arXiv subjects

Andreas Rößler

Publications and source records attributed to Andreas Rößler.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

The multilevel Monte Carlo path simulation method introduced by Giles ({\it Operations Research}, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.

q-fin.CP↗

arXiv2025

A Class of Stochastic Runge-Kutta Methods for Stochastic Differential Equations Converging with Order 1 in $L^p$-Norm

For the approximation of solutions for Itô and Stratonovich stochastic differential equations (SDEs)a new class of efficient stochastic Runge-Kutta (SRK) methods is developed. As the main novelty only two stages are necessary for the proposed SRK methods of order 1 that can be applied to SDEs with non-commutative or with commutative noise. In addition, a variant of the SRK method for SDEs with additive noise is presented. All proposed SRK methods cover also the case of drift-implicit schemes and general order conditions for the coefficients are calculated explicitly. The new class of SRK methods is highly efficient in the sense that it features computational cost depending only linearly on the dimension of the SDE and on the dimension of the driving Wiener process. For all proposed SRK methods strong convergence with order 1 in $L^p$-norm for any $p \geq 2$ is proved. Moreover, sufficient conditions for approximated iterated stochastic integrals are established such that convergence with order 1 in $L^p$-norm is preserved if they are applied for the SRK method. The presented theoretical results are confirmed by numerical experiments.

math.NA↗

arXiv2024

An Exponential Stochastic Runge-Kutta Type Method of Order up to 1.5 for SPDEs of Nemytskii-type

For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge-Kutta type that allows for convergence with a temporal order of up to 3/2 and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

math.NA↗

arXiv2023

An Analysis of Approximation Algorithms for Iterated Stochastic Integrals and a Julia and MATLAB Simulation Toolbox

For the approximation and simulation of twofold iterated stochastic integrals and the corresponding Lévy areas w.r.t. a multi-dimensional Wiener process, we review four algorithms based on a Fourier series approach. Especially, the very efficient algorithm due to Wiktorsson and a newly proposed algorithm due to Mrongowius and Rössler are considered. To put recent advances into context, we analyse the four Fourier-based algorithms in a unified framework to highlight differences and similarities in their derivation. A comparison of theoretical properties is complemented by a numerical simulation that reveals the order of convergence for each algorithm. Further, concrete instructions for the choice of the optimal algorithm and parameters for the simulation of solutions for stochastic (partial) differential equations are given. Additionally, we provide advice for an efficient implementation of the considered algorithms and incorporated these insights into an open source toolbox that is freely available for both Julia and MATLAB programming languages. The performance of this toolbox is analysed by comparing it to some existing implementations, where we observe a significant speed-up.

math.NA↗

arXiv2021

On the Approximation and Simulation of Iterated Stochastic Integrals and the Corresponding Lévy Areas in Terms of a Multidimensional Brownian Motion

A new algorithm for the approximation and simulation of twofold iterated stochastic integrals together with the corresponding Lévy areas driven by a multidimensional Brownian motion is proposed. The algorithm is based on a truncated Fourier series approach. However, the approximation of the remainder terms differs from the approach considered by Wiktrosson (2001). As the main advantage, the presented algorithm makes use of a diagonal covariance matrix for the approximation of one part of the remainder term and has a higher accuracy due to an exact approximation of the other part of the remainder. This results in a significant reduction of the computational cost compared to, e.g., the algorithm introduced by Wiktorsson. Convergence in $L^p(Ω)$-norm with $p \geq 2$ for the approximations calculated with the new algorithm as well as for approximations calculated by the basic truncated Fourier series algorithm is proved and the efficiency of the new algorithm is analyzed.

math.PR↗

arXiv2021

A Note on Some Martingale Inequalities

We derive inequalities for time-discrete and time-continuous martingales that are similar to the well-known Burkholder inequalities. For the time-discrete case arbitrary martingales in $L^p(Ω)$ are treated, whereas in the time-continuous case martingales defined by Itô integrals w.r.t. a multi-dimensional Wiener process are considered. The estimates for the time-discrete martingales are related to the more general results by I. Pinelis (1994) and are proved to be sharp by a different and more elementary proof for this special setting. Further, for time-continuous martingales the presented inequalities are generalizations of similar estimates proved by M. Zakai (1967) and E. Rio (2009) to the general multi-dimensional case. Especially, these inequalities possess smaller constants compared to the ones that result if the original Burkholder inequalities would be applied for such estimates. Therefore, the presented inequalities are highly valuable in, e.g., stochastic analysis and stochastic numerics.

math.PR↗

arXiv2020

High order numerical integrators for single integrand Stratonovich SDEs

We show that applying any deterministic B-series method of order $p_d$ with a random step size to single integrand SDEs gives a numerical method converging in the mean-square and weak sense with order $\lfloor p_d/2\rfloor$.As an application, we derive high order energy-preserving methods for stochastic Poisson systems as well as further geometric numerical schemes for this wide class of Stratonovich SDEs.

math.NA↗

arXiv2020

A Derivative-Free Milstein Type Approximation Method for SPDEs covering the Non-Commutative Noise case

Higher order schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations that need not to fulfill a commutativity condition for the noise term and which can flexibly be combined with some approximation method for the involved iterated integrals. Recently, the authors introduced two algorithms to simulate such iterated stochastic integrals; these clear the way for the implementation of the proposed higher order scheme. We prove the mean-square convergence of the introduced derivative-free Milstein type scheme which attains the same order as the original Milstein scheme. The original scheme, however, is definitely outperformed when the computational cost is taken into account additionally, that is, in terms of the effective order of convergence. We derive the effective order of convergence for the derivative-free Milstein type scheme analytically in the case that one of the recently proposed algorithms for the approximation of the iterated stochastic integrals is applied. Compared to the exponential Euler scheme and the original Milstein scheme, the proposed derivative-free Milstein type scheme possesses at least the same and in most cases even a higher effective order of convergence depending on the particular SPDE under consideration. These analytical results are illustrated and confirmed with numerical simulations.

math.PR↗

arXiv2019

An Analysis of the Milstein Scheme for SPDEs without a Commutative Noise Condition

In order to approximate solutions of stochastic partial differential equations (SPDEs) that do not possess commutative noise, one has to simulate the involved iterated stochastic integrals. Recently, two approximation methods for iterated stochastic integrals in infinite dimensions were introduced in C. Leonhard and A. Rößler: Iterated stochastic integrals in infinite dimensions: approximation and error estimates, Stoch. Partial Differ. Equ. Anal. Comput., 7(2): 209-239 (2019). As a result of this, it is now possible to apply the Milstein scheme by Jentzen and Röckner: A Milstein scheme for SPDEs, Found. Comput. Math., 15(2): 313-362 (2015) to equations that need not fulfill the commutativity condition. We prove that the order of convergence of the Milstein scheme can be maintained when combined with one of the two approximation methods for iterated stochastic integrals. However, we also have to consider the computational cost and the corresponding effective order of convergence for a meaningful comparison with other schemes. An analysis of the computational cost shows that, in dependence on the equation, a combination of the Milstein scheme with both of the two methods may be the preferred choice. Further, the Milstein scheme is compared to the exponential Euler scheme and we show for different SPDEs depending on the parameters describing, e.g., the regularity of the equation, which one of the schemes achieves the highest effective order of convergence.

math.PR↗

arXiv2018

Enhancing the Order of the Milstein Scheme for Stochastic Partial Differential Equations with Commutative Noise

We consider a higher-order Milstein scheme for stochastic partial differential equations with trace class noise which fulfill a certain commutativity condition. A novel technique to generally improve the order of convergence of Taylor schemes for stochastic partial differential equations is introduced. The key tool is an efficient approximation of the Milstein term by particularly tailored nested derivative-free terms. For the resulting derivative-free Milstein scheme the computational cost is, in general, considerably reduced by some power. Further, a rigorous computational cost model is considered and the so called effective order of convergence is introduced which allows to directly compare various numerical schemes in terms of their efficiency. As the main result, we prove for a broad class of stochastic partial differential equations, including equations with operators that do not need to be pointwise multiplicative, that the effective order of convergence of the proposed derivative-free Milstein scheme is significantly higher than for the original Milstein scheme. In this case, the derivative-free Milstein scheme outperforms the Euler scheme as well as the original Milstein scheme due to the reduction of the computational cost. Finally, we present some numerical examples that confirm the theoretical results.

math.NA↗

arXiv2017

Iterated Stochastic Integrals in Infinite Dimensions - Approximation and Error Estimates

Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden, Platen, and Wright (1992) and by Wiktorsson (2001) for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional $Q$-Wiener process in case of a trace class operator $Q$ given the increments of the $Q$-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator $Q$. This difference arises as the stochastic process is of infinite dimension.

math.PR↗

arXiv2016

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 Rößler 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.

math.NA↗

arXiv2014

On the Acceleration of the Multi-Level Monte Carlo Method

The multi-level Monte Carlo method proposed by M. Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper, a modified multi-level Monte Carlo estimator is proposed with significantly reduced computational costs. As the main result, it is proved that the modified estimator reduces the computational costs asymptotically by a factor $(p/α)^2$ if weak approximation methods of orders $α$ and $p$ are applied in case of computational costs growing with same order as variances decay.

math.PR↗

arXiv2013

Stability Analysis and Classification of Runge-Kutta Methods for Index 1 Stochastic Differential-Algebraic Equations with Scalar Noise

The problem of solving stochastic differential-algebraic equations (SDAEs) of index one with a scalar driving Brownian motion is considered. Recently, the authors proposed a class of stiffly accurate stochastic Runge-Kutta (SRK) methods that do not involve any pseudo-inverses or projectors for the numerical solution of the problem. Based on this class of approximation methods, a classification for the coefficients of stiffly accurate SRK methods attaining strong order 0.5 as well as strong order 1.0 are calculated. Further, the mean-square stability for the considered class of SRK methods is analysed. As the main result, families of A-stable efficient order 0.5 and 1.0 stiffly accurate SRK methods with a minimal number of stages for SDEs as well as for SDAEs are presented.

math.NA↗

arXiv2013

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.

math.NA↗

arXiv2013

Stochastic Taylor Expansions for Functionals of Diffusion Processes

In the present paper, a stochastic Taylor expansion of some functional applied to the solution process of an Itô or Stratonovich stochastic differential equation with a multi-dimensional driving Wiener process is given. Therefore, the multi-colored rooted tree analysis is applied in order to obtain a transparent representation of the expansion which is similar to the B-series expansion for solutions of ordinary differential equations in the deterministic setting. Further, some estimates for the mean--square and the mean truncation errors are given.

math.PR↗

arXiv2013

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.

math.PR↗

arXiv2013

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 Itô stochastic differential equations with a multidimensional Wiener process was introduced by Rößler. 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 to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.

math.NA↗