System Of Stochastic Differential Equations Research Articles

Most probable transition paths in piecewise-smooth stochastic differential equations

We develop a path integral framework for determining most probable paths for a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin–Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an n-dimensional system with a switching manifold in the drift that forms an (n−1)−dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the drift and use Γ−convergence to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin–Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of the derived functional through two case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.

Convolutional neural network based simulation and analysis for backward stochastic partial differential equations

We develop a generic convolutional neural network (CNN) based numerical scheme to simulate the 2-tuple adapted strong solution to a unified system of backward stochastic partial differential equations (B-SPDEs) driven by Brownian motions, which can be used to model many real-world system dynamics such as optimal control and differential game problems. The dynamics of the scheme is modeled by a CNN through conditional expectation projection. It consists of two convolution parts: W layers of backward networks and L layers of reinforcement iterations. Furthermore, it is a completely discrete and iterative algorithm in terms of both time and space with mean-square error estimation and almost sure (a.s.) convergence supported by both theoretical proof and numerical examples. In doing so, we need to prove the unique existence of the 2-tuple adapted strong solution to the system under both conventional and Malliavin derivatives with general local Lipschitz and linear growth conditions.

A variational quantum algorithm for the Feynman-Kac formula

We propose an algorithm based on variational quantum imaginary time evolution for solving the Feynman-Kac partial differential equation resulting from a multidimensional system of stochastic differential equations. We utilize the correspondence between the Feynman-Kac partial differential equation (PDE) and the Wick-rotated Schrödinger equation for this purpose. The results for a (2+1) dimensional Feynman-Kac system obtained through the variational quantum algorithm are then compared against classical ODE solvers and Monte Carlo simulation. We see a remarkable agreement between the classical methods and the quantum variational method for an illustrative example on six and eight qubits. In the non-trivial case of PDEs which are preserving probability distributions – rather than preserving the ℓ2-norm – we introduce a proxy norm which is efficient in keeping the solution approximately normalized throughout the evolution. The algorithmic complexity and costs associated to this methodology, in particular for the extraction of properties of the solution, are investigated. Future research topics in the areas of quantitative finance and other types of PDEs are also discussed.

On first order mean field game systems with a common noise

We consider mean field games without idiosyncratic but with Brownian type common noise. We introduce a notion of solutions of the associated backward-forward system of stochastic partial differential equations. We show that the solution exists and is unique for monotone coupling functions. We also use the solution to find approximate optimal strategies (Nash equilibria) for N-player differential games with common but no idiosyncratic noise. An important step in the analysis is the study of the well-posedness of a stochastic backward Hamilton–Jacobi equation.

Modeling and analysis of the Haldane genetic model under Brownian motion using stochastic differential equation

Heterozygote advantage as a natural consequence of adaptation in diploid organisms is an attractive mechanism by which two alleles are maintained in natural populations. It has significant effects on biodiversity conservation and plant and animal breeding programs. The mathematical modeling of this biological mechanism is important for eco-evolutionary dynamics studies and genetics investigations. In this paper, I aimed to formalize the changes of gene frequency in time v(t), and in time and space v(t,x) with additive effects in a birth and death process of the Haldane genetic model using Brownian motion under fluctuations of habitat. In addition, the gene-environment interactions were evaluated under the mechanism. The mathematical model was investigated in both deterministic and white noise forms. It was shown that if the environmental random processes in the Haldane genetic model changed quickly and smoothly, then the diffusion approximation of the allele frequencies could be modeled and analyzed by a stochastic partial differential equation. It was revealed that the mathematical model used in this paper belonged to a more general model. The mathematical model was analyzed and since the modeling by the Cauchy problem had not had a usual global solution, the qualitative behavior of the solutions was considered. Besides, the generalizations of ItÔ integral were defined as the integrals of Wick products of random parameters and noise components. It was found that if v(t,x) behaved like a super-Brownian motion and the fatal mutations took place, as a consequence a tiny group of alleles was quickly disappeared. The v(t,x) was unstable when it was close to one. The stationary phase appeared and v(t,x) tended to the stationary situation in the intermediate region under the stabilizing selection. This was a condition under additive gene effect, but with the presence of dominance gene effect, it might be ambidirectional without considering the epistatic effects. The emergence of the dominance and epistatic effects was due to the directional selection. Since Falconer and MacKay had already introduced a deterministic model to study the frequency of genes with no spatial spreading of the population and no stochastic processes, another model was explained to study their equation in the case of heterozygote intermediate for diffusion approximation of frequency of genes, including white noise. It was shown that if the rates of mutation and selection became very small, then the model would be more deterministic and predictable. On the other hand, if the rates of mutation and selection became large, then the model would be more stochastic, and more fluctuations occurred because of the strong effective noise strength. In this case, the stationary situation did not take place. The outlook can help to model the similar biological mechanisms in eco-evolutionary community genetics for studying the indirect genetic effects via the systems of stochastic partial differential equations, and white noise calculus.

Zero-Inertia Limit: From Particle Swarm Optimization to Consensus-Based Optimization

Recently a continuous description of particle swarm optimization (PSO) based on a system of stochastic differential equations was proposed by Grassi and Pareschi in [Math. Models Methods Appl. Sci., 31 (2021), pp. 1625--1657] where the authors formally showed the link between PSO and the consensus-based optimization (CBO) through the zero-inertia limit. This paper is devoted to solving this theoretical open problem proposed in [S. Grassi and L. Pareschi, Math. Methods Appl. Sci., 31 (2021), pp. 1625--1657] by providing a rigorous derivation of CBO from PSO through the limit of zero inertia, and a quantified convergence rate is obtained as well. The proofs are based on a probabilistic approach by investigating both the weak and strong convergence of the corresponding stochastic differential equations of Mckean type in the continuous path space and the results are illustrated with some numerical examples.

Long-term properties of finite-correlation-time isotropic stochastic systems.

We consider finite-dimensional systems of linear stochastic differential equations ∂_{t}x_{k}(t)=A_{kp}(t)x_{p}(t), A(t) being a stationary continuous statistically isotropic stochastic process with values in real d×d matrices. We suppose that the laws of A(t) satisfy the large-deviation principle. For these systems, we find exact expressions for the Lyapunov and generalized Lyapunov exponents and show that they are determined in a precise way only by the rate function of the diagonal elements of A.

The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case

This study concentrates on the analysis of a stochastic SIC epidemic system with an enhanced and general perturbation. Given the intricacy of some impulses caused by external disturbances, we integrate the quadratic Lévy noise into our model. We assort the long-run behavior of a perturbed SIC epidemic model presented in the form of a system of stochastic differential equations driven by second-order jumps. By ameliorating the hypotheses and using some new analytical techniques, we find the exact threshold value between extinction and ergodicity (persistence) of our system. The idea and analysis used in this paper generalize the work of N. T. Dieu et al. (2020), and offer an innovative approach to dealing with other random population models. Comparing our results with those of previous studies reveals that quadratic jump-diffusion has no impact on the threshold value, but it remarkably influences the dynamics of the infection and may worsen the pandemic situation. In order to illustrate this comparison and confirm our analysis, we perform numerical simulations with some real data of COVID-19 in Morocco. Furthermore, we arrive at the following results: (i) the time average of the different classes depends on the intensity of the noise (ii) the quadratic noise has a negative effect on disease duration (iii) the stationary density function of the population abruptly changes its shape at some values of the noise intensity.Mathematics Subject Classification 2020: 34A26; 34A12; 92D30; 37C10; 60H30; 60H10.

On a stochastic nonlocal system with discrete diffusion modeling life tables

In this paper, we study a stochastic system of differential equations with nonlocal discrete diffusion. For two types of noises, we study the existence of either positive or probability solutions. Also, we analyze the asymptotic behavior of solutions in the long term, showing that under suitable assumptions they tend to a neighborhood of the unique deterministic fixed point. Finally, we perform numerical simulations and discuss the application of the results to life tables for mortality in Spain.

On the behaviour of solutions to a kind of third order neutral stochastic differential equation with delay

This article demonstrates the behaviour of solutions to a kind of nonlinear third order neutral stochastic differential equations. Setting x^{prime }(t)=y(t), y^{prime }(t) =z(t) the third order differential equation is ablated to a system of first order differential equations together with its equivalent quadratic function to derive a suitable downright Lyapunov functional. This functional is utilised to obtain criteria which guarantee stochastic stability of the trivial solution and stochastic boundedness of the nontrivial solutions of the discussed equations. Furthermore, special cases are provided to verify the effectiveness and reliability of our hypotheses. The results of this paper complement the existing decisions on system of nonlinear neutral stochastic differential equations with delay and extend many results on third order neutral and stochastic differential equations with and without delay in the literature.

Rare event asymptotics for exploration processes for random graphs

Large deviations for random graph models has been a topic of significant recent research activity. Much work in this area is focused on the class of dense random graph models (number of edges in the graph scale as n2, where n is the number of vertices) where the theory of graphons has emerged as a principal tool in the study of large deviation properties. These tools do not give a good approach to large deviation problems for random graph models in the sparse regime. The aim of this paper is to study an approach for large deviation problems in this regime by establishing large deviation principles (LDP) on suitable path spaces for certain exploration processes of the associated random graph sequence. Exploration processes are an important tool in the study of sparse random graph models and have been used to understand detailed asymptotics of many functionals of sparse random graphs, such as component sizes, surplus, deviations from trees, etc. In the context of rare event asymptotics of interest here, the point of view of exploration process transforms a large deviation analysis of a static random combinatorial structure to the study of a small noise LDP for certain stochastic dynamical systems with jumps. Our work focuses on one particular class of random graph models, namely the configuration model; however, the general approach of using exploration processes for studying large deviation properties of sparse random graph models has broader applicability. The goal is to study asymptotics of probabilities of nontypical behavior in the large network limit. The first key step for this is to establish a LDP for an exploration process associated with the configuration model. A suitable exploration process here turns out to be an infinite-dimensional Markov process with transition probability rates that diminish to zero in certain parts of the state space. Large deviation properties of such Markovian models is challenging due to poor regularity behavior of the associated local rate functions. Our proof of the LDP relies on a representation of the exploration process in terms of a system of stochastic differential equations driven by Poisson random measures and variational formulas for moments of nonnegative functionals of Poisson random measures. Uniqueness results for certain controlled systems of deterministic equations play a key role in the analysis. Next, using the rate function in the LDP for the exploration process we formulate a calculus of variations problem associated with the asymptotics of component degree distributions. The second key ingredient in our study is a careful analysis of the infinite-dimensional Euler–Lagrange equations associated with this calculus of variations problem. Exact solutions of these systems of nonlinear differential equations are identified which then provide explicit formulas for decay rates of probabilities of nontypical component degree distributions and related quantities.

An efficient computational scheme to solve a class of fractional stochastic systems with mixed delays

Providing effective numerical methods to approximate the solution of fractional order stochastic differential equations is of great importance, since the exact solution of this type of equations is not available in many cases. In this paper, a stepwise collocation method for solving a system of nonlinear stochastic fractional differential equations (NSFDEs) with mixed delays is presented. First, an approximation of the white noise term is considered and the convergence of the solution of the problem with this approximated white noise term to the solution of the main problem is proved. Then, a combination of a stepwise scheme and a Legendre collocation technique is introduced to solve the stochastic system. In each step, the problem is studied in a subdomain and the proposed method transforms the NSFDE with delays into a system of nonlinear algebraic equations. Moreover, the convergence analysis of the proposed numerical method is described. Two numerical test examples are provided to verify the efficiency of the numerical technique. Finally, a practical epidemic model is examined to show the applicability of this scheme.

Stochastic serotonin model with discontinuous drift

In this paper, motivated by the significance of serotonin in many areas of science, such as biology, neurology, psychiatry, among others, a mathematical serotonin model is proposed. This model incorporates deterministic as well as random influences on the serotonin dynamics. Because of that, the model is based on a system of stochastic differential equations with discontinuous drift coefficient. The main result of the paper is the existence and uniqueness of the solution of that system and due to the nature of the drift coefficient it requires nonstandard approach.

Numerical solution of the stochastic neural field equation with applications to working memory

The main goal of the present work is to investigate the effect of noise in some neural fields, used to simulate working memory processes. The underlying mathematical model is a stochastic integro-differential equation. In order to approximate this equation we apply a numerical scheme which uses the Galerkin method for the space discretization. In this way we obtain a system of stochastic differential equations, which are then approximated in two different ways, using the Euler–Maruyama and the Itô–Taylor methods. We apply this numerical scheme to explain how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events. Numerical examples are presented and their results are discussed.

Analysis and robust [formula omitted] control for systems of stochastic differential equations with piecewise constant arguments

This paper deals with systems of stochastic differential equations with piecewise constant arguments (SEPCA), where the arguments are of a delay type. To reduce the complexity of the proposed system, SEPCA is treated as a hybrid (or switched) system. This effective approach motivates the applicability of the classical theory of ordinary stochastic differential equations and the design of a switching law. We are mainly concerned with establishing some fundamental properties including the existence of a unique solution, mean-square (m.s.) stability and the stabilization of the system. The existence-uniqueness result is obtained under the Lipschitz condition and linear growth of the systems coefficients. Moreover, the stability property is developed by two techniques, namely the comparison principle and Razumikhin methodology in which we use the Lyapunov function rather than Lyapunov functional. To enhance these theoretical results, the problem of the input-to-state stabilization (ISS) is achieved by a robust H∞ control for systems with admissible parameter uncertainty and is subjected to time-varying disturbance input. Numerical examples with simulations are presented to clarify the proposed approaches.

Kernel learning backward SDE filter for data assimilation

In this paper, we develop a kernel learning backward SDE filter method to estimate the state of a stochastic dynamical system based on its partial noisy observations. A system of forward backward stochastic differential equations is used to propagate the state of the target dynamical model, and Bayesian inference is applied to incorporate the observational information. To characterize the dynamical model in the entire state space, we introduce a kernel learning method to learn a continuous global approximation for the conditional probability density function of the target state by using discrete approximated density values as training data. Numerical experiments demonstrate that the kernel learning backward SDE is highly effective.

Mean Field Models to Regulate Carbon Emissions in Electricity Production.

The most serious threat to ecosystems is the global climate change fueled by the uncontrolled increase in carbon emissions. In this project, we use mean field control and mean field game models to analyze and inform the decisions of electricity producers on how much renewable sources of production ought to be used in the presence of a carbon tax. The trade-off between higher revenues from production and the negative externality of carbon emissions is quantified for each producer who needs to balance in real time reliance on reliable but polluting (fossil fuel) thermal power stations versus investing in and depending upon clean production from uncertain wind and solar technologies. We compare the impacts of these decisions in two different scenarios: (1) the producers are competitive and hopefully reach a Nash equilibrium; (2) they cooperate and reach a social optimum. In the model, the producers have both time dependent and independent controls. We first propose nonstandard forward–backward stochastic differential equation systems that characterize the Nash equilibrium and the social optimum. Then, we prove that both problems have a unique solution using these equations. We then illustrate with numerical experiments the producers’ behavior in each scenario. We further introduce and analyze the impact of a regulator in control of the carbon tax policy, and we study the resulting Stackelberg equilibrium with the field of producers.

Stochastic perturbation of a cubic anharmonic oscillator

<p style='text-indent:20px;'>We perturb with an additive noise the Hamiltonian system associated to a cubic anharmonic oscillator. This gives rise to a system of stochastic differential equations with quadratic drift and degenerate diffusion matrix. Firstly, we show that such systems possess explosive solutions for certain initial conditions. Then, we carry a small noise expansion's analysis of the stochastic system which is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We then investigate the probabilistic properties of the sequence of coefficients which turn out to be the unique strong solutions of stochastic perturbations of the well-known Lamé's equation. We also obtain explicit expressions of these in terms of Jacobi elliptic functions. Furthermore, we prove, in the case of Brownian noise, a lower bound for the probability that the truncated expansion stays close to the solution of the deterministic problem. Lastly, when the noise is bounded, we provide conditions for the almost sure convergence of the global expansion.</p>

Balanced-Euler approximation schemes for stiff systems of stochastic differential equations

This paper aims to design new families of balanced-Euler approximation schemes for the solutions of stiff stochastic differential systems. To prove the mean-square convergence, we use some fundamental inequalities such as the global Lipschitz condition and linear growth bound. The meansquare stability properties of our new schemes are analyzed. Also, numerical examples illustrate the accuracy and efficiency of the proposed schemes.

Стохастична модель хижак-жертва, що залежить від щільності популяції хижака

The system of stochastic differential equations describing a nonautonomous stochastic density-dependent predator-prey model with Holling-type II functional response disturbed by white noise, centered and non-centered Poisson noises is considered. So, in this model we take into account levels of predator density dependence and jumps, corresponding to the centered and non-centered Poisson measures. The existence and uniqueness theorem for the positive, global (no explosions in the finite time) solution of the considered system is proved. We obtain sufficient conditions of stochastic ultimate boundedness and stochastic permanence in the considered stochastic predator-prey model.