Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths

Abstract

We consider Hamiltonian systems driven by multi-dimensional Gaussian processes in rough path sense, which include fractional Brownian motions with Hurst parameter $H\in (1/4,1/2]$. We indicate that the phase flow preserves the symplectic structure almost surely and this property could be inherited by symplectic Runge--Kutta methods, which are implicit methods in general. If the vector fields belong to $Li{p}^{\gamma }$, we obtain the solvability of Runge--Kutta methods and the pathwise convergence rates. For linear and skew symmetric vector fields, we focus on the midpoint scheme to give corresponding results. Numerical experiments verify our theoretical analysis.

References in corpus (1)

• Crank-Nicolson scheme for stochastic differential equations driven by fractional Brownian motions

Cited by in corpus (3)

• Strong convergence rate of Runge--Kutta methods and simplified step-$N$ Euler schemes for SDEs driven by fractional Brownian motions
• Semi-implicit Taylor schemes for stiff rough differential equations
• Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions