Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations

Rößler, Andreas

doi:10.1137/060673308

Cited by 86 publications

(63 citation statements)

References 20 publications

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“…The system of equations (7), (8) was solved using the stochastic Runge-Kutta method RI1 approximating multidimensional Wiener process Itô equations with weak order (3,2) [18]. Naturally, in numerical simulation the given system was reduced to the Itô form using the technique described in Appendix A.…”

Section: Results Of Simulation Conclusionmentioning

confidence: 99%

Lévy Random Walks as Nonlinear Brownian Motion

Lubashevsky

Saitô

Uchimura

2013

Stochastic Systems Theory and its Applications (SSS)

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For describing irregular movement of animals in random search as well as patterns of human travel during daily activity an original model of Brownian nonlinear motion is proposed. The wandering particle is represented as a point in the extended phase space comprising its position, velocity, and, in addition, the acceleration. The acceleration is assumed to be governed by a certain random process with stochastic selfacceleration. The acceleration dynamics is described by a nonlinear stochastic differential equation of the Hänggi-Klimontovich type. Its regular component represents the preference of the particle moving with a certain fixed velocity. The stochastic component with the noise intensity growing with the acceleration is related to the active behavior of the wandering particle in changing the motion direction. The model is studied numerically. The obtained results allow us to state that the developed model generates motion trajectories that can be treated as Lévy random walks. The latter statement can be regarded as the main original point of the present work demonstrating a new approach to modeling the random searching based on continuous Markovian processes.

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Section: Results Of Simulation Conclusionmentioning

confidence: 99%

Lévy Random Walks as Nonlinear Brownian Motion

Lubashevsky

Saitô

Uchimura

2013

Stochastic Systems Theory and its Applications (SSS)

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“…Another method of strong order 1 and weak order 1 has been considered in [4]. Using the approximation (20) from [31], a multi-dimensional derivative free version, denoted S-ROCK(1,1), can be obtained straightforwardly by replacing the last line in (37) by…”

Section: Weak Order One S-rock Methods [4]mentioning

confidence: 99%

“…Next, we use the following approximation first proposed in [31] to construct efficient derivative free second order methods,…”

Section: Efficient Derivative Free Explicit Milstein-talay Methodsmentioning

confidence: 99%

“…The main feature of the algorithm is that it has an arbitrarily large mean-square stability region that grows quadratically with respect to the number of drift function evaluations. For an efficient implementation, the integrators are derivative free, and the number of diffusion function evaluations is independent of the number of Weiner processes involved, similarly to the methods introduced in [31] . As an additional feature, the proposed methods have a large asymptotic stability region close to the origin.…”

Section: Introductionmentioning

confidence: 99%

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Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

Abdulle¹,

Vilmart²,

Zygalakis³

2013

SIAM J. Sci. Comput.

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To cite this version:Assyr Abdulle, Gilles Vilmart, Konstantinos Zygalakis. Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput., Sociey for Industrial and Applied Mathematics, 2013, 35 (4) We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the stepsize reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta Chebyshev methods (ROCK2) for deterministic problems. The convergence, mean-square and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results.

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“…On the other hand, we consider the following Runge-Kutta type scheme of weak order two, which is a derivative free version of the Milstein-Talay method [32] derived in [3, Lemma 3.1] using an idea in [28] to make the number of evaluations of each diffusion function g r , r = 1, . .…”

Section: A Test Problem With Non-commutative Noisementioning

confidence: 99%

Weak Second Order Multirevolution Composition Methods for Highly Oscillatory Stochastic Differential Equations with Additive or Multiplicative Noise

Vilmart¹

2014

SIAM J. Sci. Comput.

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To cite this version:Gilles Vilmart. Weak second order multi-revolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise. SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2014, 36 (4) AbstractWe introduce a class of numerical methods for highly oscillatory systems of stochastic differential equations with general noncommutative noise. We prove global weak error bounds of order two uniformly with respect to the stiffness of the oscillations, which permits to use large time steps. The approach is based on the micro-macro framework of multi-revolution composition methods recently introduced for deterministic problems and inherits its geometric features, in particular to design integrators preserving exactly quadratic first integral. Numerical experiments, including the stochastic nonlinear Schrödinger equation with space-time multiplicative noise, illustrate the performance and versatility of the approach.

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Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations

Cited by 86 publications

References 20 publications

Lévy Random Walks as Nonlinear Brownian Motion

Lévy Random Walks as Nonlinear Brownian Motion

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

Weak Second Order Multirevolution Composition Methods for Highly Oscillatory Stochastic Differential Equations with Additive or Multiplicative Noise

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