Numerical scheme for the invariant measure of highly nonlinear McKean-Vlasov stochastic differential equation

A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.

In this work, we consider a one-dimensional Itô diffusion process Xt with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by the observation of the expectation of the process during a small time interval, and starting from any value X0 in a given subset of . With the same type of observation, and given the drift coefficient, we also show that the diffusion coefficient is uniquely determined. When both coefficients are unknown, we show that they are simultaneously uniquely determined by the observation of the expectation and variance of the process, during a small time interval, and starting again from any value X0 in a given subset of . To derive these results, we apply the Feynman-Kac theorem which leads to a linear parabolic equation with unknown coefficients in front of the first and second order terms. We then solve the corresponding inverse problem with PDE technics which are mainly based on the strong parabolic maximum principle.

Strong convergence of the split-step backward Euler method for stochastic delay differential equations with a nonlinear diffusion coefficient

Closed-form approximate solutions for a class of coupled nonlinear stochastic differential equations

This paper examines exponential mean square stability of the split-step theta approximation and the stochastic theta method for the stochastic differential delay equations and stochastic ordinary differential equations (SODEs) under a coupled monotone condition on drift and diffusion coefficients. It is shown that for the two classes of the theta approximations can preserve the exponential mean square stability when some conditions on the stepsize and drift coefficient are imposed, but for , without the globally Lipschitz continuity, these two classes of theta methods show exponentially mean square stability unconditionally. Moreover, for sufficiently small stepsize, the decay rate as measured by the bound of the Lyapunov exponent can be reproduced arbitrarily accurately. Some results in this paper extend the existing results for linear SODEs to nonlinear stochastic differential equations (SDEs), and also improve our previous results of numerical stability of nonlinear SDEs.

Projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the considered Kolmogorov equations are $n$-times continuously Fr\'{e}chet differentiable, then so are the generalized solutions at every positive time. In addition, a key contribution of this work is to prove suitable enhanced regularity properties for the derivatives of the generalized solutions of the Kolmogorov equations in the sense that the dominating linear operator in the drift coefficient of the Kolmogorov equation regularizes the higher order derivatives of the solutions. Such enhanced regularity properties are of major importance for establishing weak convergence rates for spatial and temporal numerical approximations of stochastic partial differential equations.

Stochastic differential equations can always simulate the scientific problem in practical truthfully. They have been widely used in Physics, Chemistry, Cybernetics, Finance, Neural Networks, Bionomics, etc. So far there are not many results on the numerical stability of nonlinear neutral stochastic delay differential equations. The purpose of our work is to show that the Euler method applied to the nonlinear neutral stochastic delay differential equations is mean square stable under the condition which guarantees the stability of the analytical solution. The main aim of this paper is to establish new results on the numerical stability. It is proved that the Euler method is mean-square stable under suitable condition, i.e., assume the some conditions are satisfied, then, the Euler method applied to the nonlinear neutral stochastic delay differential equations with initial data is mean-square stable. Moreover, the theoretical result is also verified by a numerical example.

On solutions to set-valued and fuzzy stochastic differential equations

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and super-linear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models.

In this paper, a class of nonlinear stochastic neutral differential equations with delays is investigated. By using the properties of $${\mathcal{M}}$$ -matrix, a differential-difference inequality is established. Basing on the differential-difference inequality, we develop a $${\mathcal{L}}$$ -operator-difference inequality such that it is effective for stochastic neutral differential equations. By using the $${\mathcal{L}}$$ -operator-difference inequality, we obtain the global attracting and invariant sets of nonlinear stochastic neutral differential equations with delays. In addition, we derive the sufficient condition ensuring the exponential p-stability of the zero solution of nonlinear stochastic neutral differential equations with delays. One example is presented to illustrate the effectiveness of our conclusion.

Stability criteria for stochastic differential delay equations (SDDEs) have been studied intensively for the past few decades. However, most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability of highly nonlinear hybrid stochastic differential equations with a single delay is investigated in [Fei, Hu, Mao and Shen, Automatica, 2017], whose work, in this paper, is extended to highly nonlinear hybrid stochastic differential equations with variable multiple delays. In other words, this paper establishes the stability criteria of highly nonlinear hybrid variable multiple-delay stochastic differential equations. We also discuss an example to illustrate our results.

In this paper, we consider stability and ergodicity of nonlinear stochastic delay differential equations with infinite delay on C r . Existing results depend on linear structure or finite delay. Under the local Lipschitz condition and modified dissipative condition, we obtain the stability in distribution of the solution map x t with nonlinear distribution delay, which also admits a unique invariant measure under slightly stronger conditions. We show that the convergence rate to the invariant measure is exponential under the Wasserstein distance.

An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators