Stochastic Functional Equations Research Articles

Analytical and Computational Investigations of Stochastic Functional Integral Equations: Solution Existence and Euler–Karhunen–Loève Simulation

This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn’s fixed-point theorem adapted for stochastic processes, a robust analytical framework is developed. Additionally, this paper introduces the Euler–Karhunen–Loève method, which utilizes the Karhunen–Loève expansion to represent stochastic processes, particularly suited for handling continuous-time processes with an infinite number of random variables. By conducting thorough analysis and computational simulations, which also involve implementing the Euler–Karhunen–Loève method, this paper effectively highlights the practical relevance of the proposed methodology. Two specific instances, namely, the Delay Cox–Ingersoll–Ross process and modified Black–Scholes with proportional delay model, are utilized as illustrative examples to underscore the effectiveness of this approach in tackling real-world challenges encountered in the realms of finance and stochastic dynamics.

Artificial neural networks to solve dynamic programming problems: A bias-corrected Monte Carlo operator

Artificial Neural Networks (ANNs) are powerful tools that can solve dynamic programming problems arising in economics. In this context, estimating ANN parameters involves minimizing a loss function based on the model's stochastic functional equations. In general, the expectations appearing in the loss function admit no closed-form solution, so numerical approximation techniques must be used. In this paper, I analyze a bias-corrected Monte Carlo operator (bc-MC) that approximates expectations by Monte Carlo. I show that the bc-MC operator is a generalization of the all-in-one expectation operator, already proposed in the literature. I demonstrate that, under some conditions on the primitives of the economic model, the bc-MC operator is the unbiased estimator of the loss function with the minimum variance. I propose a method to optimally set the hyperparameters defining the bc-MC operator, and illustrate the findings numerically with well-known economic models. I also demonstrate that the bc-MC operator can scale to high-dimensional models. With just approximately a minute of computing time, I find a global solution to an economic model with a kink in the decision function and more than 100 dimensions.

Application of fixed point theorem on the study of the existence of solutions in some fractional stochastic functional integral equations

Application of fixed point theorem on the study of the existence of solutions in some fractional stochastic functional integral equations

Large deviations for the stochastic functional integral equation with nonlocal condition

PurposeThe purpose of this paper is to study large deviations for the solution processes of a stochastic equation incorporated with the effects of nonlocal condition.Design/methodology/approachA weak convergence approach is adopted to establish the Laplace principle, which is same as the large deviation principle in a Polish space. The sufficient condition for any family of solutions to satisfy the Laplace principle formulated by Budhiraja and Dupuis is used in this work.FindingsFreidlin–Wentzell type large deviation principle holds good for the solution processes of the stochastic functional integral equation with nonlocal condition.Originality/valueThe asymptotic exponential decay rate of the solution processes of the considered equation towards its deterministic counterpart can be estimated using the established results.

On a unique solution and stability analysis of a class of stochastic functional equations arising in learning theory

Abstract Numerous computational and learning theory models have been studied using probabilistic functional equations. Especially in two-choice scenarios, the vast bulk of animal behavior research divides such situations into two different events. They split these actions into two possibilities according to the animals’ progress toward a particular decision. However, reward plays a crucial role in such experiments because, based on the selected side and the food placement, such scenarios may be classified into four distinct categories. This article aims to explore the animals’ reactions to such circumstances by presenting a generic stochastic functional equation. By using the well-known fixed point theory results, we examine the existence, uniqueness, and stability of solutions to the suggested functional equation. Moreover, an example is included to emphasize the significance of our findings.

On Solutions and Stability of Stochastic Functional Equations Emerging in Psychological Theory of Learning

We show how to apply the well-known fixed-point approach in the study of the existence, uniqueness, and stability of solutions to some particular types of functional equations. Moreover, we also obtain the Ulam stability result for them. The functional equations that we consider can be used to explain various experiments in mathematical psychology and arise in a natural way in the stochastic approach to the processes of perception, learning, reasoning, and cognition.

Optimal controls problems for some impulsive stochastic integro-differential equations with state-dependent delay

ABSTRACT In this paper, optimal control problems for a class of stochastic functional integral-differential equations in Hilbert spaces are investigated. First, the existence of mild solutions is investigated using stochastic analysis theory, fixed point theorems, and Grimmer's resolvent operator theory. Following that, the existence requirements of optimal pairs of systems governed by stochastic partial integro-differential equations with infinite delay are discussed. The results are achieved by the use of a combination of Lipschitz and Carathéodory conditions. At the end of the paper, an illustration is supplied to help highlight the key findings.

On a Unique Solution of the Stochastic Functional Equation Arising in Gambling Theory and Human Learning Process

The term “learning” is often used to refer to a generally stable behavioral change resulting from practice. However, it is a fundamental biological capacity far more developed in humans than in other living beings. In an animal or human being, the learning phase may often be viewed as a series of choices between multiple possible reactions. Here, we analyze a specific type of human learning process related to gambling in which a subject inserts a poker chip to operate a two-armed bandit device and then presses one of the two keys. Through the use of an electromagnet, one or more poker chips are given to the individual in a container located in the apparatus’s center. If a chip is provided, it is declared a winner; otherwise, it is considered a loser. The goal of this paper is to look at the subject’s actions in such situations and provide a mathematical model that is appropriate for it. The existence of a unique solution to the suggested human learning model is examined using relevant fixed point results.

Exponential behavior of nonlinear stochastic partial functional equations driven by Poisson jumps and Rosenblatt process

In this article, we discuss the Asymptotic behaviour of mild solutions of nonlinear stochastic partial functional equations driven by Poisson jumps and Rosenblatt process. The Banach fixed point theorem and the theory of resolvent operator devolped by Grimmer are used. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained results.

A unified fixed point approach to study the existence and uniqueness of solutions to the generalized stochastic functional equation emerging in the psychological theory of learning

<abstract><p>The model of decision practice reflects the evolution of moral judgment in mathematical psychology, which is concerned with determining the significance of different options and choosing one of them to utilize. Most studies on animals behavior, especially in a two-choice situation, divide such circumstances into two events. Their approach to dividing these behaviors into two events is mainly based on the movement of the animals towards a specific choice. However, such situations can generally be divided into four events depending on the chosen side and placement of the food. This article aims to fill such gaps by proposing a generic stochastic functional equation that can be used to describe several psychological and learning theory experiments. The existence, uniqueness, and stability analysis of the suggested stochastic equation are examined by utilizing the notable fixed point theory tools. Finally, we offer two examples to substantiate our key findings.</p></abstract>

The stability with the general decay rate of the solution for stochastic functional Navier-Stokes equations

<p style='text-indent:20px;'>This paper is concerned with the general stability of the solution to a stochastic functional 2D Navier-Stokes equation driven by a multiplicative white noise when the viscosity coefficient is time varying. First we give some sufficient conditions ensuring the existence and uniqueness of global solutions. Then the general stability of the solution in the sense of p-th (<inline-formula><tex-math id="M1">\begin{document}$ p\geq2 $\end{document}</tex-math></inline-formula>) moment is established. From this fact we further prove that the null solution is almost surely stable with the general decay rate. The convergence in probability of the solution is also analyzed.</p>

Application of fixed point theorem to solvability of functional stochastic integral equations

In this paper, the existence of the solution of a class of nonlinear stochastic functional integral equations is investigated. Such that the concept of measures of noncompactness, and Petryshyn’s fixed point theorem in Banach space has been used. In addition to proving the theorems of the existence of the solution, by solving some examples, it is shown that the method is efficient. We also give an example that satisfies the conditions of our main theorem, while the conditions described in some other papers do not.

Stochastic functional Kolmogorov equations, I: Persistence

This work (Part (I)) together with its companion (Part (II)) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the results, it seems to be instructive to divide our contributions to two parts. In contrast to the existing literature, our effort is to advance the knowledge by allowing delay and past dependence, yielding essential utility to a wide range of applications. A long-standing question of fundamental importance pertaining to biology and ecology is: What are the minimal necessary and sufficient conditions for long-term persistence and extinction (or for long-term coexistence of interacting species) of a population? Regardless of the particular applications encountered, persistence and extinction are properties shared by Kolmogorov systems. While there are many excellent treaties of stochastic-differential-equation-based Kolmogorov equations, the work on stochastic Kolmogorov equations with past dependence is still scarce. Our aim here is to answer the aforementioned basic question. This work, Part (I), is devoted to characterization of persistence, whereas its companion, Part (II) is devoted to extinction. The main techniques used in this paper include the newly developed functional Itô formula and asymptotic coupling and Harris-like theory for infinite dimensional systems specialized to functional equations. General theorems for stochastic functional Kolmogorov equations are developed first. Then a number of applications are examined covering, improving, and substantially extending the existing literature. Furthermore, our results reduce to that in the existing literature of Kolmogorov systems when there is no past dependence.

Stochastic functional Kolmogorov equations II: Extinction

This work, Part II, together with its companion-Part I develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the problems, it is natural to divide our contributions into two parts to answer a long-standing question in biology and ecology. What are the minimal conditions for long-term persistence and extinction of a population? Part I of our work provides characterization of persistence, whereas in this part, extinction is the main focus. The techniques used in this paper are combination of the newly developed functional Itô formula and a dynamic system approach. Compared to the study of stochastic Kolmogorov systems without delays, the main difficulty is that infinite dimensional systems have to be treated. The extinction is characterized after investigating random occupation measures and examining behavior of functional systems around boundaries. Our characterizations of long-term behavior of the systems reduce to that of Kolmogorov systems without delay when there is no past dependence. A number of applications are also examined.

On the solvability of a general class of a coupled system of stochastic functional integral equations

The main objective in this work is to find some weaker conditions that guarantee the existence of continuous solutions of some stochastic coupled systems of the Urysohn–Volterra–Itô–Doob type. The uniqueness of the solution for these systems is investigated as well. It is worth mentioning that the results in this work include many previously published results as special cases.

Leap-frog method for stochastic functional wave equations

Leap-frog method for stochastic functional wave equations

Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results.

Stochastic systems with memory and jumps

Stochastic systems with memory naturally appear in life science, economy, and finance. We take the modelling point of view of stochastic functional delay equations and we study these structures when the driving noises admit jumps. Our results concern existence and uniqueness of strong solutions, estimates for the moments and the fundamental tools of calculus, such as the Itô formula. We study the robustness of the solution to the change of noises. Specifically, we consider the noises with infinite activity jumps versus an adequately corrected Gaussian noise. The study is presented in two different frameworks: we work with random variables in infinite dimensions, where the values are considered either in an appropriate Lp-type space or in the space of càdlàg paths. The choice of the value space is crucial from the modelling point of view, as the different settings allow for the treatment of different models of memory or delay. Our techniques involve tools of infinite dimensional calculus and the stochastic calculus via regularisation.

Well-Posedness of the Cauchy Problem for Stochastic Evolution Functional Equations

We prove the existence, uniqueness, and continuous dependence on the initial data of the solutions of the Cauchy problem for stochastic evolution functional equations with random coefficients in Hilbert spaces. We propose a method for constructing an approximating sequence for the solution of the Cauchy problem and obtain an estimate for the rate of convergence to the exact solution.

Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHugh–Nagumo equation

In this work we introduce and analyse a stochastic functional equation, which contains both delayed and advanced arguments. This equation results from adding a stochastic term to the discrete FitzHugh–Nagumo equation which arises in mathematical models of nerve conduction. A numerical method is introduced to compute approximate solutions and some numerical experiments are carried out to investigate their dynamical behaviour and compare them with the solutions of the corresponding deterministic equation.