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The stochastic resonance phenomenon implies “positive” changing of a system behaviour when noise is added to the system. The phenomenon has found numerous applications in physics, neuroscience, biology, medicine, mechanics and other fields. The present paper concerns this phenomenon for parametrically excited stochastic systems, i.e. systems that feature deterministic input signals that affect their parameters, e.g. stiffness, damping or mass properties. Parametrically excited systems are now widely used for signal sensing, filtering and amplification, particularly in micro- and nanoscale applications. And noise and uncertainty can be essential for systems at this scale. Thus, these systems potentially can exhibit stochastic resonance. In the present paper, we use a “deterministic” approach to describe the stochastic resonance phenomenon that implies replacing noise by deterministic high-frequency excitations. By means of the approach, we show that stochastic resonance can occur for parametrically excited systems and determine the corresponding resonance conditions.
Metastable transitions in Langevin dynamics can exhibit rich behaviours that are markedly different from its overdamped limit. In addition to local alterations of the transition path geometry, more fundamental global changes may exist. For instance, when the dissipation is weak, heteroclinic connections that exist in the overdamped limit do not necessarily have a counterpart in the Langevin system, potentially leading to different transition rates. Furthermore, when the friction coefficient depends on the velocity, the overdamped limit no longer exists, but it is still possible to efficiently find instantons. The approach, we employed for these discoveries, was based on (i) a simple rewriting of the Freidlin–Wentzell action in terms of time-reversed dynamics and (ii) an adaptation of the string method, which was originally designed for gradient systems, to this specific non-gradient system.
Results on the behaviour of a pendulum which is parametrically excited by large amplitude random loads at its pivot are presented, including a novel experimental case study. Thereby, it is dealt with a random excitation by a non-white Gaussian stochastic process with prescribed spectral density. Special focus is devoted to stochastic processes resulting from random sea wave elevation and the question whether random sea waves can lead to rotational motion of the parametrically excited pendulum. The motivation for such an experimental study is energy harvesting from ocean waves.
This study investigates the phenomenon of targeted energy transfer (TET) from a linear oscillator to a nonlinear attachment behaving as a nonlinear energy sink for both transient and stochastic excitations. First, the dynamics of the underlying Hamiltonian system under deterministic transient loading is studied. Assuming that the transient dynamics can be partitioned into slow and fast components, the governing equations of motion corresponding to the slow flow dynamics are derived and the behaviour of the system is analysed. Subsequently, the effect of noise on the slow flow dynamics of the system is investigated. The Itô stochastic differential equations for the noisy system are derived and the corresponding Fokker–Planck equations are numerically solved to gain insights into the behaviour of the system on TET. The effects of the system parameters as well as noise intensity on the optimal regime of TET are studied. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis.
An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.
In the paper we study the Zakai and Kushner–Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system. Moreover, we prove that under some general assumption, the Zakai equation has pathwise uniqueness and uniqueness in joint law, and the Kushner–Stratonovich equation is unique in joint law.
This paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.
We consider a general class of epidemic models obtained by applying the random time changes of Ethier and Kurtz (2005) to a collection of Poisson processes and we show the large deviation principle for such models. We generalise the approach followed by Dolgoarshinnykh (2009) in the case of the SIR epidemic model. Thanks to an additional assumption which is satisfied in many examples, we simplify the recent work of Kratz and Pardoux (2017).
The deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement of randomness and the coupling of the FSDEs and BSDEs. This paper will describe how to use the simplest Euler method in each DC step–leading to simple computational complexity–to achieve high order rate of convergence.
In this paper, we investigate the mean-square convergence of the split-step θ-scheme for nonlinear stochastic differential equations with jumps. Under some standard assumptions, we rigorously prove that the strong rate of convergence of the split-step θ-scheme in strong sense is one half. Some numerical experiments are carried out to assert our theoretical result.
By introducing a new Gaussian process and a new compensated Poisson random measure, we propose an explicit prediction-correction scheme for solving decoupled forward backward stochastic differential equations with jumps (FBSDEJs). For this scheme, we first theoretically obtain a general error estimate result, which implies that the scheme is stable. Then using this result, we rigorously prove that the accuracy of the explicit scheme can be of second order. Finally, we carry out some numerical experiments to verify our theoretical results.
Motivated by a problem in neural encoding, we introduce an adaptive (or real-time) parameter estimation algorithm driven by a counting process. Despite the long history of adaptive algorithms, this kind of algorithm is relatively new. We develop a finite-time averaging analysis which is nonstandard partly because of the point process setting and partly because we have sought to avoid requiring mixing conditions. This is significant since mixing conditions often place restrictive history-dependent requirements on algorithm convergence.
Upon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.
This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons in 1998. Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p, and the theory of integration of Lipγ one-forms (γ > p–1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and give sufficient conditions for existence and uniqueness of solution.
We study the asymptotic behavior in Sobolev norm of the local time of the d-dimensional fractional Brownian motion with N-parameters when the space variable tends to zero, both for the fixed time case and when simultaneously time tends to infinity and space variable to zero.
Convergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.
We extend the Kalman-Bucy filter to the case where both the system and observation processes are driven by finite dimensional Lévy processes, but whereas the process driving the system dynamics is square-integrable, that driving the observations is not; however it remains integrable. The main result is that the components of the observation noise that have infinite variance make no contribution to the filtering equations. The key technique used is approximation by processes having bounded jumps.
Chladni figures are formed when particles scattered across a plate move due to an external harmonic force resonating with one of the natural frequencies of the plate. Chladni figures are precisely the nodal set of the vibrational mode corresponding to the frequency resonating with the external force. We propose a plausible model for the movement of the particles that explains the formation of Chladni figures in terms of the stochastic stability of the equilibrium solutions of stochastic differential equations.
In this paper we study Nash equilibrium payoffs for nonzero-sum stochastic differential games with two reflecting barriers. We obtain an existence and a characterization of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations with two reflecting barriers.
A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.