Stochastic Differential Operators Research Articles

Theoretical and numerical analysis of a chaotic model with nonlocal and stochastic differential operators

A set of nonlinear ordinary differential equations has been considered in this paper. The work tries to establish some theoretical and analytical insights when the usual time-deferential operator is replaced with the Caputo fractional derivative. Using the Caratheodory principle and other additional conditions, we established that the system has a unique system of solutions. A variety of well-known approaches were used to investigate the system. The stochastic version of this system was solved using a numerical approach based on Lagrange interpolation, and numerical simulation results were produced.

On a SEIR-type model of COVID-19 using piecewise and stochastic differential operators undertaking management strategies

<abstract><p>In this work, an epidemic model of a susceptible, exposed, infected and recovered SEIR-type is established for the distinctive dynamic compartments and epidemic characteristics of COVID-19 as it spreads across a population with a heterogeneous rate. The proposed model is investigated using a novel approach of fractional calculus known as piecewise derivatives. The existence theory is demonstrated through the establishment of sufficient conditions. In addition, result related to Hyers-Ulam stability is also derived for the considered model. A numerical method based on modified Euler procedure is also constructed to simulate the approximate solutions of the proposed model by employing various values of fractional orders. We testified the numerical results by using real available data of Japan. In addition, some results for the SEIR-type model are also presented graphically using the stochastic process, and the obtained results are discussed.</p></abstract>

Wick calculus on spaces of regular generalized functions of Levy white noise analysis

Many objects of the Gaussian white noise analysis (spaces of test and generalized functions, stochastic integrals and derivatives, etc.) can be constructed and studied in terms of so-called chaotic decompositions, based on a chaotic representation property (CRP): roughly speaking, any square integrable with respect to the Gaussian measure random variable can be decomposed in a series of Ito's stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP (except the Gaussian and Poissonian particular cases). Nevertheless, there are different generalizations of this property. Using these generalizations, one can construct different spaces of test and generalized functions. And in any case it is necessary to introduce a natural product on spaces of generalized functions, and to study related topics. This product is called a Wick product, as in the Gaussian analysis.
 The construction of the Wick product in the Levy analysis depends, in particular, on the selected generalization of the CRP. In this paper we deal with Lytvynov's generalization of the CRP and with the corresponding spaces of regular generalized functions. The goal of the paper is to introduce and to study the Wick product on these spaces, and to consider some related topics (Wick versions of holomorphic functions, interconnection of the Wick calculus with operators of stochastic differentiation). Main results of the paper consist in study of properties of the Wick product and of the Wick versions of holomorphic functions. In particular, we proved that an operator of stochastic differentiation is a differentiation (satisfies the Leibniz rule) with respect to the Wick multiplication.

Bounded Operators of Stochastic Differentiation on Spaces of Nonregular Generalized Functions in the Lévy White Noise Analysis

Background. Operators of stochastic differentiation play an important role in the Gaussian white noise analysis. In particular, they can be used in order to study properties of the extended stochastic integral and of solutions of normally ordered stochastic equations. Although the Gaussian analysis is a developed theory with numerous applications, in problems of mathematics not only Gaussian random processes arise. In particular, an important role in modern researches belongs to Lévy processes. So, it is necessary to develop a Lévy analysis, including the theory of operators of stochastic differentiation.Objective. During recent years the operators of stochastic differentiation were introduced and studied, in particular, on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Lévy analysis. In this paper, we make the next step: introduce and study such operators on spaces of nonregular generalized functions.Methods. We use, in particular, the theory of Hilbert equipments and Lytvynov’s generalization of the chaotic representation property.Results. The main result is a theorem about properties of operators of stochastic differentiation.Conclusions. The operators of stochastic differentiation are considered on the spaces of nonregular generalized functions of the Lévy white noise analysis. This can be interpreted as a contribution in a further development of the Lévy analysis. Applications of the introduced operators are quite analogous to the applications of the corresponding ope­rators in the Gaussian analysis.

Operators of stochastic differentiation on spaces of nonregular generalized functions of Levy white noise analysis

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study some properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, in the same way as on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Levy white noise analysis. In the present paper we make the next natural step: introduce and study operators of stochastic differentiation on spaces of nonregular generalized functions of the Levy white noise analysis (i.e., on spaces of generalized functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions). In so doing, we use Lytvynov's generalization of the chaotic representation property. The researches of the present paper can be considered as a contribution in a further development of the Levy white noise analysis.

On operators of stochastic differentiation on spaces of regular test and generalized functions of Lévy white noise analysis

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. In this paper we introduce and study bounded and unbounded operators of stochastic differentiation in the Levy white noise analysis. More exactly, we consider these operators on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using the Lytvynov's generalization of the chaotic representation property. This gives a possibility to extend to the Levy white noise analysis and to deepen the corresponding results of the classical white noise analysis.

Low-temperature random matrix theory at the soft edge

“Low temperature” random matrix theory is the study of random eigenvalues as energy is removed. In standard notation, β is identified with inverse temperature, and low temperatures are achieved through the limit β → ∞. In this paper, we derive statistics for low-temperature random matrices at the “soft edge,” which describes the extreme eigenvalues for many random matrix distributions. Specifically, new asymptotics are found for the expected value and standard deviation of the general-β Tracy-Widom distribution. The new techniques utilize beta ensembles, stochastic differential operators, and Riccati diffusions. The asymptotics fit known high-temperature statistics curiously well and contribute to the larger program of general-β random matrix theory.

Invariant Representation for Stochastic Differential Operator by Bsdes with Uniformly Continuous Coefficients and its Applications

In this paper, we prove that a kind of second order stochastic differential operator can be represented by the limit of solutions of BSDEs with uniformly continuous coefficients. This result is a generalization of the representation for the uniformly continuous generator. With the help of this representation, we obtain the corresponding converse comparison theorem for the BSDEs with uniformly continuous coefficients, and get some equivalent relationships between the properties of the generator g and the associated solutions of BSDEs. Moreover, we give a new proof about g-convexity.

Clamped nano-beams as adsorption induced sensors: Linear and non-linear effects

This work analyzes the deflection of clamped nano-beam due to stochastic surface stresses, induced by adsorption/desorption of surrounding particles. Both linear and non-linear effects (mid-plane stretching) are considered. A mechanical model for 1D nano-beam is first introduced and includes the surface effects via their effective cross sectional residual force and moment. The model considers local non-stationary surface residual stresses, governed by Langmuir’s interaction model. Local adsorption relations are described by non-deterministic model, from which the statistics of the surface residual stresses are extracted and their time-space correlations are calculated. A straightforward perturbation method is used to evaluate the non-linear effects. In each order of the approximated solution, the nano-beam deflections are governed by a stochastic differential operator with a non-deterministic load. Equations are solved analytically by the Functional Perturbation method (FPM) and validated by Monte-Carlo simulations. It is found that the non-deterministic nature of the nano-beam deflections can be used for in situ sensing applications, in cases of very fast or slow adsorption schemes, for which the microscopic sensors are not sufficient. Geometric non-linear effects can be used in order to achieve fine tuning of the sensitivity.

Distributed Control of Uncertain Systems Using Superpositions of Linear Operators

Control in the natural environment is difficult in part because of uncertainty in the effect of actions. Uncertainty can be due to added motor or sensory noise, unmodeled dynamics, or quantization of sensory feedback. Biological systems are faced with further difficulties, since control must be performed by networks of cooperating neurons and neural subsystems. Here, we propose a new mathematical framework for modeling and simulation of distributed control systems operating in an uncertain environment. Stochastic differential operators can be derived from the stochastic differential equation describing a system, and they map the current state density into the differential of the state density. Unlike discrete-time Markov update operators, stochastic differential operators combine linearly for a large class of linear and nonlinear systems, and therefore the combined effects of multiple controllable and uncontrollable subsystems can be predicted. Design using these operators yields systems whose statistical behavior can be specified throughout state-space. The relationship to Bayesian estimation and discrete-time Markov processes is described.

Elements of a non-gaussian analysis on the spaces of functions of infinitely many variables

We present a review of some results of the non-Gaussian analysis in the biorthogonal approach and consider elements of the analysis associated with the generalized Meixner measure. The main objects of our interest are stochastic integrals, operators of stochastic differentiation, elements of theWick calculus, and related topics.

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.

Hydrofoil vibration induced by a random flow: a stochastic perturbation approach

The problem of an elastic lifting hydrofoil in a randomly perturbed flow is considered. It appears that in these conditions the phenomenon of hydroelastic-induced vibrations is controlled by a stochastic differential operator. By using the theory of stochastic perturbation, a technique of solution for this class of problems is proposed, leading to an effective numerical solution. The fundamentals of the method are given and it is applied to the general problem of hydroelastic vibrations. A numerical application to the case of an elastic control surface for a prototype-high- speed marine vehicle is presented. Comparisons between the results obtained by the Stochastic Perturbation Method (SPM) and those provided by standard Monte Carlo simulations (MCS) show the accuracy of the proposed method and a useful saving in computational time. A method is given for comparing the computational time required by the two methods, for a given statistical accuracy.

Statistical hydroelasticity of a wing in a random flow: A stochastic perturbation approach

The problem of an elastic lifting body in a randomly perturbed flow is considered. It appears that the phenomenon of hydro-elastic induced vibrations is controlled by a stochastic differential operator. By using the theory of stochastic perturbation, a new technique of solution for this class of problems is proposed leading to a very effective numerical solution.

An approach to the stochastic calculus in the non‐Gaussian case

We introduce and study a class of operators of stochastic differentiation and integration for non‐Gaussian processes. As an application, we establish an analog of the Itô formula.

Stochastic analysis of estuarine hydraulics

A new methodology is presented for the solution of the stochastic hydraulic equations characterizing steady, one-dimensional estuarine flow. The methodology is predicated on quasi-linearization, perturbation methods, and the finite difference approximation of the stochastic differential operators. Assuming Manning's roughness coefficient is the principal source of uncertainty in the model, stochastic equations are presented for the water depths and flow rates in the estuarine system. Moment equations are developed for the mean and variance of the water depths. The moment equations are compared with the results of Monte Carlo simulation experiments. The results confirm that for any spatial location in the estuary that (1) as the uncertainty in the channel roughness increases, the uncertainty in mean depth increases, and (2) the predicted mean depth will decrease with increasing uncertainty in Manning'sn. The quasi-analytical approach requires significantly less computer time than Monte Carlo simulations and provides explicit

Stochastic differential operator equations with random initial conditions

The iterative solution of the stochastic differential operator equation is considered for the case of accompanying random, rather than zero, initial conditions. The mean solution and the correlation are determined.

Invariant imbedding and the numerical determination of Green's functions

A Cauchy system for a Green's function is derived. This is investigated numerically, and applications to the determination of eigenvalues and stochastic differential operators are sketched.