DeepAI AI Chat
Log In Sign Up

Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems based on the Wong--Zakai approximation

by   Chuchu Chen, et al.
Chinese Academy of Science

We investigate the stochastic modified equation which plays an important role in the stochastic backward error analysis for explaining the mathematical mechanism of a numerical method. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation, which is a perturbation of the Wong--Zakai approximation of the rough differential equation. For a symplectic method applied to a rough Hamiltonian system, the associated stochastic modified equation is proved to have a Hamiltonian formulation. Second, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in the rough path theory. Third, if increments of noises are simulated by truncated random variables, we show that the one-step error can be made exponentially small with respect to the time step size. Numerical experiments verify our theoretical results.


page 1

page 2

page 3

page 4


Backward error analysis for conjugate symplectic methods

The numerical solution of an ordinary differential equation can be inter...

Runge-Kutta methods for rough differential equations

We study Runge-Kutta methods for rough differential equations which can ...

On the discrete-time simulation of the rough Heston model

We study Euler-type discrete-time schemes for the rough Heston model, wh...

A stochastic Hamiltonian formulation applied to dissipative particle dynamics

In this paper, a stochastic Hamiltonian formulation (SHF) is proposed an...

Revisiting the Effects of Stochasticity for Hamiltonian Samplers

We revisit the theoretical properties of Hamiltonian stochastic differen...

Approximation of Stochastic Volterra Equations with kernels of completely monotone type

In this work, we develop a multi-factor approximation for Stochastic Vol...

Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states

We consider numerical schemes for computing the linear response of stead...