Class Of Stochastic Differential Equations Research Articles

Stationary distributions for stochastic differential equations with memory driven by [formula omitted]-stable processes

In this work, we consider the stationarity of a class of stochastic differential equations with memory driven by α-stable processes. Sufficient conditions are given to guarantee the existence and uniqueness of the stationary distribution for the stochastic system. Last, an example is given to illustrate the results.

Drift-implicit Euler scheme for sandwiched processes driven by Hölder noises

In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary λ-Hölder continuous process, λ ∈ (0,1). We prove that, under some mild moment assumptions on the Hölder constant of the noise, the L^{r}({Omega };L^{infty }([0,T]))-approximation error converges to 0 as O(Δλ), Δ → 0. To exemplify, we consider numerical schemes for the generalized Cox–Ingersoll–Ross and Tsallis–Stariolo–Borland models. The results are illustrated by simulations.

Stability Analysis of Stochastic Differential Equation Driven by G-Brownian Motion under Non-Lipschitz Condition

This paper is devoted to studying the p-th moment exponential stability for a class of stochastic differential equation (SDE) driven by G-Brownian motion under non-Lipschitz condition. The delays considered in this paper are time-varying delays τ i t 1 ≤ i ≤ 3 . Since the coefficients are non-Lipschitz, the normal enlargement on the coefficients is not available and the Gronwall inequality is not suitable in this case. By Bihari inequality and Itô integral formula, it is pointed out that there exists a constant τ ∗ such that the p-th moment exponential stability holds if the time-varying delays are smaller than τ ∗ .

Trajectory fitting estimation for a class of SDEs with small Lévy noises

In this paper, we consider the problem of trajectory fitting estimation for a class of stochastic differential equations with small Lévy noises based on continuous-time observations. The consistency, the rate of convergence, and asymptotic distribution of the trajectory fitting estimator are established as a small dispersion coefficient tends to zero.

Bilateral Harnack Inequalities for Stochastic Differential Equation with Multiplicative Noise

By constructing a coupling with unbounded time-dependent drift, a lower bound estimate of dimension-free Harnack inequality with power is obtained for a large class of stochastic differential equation with multiplicative noise. The key is an application of the inverse Hölder inequality. Combining this with the well-known upper bound, bilateral dimension-free Harnack inequality with power is established. As a dual inequality, the bilateral shift-Harnack inequalities with power are also investigated for stochastic differential equation with additive noise. Applications to the study of heat kernel inequalities are provided to illustrate the obtained inequalities.

On Irreversible Metropolis Sampling Related to Langevin Dynamics

There has been considerable interest in designing Markov chain Monte Carlo algorithms by exploiting numerical methods for Langevin dynamics, which includes Hamiltonian dynamics as a deterministic case. A prominent approach is Hamiltonian Monte Carlo (HMC), where a leapfrog discretization of Hamiltonian dynamics is employed. We investigate a recently proposed class of irreversible sampling algorithms, called Hamiltonian assisted Metropolis sampling (HAMS), which uses an augmented target density similarly as in HMC but involves a flexible proposal scheme and a carefully formulated acceptance-rejection scheme to achieve generalized reversibility. We show that as the step size tends to 0, the HAMS proposal satisfies a class of stochastic differential equations including Langevin dynamics as a special case. We provide theoretical results for HAMS, including algebraic properties of the acceptance probability, the stationary variance from the HAMS proposal, and the expected acceptance rate under a product Gaussian target distribution and the convergence rate under standard Gaussian. From these results, we derive default choices of tuning parameters for HAMS such that only the step size needs to be tuned in applications. Various relatively recent algorithms for Langevin dynamics are also shown to fall in the class of HAMS proposals up to negligible differences. Our numerical experiments on sampling high-dimensional latent variables confirm that the HAMS algorithms consistently achieve superior performance compared with several Metropolis-adjusted algorithms based on popular integrators of Langevin dynamics.

Weighted maximum likelihood estimation for individual growth models

We apply a class of stochastic differential equations to model individual growth in a randomly fluctuating environment using cattle weight data. We have used maximum likelihood theory to estimate the parameters. However, for cattle data, it is often not feasible to obtain animal's observations at equally spaced ages nor even at the same ages for different animals and there is typically a small number of observations at older ages. For these reasons, maximum likelihood estimates can be quite inaccurate, being interesting to consider in the likelihood function a weight function associated to the elapsed times between two consecutive observations of each animal, which results in the weighted maximum likelihood method. We compare the results obtained from both methods in several data structures and conclude that the weighted maximum likelihood improves the estimation when observations at older ages are scarce and the observation instants are unequally spaced, whereas the maximum likelihood estimates are recommended when animals are weighted at equally spaced ages. For unequally spaced observations, a bootstrap estimation method was also applied to correct the bias of the maximum likelihood estimates; it revealed to be a more precise alternative, except when the available data only has young animals.

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

Viability for mixed stochastic differential equations driven by fractional Brownian motion and its application

ABSTRACT In this paper, we focus on a class of stochastic differential equations driven by the standard Brownian motion and fractional Brownian motion with Hurst parameter under a weaker conditions than Lipschitz one. In the sense of the pathwise Riemann–Stieltjes integral, we prove the convergence of solutions for the considered equations. By making use of some transformation techniques and approximation means, we obtain some sufficient conditions on the viability for the stochastic systems under investigation. Subsequently, by using some distance functions, we establish some necessary conditions and some sufficient conditions for the viability property with respect to a given non-empty closed set K, but K can be not smooth. As applications of the results, we explore the comparison theorems about the mixed stochastic differential equations driven by fractional Brownian motion under some non-Lipschitz conditions.

Backward doubly stochastic differential equations (BDSDEs): Existence and Uniqueness

In this paper, we present a class of stochastic differential equations with terminal condition, called backward doubly stochastic differential equations (BDSDEs). Precisely, we will prove the existence and uniqueness of the solutions of FBDSDEs but under weaker conditions

The perfection of local semi-flows and local random dynamical systems with applications to SDEs

We provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dynamical system. Such a (local) semi-flow induces a (local) random dynamical system (RDS). Then we show that this result can be applied to several classes of stochastic differential equations driven by semimartingales with stationary increments such as equations with locally monotone coefficients and equations with singular drift. For these examples it was previously unknown whether they generate a (local) RDS or not.

Optimal index and averaging principle for Itô–Doob stochastic fractional differential equations

In this paper, a class of Itô–Doob stochastic fractional differential equations (Itô–Doob SFDEs) models are discussed. Using the time scale transformation method, we consider the averaging principle of the transformed equations and establish the relevant results. At the same time, we find that the optimal index for the original Itô–Doob SFDEs can be determined, the selection of such index is similar to the classical stochastic differential equations model.

Carathéodory approximate solutions for a class of stochastic differential equations involving the local time at point zero with one-sided Lipschitz continuous drift coefficients

Abstract In this paper, we study the Carathéodory approximate solution for a class of stochastic differential equations involving the local time at point zero. Based on the Carathéodory approximation procedure, we prove that stochastic differential equations involving the local time at point zero have a unique solution, and we show that the Carathéodory approximate solution converges to the solution of stochastic differential equations involving the local time at point zero with one-sided Lipschitz drift coefficient.

On Asymptotic Preserving Schemes for a Class of Stochastic Differential Equations in Averaging and Diffusion Approximation Regimes

On Asymptotic Preserving Schemes for a Class of Stochastic Differential Equations in Averaging and Diffusion Approximation Regimes

Strong and weak convergence rates for slow–fast stochastic differential equations driven by α-stable process

In this paper, we study the averaging principle for a class of stochastic differential equations driven by α-stable processes with slow and fast time-scales, where α∈(1,2). We prove that the strong and weak convergence order are 1−1/α and 1 respectively. We show, by a simple example, that 1−1/α is the optimal strong convergence rate.

Neuromorphic scaling advantages for energy-efficient random walk computations

Computing stands to be radically improved by neuromorphic computing (NMC) approaches inspired by the brain's incredible efficiency and capabilities. Most NMC research, which aims to replicate the brain's computational structure and architecture in man-made hardware, has focused on artificial intelligence; however, less explored is whether this brain-inspired hardware can provide value beyond cognitive tasks. We demonstrate that high-degree parallelism and configurability of spiking neuromorphic architectures makes them well-suited to implement random walks via discrete time Markov chains. Such random walks are useful in Monte Carlo methods, which represent a fundamental computational tool for solving a wide range of numerical computing tasks. Additionally, we show how the mathematical basis for a probabilistic solution involving a class of stochastic differential equations can leverage those simulations to provide solutions for a range of broadly applicable computational tasks. Despite being in an early development stage, we find that NMC platforms, at a sufficient scale, can drastically reduce the energy demands of high-performance computing (HPC) platforms.

Anticipated Generalized Backward Doubly Stochastic Differential Equations

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

Weak convergence of delay SDEs with applications to Carathéodory approximation

<p style='text-indent:20px;'>In this paper, we consider a fundamental class of stochastic differential equations with time delays. Our aim is to investigate the weak convergence with respect to delay parameter of the solutions. Based on the techniques of Malliavin calculus, we obtain an explicit estimate for the rate of convergence. An application to the Carathéodory approximation scheme of stochastic differential equations is provided as well.</p>

Iнварiантнi поверхнi для певних класiвсистем другого порядку стохастичних диференцiальних рiвнянь зi стрибками

In this paper, we consider the concept of invariant sets of inhomogeneous stochastic differential equations with jumps. For certain classes of systems of the second order of inhomogeneous stochastic differential equations with jumps the necessary and sufficient conditions for the invariance of the corresponding surfaces are established. The obtained results provide opportunities to find the invariant surfaces and conditions of their invariance for the specified classes of stochastic differential equations.

Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes

<abstract><p>The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens <sup>[<xref ref-type="bibr" rid="b21">21</xref>]</sup>). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.</p></abstract>