Abstract
In this paper, our aim is to develop a compensated split-step θ (CSSθ) method for nonlinear jump-diffusion systems. First, we prove the convergence of the proposed method under a one-sided Lipschitz condition on the drift coefficient, and global Lipschitz condition on the diffusion and jump coefficients. Then we further show that the optimal strong convergence rate of CSSθ can be recovered, if the drift coefficient satisfies a polynomial growth condition. At last, a nonlinear test equation is simulated to verify the results obtained from theory. The results show that the CSSθ method is efficient for simulating the nonlinear jump-diffusion systems.
Highlights
The aim of this paper is to study the strong convergence of the compensated split-step θ (CSSθ) method for the following nonlinear jump-diffusion systems: dX(t) = f X t– dt + g X t– dW (t) + h X t– dN(t) ( . )for t >, with X( –) = X ∈ Rn, where X(t–) denotes lims→t– X(s), f : Rn → Rn, g : Rn → Rn×m, h : Rn → Rn, W (t) is an m-dimensional Wiener process, and N(t) is a scalar Poisson process with intensity λ.Most of the studies concerned with numerical analysis for stochastic differential equations with jumps (SDEwJs) are based on the assumption of globally Lipschitz continuous coefficients, for example, [ – ]
1 Introduction The aim of this paper is to study the strong convergence of the CSSθ method for the following nonlinear jump-diffusion systems: dX(t) = f X t– dt + g X t– dW (t) + h X t– dN(t) for t >, with X( –) = X ∈ Rn, where X(t–) denotes lims→t– X(s), f : Rn → Rn, g : Rn → Rn×m, h : Rn → Rn, W (t) is an m-dimensional Wiener process, and N(t) is a scalar Poisson process with intensity λ
To the best of our knowledge, there is no result about the strong convergence of the CSSθ method for SDEwJs with non-globally Lipschitz continuous coefficients
Summary
The development of numerical methods for nonlinear jump-diffusion systems with non-globally Lipschitz continuous coefficients is not as fast as nonlinear SDEs. There To the best of our knowledge, there is no result about the strong convergence of the CSSθ method for SDEwJs with non-globally Lipschitz continuous coefficients. ) are C , there exist constants K , Lg and Lh > , such that the drift coefficient f satisfies a one-sided Lipschitz condition, x – y, f (x) – f (y) ≤ K|x – y| , ∀x, y ∈ Rn, H(x) – h(y) ≤ Lh|x – y| , ∀x, y ∈ Rn. We assume that all moments of the initial solution are bounded, that is, for any p ∈ [ , +∞) there exists a positive constant C, such that
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