Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noises
SIAM Journal on Scientific Computing
Weak Approximations and Extrapolations of Stochastic Differential Equations with Jumps
SIAM Journal on Numerical Analysis
Stochastic differential algebraic equations of index 1 and applications in circuit simulation
Journal of Computational and Applied Mathematics
A Jump-Diffusion Model for Option Pricing
Management Science
Numerical methods for nonlinear stochastic differential equations with jumps
Numerische Mathematik
Multistep methods for SDEs and their application to problems with small noise
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems
Journal of Computational and Applied Mathematics
Strong approximations of stochastic differential equations with jumps
Journal of Computational and Applied Mathematics
Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation (Advances in Design and Control)
Adaptive Weak Approximation of Diffusions with Jumps
SIAM Journal on Numerical Analysis
Multilevel Monte Carlo Path Simulation
Operations Research
Numerical simulation of stochastic PDEs for excitable media
Journal of Computational and Applied Mathematics
Stochastic Runge-Kutta Methods for Itô SODEs with Small Noise
SIAM Journal on Scientific Computing
Almost sure convergence of numerical approximations for Piecewise Deterministic Markov Processes
Journal of Computational and Applied Mathematics
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In this paper we consider Runge-Kutta methods for jump-diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge-Kutta methods. First, we analyse schemes where the drift is approximated by a Runge-Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge-Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge-Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.