Abstract
In this paper, we propose a new weak second-order numerical scheme for solving stochastic differential equations with jumps. By using trapezoidal rule and the integration-by-parts formula of Malliavin calculus, we theoretically prove that the numerical scheme has second-order convergence rate. To demonstrate the effectiveness and the second-order convergence rate, three numerical experiments are given.
Highlights
Using a new proof method, i.e. the trapezoidal rule and integration-by-parts formula of Malliavin calculus theory, we rigorously prove that the new scheme has secondorder convergence rate
Which is the kth component n +1 of Xttnn+,X1. It follows from Itô–Taylor formula and trapezoidal rule that
We propose a new simplified weak second-order numerical scheme for solving stochastic differential equations with jumps
Summary
Hu and Gan [17] studied numerical stability of balanced methods for solving SDEwJs. Mil’shtein [18] studied the second-order accuracy integration of stochastic differential equations. The higher order of Runge–Kutta methods for jump-diffusion differential equations can be found in [20]. These numerical schemes include multiple stochastic integrals, which are difficult to accurately compute and simulate. Liu and Li [21] proposed the weak stochastic Taylor order 2.0 (WST2) scheme of SDEwJs with three-point distribution random variables, obtained the convergence rate of the Itô–Taylor scheme, i.e. using the Itô–Taylor expansion and product rule.
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