Andreas Rößler

Runge-Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations

Some new stochastic Runge-Kutta (SRK) methods for the strong approximation of solutions of stochastic differential equations (SDEs) with improved efficiency are introduced. Their convergence is proved by applying multicolored rooted tree analysis. Order conditions for the coefficients of explicit and implicit SRK methods are calculated. As the main novelty, order 1.0 strong SRK methods with significantly reduced computational complexity for Ito as well as for Stratonovich SDEs with a multidimensional driving Wiener process are presented where the number of stages is independent of the dimension of the Wiener process. Further, an order 1.0 strong SRK method customized for Ito SDEs with commutative noise is introduced. Finally, some order 1.5 strong SRK methods for SDEs with scalar, diagonal, and additive noise are proposed. All introduced SRK methods feature significantly reduced computational complexity compared to well-known schemes.

Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order.Abstract A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order.

Second Order Runge-Kutta Methods for Itô Stochastic Differential Equations

A new class of stochastic Runge-Kutta methods for the weak approximation of the solution of Ito stochastic differential equation systems with a multidimensional Wiener process is introduced. As the main innovation, the number of stages of the methods does not depend on the dimension of the driving Wiener process, and the number of necessary random variables which have to be simulated is reduced considerably. Compared to well-known schemes, this reduces the computational effort significantly. Order conditions for the stochastic Runge-Kutta methods assuring weak convergence with order two are calculated by applying the colored rooted tree analysis due to the author. Further, some coefficients for explicit second order stochastic Runge-Kutta schemes are presented.

Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.Abstract Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.

Diagonally drift-implicit Runge--Kutta methods of weak order one and two for Itô SDEs and stability analysis

The class of stochastic Runge-Kutta methods for stochastic differential equations due to Roszler is considered. Coefficient families of diagonally drift-implicit stochastic Runge-Kutta (DDISRK) methods of weak order one and two are calculated. Their asymptotic stability as well as mean-square stability (MS-stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.

Adaptive schemes for the numerical solution of SDEs: a comparison

The efficient numerical solution of stochastic differential equations is important for applications in many fields. Adaptive schemes, well developed in the deterministic setting, may be one possible way to reduce computational cost. We review the two main step size control algorithms that have been proposed in recent years for stochastic differential systems and compare their efficiency in a simulation study.

Method of lines for stochastic boundary-value problems with additive noise

Abstract We propose a new numerical method for solving stochastic boundary-value problems. The method uses the deterministic method of lines to treat the time, space and randomness separately. The emphasis in the present study is given to stochastic partial differential equations with forced additive noise. The spatial discretization is carried out using a second-order finite volume method, while the associated stochastic differential system is numerically solved using a class of stochastic Runge–Kutta methods. The performance of the proposed methods is tested for a stochastic advection–diffusion problem and a stochastic Burgers equation driven with white noise. Numerical results are presented in both one and two space dimensions.

Families of efficient second order Runge--Kutta methods for the weak approximation of Itô stochastic differential equations

Recently, a new class of second order Runge-Kutta methods for Ito stochastic differential equations with a multidimensional Wiener process was introduced by Roszler [A. Roszler, Second order Runge-Kutta methods for Ito stochastic differential equations, Preprint No. 2479, TU Darmstadt, 2006]. In contrast to second order methods earlier proposed by other authors, this class has the advantage that the number of function evaluations depends only linearly on the number of Wiener processes and not quadratically. In this paper, we give a full classification of the coefficients of all explicit methods with minimal stage number. Based on this classification, we calculate the coefficients of an extension with minimized error constant of the well-known RK32 method [J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, West Sussex, 2003] to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.

Stochastic Runge-Kutta Methods for Itô SODEs with Small Noise

We consider stochastic Runge-Kutta methods for Ito stochastic ordinary differential equations, and study their mean-square convergence properties for problems with small multiplicative noise or additive noise. First we present schemes where the drift part is approximated by well-known methods for deterministic ordinary differential equations, and a Maruyama term is used to discretize the diffusion. Further, we suggest improving the discretization of the diffusion part by taking into account also mixed classical-stochastic integrals, and we present a suitable class of fully derivative-free methods. We show that the relation of the applied step-sizes to the smallness of the noise is essential to decide whether the new methods are worth the effort. Simulation results illustrate the theoretical findings.

Trees and asymptotic expansions for fractional stochastic differential equations

In this article, we consider an n-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameterH >1/3. We derive an expansion for E[f (Xt )] in terms of t, where X denotes the solution to the SDE and f :Rn →R is a regular function. Comparing to F. Baudoin and L. Coutin, Stochastic Process. Appl. 117 (2007) 550-574, where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift, we parametrize our expansion with trees, which makes it easier to use, and we obtain a sharp estimate of the remainder for the case H >1/2.