Institut fiir Angewandt~ Analysis und Stochastik Balanced implicit methods for stiff stochastic systems: An introduction and numerical experiments (original) (raw)

Stiffly accurate Runge–Kutta methods for stiff stochastic differential equations

2001

In this paper we discuss implicit methods based on stiffly accurate Runge–Kutta methods and splitting techniques for solving Stratonovich stochastic differential equations (SDEs). Two splitting techniques: the balanced splitting technique and the deterministic splitting technique, are used in this paper. We construct a two-stage implicit Runge–Kutta method with strong order 1.0 which is corrected twice and no update is needed. The stability properties and numerical results show that this approach is suitable for solving stiff SDEs.

First-order weak balanced schemes for bilinear stochastic differential equations

This paper starts to develop balanced schemes for stochastic differential equations (SDEs) with multiplicative noise based on the addition of stabilizing functions to the drift terms. First, we use the linear scalar SDE as a test problem to show that it is possible to construct efficient almost sure stable first-order weak balanced schemes. Second, we design balanced schemes for bilinear SDEs that achieve the first order of weak convergence, and do not involve the simulation of multiple stochastic integrals. Numerical experiments show a promising performance of the new numerical methods.

Qualitative behavior of stiff ODEs through a stochastic approach

An International Journal of Optimization and Control: Theories & Applications (IJOCTA)

In the last few decades, stiff differential equations have attracted a great deal of interest from academic society, because much of the real life is covered by stiff behavior. In addition to importance of producing model equations, capturing an exact behavior of the problem by dealing with a solution method is also handling issue. Although there are many explicit and implicit numerical methods for solving them, those methods cannot be properly applied due to their computational time, computational error or effort spent for construction of a structure. Therefore, simulation techniques can be taken into account in capturing the stiff behavior. In this respect, this study aims at analyzing stiff processes through stochastic approaches. Thus, a Monte Carlo based algorithm has been presented for solving some stiff ordinary differential equations and system of stiff linear ordinary differential equations. The produced results have been qualitatively and quantitatively discussed.

The numerical solution of stochastic differential equations

The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1977

A method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge–Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.

Weak convergence of a numerical scheme for stochastic differential equations

Probability and Mathematical Statistics

WEAK CONVERGENCE OF A NUMERICAL SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONSIn this paper a numerical scheme approximating the solution to a stochastic differential equation is presented. On bounded subsets of time, this scheme has a finite state space, which allows us to decrease the round-off error when the algorithm is implemented. At the same time, the scheme introduced turns out locally consistent for any step size of time. Weak convergence of the scheme to the solution of the stochastic differential equation is shown.