Theorem For Stochastic Differential Equations Research Articles

Physics-Informed Representation and Learning: Control and Risk Quantification

Optimal and safety-critical control are fundamental problems for stochastic systems, and are widely considered in real-world scenarios such as robotic manipulation and autonomous driving. In this paper, we consider the problem of efficiently finding optimal and safe control for high-dimensional systems. Specifically, we propose to use dimensionality reduction techniques from a comparison theorem for stochastic differential equations together with a generalizable physics-informed neural network to estimate the optimal value function and the safety probability of the system. The proposed framework results in substantial sample efficiency improvement compared to existing methods. We further develop an autoencoder-like neural network to automatically identify the low-dimensional features in the system to enhance the ease of design for system integration. We also provide experiments and quantitative analysis to validate the efficacy of the proposed method. Source code is available at https://github.com/jacobwang925/path-integral-PINN.

A Support Theorem for Stochastic Differential Equations Driven by a Fractional Brownian Motion

A Support Theorem for Stochastic Differential Equations Driven by a Fractional Brownian Motion

Domain recurrence and probabilistic analysis of residence time of stochastic systems and domain aiming control

SummaryThe problem of domain aiming control is formulated for a controlled stochastic nonlinear system, which involves regularity of the solution to the resulting closed‐loop stochastic system. To begin with, an extended existence and uniqueness theorem for stochastic differential equations with local Lipschitz coefficients is proven by using a Lyapunov‐type function. A Lyapunov‐based sufficient condition on nonregularity is also given. The notions of domain recurrence and residence time for stochastic nonlinear systems are introduced, and various criteria for the recurrence and nonrecurrence relative to a bounded open domain or an unbounded domain are provided. Furthermore, upper bounds of expectation and moment‐generating function of the residence time are derived. In particular, a connection between the mean residence time and a Dirichlet problem is investigated and illustrated with a numerical example. Finally, the problem of domain aiming control is considered for certain types of nonlinear and linear stochastic systems. Several examples are provided to illustrate the theoretical results.

On Stability and Comparison Theorems for Systems of Stochastic Differential Equations

We prove comparison theorems for stochastic differential equations (briefly, SDE) with respect to a standard multidimensional Wiener process as well as for components of systems of SDE with respect to a multidimensional Wiener process. The obtained results are applied to the study of the stability with probability 1 of the perturbed solutions to the SDE.

Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

This paper investigates dynamic behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte (CTL) to the growth of an immunogenic tumour. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs) and strong ergodicity theorem. Firstly, we prove the existence and uniqueness of the global positive solution for the tumor-immune system. Then we go a further step to study the boundaries of moments for tumor cells and effector cells and the asymptotic behavior in the boundary equilibrium points. Furthermore, we discuss the existence and uniqueness of stationary distribution and stochastic permanence of the tumor-immune system. Finally, we give several examples and numerical simulations to verify our results.

Support Theorem for Stochastic Differential Equations with Sobolev Coefficients

In this paper we prove a support theorem for stochastic differential equations with Sobolev coefficients in the framework of DiPerna-Lions theory.

Extinction in stochastic predator-prey population model with Allee effect on prey

This paper presents the analysis of the conditions which lead the stochastic predator-prey model with Allee effect on prey population to extinction. In order to find these conditions we first prove the existence and uniqueness of global positive solution of considered model using the comparison theorem for stochastic differential equations. Then, we establish the conditions under which extinction of predator and prey populations occur. We also find the conditions for parameters of the model under which the solution of the system is globally attractive in mean. Finally, the numerical illustration with real life example is carried out to confirm our theoretical results.

Competition in the chemostat: A stochastic multi-species model and its asymptotic behavior

In this paper, a stochastic chemostat model in which n-species compete for a single growth-limiting substrate is considered. We first prove that the stochastic model has an unique global positive solution by using the comparison theorem for stochastic differential equations. Then we show that when the noise intensities are small, the competition outcome in the chemostat is completely determined by the species’ stochastic break-even concentrations: the species with the lowest stochastic break-even concentration survives and all other species will go to extinction in the chemostat. In other words, the competitive exclusion principle holds for stochastic competition chemostat model when the noise intensities are small. Moreover, we find that noise may change the destiny of the species. Numerical simulations illustrate the obtained results.

On the solvability of nonautonomous stochastic differential equations with current velocities

Under natural conditions, we prove an existence theorem for stochastic differential equations with current velocities (mean derivatives) and with nonautonomous right-hand side.

Favard separation method for almost periodic stochastic differential equations

Favard separation method is an important means to study almost periodic solutions to linear differential equations; later, Amerio applied Favard's idea to nonlinear differential equations. In this paper, by appropriate choosing separation and almost periodicity in distribution sense, we obtain the Favard and Amerio type theorems for stochastic differential equations.

The Yamada-Watanabe theorem for mild solutions to stochastic partial differential equations

We prove the Yamada-Watanabe theorem for semilinear stochastic partial differential equations with path-dependent coefficients. The so-called method of the moving frame allows us to reduce the proof to the Yamada-Watanabe theorem for stochastic differential equations in infinite dimensions.

A Peano-Like Theorem for Stochastic Differential Equations with Nonlocal Sample Dependence

The existence of a not-necessarily-unique strong solution for a stochastic differential equations with nonlocal sample dependence is established under the assumption that the coefficients satisfy an asymptotically local boundedness condition in addition to continuity. The proof is by an Euler-like construction of approximations. These equations include mean-field stochastic differential equations, but the nonlocal sample dependence can be more general than just the dependence on moments of the solution.

Existence and Uniqueness Theorem for Stochastic Differential Equations with Self-Exciting Switching

We introduce a new kind of equation, stochastic differential equations with self-exciting switching. Firstly, we give some preliminaries for this kind of equation, and then, we get the main results of our paper; that is, we gave the sufficient condition which can guarantee the existence and uniqueness of the solution.

Limit Theorems for Stochastic Differential Equations with Discontinuous Coefficients

In this article we prove a transference principle for stochastic differential equations (SDEs) with discontinuous coefficients. In particular, the well-posedness of SDEs and Fokker–Planck equations with irregular coefficients is established.

Barbashin-Krasovskii theorem for stochastic differential equations

A system of stochastic differential equations which has a zero solution is considered. It is assumed that there exists a positive definite function such that the corresponding operator is nonpositive. It is proved that if the set does not include entire semitrajectories of the system almost surely, then the zero solution is asymptotically stable in probability.

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.

Some New Aspects of Lyapunov-Type Theorems for Stochastic Differential Equations of Neutral Type

Some new Lyapunov-type theorems for stochastic differential equations of neutral type are proved. It is shown that these theorems simplify an application of Kolmanovskii and Shaikhet's general method of Lyapunov functionals construction for stability investigation of different mathematical models.

A generalization of Mil’shtein’s theorem for stochastic differential equations

A generalization of Mil’shtein’s theorem for stochastic differential equations

A martingale on the zero-set of a holomorphic function

We give a simple probabilistic proof of the classical fact from complex analysis that the zeros of a holomorphic function of several variables are never isolated and that they are not contained in any compact set. No facts from complex analysis are assumed other than the Cauchy-Riemann definition. From stochastic analysis only the Ito formula and the standard existence theorem for stochastic differential equations are required.

Almost Sure Comparisons for First Passage Times of Diffusion Processes through Boundaries

Conditions on the boundary and parameters that produce ordering in the first passage time distributions of two different diffusion processes are proved making use of comparison theorems for stochastic differential equations. Three applications of interest in stochastic modeling are presented: a sensitivity analysis for diffusion models characterized by means of first passage times, the comparison of different diffusion models where first passage times represent an important feature and the determination of upper and lower bounds for first passage time distributions.