial value (1, 1), the exact solution is on the circle with center at the origin and radius r = √ 2, which implies another invariant of this system (see also [1,Example 1]). We present the phase trajectory for three sample paths in Figure 3.…”

“…For the Lip γ case, we obtain that the convergence rate of the midpoint scheme is ( 3 2ρ −1−ε), for arbitrary small ε and ρ ∈ [1, 3/2). The convergence rate of another two symplectic Runge-Kutta methods, whose stages are higher, is ( 1 ρ − 1 2 − ε) for ρ ∈ [1,2). This is limited by the convergence rate of Wong-Zakai approximations, similar to the simplified step-3 Euler scheme in [6].…”

Central limit theorems for multilevel Monte Carlo methods

“…Similar reductions in the variance decay rate may occur if the SDE coefficients have low regularity or if its driving path has lower regularity than a Wiener process, cf. [6,17].…”