Compensated Poisson Process Research Articles

Influence of fractal–fractional differentiation and independent quadratic Lévy jumps on the dynamics of a general epidemic model with vaccination strategy

The emergence and reemergence of various infectious diseases have adversely affected human life both from a social and economic point of view, and thus the study of epidemic diseases has diverted the attention of scientists from different fields. Particularly, to understand the dynamics and control mechanisms of the novel infections, one must focus on mathematical modeling, which is assumed to be the most important tool for analyzing the new epidemics. In this sense, the most common strategy used in the literature is to characterize the dynamics of infections via compartmental models. In this work, we formulated a general mathematical model along with a vaccination strategy to reveal the novel findings on the asymptotic properties of the underlying model. The complicated rapid fluctuations of the problem are taken into consideration, and second-order quadratic jumps with four independent compensated Poisson processes are studied. The model was further enhanced by including the two general interference functions. The key threshold determinant R0⋆ is calculated and it is proved that for R0⋆>1, the model has the properties of stationarity and ergodicity (i.e., permanence scenario). However, if R0⋆ is less than one, the infection will disappear exponentially (i.e., disappearance scenario). These results exhibit that noise sources play a dominant role in explaining the asymptotic behavior of any biological phenomenon. Numerically, we confirm the above-mentioned illustrations, and the simulation suggests the following results: (a) the mean-time of the solution depends on the noise intensities (b) the second-order Poisson jumps produce a negative effect on the time required for the survival of the infection. The study was further extended by considering non-integer differential equations, and the underlying model was studied both from fractional and fractal dimensions. This combination tries to give a physical explanation of infectious pathways. The stability of fractional ordered model was studied by using the Hyers–Ulam (HU) approach. To find the numerical solution of the fractional model, we presented and developed a fractal–fractional numerical scheme by employing the Adams Bashforth’s method. We find that studying the models from a fractal and fractional point of view has a major impact on the infection’s development. The dynamics of the times-series and chaotic behaviors vary by merely modifying the fractal or fractional orders of the model. More generally, the theoretical and numerical results of this paper provide excellent insight into the long-run behavior of the epidemic under a vaccination strategy.

Large deviation principle for a mixed fractional and jump diffusion process

Abstract We study the asymptotic behavior of a solution of a mixed differential equation driven by independent fractional Brownian motion with Hurst index H ∈ ( 0 ; 1 ) {H\in(0;1)} and compensated Poisson process. This study consists in determining the uniform Freidlin–Wentzell estimates in a temporal distribution space.

Risk-sensitive zero-sum stochastic differential game for jump–diffusions

This article is concerned with the infinite horizon risk-sensitive zero-sum stochastic differential game problem for a class of jump–diffusions controlled through the drift, and driven by a compensated Poisson process and a Wiener process. Under certain geometric stability assumption on the dynamics, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. We obtain our result by exploiting the stochastic representation of the principal eigenfunction of the associated Hamilton–Jacobi–Isaac (HJI) equation, which is a semilinear integro-partial differential equation.

Exponential stability of implicit numerical solution for nonlinear neutral stochastic differential equations with time‐varying delay and poisson jumps

The aim of this work is to investigate the exponential mean‐square stability for neutral stochastic differential equations with time‐varying delay and Poisson jumps. When all the drift, diffusion, and jumps coefficients are allowed to be nonlinear, the exponential mean‐square stability of the analytic solution to the equation is obtained. It is revealed that the implicit backward Euler–Maruyama numerical solution can reproduce the corresponding stability of the analytic solution under some given nonlinear conditions. It is different from the explicit Euler–Maruyama numerical solution whose stability depends on the linear growth condition. With some requirements related to the delayed function and the property of compensated Poisson process, we deal with time‐varying delay and Poisson jumps. One highly nonlinear example is given to confirm the effectiveness of our theory.

Formula omitted] control for Poisson-driven stochastic systems

The paper aims to investigate the H∞ control problem for systems perturbed by jump random noise, i.e., Poisson-driven stochastic systems (SSs). Firstly, this paper uses the Doob-Meyer decomposition and measure theory to give a model transformation method, and Poisson-driven SSs, which are SSs driven by semi-martingale, are transformed into SSs driven by compensated Poisson process. Since SSs driven by compensated Poisson process are SSs driven by martingale, we can use effective properties and tools in martingale theory to investigate the H∞ control problem. Secondly, this paper utilizes martingale theory to deal with the jump item and the sum of stochastic integrals (SIs) with respect to (w.r.t.) the continuous part of states in Itô formula, and then gives an equivalent Itô formula for SDEs driven by compensated Poisson process. Thirdly, on the basis of these, this paper presents a simple H∞ controller design method. Furthermore, the design criterion contains information about the average number of jump random events in a unit time, and one can utilize convex optimization algorithm to estimate the maximum average number of jump random events in a unit time, which the closed-loop system can tolerate to achieve the stability and H∞ performance. Finally, the usefulness of the design result is verified by two numerical examples.

Backstepping control design for stochastic systems driven by Lévy processes

This paper presents a constructive method to design control inputs that almost surely globally -exponentially stabilise strict-feedback systems driven by Lévy processes, which consist of Wiener and compensated Poisson processes, at the origin. The control design is based on a recently developed Lyapunov-type theorem for the study of well-posedness and almost sure -exponential stability of SDEs driven by Lévy processes, the backstepping method, and new Lyapunov functions.

Stability in probability and inverse optimal control of evolution systems driven by Levy processes

This paper first develops a Lyapunov-type theorem to study global well-posedness &#x0028 existence and uniqueness of the strong variational solution &#x0029 and asymptotic stability in probability of nonlinear stochastic evolution systems &#x0028 SESs &#x0029 driven by a special class of Levy processes, which consist of Wiener and compensated Poisson processes. This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes. The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system, and minimizes a meaningful cost functional that penalizes both states and control. The approach does not require to solve a Hamilton-Jacobi-Bellman equation &#x0028 HJBE &#x0029. An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.

Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles

We study a discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation (DBBSDE in short) with jumps, driven by a Brownian motion and an independent compensated Poisson process. Moreover, we suppose that the obstacles are right continuous and left limited (RCLL) processes with predictable and totally inaccessible jumps and satisfy Mokobodzki's condition. Our main contribution consists in the construction of an implementable numerical scheme, based on two random binomial trees and the penalization method, which is shown to converge to the solution of the DBBSDE. Finally, we illustrate the theoretical results with some numerical examples in the case of general jumps.

The Predictable Representation Property of Compensated-Covariation Stable Families of Martingales

We study the predictable representation property for certain families of square integrable martingales, which we call compensated-covariation stable families. The definition of the predictable representation property is given by means of stable subspaces. The main result is that compensated-covariation stable families of martingales which satisfy some further conditions possess the predictable representation property. As first examples, we consider continuous Gaussian families of martingales and independent families of compensated Poisson processes. Then we apply the result to the case of Lévy processes. We shall construct families of martingales relative to a Lévy filtration which possess the predictable representation property. We give several examples including Teugels martingales.

Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles

We introduce a discrete time reflected scheme to solve doubly reflected Backward Stochastic Differential Equations with jumps (in short DRBSDEs), driven by a Brownian motion and an independent compensated Poisson process. As in Dumitrescu and Labart (2014), we approximate the Brownian motion and the Poisson process by two random walks, but contrary to this paper, we discretize directly the DRBSDE, without using a penalization step. This gives us a fully implementable scheme, which only depends on one parameter of approximation: the number of time steps n (contrary to the scheme proposed in Dumitrescu and Labart (2014), which also depends on the penalization parameter). We prove the convergence of the scheme, and give some numerical examples.

The chaotic representation property of compensated-covariation stable families of martingales

In the present paper, we study the chaotic representation property for certain families XX of square integrable martingales on a finite time interval [0,T][0,T]. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family XX of square integrable martingales having deterministic mutual predictable covariation ⟨X,Y⟩⟨X,Y⟩ for all X,Y∈XX,Y∈X. The main result of the present paper is stated in Theorem 5.8 below: If XX is a compensated-covariation stable family of square integrable martingales such that ⟨X,Y⟩⟨X,Y⟩ is deterministic for all X,Y∈XX,Y∈X and, furthermore, the system of monomials generated by XX is total in L2(Ω,FXT,P)L2(Ω,FTX,P), then XX possesses the chaotic representation property with respect to the σσ-field FXTFTX. We shall apply this result to the case of Levy processes. Relative to the filtration FLFL generated by a Levy process LL, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Levy processes, several examples of concrete families XX of martingales including Teugels martingales.

LAN property for a simple Lévy process

In this paper, we consider a simple Lévy process given by a Brownian motion and a compensated Poisson process, whose drift and diffusion parameters as well as its intensity are unknown. Supposing that the process is observed discretely at high frequency, we derive the local asymptotic normality (LAN) property. In order to obtain this result, Malliavin calculus and Girsanov's theorem are applied in order to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied.

Higher order approximations via Stein's method

This paper is a sequel of \cite{CD:2012}. We show how to establish a functional Edgeworth expansion of any order thanks to the Stein method. We apply the procedure to the Brownian approximation of compensated Poisson process and to the linear interpolation of the Brownian motion. It is then apparent that these two expansions are of rather different form.

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a PoissonU-statistic. The approach is based on recent results of Peccatiet al.(2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process ofk-dimensional flats in.

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a PoissonU-statistic. The approach is based on recent results of Peccatiet al.(2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process ofk-dimensional flats in.

A new family of time-space harmonic polynomials with respect to Lévy processes

By means of a symbolic method, a new family of time-space harmonic polynomials with respect to L\'evy processes is given. The coefficients of these polynomials involve a formal expression of L\'evy processes by which many identities are stated. We show that this family includes classical families of polynomials such as Hermite polynomials. Poisson-Charlier polynomials result to be a linear combinations of these new polynomials, when they have the property to be time-space harmonic with respect to the compensated Poisson process. The more general class of L\'evy-Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to L\'evy processes of very general form. We show the role played by cumulants of L\'evy processes so that connections with boolean and free cumulants are also stated.

Numerical simulations for the pricing of options in jump diffusion markets

In this paper we find numerical solutions for the pricing problem in jump diffusion markets. We utilize a model in which the underlying asset price is generated by a process that consists of a Brownian motion and an independent compensated Poisson process. By risk neutral pricing the option price can be expressed as an expectation. We simulate the option price numerically using the Monte Carlo method.

Compensated stochastic theta methods for stochastic differential equations with jumps

In this paper, a family of compensated stochastic theta methods (CSTM), as opposed to stochastic theta methods (STM) are proposed after the introduction of a compensated Poisson process. These methods are justified to have a strong convergence order of 1/2. Further we investigate mean-square stability of the proposed methods. For a linear test equation, we show that an extension of the deterministic A-stability property holds for CSTM, if and only if 1 / 2 ⩽ θ ⩽ 1 . For a general nonlinear problem, of which the drift term f has a negative one-sided Lipschitz constant and the diffusion terms g , h satisfy global Lipschitz condition, we find that backward Euler method (STM with θ = 1 ) preserves stability under a stepsize constraint, while compensated backward Euler method (CSTM with θ = 1 ) gives a generalization of the deterministic B-stability. Those stability results indicate that CSTM achieve superiority over STM in terms of stability.

An Extension of the Ocone–Haussmann–Clark Formula for the Compensated Poisson Processes

The Sobolev-type spaces $D_{p,1,\alpha }^{CP} (1\le p\le2)$ are defined for the compensated Poisson process, and the stochastic integral representation (analogous to the Ocone–Haussmann–Clark formula) is derived for the functionals from these spaces. The formula is given for the computation of the predictable projections of the stochastic derivatives of the above-mentioned functionals.

Anticipating integrals and martingales on the Poisson space

Let Ñ t be a standard compensated Poisson process on [0, 1]. We prove a new characterization of anticipating integrals of the Skorokhod type with respect to Ñ , and use it to obtain several counterparts to well established properties of semimartingale stochastic integrals. In particular we show that, if the integrand is sufficiently regular, anticipating Skorokhod integral processes with respect to Ñ admit a pointwise representation as usual Itô integrals in an independently enlarged filtration. We apply such a result to: (i) characterize Skorokhod integral processes in terms of products of backward and forward Poisson martingales, (ii) develop a new Itô-type calculus for anticipating integrals on the Poisson space, and (iii) write Burkholder-type inequalities for Skorokhod integrals.