Stochastic Differential Equations Of Ito Type Research Articles

Weak convergence of balanced implicit methods for Ito-type stochastic differential equations with variable step sizes

Weak convergence of balanced implicit methods for Ito-type stochastic differential equations with variable step sizes

Instabilizability Conditions for Continuous-Time Stochastic Systems Under Control Input Constraints

In this letter, we investigate constrained control of continuous-time linear stochastic systems. We show that for certain system parameter settings, constrained control policies can never achieve stabilization. Specifically, we explore a class of control policies that are constrained to have a bounded average second moment for Ito-type stochastic differential equations with additive and multiplicative noise. We prove that in certain settings of the system parameters and the bounding constant of the control constraint, divergence of the second moment of the system state is inevitable regardless of the initial state value and regardless of how the control policy is designed.

Modeling of adulthood obesity in Spain using Itô-type stochastic differential equations

Abstract Obesity is growing riskily in developed and developing countries. This should pose major concerns for the countries, not only from the health point, but also from the economic perspective. Our case study relies on the excess weight dynamics in Spain. The Spanish National Health Survey (ENSE) 2017 collects the percentage of overweight and obese adults in Spain from the year 1987 to 2017. A recent contribution proposed a nonautonomous compartmental system of ordinary differential equations to calibrate the incidence of excess weight in the Spanish adulthood population. Essentially, three principles were followed: the total adulthood population is time-dependent, the subpopulations interact homogeneously along the country, and excess weight plays the role of an infectious disease that is transmitted through contact by social pressure. Accounting for both data and model errors, frequentist nonlinear regression and Bayesian inference were conducted. The methods agreed well in terms of fit, prediction, bands and sensitivity analysis. In the present paper, the deterministic compartmental system of ordinary differential equations is randomized in a different manner, by employing Ito-type stochastic differential equations. The derivatives of the compartments are perturbed by Gaussian white noise-type pure errors that have a rough and unpredictable structure. From the Euler-Maruyama discretization, several strategies are utilized for estimating the parameters, based on the moments method and maximum likelihood estimation. Comparison is performed numerically by assessing the fit to the data.

An Efficient Numerical Algorithm for Solving Data Driven Feedback Control Problems

The goal of this paper is to solve a class of stochastic optimal control problems numerically, in which the state process is governed by an Ito type stochastic differential equation with control process entering both in the drift and the diffusion, and is observed partially. The optimal control of feedback form is determined based on the available observational data. We call this type of control problems the data driven feedback control. The computational framework that we introduce to solve such type of problems aims to find the best estimate for the optimal control as a conditional expectation given the observational information. To make our method feasible in providing timely feedback to the controlled system from data, we develop an efficient stochastic optimization algorithm to implement our computational framework.

Regulation of Continuous-Time Stochastic Systems under State and Control Constraints

In this paper the regulation problem for continuous-time stochastic systems by linear state-feedback under state and/or control constraints is investigated. Nonlinear dynamical systems affected by stochastic disturbances modelled as q-dimensional Wiener processes with independent components are considered. Linear inequalities on the mean values of the state and the control are considered as control constraints. The analysis is based on the use of monotone comparison systems for Ito type stochastic differential equations and the positive invariance of polyhedral sets. This enables one to derive stability properties in terms of the convergence of the state vector mean values, thus avoiding complicated computations faced when stability properties are obtained by studying the behavior of higher order moments. For the class of linear systems a design approach based on linear programming is proposed.

Iterative methods for nonlinear systems associated with finite difference approach in stochastic differential equations

We present some iterative methods of different convergence orders for solving systems of nonlinear equations. Their computational complexities are studies. Then, we introduce the method of finite difference for solving stochastic differential equations of Ito-type. Subsequently, our multi-step iterative schemes are employed in this procedure. Several experiments are finally taken into account to show that the presented approach and methods work well.

$$H_\infty $$ H ∞ filtering for stochastic singular fuzzy systems with time-varying delay

This paper considers the \(H_\infty \) filtering problem for stochastic singular fuzzy systems with time-varying delay. We assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by Ito-type stochastic differential equations. Based on an auxiliary vector and an integral inequality, a set of delay-dependent sufficient conditions is established, which ensures that the filtering error system is \({\mathrm{e}^{\lambda t}}\)-weighted integral input-to-state stable in mean (iISSiM). A fuzzy filter is designed such that the filtering error system is impulse-free, \({\mathrm{e}^{\lambda t}}\)-weighted iISSiM and the \(H_\infty \) attenuation level from disturbance to estimation error is below a prescribed scalar. A set of sufficient conditions for the solvability of the \(H_\infty \) filtering problem is obtained in terms of a new type of Lyapunov function and a set of linear matrix inequalities. Simulation examples are provided to illustrate the effectiveness of the proposed filtering approach developed in this paper.

Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering

This paper is concerned with parameter estimation in linear and non-linear Ito type stochastic differential equations using Markov chain Monte Carlo (MCMC) methods. The MCMC methods studied in this paper are the Metropolis–Hastings and Hamiltonian Monte Carlo (HMC) algorithms. In these kind of models, the computation of the energy function gradient needed by HMC and gradient based optimization methods is non-trivial, and here we show how the gradient can be computed with a linear or non-linear Kalman filter-like recursion. We shall also show how in the linear case the differential equations in the gradient recursion equations can be solved using the matrix fraction decomposition. Numerical results for simulated examples are presented and discussed in detail.

Almost sure and moment exponential stability of predictor-corrector methods for stochastic differential equations

This paper deals with almost sure and moment exponential stability of a class of predictor-corrector methods applied to the stochastic differential equations of Ito-type. Stability criteria for this type of methods are derived. The methods are shown to maintain almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions. A numerical experiment further testifies these theoretical results.

Stochastic mechanics of electron transfer in graphene

In this paper, we present a new approach to electron transports in nano-scale devices by using Nelson's quantum stochastic mechanical simulations. Transferring electrons in pure and doped graphene ribbons (GRs) are numerically investigated as quantum Brownian motions with wavefunctions of isolated GRs. The time evolution of the quantum stochastic process is given by Ito type stochastic differential equation. From the analysis of the electron motion, we directly obtain the electron mobilities in GRs under some electric fields. We show the electron motion depends on the stochastic distribution of the eigenfunction or the molecular orbital and then the electron mobility is affected by the dopant atom.

Stochastic methods for Dirichlet problems

Using an equivalent expression for solutions of second order Dirichlet problems in terms of Ito type stochastic differential equations, we develop a numerical solution method for Dirichlet boundary value problems. It is possible with this idea to solve for solution values of a partial differential equation at isolated points without having to construct any kind of mesh and without knowing approximations for the solution at any other points. Our method is similar to a recently published approach, but differs primarily in the handling of the boundary. Some numerical examples are presented, applying these techniques to model Laplace and Poisson equations on the unit disk.

Direct Derivation of Corrective Terms in SDE Through Nonlinear Transformation on Fokker–Planck Equation

This paper examines the problem of probabilistic characterization of nonlinear systems driven by normal and Poissonian white noise. By means of classical nonlinear transformation the stochastic differential equation driven by external input is transformed into a parametric-type stochastic differential equation. Such equations are commonly handled with Ito-type stochastic differential equations and Ito's rule is used to find the response statistics. Here a different approach is proposed, which mainly consists in transforming the Fokker–Planck equation for the original system driven by external input, in the transformed probability density function of the new state variable. It will be shown that the Wong–Zakay or Stratonovich corrective term and the hierarchy of correction terms in the case of Poissonian white noise arise in a natural way.

Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients

In the present paper, using a Picard Type method of approximation, we investigate the global existence of mild solutions for a class of Ito Type stochastic differential equations whose coefficients satisfy conditions more general than the Lipschitz and linear growth ones.

A Leader-Follower Stochastic Linear Quadratic Differential Game

A leader-follower stochastic differential game is considered with the state equation being a linear Ito-type stochastic differential equation and the cost functionals being quadratic. We allow that the coefficients of the system and those of the cost functionals are random, the controls enter the diffusion of the state equation, and the weight matrices for the controls in the cost functionals are not necessarily positive definite. The so-called open-loop strategies are considered only. Thus, the follower first solves a stochastic linear quadratic (LQ) optimal control problem with the aid of a stochastic Riccati equation. Then the leader turns to solve a stochastic LQ problem for a forward-backward stochastic differential equation. If such an LQ problem is solvable, one obtains an open-loop solution to the two-person leader-follower stochastic differential game. Moreover, it is shown that the open-loop solution admits a state feedback representation if a new stochastic Riccati equation is solvable.

Sampling implicit surfaces based on stochastic differential equations with converging constraint

We propose a new stochastic sampling method for implicit surfaces based on an Ito-type stochastic differential equation with the converging constraint. The equation can realize a stationary state in which the probability distribution becomes uniform over the whole area of a constraint surface. A hypothetical stochastic particle described by the equation performs Brownian motion, i.e. random walk, being confined on a constraint surface, i.e. an implicit surface. Therefore its trajectory gives us sample points on it. The stochastic sampling method is fast and widely applicable to implicit surfaces defined with twice differentiable constraint functions. It is applicable to non-polynomial surfaces and surfaces with branches, too. It does not require an initial sample point on a sampled surface to start sampling. It works well with a single-particle system without necessity of introducing interaction between particles. It also has advantageous features at the rendering stage.

Cone-valued lyapunov functions and the stability of stochastic differential equations

Stochastic differential equations of Ito-type are considered and the theory of stochastic differential inequalities is systematically developed. Sufficient conditions for stability in probability, with probability one and in the mean of the Ito-type stochastic differential equations are given, using the method of cone-valued Lyapunov functions. Necessary and sufficient conditions for the construction of stochastic cone-valued Lyapunov functions are obtained for the cases where the Ito-type stochastic differential equations have uniform asymptotic stability in probability and uniform asymptotic stability in the mean. The results obtained are applied to the study of connective stability of multispecies community in a stochastic environment

A Lagrangian stochastic model for nonpassive particle diffusion in turbulent flows

In this paper, we present a Lagrangian stochastic model for heavy particle dispersion in turbulence. The model includes the equation of motion for a heavy particle and a stochastic approach to predicting the velocity of fluid elements along the heavy particle trajectory. The trajectory crossing effect of heavy particles is described by using an Ito type stochastic differential equation combined with a fractional Langevin equation. The comparison of the predicted dispersion of four heavy particles with the observations shows that the model is potentially useful but requires further development.

Nonlinearly damped systems under simultaneous broad-band and harmonic excitations

The probability distribution of the response of a nonlinearly damped system subjected to both broad-band and harmonic excitations is investigated. The broad-band excitation is additive, and the harmonic excitations can be either additive or multiplicative. The frequency of a harmonic excitation can be either near or far from a resonance frequency of the system. The stochastic averaging method is applied to obtain the Ito type stochastic differential equations for an averaged system described by a set of slowly varying variables, which are approximated as components of a Markov vector. Then, a procedure based on the concept of stationary potential is used to obtain the exact stationary probability density for a class of such averaged systems. For those systems not belonging to this class, approximate solutions are obtained using the method of weighted residuals. Application of the exact and approximate solution procedures are illustrated in two specific cases, and the results are compared with those obtained from Monte Carlo simulations.

On semimartingale diffusions and stochastic differential equations

Given a regular diffusion X on the real axis which is a semimartingale we describe the semimartingale decomposition of X. We then give necessary and sufficient conditions in terms of the scale and the speed measure for X being a solution of an Ito type stochastic differential equation driven by a Wiener process and with classical drift or a drift term involving the local time of X. A regular diffusion is also characterized as unique solution of a certain martingale problem. Finally we discuss an example related to skew Brownian motion

Diffusion equations in duals of nuclear spaces

A stochastic Galerkin method is used to establish the existence of a solution to a martingale problem posed by an Ito type stochastic differential equation for processes taking values in the dual of a nuclear space. Uniqueness of the strong solution is also shown using the monotonicity condition. An application to the motion of random strings is discussed.