Robust defect control with Runge-Kutta schemes
SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods
ACM Transactions on Mathematical Software (TOMS)
Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods
ACM Transactions on Mathematical Software (TOMS)
Control Strategies for the Iterative Solution of Nonlinear Equations in ODE Solvers
SIAM Journal on Scientific Computing
Variable step size control in the numerical solution of stochastic differential equations
SIAM Journal on Applied Mathematics
Balanced Implicit Methods for Stiff Stochastic Systems
SIAM Journal on Numerical Analysis
Step size control in the numerical solution of stochastic differential equations
Journal of Computational and Applied Mathematics
Physica D - Special issue originating from the 18th Annual International Conference of the Center for Nonlinear Studies, Los Alamos, NM, May 11&mdash ;15, 1998
A Variable Stepsize Implementation for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Digital filters in adaptive time-stepping
ACM Transactions on Mathematical Software (TOMS)
SDELab: A package for solving stochastic differential equations in MATLAB
Journal of Computational and Applied Mathematics
Stochastic simulation of chemical reactions in spatially complex media
Computers & Mathematics with Applications
The fully implicit stochastic-α method for stiff stochastic differential equations
Journal of Computational Physics
Stabilized explicit Runge-Kutta methods for multi-asset American options
Computers & Mathematics with Applications
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The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the "best" choice for an initial stepsize, as well as developing effective strategies for stepsize control-- the same, of course, must be carried out in the stochastic case.In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy.