Stochastic stabilization of first-passage failure of Rayleigh oscillator under Gaussian White-Noise parametric excitations

Stochastic stabilization of first-passage failure of Rayleigh oscillator under Gaussian White-Noise parametric excitation is studied. The equation of motion of the system is first reduced to an averaged Itô stochastic differential equation by using the stochastic averaging method. Then, a backward Kolmogorov equation governing the conditional reliability function of first-passage failure is established. The conditional reliability function, and the conditional probability density are obtained by solving the backward Kolmogorov equation with boundary conditions. Finally, the cost function and optimal control forces are determined by the requirements of stabilizing the system by evaluating the maximal Lyapunov exponent. The numerical results show that the procedure is effective and efficiency.     

Stochastic optimal control of first-passage failure for coupled Duffing–van der Pol system under Gaussian white noise excitations

In this paper, the stochastic averaging method of quasi-non-integrable-Hamiltonian systems is applied to Duffing–van der Pol system to obtain partially averaged Ito stochastic differential equations. On the basis of the stochastic dynamical programming principle and the partially averaged Ito equation, dynamical programming equations for the reliability function and the mean first-passage time of controlled system are established. Then a non-linear stochastic optimal control strategy for coupled Duffing–van der Pol system subject to Gaussian white noise excitation is taken for investigating feedback minimization of first-passage failure. By averaging the terms involving control forces and replacing control forces by the optimal ones, the fully averaged Ito equation is derived. Thus, the feedback minimization for first-passage failure of controlled system can be obtained by solving the final dynamical programming equations. Numerical results for first-passage reliability function and mean first-passage time of the controlled and uncontrolled systems are compared in illustrative figures to show effectiveness and efficiency of the proposed method.     

Response of Duffing system with delayed feedback control under combined harmonic and real noise excitations

This paper presents a procedure for predicting the response of Duffing system with delayed feedback bang–bang control under combined harmonic and real noise excitations by using the stochastic averaging method. First, the time-delayed feedback bang–bang control force is expressed approximately in terms of the system state variables without time delay. Then the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. Finally, the response of the system is obtained by solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equations. It is shown that the time delay in feedback control can deteriorate the control effectiveness and cause bifurcation of stochastic jump of Duffing system. The validity of the proposed method is confirmed by digital simulation.     

Research on nonlinear stochastic dynamical price model

In consideration of many uncertain factors existing in economic system, nonlinear stochastic dynamical price model which is subjected to Gaussian white noise excitation is proposed based on deterministic model. One-dimensional averaged Itô stochastic differential equation for the model is derived by using the stochastic averaging method, and applied to investigate the stability of the trivial solution and the first-passage failure of the stochastic price model. The stochastic price model and the methods presented in this paper are verified by numerical studies.     

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.     

A “direct” method to prove the generalized Itô–Venttsel’ formula for a generalized stochastic differential equation

For the first time we present a complete proof (from the standpoint of stochastic analysis) of the generalized Itô–Venttsel’ formula whose ideas were adduced in [8]. The proposed proof is an approach to construct the generalized Itô–Venttsel’ formula based on the direct application of the generalized Itô formula and the theory of stochastic approximation in contrast to the proof presented in [17] and based on the method of the integral invariants of a stochastic differential equation.     

New exact solutions to the Fokker–Planck–Kolmogorov equation

A generalized Liouville theorem has been proven for Itô systems. This allows us to show that the conserved quantities of the deterministic part of the Itô systems lead to the solution of the Fokker–Planck–Kolmogorov equation. The results have been applied to a stochastic 3-species Lotka Volterra system and the semi-classical Jaynes–Cummings system.     

On the Itô equation

The Itô equation is shown to be a singular case of a physical system representable in terms of general stochastic processes.     

Optimal control over nonlinear stochastic systems

The synthesis of optimal control over nonlinear stochastic systems that are described by the Itô equations is reduced to the solution of recurrence relations derived from the Bellman stochastic equation.     

Stochastic model for colored noise

The input-output point of view for the Itô equation to avoid stochastic integration is related to the stochastic-operator approach for a physically reasonable robust mathematical model.     

Kinetic energy of the Langevin particle

We compute the kinetic energy of the Langevin particle using different approaches. We build stochastic differential equations that describe this physical quantity based on both the Itô and Stratonovich stochastic integrals. It is shown that the Itô equation possesses a unique solution, whereas the Stratonovich one possesses infinitely many, all but one absent of physical meaning. We discuss how this fact matches with the existent discussion on the Itô vs Stratonovich dilemma and the apparent preference toward the Stratonovich interpretation in the physical literature.     

Nonlinear Filtering of Stochastic Navier-Stokes Equation with Itô-Lévy Noise

In this article, we study the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Itô-Lévy noise. Nonlinear filtering problem is formulated for the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data. The corresponding Fujisaki-Kallianpur-Kunita and Zakai equations describing the time evolution of the nonlinear filter are derived. Existence and uniqueness of measure-valued solutions are proven for these filtering equations.     

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.     

Construction of Equivalent Stochastic Differential Equation Models

Abstract

It is shown how different but equivalent Itô stochastic differential equation models of random dynamical systems can be constructed. Advantages and disadvantages of the different models are described. Stochastic differential equation models are derived for problems in chemistry, textile engineering, and epidemiology. Computational comparisons are made between the different stochastic models.     

The Numerical Approximation of Stochastic Partial Differential Equations

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itô formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered.     

Estimation of parameters of one stochastic differential equation

A problem of estimation of the unknown parameters of the solution of an Itô stochastic differential equation is considered from a process observed at a finite number of points.     

Estimating parameters in diffusion processes using an approximate maximum likelihood approach

We present an approximate Maximum Likelihood estimator for univariate Itô stochastic differential equations driven by Brownian motion, based on numerical calculation of the likelihood function. The transition probability density of a stochastic differential equation is given by the Kolmogorov forward equation, known as the Fokker-Planck equation. This partial differential equation can only be solved analytically for a limited number of models, which is the reason for applying numerical methods based on higher order finite differences.The approximate likelihood converges to the true likelihood, both theoretically and in our simulations, implying that the estimator has many nice properties. The estimator is evaluated on simulated data from the Cox-Ingersoll-Ross model and a non-linear extension of the Chan-Karolyi-Longstaff-Sanders model. The estimates are similar to the Maximum Likelihood estimates when these can be calculated and converge to the true Maximum Likelihood estimates as the accuracy of the numerical scheme is increased. The estimator is also compared to two benchmarks; a simulation-based estimator and a Crank-Nicholson scheme applied to the Fokker-Planck equation, and the proposed estimator is still competitive.     

Use of stochastic control theory to model a forest management system

A forest management problem due to Hellman has been modelled as a stochastic control problem with one state variable (inventory level) and one control variable (consumption rate of wood by the factories). The stochastic process governing the evolution of the inventory level is transformed into an Itô stoachastic differential equation by approximating the compound Poisson process of wood arrivals into the depot as a Wiener process. The resulting stochastic control problem is solved by using the Hamilton-Jacobi-Bellman equation of stochastic dynamic programming. Two numerical examples illustrate the results.     

Derivation of Several SDE Systems in One- and Two-Locus Population Genetics

A direct approach is described for deriving stochastic differential equations (SDEs) for the dynamics of evolving populations. Itô SDEs are presented and compared for populations of haploid and diploid individuals with one or more alleles at one locus undergoing pure genetic drift. The results agree with previous investigations in mathematical genetics using diffusion approximations. Furthermore, a stochastic differential equation model is derived for diploid populations with two alleles at two loci. The derived SDE systems provide unifying, consistent models.