Stochastic discrete Hamiltonian variational integrators

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.     

A class of orthogonal integrators for stochastic differential equations

The purpose of this paper is to construct a class of orthogonal integrators for stochastic differential equations (SDEs). The family of SDEs with orthogonal solutions is univocally characterized. For this, a class of orthogonal integrators is introduced by imposing constraints to Runge–Kutta (RK) matrices and weights of the standard stochastic RK schemes.The performance of the method is illustrated by means of numerical simulations.     

Why are there only three quantum noises?

In quantum stochastic calculus on the symmetric Fock space over L ²(ℝ₊), adapted processes of operators are integrated with respect to creation, annihilation and number processes. The main property which allows this integration is that the increments of integrators between s and t act only on Fock space over L ²([s, t]). In this article, we prove that there are no other process of closable operators on coherent vectors with this property. Thus the only possible integrators in quantum stochastic calculus are the creation, annihilation and number processes.     

Weak local linear discretizations for stochastic differential equations: Convergence and numerical schemes

Weak local linear (WLL) discretizations are playing an increasing role in the construction of effective numerical integrators and inference methods for stochastic differential equations (SDEs) with additive noise. However, due to limitations in the existing numerical implementations of WLL discretizations, the resulting integrators and inference methods have either been restricted to particular classes of autonomous SDEs or showed low computational efficiency. Another limitation is the absence of a systematic theoretical study of the rate of convergence of the WLL discretizations and numerical integratos. This task is the main purpose of the present paper. A second goal is introducing a new WLL scheme that overcomes the numerical limitations mentioned above. Additionally, a comparative analysis between the new WLL scheme and some conventional weak integrators is also presented.     

Pathwise accuracy and ergodicity of metropolized integrators for SDEs

Metropolized integrators for ergodic stochastic differential equations (SDEs) are proposed that (1) are ergodic with respect to the (known) equilibrium distribution of the SDEs and (2) approximate pathwise the solutions of the SDEs on finite‐time intervals. Both these properties are demonstrated in the paper, and precise strong error estimates are obtained. It is also shown that the Metropolized integrator retains these properties even in situations where the drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for SDEs typically become unstable and fail to be ergodic. © 2009 Wiley Periodicals, Inc.     

Compact derivations relative to semifinite von Neumann algebras

A non-commutative theory of stochastic integration is constructed in which the integrators are the components of the quantum Brownian motion with non-unit variance. Unlike the unit variance (Fock) case, there is a Kunita-Watanabe type representation theorem for processes which are martingales with respect to the generated filtration.     

Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations

We study a class of numerical methods for a system of second-order SDE driven by a linear fast force generating high frequency oscillatory solutions. The proposed schemes permit the use of large step sizes, have uniform global error bounds in the position (i.e. independent of the large frequencies present in the SDE) and offer various additional properties. This new family of numerical integrators for SDE can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problems.     

Conformal structure-preserving schemes for damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise

Since many physical phenomena are often influenced by dispersive medium, energy compensation and random perturbation, exploring the dynamic behaviors of the damped-driven stochastic system has becoming a hot topic in mathematical physics in recent years. In this paper, inspired by the stochastic conformal structure, we investigate the geometric numerical integrators for the damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise. To preserve the conformal structures of the system, by using symplectic Euler method, implicit midpoint method and Fourier pseudospectral method, we propose three kinds of stochastic conformal schemes satisfying corresponding discrete stochastic multiconformal-symplectic conservation laws and discrete global/local charge conservation laws. Numerical experiments illustrate the structure-preserving properties of the proposed schemes, as well as favorable results over traditional nonconformal schemes, which are consistent with our theoretical analysis.     

Laplace’s method for iterated complex Brownian integrals

A kind of Laplace’s method is developped for iterated stochastic integrals where integrators are complex standard Brownian motions. Then it is used to extend properties of Bougerol and Jeulin’s path transform in the random case when simple representations of complex semisimple Lie algebras are not supposed to be minuscule.     

SDELab: A package for solving stochastic differential equations in MATLAB

We introduce SDELab, a package for solving stochastic differential equations (SDEs) within MATLAB. SDELab features explicit and implicit integrators for a general class of Itô and Stratonovich SDEs, including Milstein's method, sophisticated algorithms for iterated stochastic integrals, and flexible plotting facilities.     

Noncommutative stochastic integration through decoupling

We present an abstract approach to noncommutative stochastic integration in the context of a finite von Neumann algebra equipped with a normal, faithful, tracial state, with respect to processes with tensor or freely independent increments satisfying a stationarity condition, using a decoupling technique. We obtain necessary and sufficient conditions for stochastic integrability of Lp-processes with respect to such integrators. We apply the theory to stochastic integration with respect to Boson and free Brownian motion.     

Skorohod integration and stochastic calculus beyond the fractional Brownian scale

We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative stochastic differential equations are solved; an analysis of existence for the stochastic heat equation is given.     

MULTISYMPLECTIC COMPOSITION INTEGRATORS OF HIGH ORDER

A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a general formula for composition method.     

Stopping a two parameter weak martingale

Summary This paper deals with the following problem, given a two parameter stochastic process, under what conditions is it possible to stop the process at any stopping line? It is shown that the class of stoppable processes is strictly larger than the class of two parameter integrators. Sufficient conditions for a weak martingale to be stoppable are derived and the stopped r.v. is represented as a one parameter optional dual projection.Work partially supported by a grant from the Research Authority at Bar-Ilan UniversityWork supported by the fund for promotion of research at the Technion     

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.     

New open modified Newton Cotes type formulae as multilayer symplectic integrators

New modified open Newton Cotes integrators are introduced in this paper. For the new proposed integrators the connection between these new algorithms, differential methods and symplectic integrators is studied. Much research has been done on one step symplectic integrators and several of them have obtained based on symplectic geometry. However, the research on multistep symplectic integrators is very poor. Zhu et al. [1] studied the well known open Newton Cotes differential methods and they presented them as multilayer symplectic integrators. Chiou and Wu [2] studied the development of multistep symplectic integrators based on the open Newton Cotes integration methods. In this paper we introduce a new open modified numerical method of Newton Cotes type and we present it as symplectic multilayer structure. The new obtained symplectic schemes are applied for the solution of Hamilton’s equations of motion which are linear in position and momentum. An important remark is that the Hamiltonian energy of the system remains almost constant as integration proceeds. We have applied also efficiently the new proposed method to a nonlinear orbital problem and an almost periodic orbital problem.     

Mean-square A-stable diagonally drift-implicit integrators of weak second order for stiff Itô stochastic differential equations

We introduce two drift-diagonally-implicit and derivative-free integrators for stiff systems of Itô stochastic differential equations with general non-commutative noise which have weak order 2 and deterministic order 2, 3, respectively. The methods are shown to be mean-square A-stable for the usual complex scalar linear test problem with multiplicative noise and improve significantly the stability properties of the drift-diagonally-implicit methods previously introduced (Debrabant and Rößler, Appl. Numer. Math. 59(3–4):595–607, 2009).     

Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge–Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behaviour of the new exponential integrators in comparison with some symmetric and symplectic Runge–Kutta methods in the literature.