Second order weak Runge–Kutta type methods for Itô equations

Mackevičius, Vigirdas; Navikas, Jurgis

doi:10.1016/s0378-4754(00)00284-6

Cited by 11 publications

(9 citation statements)

References 6 publications

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“…to the nonlinear SDE [16,21] (7.1) dX(t) = 1 2 X(t) + X(t) 2 + 1 dt + X(t) 2 + 1 dW (t), X(0) = 0, on the time interval I = [0, 1] with the solution X(t) = sinh(t + W (t)).…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Many stochastic schemes fall into the class of stochastic Runge-Kutta (SRK) methods. SRK methods have been studied both for strong approximation [1,10,11,16], where one is interested in obtaining good pathwise solutions, and for weak approximation [8,9,16,19,21,32], which focuses on the expectation of functionals of solutions. Order conditions for these methods are found by comparing series of the exact and the numerical solutions.…”

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confidence: 99%

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B–Series Analysis of Stochastic Runge–Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values

Debrabant¹,

Kværnø²

2009

SIAM J. Numer. Anal.

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In recent years, implicit stochastic Runge-Kutta (SRK) methods have been developed both for strong and weak approximations. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes such as simple iteration, modified Newton iteration or full Newton iteration. We employ a unifying approach for the construction of stochastic B-series which is valid both for Itô-and Stratonovich-stochastic differential equations (SDEs) and applicable both for weak and strong convergence to analyze the order of the iterated Runge-Kutta method. Moreover, the analytical techniques applied in this paper can be of use in many other similar contexts.

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“…to the nonlinear SDE [16,21] (7.1) dX(t) = 1 2 X(t) + X(t) 2 + 1 dt + X(t) 2 + 1 dW (t), X(0) = 0, on the time interval I = [0, 1] with the solution X(t) = sinh(t + W (t)).…”

Section: Numerical Examplesmentioning

confidence: 99%

mentioning

confidence: 99%

B–Series Analysis of Stochastic Runge–Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values

Debrabant¹,

Kværnø²

2009

SIAM J. Numer. Anal.

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“…Thus the RHS of Equation (24) is O(h 4 ). Similarly, inequality (20) for p = 3 may also be proved. Inequality (21) for p = 1 is also readily proved, given that E(…”

Section: Propositionmentioning

confidence: 84%

“…To prove inequality (20) for p = 2 (i.e. for the second moments of displacement increments), it is first noted (via Equation (19)) that:…”

Section: Propositionmentioning

confidence: 99%

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Roy

2006

Int. J. Numer. Meth. Engng

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SUMMARYEngineers are often more concerned with the computation of statistical moments (or mathematical expectations) of the response of stochastically driven dynamical systems than with the determination of path-wise response histories. With this in perspective, weak stochastic solutions of dynamical systems, modelled as n degrees-of-freedom (DOF) mechanical oscillators and driven by additive and/or multiplicative white noise (or, filtered white noise) processes, are considered in this study. While weak stochastic solutions are simpler and quicker to compute than strong (sample path-wise) solutions, it must be emphasized that sample realizations of weak solutions have no path-wise similarity with strong solutions. However, the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is 'close' to that of the true response (if it exists) within a certain order of a given time step size. Computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems (oscillators) of engineering interest. To efficiently generate such weak responses, a novel class of weak stochastic Newmark methods (WSNMs), based on implicit Ito-Taylor expansions of displacement and velocity fields, is proposed. The resulting multiple stochastic integrals (MSIs) in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities. In fact, yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher-level MSIs. Estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided. Numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy, simplicity and higher computational speed of WSNMs vis-à-vis a few other popularly used stochastic integration schemes as well as the path-wise versions of the stochastic Newmark scheme.

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“…Whereas strong approximation methods are designed to obtain good pathwise solutions [1], weak approximation focuses on the expectation of functionals of the solution. Second order stochastic Runge-Kutta (SRK) methods for the weak approximation of SDEs were proposed by Kloeden and Platen [4], Komori [5], Mackevicius and Navikas [6], Tocino and Vigo-Aguiar [13], and the authors [3,9]. However, these methods were not suitable for problems with high numbers m of Wiener processes, because for these methods the number of function evaluations per step increases quadratically in m. Recently, new classes of SRK methods were introduced by Rößler [10,?…”

Section: Introductionmentioning

confidence: 99%

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

Debrabant

Rößler

2009

Applied Numerical Mathematics

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Recently, a new class of second order Runge-Kutta methods for Itô stochastic differential equations with a multidimensional Wiener process was introduced by Rößler [10]. 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 [2] to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.

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Second order weak Runge–Kutta type methods for Itô equations

Cited by 11 publications

References 6 publications

B–Series Analysis of Stochastic Runge–Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values

B–Series Analysis of Stochastic Runge–Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

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

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