Balanced Implicit Methods for Stiff Stochastic Systems
SIAM Journal on Numerical Analysis
Numerical solutions of stochastic differential equations — implementation and stability issues
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
Mean-Square and Asymptotic Stability of the Stochastic Theta Method
SIAM Journal on Numerical Analysis
A Jump-Diffusion Model for Option Pricing
Management Science
Numerical methods for nonlinear stochastic differential equations with jumps
Numerische Mathematik
Split-step backward balanced Milstein methods for stiff stochastic systems
Applied Numerical Mathematics
Asymptotic Stability of a Jump-Diffusion Equation and Its Numerical Approximation
SIAM Journal on Scientific Computing
Compensated stochastic theta methods for stochastic differential equations with jumps
Applied Numerical Mathematics
Convergence and stability of the balanced methods for stochastic differential equations with jumps
International Journal of Computer Mathematics
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Positive results are proved here about the ability of balanced methods to reproduce the asymptotic stability of the stochastic differential equation with jumps. Balanced methods including strong balanced methods and weak balanced methods, which possess implicitness in the diffusion term, have the potential to overcome some of the numerical instabilities that are often experienced when using the explicit methods. The paper shows that the asymptotic stability for stochastic jump-diffusion differential equations is inherited by the two kinds of balanced methods with sufficiently small stepsizes. Some numerical experiments included in the paper illustrate the theoretical results.