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We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large deviation principle of the stochastic reaction-diffusion lattice systems defined on the $ N $-dimensional integer set, where the nonlinear drift term is locally Lipschitz continuous with polynomial growth of any degree and the nonlinear diffusion term is locally Lipschitz continuous with linear growth. We first prove the convergence of the solutions of the controlled stochastic lattice systems, and then establish the large deviations by the weak convergence method based on the equivalence of the large deviation principle and the Laplace principle. [ABSTRACT FROM AUTHOR]

We provide a unified treatment of pathwise large and moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution form. Our methodology is based on the weak convergence approach by Budhiraja and Dupuis (2019); Dupuis and Ellis (1997). We show in particular how this framework encompasses most rough volatility models used in mathematical finance, yields pathwise moderate deviations for the first time and generalises many recent results in the literature. [ABSTRACT FROM AUTHOR]

We consider a nonlinear filtering problem of multiscale non-Gaussian signal processes and observation processes with jumps. First, we prove that the dimension for the signal system can be reduced by a homogenized approach. Second, convergence of the corresponding nonlinear filtering to the homogenized filtering is shown by a weak convergence technique. Finally, we give an example to explain our result. [ABSTRACT FROM AUTHOR]

We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments. [ABSTRACT FROM AUTHOR]

Stochastic models of chemical reaction networks are an important tool to describe and analyze noise effects in cell biology. When chemical species and reaction rates in a reaction system have different orders of magnitude, the associated stochastic system is often modeled in a multiscale regime. It is known that multiscale models can be approximated with a reduced system such as mean field dynamics or hybrid systems, but the accuracy of the approximation remains unknown. In this paper, we estimate the probability distribution of low copy species in multiscale stochastic reaction systems under a short timescale. We also establish an error bound for this approximation. Throughout the paper, typical mass-action systems are mainly handled, but we also show that the main theorem can be extended to general kinetics, which generalizes existing results in the literature. Our approach is based on a direct analysis of the Kolmogorov equation, in contrast to classical approaches in the existing literature. [ABSTRACT FROM AUTHOR]

This paper focuses on stability of stochastic hybrid systems with random delay driven by a singularly perturbed Markov chain. By weak convergence and the martingale method, the limit system is obtained. Using the limit system as a bridge, this paper establishes the moment exponential stability of the original system by Razumikhin-type techniques. Finally, an example is given to illustrate this result. [ABSTRACT FROM AUTHOR]

We consider a slow–fast stochastic differential system with Lévy noise. We will employ the perturbed test function method to study the normal deviation of the slow–fast system. Our main result states that the deviation can be approximated by a Gaussian process and the central limit theorem is obtained for the system. [ABSTRACT FROM AUTHOR]

In many problems arising in statistical physics, statistical mechanics, and many related fields, one needs to deal with such nonlinear stochastic differential equations as Ginzburg–Landau equations and Lotka–Volterra equations, etc. Such equations all belong to the class of stochastic Kolmogorov systems. Because of their importance and wide range of applications, these systems have received much attention in recent years. Devoted to stochastic Kolmogorov systems, in contrast to the usual setup of using a Brownian motion as a driving force, in this paper, the underlying system is assumed to be subject to wideband type of noise perturbations. The main thought is that Brownian motion is an idealization used in a wide range of applications, whereas the wideband noise processes is much easier to be realized in the actual applications. Although it is a good approximation to a diffusion process, the process under wideband noise becomes non-Markovian. Using weak convergence methods, we show that the limits are the desired Kolmogorov systems driven by Brownian motions. • In contrast to the existing work, this paper treats Kolmogorov systems under wideband noise perturbations. • This paper shows under suitable conditions, the systems under wideband noises converge weakly to systems driven by a white noise using a martingale problem formulation. • The systems under wideband noises are generally non-Markovian, which is suited for the actual systems. The limit, however, becomes Markovian. [ABSTRACT FROM AUTHOR]