Analysis of estimation accuracy of the first moments of a Monte Carlo solution to an SDE with Wiener and Poisson components

In this paper, the estimation accuracy of the first moments of a numerical solution to an SDE with Wiener and Poisson components is investigated by a generalized explicit Euler method. Exact expressions for the mathematical expectation and variance of a test SDE solution are obtained. These expressions allow us to investigate the estimation accuracy obtained by a Monte Carlo method versus the SDE parameters, the integration step, and the size of the ensemble of simulated trajectories of the solution. The results of test numerical experiments are presented.     

Parametric analysis of the oscillatory solutions to stochastic differential equations with the Wiener and Poisson components by the Monte Carlo method

Using the Monte Carlo method, we address the influence of the Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs). For the linear and Van der Pol oscillators, we study the accuracy of estimates of the functionals of numerical solutions to SDEs obtained by the generalized explicit Euler method. For a linear oscillator, we obtain the exact analytical expressions for the mathematical expectation and the variance of the SDE solution. These expressions allow us to investigate the dependence of the accuracy of estimates of the solution moments on the values of SDE parameters, the size of meshsize, and the ensemble of simulated trajectories of the solution. For the Van der Pol oscillator, we study the dependence of the frequency and the damping rate of the oscillations of the mathematical expectation of SDE solution on the values of parameters of the Poisson component. The results of the numerical experiments are presented.     

Critical Homogenization of SDEs Driven by a Levy Process in Random Medium

We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is, when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations.     

Existence and pathwise uniqueness to an SPDE driven by α-stable colored noise

In this paper we study a stochastic partial differential equation (SPDE) with Hölder continuous coefficient driven by an $\alpha$-stable colored noise. The pathwise uniqueness is proved by using a backward doubly stochastic differential equation backward (SDE) to take care of the Laplacian. The existence of solution is shown by considering the weak limit of a sequence of SDE system which is obtained by replacing the Laplacian operator in the SPDE by its discrete version. We also study an SDE system driven by Poisson random measures.     

线性混合模型参数的部分岭型谱分解估计

对于随机效应部分为一般平衡多项分类的线性模型, 将王松桂等\ucite{1}提出的一种称之为谱分解估计(SDE)的参数估计新方法推广到具有病态设计阵的线性模型, 提出了部分岭型谱分解估计, 通过类似于主成分估计的降维模型变换, 可以很方便的研究它的抗干扰性和其它重要性质.本文的结果可以很方便的应用于Panel模型.     

Analysis of a stochastic predator-prey system with foraging arena scheme

ABSTRACT

This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results.     

Two-component Poisson mixture regression modelling of count data with bivariate random effects

Two-component Poisson mixture regression is typically used to model heterogeneous count outcomes that arise from two underlying sub-populations. Furthermore, a random component can be incorporated into the linear predictor to account for the clustering data structure. However, when including random effects in both components of the mixture model, the two random effects are often assumed to be independent for simplicity. A two-component Poisson mixture regression model with bivariate random effects is proposed to deal with the correlated situation. A restricted maximum quasi-likelihood estimation procedure is provided to obtain the parameter estimates of the model. A simulation study shows both fixed effects and variance component estimates perform well under different conditions. An application to childhood gastroenteritis data demonstrates the usefulness of the proposed methodology, and suggests that neglecting the inherent correlation between random effects may lead to incorrect inferences concerning the count outcomes.     

On Random Marked Sets with a Smaller Integer Dimension

The paper deals with random marked sets in ${\mathbb R}^d$ which have integer dimension smaller than d. Statistical analysis is developed which involves the random-field model test and estimation of first and second-order characteristics. Special models are presented based on tessellations and solutions of stochastic differential equations (SDE). The simulation of these sets makes use of marking by means of Gaussian random fields. A space-time nature of the model based on SDE is taken into account. Numerical results of the estimation and testing are discussed. Real data analysis from the materials research investigating a grain microstructure with disorientations of faces as marks is presented.     

Long time behavior of a mean-field model of interacting neurons

We study the long time behavior of the solution to some McKean–Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant probability measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant probability measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean–Vlasov equation.     

Normal convergence using Malliavin calculus with applications and examples

We prove the chain rule in the more general framework of the Wiener–Poisson space, allowing us to obtain the so-called Nourdin–Peccati bound. From this bound, we obtain a second-order Poincaré-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener–Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein–Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to infinitely many “small” jumps (particularly fractional Lévy processes), and the product of two Ornstein–Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality.     

Optimal sampling design for global approximation of jump diffusion stochastic differential equations

The paper deals with strong global approximation of stochastic differential equations (SDEs) driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition (see Chapter 6.3 in [21]). We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient, with respect to asymptotic constants, than those based on the equidistant mesh.     

Coordinated maintenance in a multi-component system with compound Poisson deterioration

This paper proposes a coordinated maintenance model in a multi-component system with compound Poisson deterioration. The main contribution is a policy-iteration approach for Semi-Markov processes that optimizes the threshold at which the component is eligible for preventive maintenance if another component requires corrective maintenance. The methodology is novel as we develop explicit expressions for the policy evaluation and prove these expressions to satisfy the set of linear equations which characterize traditional policy evaluation. By doing so, long-run average cost savings are achieved, since setup costs can be shared.     

A delay-time-based maintenance model of a multi-component system

There is a well established literature on delay-time modelsof regular inspection policies where inspections may or maynotbe perfect, and where the initial point u of a defect arisesas a homogeneous Poisson process. This paper extends the modellingin two ways. The first is to include the observed practice wherethe multi-component system is inspected not only on a plannedbasis, but also when a component fails. The second extensionis to use a nonhomogeneous Poisson process to describe defectarrivals in the system. An inspection–replacement modelbased upon these two extensions is then developed for a multi-componentsystem. The total expected cost per unit time is minimized withrespect to theinspection intervals and the system replacementtime. The likelihood function of the time of failures and thenumber of defects found at inspections is established, in orderto estimate model parameters based upon routinely collectedmaintenance data. As a special case of the general model, aninspection model—based upon a homogeneous Poisson processof defects arising—is also proposed, which has a relativelysimple structure. Both simulated and real-life data of failuresand defects identified at inspections are used to test the modelsand parameter-estimating procedure.     

Confined steady states of a Vlasov‐Poisson plasma in an infinitely long cylinder

We consider the two‐dimensional Vlasov‐Poisson system to model a two‐component plasma whose distribution function is constant with respect to the third space dimension. First, we show how this two‐dimensional Vlasov‐Poisson system can be derived from the full three‐dimensional model. The existence of compactly supported steady states with vanishing electric potential in a three‐dimensional setting has already been investigated in the literature. We show that these results can easily be adapted to the two‐dimensional system. However, our main result is to prove the existence of compactly supported steady states even with a nontrivial self‐consistent electric potential.     

Sieve maximum likelihood estimation for doubly semiparametric zero-inflated Poisson models

For nonnegative measurements such as income or sick days, zero counts often have special status. Furthermore, the incidence of zero counts is often greater than expected for the Poisson model. This article considers a doubly semiparametric zero-inflated Poisson model to fit data of this type, which assumes two partially linear link functions in both the mean of the Poisson component and the probability of zero. We study a sieve maximum likelihood estimator for both the regression parameters and the nonparametric functions. We show, under routine conditions, that the estimators are strongly consistent. Moreover, the parameter estimators are asymptotically normal and first order efficient, while the nonparametric components achieve the optimal convergence rates. Simulation studies suggest that the extra flexibility inherent from the doubly semiparametric model is gained with little loss in statistical efficiency. We also illustrate our approach with a dataset from a public health study.     

SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations

It is known that the transition probabilities of a solution to a classical It? stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Time-fractional Kolmogorov-type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Lévy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman–Kac formula.     

一类线性混合模型的谱分解估计

谱分解估计(SDE)是新近提出的关于线性混合模型参数的一种新的估计方法,此方法的一个突出特点是同时给出固定效应参数和方差分量的显式解估计．本文就含两个方差分量的线性混合模型,对谱分解估计的性质做了进一步的研究,获得了方差分量的SDE和方差分析估计相等的充分必要条件,证明了在一定的条件下方差分量的SDE为一致最小方差无偏估计．     

Asymptotic expansion for forward-backward SDEs with jumps

This work provides a semi-analytic approximation method for decoupled forward-backward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with σ-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.     

Model of Deformation of Monotropic Plastic Foams Parallel to the Foam Rise Direction Based on the Volume-Deformation Hypothesis

The deformation properties of monotropic plastic foams under uniaxial deformation (compression or tension) parallel to the foam rise direction is considered. The theoretical results are obtained in the case where the volume-deformation hypothesis is assumed. The validity of neglecting the fluctuation component in calculations of the effective volume strains of foams is substantiated. A tie condition permitting the sought-for semiaxes to vary not obligatorily equally is derived. The values of Young's modulus and Poisson coefficients are obtained for a wide range of model cell stretch ratios and foam space-filling coefficients. A comparison of the theoretical results with the experimental data available is performed.     

Jump-diffusions in Hilbert spaces: existence,stability and numerics

By means of an original approach, called ‘method of the moving frame’, we establish existence, uniqueness and stability results for mild and weak solutions of stochastic partial differential equations (SPDEs) with path-dependent coefficients driven by an infinite-dimensional Wiener process and a compensated Poisson random measure. Our approach is based on a time-dependent coordinate transform, which reduces a wide class of SPDEs to a class of simpler SDE (stochastic differential equation) problems. We try to present the most general results, which we can obtain in our setting, within a self-contained framework to demonstrate our approach in all details. Also, several numerical approaches to SPDEs in the spirit of this setting are presented.