Stochastic Taylor Expansion Research Articles

The synchronization, stability and stochastic resonance of stochastic global coupled system

In this paper, we aim at a global coupled system influenced by a multiplicative dichotomous noise and an additive Gaussian white noise. Since the analysis of the synchronization and stability is important to a coupled system, and there is little general method to analyze them. We provide methods to derive the synchronization and stability of particles in the system. Guaranteed by the synchronization and stability, we obtain the analytic expression of each particle’s average displacement. To verify our results, we use stochastic Taylor expansion to derive numerical iterative expressions of the displacements of particles. As for numerical simulations, we find that the numerical solutions match the analytical solutions, and we display the synchronization speed of particles, the stable region of noise intensity, resonance areas and the output amplitude gain with respect to various parameters of the system. We find that the phenomenon of stochastic resonance occurs if and only if the parameters are situated in the resonance area.

How to achieve carbon abatement in aviation with hybrid mechanism? A stochastic evolutionary game model

Aviation carbon emissions are growing as the volume of aviation traffic continues to increase, both exacerbating global greenhouse gas emissions and increasing the resistance to green aviation development. To effectively promote aviation carbon abatement, this paper, based on the hybrid mechanism composed of carbon trading and taxation, a tripartite Itô stochastic evolutionary game model is first constructed. The complex game interactions among the administration, major airlines, and minor airlines are analyzed. Second, we use Gaussian white noise as the ambient uncertainty and apply stochastic Taylor expansion to find the numerical approximation solution. Finally, through numerical simulations, the decision-making behavior of stakeholders and their sensitivity to key influencing factors are illustrated. The study shows that different variables have differential effects on stakeholders’ strategic choices in terms of convergence speed, change speed, and stability. Starting from three different regulatory paths, this study provides insights into the priority and direction of adjusting relevant variables, thereby offering guidance for policymakers and managers in effectively regulating aviation carbon abatement.

Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

Consider (X_{i}(t)) solving a system of N stochastic differential equations interacting through a random matrix {mathbf {J}} = (J_{ij}) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of (X_i(t)), initialized from some mu independent of {mathbf {J}}, are universal, i.e., only depend on the choice of the distribution mathbf {J} through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.

Numerical Analysis of a Class of Fractional Langevin Equations With the Block-by-Block Method

The fractional Langevin equation is of great scientific significance and engineering application value. Based on the classical block-by-block method, the numerical solution of a class of fractional Langevin equations with Caputo derivatives was obtained. Through introduction of the quadratic Lagrange basis function interpolation, the block-by-block convergent nonlinear equations were constructed, and the numerical solution of the Langevin equation was obtained by coupling in each block. Under the condition of 0<α<1, the stochastic Taylor expansion was used to prove that the block-by-block method is (3+α)-order convergent. Numerical experiments show that, the block-by-block method is stable and convergent under different values of α and time step <i>h</i>,and overcomes the existing methods’ disadvantages of slow speed and poor accuracy for solving fractional Langevin equations.

Bifurcation of a class of stochastic delay differential equations

In this paper, we study the bifurcation of a class of two-dimensional stochastic delay differential equations. Firstly, we translate the original system into an It? limiting diffusion system by applying stochastic Taylor expansion, small time delay expansion, polar coordinate transformation, and stochastic averaging procedure. Then we discuss the dynamical bifurcation by analyzing the qualitative changes of invariant measures, and investigate the phenomenological bifurcation by utilizing Fokker-Planck equation. The obtained conclusions are completely new, which generalize and improve some existing results.

An improved quantum projection filter

This work extends the previous quantum projection filtering scheme in Gao et al. (2019), by adding an optimality analysis result. A reformulation of the quantum projection filter is derived by minimizing the truncated Stratonovich stochastic Taylor expansion of the difference between the true quantum trajectory and its approximation on a lower-dimensional submanifold through quantum information geometric techniques. Simulation results for a qubit example demonstrate better approximation performance for the new quantum projection filter.

Design of a Quantum Projection Filter

This article develops a quantum projection filtering approach to approximating the quantum filter equation based on It $\hat{\rm o}$ stochastic Taylor expansions and quantum information geometric techniques. The proposed approximation scheme is designed so that the truncated Taylor expansion of the difference between the true quantum trajectory, and its approximation on a lower dimensional differential submanifold is minimized, through an orthogonal projection operation in the quantum Fisher metric. In addition, a convenient design for a special class of open quantum systems is formulated. Simulation results from a numerical example demonstrate the approximation capability of the proposed quantum projection filter.

Collective behaviors in globally coupled harmonic oscillators with fluctuating damping coefficient

In this paper, the collective behaviors in a globally coupled system consisting of harmonic oscillators with damping fluctuation modeled as dichotomous noise are investigated. First, the first and second moments of mean field and the fluctuation of single particle are introduced to describe the collective behaviors of the system. Then, based on the moment method and Shapiro–Loginov formula, the stability conditions for the coupled system and the analytical expressions for the output amplitude are obtained. Finally, according to the analytical expressions and numerical simulations based on the stochastic Taylor expansion, the collective behaviors are discussed, including synchronization and stochastic resonance. Stationary regime plays an important role in this coupled system. In the stationary regime, all the particles move synchronously and the output amplitudes are bounded which can be reliably estimated, whereas in the non-stationary regime, the first moment cannot be reliably estimated and the distribution of realizations is characterized by fat tail.

Stochastic and superharmonic stochastic resonances of a confined overdamped harmonic oscillator

The dynamics of many soft condensed matter and biological systems is affected by space limitations, which produce some peculiar effects on the systems' stochastic resonance (SR) behavior. In this study, we propose a model where SR can be observed: a confined overdamped harmonic oscillator that is subjected to a sinusoidal driving force and is under the influence of a multiplicative white noise. The output response of the system is a periodic signal with harmonic frequencies that are odd multiples of the driving frequency. We verify the amplitude resonances at the driving frequencies and superharmonic frequencies that are equal to three, five, and seven times the driving frequency, using a numerical method based on the stochastic Taylor expansion. The synergistic effect of the multiplicative white noise, constant boundaries, and periodic driving force that can induce a SR in the output amplitude at the driving and superharmonic frequencies is found. The SR phenomenon found in this paper is sensitive to the driving amplitude and frequency, inherent potential parameter, and boundary width, thus leading to various resonance conditions. Therefore, the mechanism found could be beneficial for the characterization of these confined systems and could constitute an important tool for controlling their basic properties.

Analyzing a stochastic process driven by Ornstein-Uhlenbeck noise.

A scalar Langevin-type process X(t) that is driven by Ornstein-Uhlenbeck noise η(t) is non-Markovian. However, the joint dynamics of X and η is described by a Markov process in two dimensions. But even though there exists a variety of techniques for the analysis of Markov processes, it is still a challenge to estimate the process parameters solely based on a given time series of X. Such a partially observed 2D process could, e.g., be analyzed in a Bayesian framework using Markov chain Monte Carlo methods. Alternatively, an embedding strategy can be applied, where first the joint dynamics of X and its temporal derivative X[over ̇] is analyzed. Subsequently, the results can be used to determine the process parameters of X and η. In this paper, we propose a more direct approach that is purely based on the moments of the increments of X, which can be estimated for different time-increments τ from a given time series. From a stochastic Taylor expansion of X, analytic expressions for these moments can be derived, which can be used to estimate the process parameters by a regression strategy.

Discrete-Time Statistical Inference for Multiscale Diffusions

We study statistical inference for small-noise-perturbed multiscale dynamical systems under the assumption that we observe a single time series from the slow process only. We construct estimators for both averaging and homogenization regimes, based on an appropriate misspecified model motivated by a second-order stochastic Taylor expansion of the slow process with respect to a function of the time-scale separation parameter. In the case of a fixed number of observations, we establish consistency, asymptotic normality, and asymptotic statistical efficiency of a minimum contrast estimator (MCE), the limiting variance having been identified explicitly; we furthermore establish consistency and asymptotic normality of a simplified MCE, which is, however, not, in general, efficient. These results are then extended to the case of high-frequency observations under a condition restricting the rate at which the number of observations may grow vis-a-vis the separation of scales. Numerical simulations illustrate the ...

Stochastic Taylor expansion of derivative-free method for stochastic differential equations

This paper demonstrates a derivation of stochastic Taylor methods for stochastic differential equations (SDEs). The stochastic Taylor series is extended and truncated at certain terms to achieve the order of convergence of stochatsic Taylor methods for SDEs. The systematic derivation of the expansion of stochastic Taylor series formula is presented. Numerical methods of Euler, Milstein scheme and stochastic Taylor methods of order 2.0 are proposed.

Numerical schemes for random ODEs with affine noise

Numerical schemes for random ordinary differential equations, abbreviated RODEs, with an affine structure can be derived in a similar way as for affine control systems using Taylor expansions that resemble stochastic Taylor expansions for Stratonovich stochastic differential equations. The driving noise processes can be quite general, such as Wiener processes or fractional Brownian motions with continuous sample paths or compound Poisson processes with piecewise constant sample paths, and even more general noises. Such affine-Taylor schemes of arbitrarily high order are constructed here. It is shown how their structure simplifies when the noise terms are additive or commutative. A derivative free counterpart is given and multi-step schemes are derived too. Numerical comparisons are provided for various explicit one-step and multi-step schemes in the context of a toggle switch model from systems biology.

On a Chen–Fliess approximation for diffusion functionals

We show that the so-called functional derivatives, as recently introduced by Dupire (Functional Ito calculus, SSRN, 2010), can provide intuitive meaning to classic expansions of path dependent functionals that appear in control theory (work of Brockett, Fliess, Sussmann et. al). We then focus on stochastic differential equations and show that vector fields can be lifted to act as derivations on such functionals. This allows to revisit and generalize the classic stochastic Taylor expansion to arrive at a Chen–Fliess approximation for smooth, path dependent functionals of SDEs with a corresponding $$L^{2}$$ -error estimate.

Stochastic Taylor Methods for Stochastic Delay Differential Equations

This paper demonstrates a systematic derivation of high order numerical methods from stochastic Taylor expansion for solving stochastic delay differential equations (SDDEs) with a constant time lag, r > 0 . The stochastic Taylor expansion of SDDEs is truncated at certain terms to achieve the order of convergence of numerical methods for SDDEs. Three different numerical schemes of Euler method, Milstein scheme and stochastic Taylor method of order 1.5 have been derived. The performance of Euler method, Milstein scheme and stochastic Taylor method of order 1.5 are investigated in a simulation study.

A Nearly Exact Reformulation of the Girsanov Linearization for Stochastically Driven Nonlinear Oscillators

The Girsanov linearization method (GLM), proposed earlier in Saha, N., and Roy, D., 2007, “The Girsanov Linearisation Method for Stochastically Driven Nonlinear Oscillators,” J. Appl. Mech.,74, pp. 885–897, is reformulated to arrive at a nearly exact, semianalytical, weak and explicit scheme for nonlinear mechanical oscillators under additive stochastic excitations. At the heart of the reformulated linearization is a temporally localized rejection sampling strategy that, combined with a resampling scheme, enables selecting from and appropriately modifying an ensemble of locally linearized trajectories while weakly applying the Girsanov correction (the Radon–Nikodym derivative) for the linearization errors. The semianalyticity is due to an explicit linearization of the nonlinear drift terms and it plays a crucial role in keeping the Radon–Nikodym derivative “nearly bounded” above by the inverse of the linearization time step (which means that only a subset of linearized trajectories with low, yet finite, probability exceeds this bound). Drift linearization is conveniently accomplished via the first few (lower order) terms in the associated stochastic (Ito) Taylor expansion to exclude (multiple) stochastic integrals from the numerical treatment. Similarly, the Radon–Nikodym derivative, which is a strictly positive, exponential (super-) martingale, is converted to a canonical form and evaluated over each time step without directly computing the stochastic integrals appearing in its argument. Through their numeric implementations for a few low-dimensional nonlinear oscillators, the proposed variants of the scheme, presently referred to as the Girsanov corrected linearization method (GCLM), are shown to exhibit remarkably higher numerical accuracy over a much larger range of the time step size than is possible with the local drift-linearization schemes on their own.

Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions

We study the Taylor expansion for the solution of a differential equation driven by a multi-dimensional Hölder path with exponent $H> 1/2$. We derive a convergence criterion that enables us to write the solution as an infinite sum of iterated integrals on a non empty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H > 1/2$. We also study the convergence in L2 of the stochastic Taylor expansion by using L2 estimates of iterated integrals and Borel-Cantelli type arguments.

Stochastic Taylor expansions and heat kernel asymptotics

These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern–Gauss–Bonnet theorem.

A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions

Nonlinear partial differential equation with random Neumann boundary conditions are considered. A stochastic Taylor expansion method is derived to simulate these stochastic systems numerically. As examples, a nonlinear parabolic equation (the real Ginzburg–Landau equation) and a nonlinear hyperbolic equation (the sine–Gordon equation) with random Neumann boundary conditions are solved numerically using a stochastic Taylor expansion method. The impact of boundary noise on the system evolution is also discussed.

Pathwise Taylor expansions for Itô random fields

In this paper we study the pathwise stochastic Taylor expansion, inthe sense of our previous work [3], for a class ofItô-type random fields in which the diffusion part is allowed to containboth the random field itself and its spatial derivatives.Random fields of such an 'self-exciting' type particularly contains thefully nonlinear stochastic PDEs of curvature driven diffusion, as well ascertain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notionof '$n$-fold' derivatives of a random field, as a fundamental device to copewith the special self-exciting nature. Unlike our previous work [3],our new expansion can be defined around any random time-space point (τ,ξ), wherethe temporal component τ does not even have to be a stopping time. Moreover, the exceptionalnull set is independent of the choice of the random point(τ,ξ). As an application, we show how this new form of pathwise Taylor expansioncould lead to a different treatment of the stochastic characteristics for a class of fullynonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hencelead to a definition of the stochastic viscosity solution for such SPDEs, which is newin the literature, and potentially of essential importance in stochastic control theory.