One-dimensional Stochastic Differential Equations Research Articles

Task-oriented machine learning surrogates for tipping points of agent-based models

We present a machine learning framework bridging manifold learning, neural networks, Gaussian processes, and Equation-Free multiscale approach, for the construction of different types of effective reduced order models from detailed agent-based simulators and the systematic multiscale numerical analysis of their emergent dynamics. The specific tasks of interest here include the detection of tipping points, and the uncertainty quantification of rare events near them. Our illustrative examples are an event-driven, stochastic financial market model describing the mimetic behavior of traders, and a compartmental stochastic epidemic model on an Erdös-Rényi network. We contrast the pros and cons of the different types of surrogate models and the effort involved in learning them. Importantly, the proposed framework reveals that, around the tipping points, the emergent dynamics of both benchmark examples can be effectively described by a one-dimensional stochastic differential equation, thus revealing the intrinsic dimensionality of the normal form of the specific type of the tipping point. This allows a significant reduction in the computational cost of the tasks of interest.

One dimensional BSDEs with critical integrable terminal values and infinite time horizon

We mainly study the existence and uniqueness of solutions to one-dimensional backward stochastic differential equations with terminal values satisfying a critical integrability and infinite time horizon. The main result is established under the assumption that the generator f exhibits a time-varying monotonicity in y and uniform continuity in z and the terminal value is required to satisfy the critical Lexp(μ02log(1+L))-integrability for a μ0>0, a condition stronger than LlogL-integrability but weaker than Lp(p>1)-integrability.

Conditional Optimization of Algorithms for Estimating Distributions of Solutions to Stochastic Differential Equations

This article discusses an alternative method for estimating marginal probability densities of the solution to stochastic differential equations (SDEs). Two algorithms for calculating the numerical–statistical projection estimate for distributions of solutions to SDEs using Legendre polynomials are proposed. The root-mean-square error of this estimate is studied as a function of the projection expansion length, while the step of a numerical method for solving SDE and the sample size for expansion coefficients are fixed. The proposed technique is successfully verified on three one-dimensional SDEs that have stationary solutions with given one-dimensional distributions and exponential correlation functions. A comparative analysis of the proposed method for calculating the numerical–statistical projection estimate and the method for constructing the histogram is carried out.

One-dimensional SDEs with LPS-type singular drift coefficients and Hölder continuous diffusion coefficients

In this paper, under the assumptions of LPS-type singular drift coefficients and Hölder continuous diffusion coefficients, the existence, uniqueness, stability, non-confluent property and strong Feller property for the solution to a one-dimensional stochastic differential equations are obtained. Our proof are based on the tools of local time and Zvonkin's transform.

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

On the limit distribution for stochastic differential equations driven by cylindrical non-symmetric α-stable Lévy processes

This paper deals with the limit distribution for a stochastic differential equation driven by a non-symmetric cylindrical [Formula: see text]-stable process. Under suitable conditions, it is proved that the solution of this equation converges weakly to that of a stochastic differential equation driven by a Brownian motion in the Skorohod space as [Formula: see text]. Also, the rate of weak convergence, which depends on [Formula: see text], for the solution towards the solution of the limit equation is obtained. For illustration, the results are applied to a simple one-dimensional stochastic differential equation, which implies the rate of weak convergence is optimal.

Sticky nonlinear SDEs and convergence of McKean–Vlasov equations without confinement

AbstractWe develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.

Stochastic differential equations with discontinuous diffusion coefficients

We study one-dimensional stochastic differential equations of the form d X t = σ ( X t ) d Y t dX_t = \sigma (X_t)dY_t , where Y Y is a suitable Hölder continuous driver such as the fractional Brownian motion B H B^H with H > 1 2 H>\frac 12 . The innovative aspect of the present paper lies in the assumptions on diffusion coefficients σ \sigma for which we assume very mild conditions. In particular, we allow σ \sigma to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.

The Large Deviation of Semi-linear Stochastic Partial Differential Equation Driven by Brownian Sheet

We prove the the large deviation principle(LDP) for the law of the one-dimensional semilinear stochastic partial differential equations driven by nonlinear multiplicative noise. Firstly, combining the energy estimate and approximation procedure, we obtain the existence of global solution. Then the large deviation principle is obtained via weak convergence method.

On weak convergence of stochastic differential equations with irregular coefficients

One-dimensional Ito stochastic differential equations and stochastic differential equations with local time that depend on a small parameter ε and have irregular coefficients (for example, the unlimited drift coefficient or the non-Lipschitz diffusion coefficient) have been considered. The available results concerning the conditions for a mutually univocal correspondence between stochastic Ito equations and stochastic equations with local time were generalized. The weak convergence of the solutions of those equations at ε → 0 were analyzed. The form of the coefficients for the limiting process was obtained. The necessary and sufficient conditions for the weak convergence of the solutions of those equations to the limit random process were proved.

On weak convergence of stochastic differential equations with irregular coefficients

One-dimensional Ito stochastic differential equations and stochastic differential equations with local time that depend on a small parameter and have irregular coefficients (for example, the unlimited drift coefficient or the non-Lipschitz diffusion coefficient) have been considered. The available results concerning the conditions for a mutually univocal correspondence between stochastic Ito equations and stochastic equations with local time were generalized. The weak convergence of the solutions of those equations at $\varepsilon \to 0$ were analyzed. The form of the coefficients for the limiting process was obtained. The necessary and sufficient conditions for the weak convergence of the solutions of those equations to the limit random process were proved.

The truncated Euler–Maruyama method of one-dimensional stochastic differential equations involving the local time at point zero

Abstract Recently, Mao developed a new explicit method, called the truncated Euler–Maruyama method for nonlinear SDEs, and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The key aim of this paper is to establish the rate of strong convergence of the truncated Euler–Maruyama method for one-dimensional stochastic differential equations involving that the local time at point zero under the drift coefficient satisfies a one-sided Lipschitz condition and plus some additional conditions.

Spiking and collapsing in large noise limits of SDEs

We analyze the strong noise limit of one-dimensional stochastic differential equations (SDEs). Our initial motivation comes from continuous measurements of open quantum systems. In this context, Bauer, Bernard and Tilloy pointed out an intriguing behavior. As the noise grows larger, the solutions exhibit locally a collapsing, that is to say, converge to pure jump processes very reminiscent of a metastability phenomenon. But surprisingly the limiting jump process is decorated by a spike process. We give a precise meaning to the convergence and completely prove these statements for a large class of one-dimensional diffusions, thanks to a robust strategy of proof.

Stationary distribution and extinction of a stochastic HLIV model with viral production and Ornstein–Uhlenbeck process

In the past few decades, the human immunodeficiency virus (HIV), as a deadly pathogen, has imposed great harmful effects on human health. In addition, from the viewpoint of the microscopic, the interference of random factor exists in the process of virus replication. However, there are few works devoted to the theoretical studies of viral infection models with biologically reasonable stochastic effects yet. To study the effects of stochastic perturbations on the pathogenesis process of HIV/AIDS, in this paper, we develop and study a stochastic HLIV model with viral production and Ornstein–Uhlenbeck process, in which we assume that the conversion rate from an eclipse-phase cell to an infected cell satisfies the Ornstein–Uhlenbeck process. Firstly, by using the existence and uniqueness theory of solutions, we prove that there is a unique global solution to the stochastic model for any initial value. Then we use stochastic Lyapunov function methods and Fatou lemma to establish sufficient conditions for the existence of a stationary distribution of positive solutions to the stochastic system, which reflects the strong persistence of the disease. In addition, by adopting the comparison theorem of one-dimensional stochastic differential equation and the theory of ergodicity, we obtain sufficient conditions for complete eradication of the infected cells and free viruses. Our theoretical results show that the bigger speed of reversion or the smaller intensity of volatility is conducive to clearing infected cells and free viruses while the larger intensity of volatility or the smaller speed of reversion is not. We also find that this way of introducing stochastic perturbations is reasonable from the perspective of mathematics and biology. Finally, numerical simulations are carried out to demonstrate our main results. The modeling and methods can be applied to study other multigroup viral infection models or epidemic model with Ornstein–Uhlenbeck process, such as S-DI-A model, DS-I-A model, DS-DI-A model, SEI epidemic model, et al.

Discussion on applicability of the complex fractional moment method in Hamiltonian system

In this paper, the complex fractional moment (CFM) method is extended from the Langevin system to the Hamiltonian system for the first time, the applicability and advantages of the CFM method in Hamiltonian systems are investigated. According to the quasi-periodic property, the stochastic averaging technique is applied to the stochastic quasi-nonintegrable Hamiltonian system, and the one-dimensional Hamiltonian stochastic differential equation is obtained. By Mellin transformation, the associated Fokker Planck Kolmogorov (FPK) equation governing the Hamilton function is transformed into a set of Ordinary Differential Equations (ODEs) of CFM. The semi-analytical solutions of the FPK equation are obtained by solving the ODEs. Furthermore, by a polynomial approximation method for the Hamiltonian case proposed in this paper, the numerical results show the influence of parameters and initial values on the evolution of the transient probability density functions (PDF) of the system. The accuracy of CFM method in the Hamiltonian case is verified by error analysis.

On density functions related to discrete time maximum of some one-dimensional diffusion processes

This article focuses on the probability density functions related to the discrete time maximum of some one-dimensional diffusion processes. Firstly, we shall consider solutions of one-dimensional stochastic differential equations and prove an integration by parts formula on the discrete time maximum of the solutions. The smoothness, expressions and upper bounds of the density function will be obtained by the formula. Secondly, Gaussian processes will be dealt with. For some Gaussian processes, we shall obtain asymptotic behaviors of the density functions related to the discrete time maximum of the processes. The Malliavin calculus and Laplace’s method play important roles for the proofs.

LAMN property for jump diffusion processes with discrete observations on a fixed time interval

We consider a one-dimensional stochastic differential equation with jumps driven by a Brownian motion and an independent Lévy process with finite Lévy measure, whose drift and diffusion coefficients depend on an unknown parameter. Under smoothness and non-degeneracy assumptions on the drift and diffusion coefficients and integrability assumption of jump size distribution, we prove the Local Asymptotic Mixed Normality property when the solution process is observed discretely at high frequency on a fixed time interval. The proof is essentially based on Malliavin calculus techniques and an analysis of the jump structure of the Lévy process.

Statistical Learning of Nonlinear Stochastic Differential Equations from Nonstationary Time Series using Variational Clustering

Parameter estimation for non-stationary stochastic differential equations (SDE) with an arbitrary non-linear drift and non-linear diffusion is accomplished in combination with a non-parametric clustering methodology. Such a model-based clustering approach includes a quadratic programming (QP) problem with equality and inequality constraints. We couple the QP problem to a closed-form likelihood function approach based on suitable Hermite-expansions to approximate the parameter values of the SDE model. The classification problem provides a smooth indicator function, which enables us to recover the underlying temporal parameter modulation of the one-dimensional SDE. As shown by the numerical examples, the clustering approach recovers a hidden functional relationship between the SDE model parameters and an additional auxiliary process. The study builds upon this functional relationship to develop closed-form, non-stationary, data-driven stochastic models for multiscale dynamical systems in real-world applications.

A numerical scheme for stochastic differential equations with distributional drift

In this paper we introduce a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a convergence rate in a suitable L1-norm and, as a by-product, a convergence rate for a numerical scheme applied to SDEs with drift in Lp-spaces with p∈(1,∞).

Quadratic [formula omitted]-BSDEs with convex generators and unbounded terminal conditions

In this paper, we first study one-dimensional quadratic backward stochastic differential equations driven by G-Brownian motions (G-BSDEs) with unbounded terminal values. With the help of a θ-method of Briand and Hu (2008) and nonlinear stochastic analysis techniques, we propose an approximation procedure to prove existence and uniqueness result when the generator is convex (or concave) and terminal value is of exponential moments of arbitrary order. Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with diagonally quadratic generators.