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In this paper, we are concerned with an operator-splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Lévy noise. More specifically, using a variant of the classical Kružkov doubling of variables approach, we show that the approximate solutions generated by the splitting scheme converge to the unique stochastic entropy solution of the underlying problems. Finally, the convergence analysis is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

In this study, we present a stochastic rumor model. The stability of the disease-free equilibrium state and instability of the free equilibrium E0{E}_{0} of stochastic epidemics model are considered with the help of Lyapunov functions. Sufficient conditions of persistence and extinction of rumor are given. Numerical simulations verify the analytical results.

We investigate a stochastic coupled model of the Cahn–Hilliard equations and the stochastic magnetohydrodynamic equations in a bounded domain of ℝ 2 {\mathbb{R}^{2}} . The model describes the flow of the mixture of two incompressible and immiscible fluids under the influence of an electromagnetic field with stochastic perturbations. We prove the existence, uniqueness and regularity of a probabilistic strong solution. The proof of the existence is based on the Galerkin approximation, the stopping time technique and some weak convergence principles in functional analysis. [ABSTRACT FROM AUTHOR]

In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 풪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball's idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a filtration sandwiched (in the sense of immersion) between two Brownian filtrations, is a weak Brownian filtration. [ABSTRACT FROM AUTHOR]

We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution. [ABSTRACT FROM AUTHOR]

The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) d ⁢ u ⁢ (t) = A ⁢ u ⁢ (t) ⁢ d ⁢ t + B ⁢ d ⁢ W ⁢ (t) , t > 0 , u ⁢ (0) = ξ . The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process { W ⁢ (t) : t ≥ 0 } . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q-Wiener processes. [ABSTRACT FROM AUTHOR]

In this paper, we have studied stochastic differential equations with unbounded delay in fractional power spaces perturbed by fractional Ornstein–Uhlenbeck process Y H , ξ ⁢ (t) with H ∈ (1 2 , 1) . Subsequently, the existence and uniqueness of mild solution of the considered equation have been proved with fixed-point theorem. Finally, we obtain the global attracting set of the considered equations by some stochastic analysis and inequality technique. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions and the approximate controllability for a class of stochastic elastic systems with structural damping and infinite delay in Hilbert spaces. The estimation of the control function is discussed, where the expression of the control function is constructed by the defined resolvent operator. Under this estimate, the existence of mild solutions for this system is obtained by the Schauder fixed point theorem and the stochastic analysis theory, and sufficient conditions for the approximate controllability are formulated and proved by using the so-called resolvent operator type condition. Finally, an example is given to illustrate the applicability of our conclusion.

In this paper, we prove the existence result for a mild solution of a fractional stochastic evolution inclusion involving the Caputo derivative in the Hilbert space driven by a fractional Brownian motion with the Hurst parameter H > 1 2 . The results are obtained by using fractional calculation, operator semigroups and the fixed point theorem for multivalued mappings. [ABSTRACT FROM AUTHOR]

In this article, the stochastic fractional Davey-Stewartson equations (SFDSEs) that result from multiplicative Brownian motion in the Stratonovich sense are discussed. We use two different approaches, namely the Riccati-Bernoulli sub-ordinary differential equations and sine-cosine methods, to obtain novel elliptic, hyperbolic, trigonometric, and rational stochastic solutions. Due to the significance of the Davey-Stewartson equations in the theory of turbulence for plasma waves, the discovered solutions are useful in explaining a number of fascinating physical phenomena. Moreover, we illustrate how the fractional derivative and Brownian motion affect the exact solutions of the SFDSEs using MATLAB tools to plot our solutions and display a number of three-dimensional graphs. We demonstrate how the multiplicative Brownian motion stabilizes the SFDSE solutions at around zero.

The optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferential and mixed fractional Brownian motion is investigated in this article. The deterministic nonlinear second-order controlled partial differential system is enriched with stochastic perturbations, non-instantaneous impulses, and Clarke subdifferential. In particular, the nonlinearities in the system that rely on the state of the solution are allowed to rely on the corresponding probability distribution of the state. The solvability of the considered system is discussed with the help of stochastic analysis, multivalued analysis, and multivalued fixed point theorem. Further, the existence of optimal control is established with the aid of Balder's theorem. Finally, an example is provided to illustrate the developed theory. [ABSTRACT FROM AUTHOR]

This paper concerns a class of fractional impulsive neutral functional differential equations with an infinite delay driven by the Rosenblatt process. A set of sufficient conditions are established for the existence of new mild solutions using fixed point theory. Finally, an illustrative example is provided to demonstrate the applicability of the theoretical result. [ABSTRACT FROM AUTHOR]

In this paper, we analyze multidimensional generalized backward stochastic differential equations with jumps in a filtration that supports a Brownian motion and an independent integer-valued random measure. Under monotonicity and linear growth assumptions on the coefficients, we give the existence and uniqueness of 핃 2 -solutions provided that the generators and the terminal condition satisfy some suitable integrability conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we take into account the stochastic Kuramoto-Sivashinsky equation forced by multiplicative noise in the Itô sense. To obtain the exact stochastic solutions of the stochastic Kuramoto-Sivashinsky equation, we apply the G′G\frac{{G}^{^{\prime} }}{G}-expansion method. Furthermore, we extend some previous results where this equation has not been previously studied in the presence of multiplicative noise. Also, we show the influence of multiplicative noise on the analytical solutions of the stochastic Kuramoto-Sivashinsky equation.

In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher-order Kirchhoff equation with variable coefficients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of Dk{{\mathcal{D}}}_{k}-absorbing sets of the stochastic dynamical system Φk{\Phi }_{k} is obtained, and the asymptotic compactness of Φk{\Phi }_{k} is proved by the decomposition method. Finally, the Dk{{\mathcal{D}}}_{k}-stochastic attractor family of the stochastic dynamical system Φk{\Phi }_{k} in Vm+k(Ω)×Vk(Ω){V}_{m+k}\left(\Omega )\times {V}_{k}\left(\Omega ) is obtained.

In this paper, we introduce a new method to analyze the convergence of the standard finite element method for Hamilton-Jacobi-Bellman equation with noncoercive operators with nonlinear source terms with the mixed boundary conditions. The method consists of combining Bensoussan-Lions algorithm with the characterization of the solution, in both the continuous and discrete contexts, as fixed point of contraction. Optimal error estimates are then derived, first between the continuous algorithm and its finite element counterpart and then between the continuous solution and the approximate solution.

In this paper, we study doubly reflected backward stochastic differential equations defined on probability spaces equipped with filtration satisfying only the usual assumptions of right continuity and completeness in the case where the barriers L and U are not necessarily right continuous. We suppose that the barriers L and U and their left limits are completely separated and we show existence and uniqueness of the solution. [ABSTRACT FROM AUTHOR]

This paper proves the existence and uniqueness theorem for generalized (reflected) backward stochastic differential equations under stochastic Lipschitz and monotone condition. The result is shown by using Picard's iteration, the Snell envelope theory and the penalization method. [ABSTRACT FROM AUTHOR]

In this paper, we consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the gradient of the log-likelihood (score function), which is an estimate that is unbiased even if the stochastic processes involved in the model must be discretized in time. To achieve this goal, we apply a doubly randomized scheme, that involves a novel coupled conditional particle filter (CCPF) on the second level of randomization. Our novel estimate helps facilitate the application of gradient-based estimation algorithms, such as stochastic-gradient Langevin descent. We illustrate our methodology in the context of stochastic gradient descent (SGD) in several numerical examples and compare with the Rhee–Glynn estimator. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate a class of nonlinear stochastic reaction-diffusion systems involving fractional Laplacian in a bounded domain. First, the existence and uniqueness of weak solutions are proved by using Galërkin’s method. Second, the existence of optimal controls for the corresponding stochastic optimal control problem is obtained. Finally, several examples are provided to demonstrate the theoretical results.

This article deals with the exact controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space under the assumption that the semigroup generated by the linear part is noncompact. Our main results are obtained by utilizing stochastic analysis technique, measure of noncompactness and the Mönch fixed point theorem. In the end, an example is presented to illustrate that our theorems guarantee the effectiveness of controllability results in the infinite dimensional spaces.

We prove a gradient estimate and the Hölder continuity of the gradient for the fundamental solution of a class of α-stable type operators with α ∈ (0, 1), which improve known results in the literature where the condition α > 1/2 is commonly assumed.

In this article, we are concerned with the neutral impulsive stochastic integro-differential equations driven by Poisson jumps and Rosenblatt process. By using resolvent operator and some analysis techniques, we ensure existence and uniqueness of solutions. Further, we investigate exponential stability of mild solutions. We have also given an example to illustrate our theoretical results.

In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {L_{\infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift. [ABSTRACT FROM AUTHOR]

We provide a large deviation principle on the stochastic differential equations with reflecting Wentzel boundary condition if δ ε {\frac{\delta}{\varepsilon}} tends to 0 when the two parameters δ (homogenization parameter) and ε (the large deviations parameter) tend to zero. Here, we suppose that the homogenization parameter converges sufficiently quickly more than the large deviations parameter. Furthermore, we will make explicit the associated rate function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimization problems of exact controllability for a new class of fractional impulsive partial stochastic differential systems with state-dependent delay in Hilbert spaces. By utilizing suitable fixed point approach without imposing severe compactness condition on the operators, the theory of analytic sectorial operators, stochastic analysis, and the Hausdorff measure of noncompactness, some sufficient conditions are derived for achieving the required results. Finally, an example is provided to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the simulation of the expectation for the solution of linear stochastic partial differential equation driven by the space-time white noise with the bounded measurable coefficient and different boundary conditions. We first propose a Monte Carlo type method for the expectation of the solution of a linear stochastic partial differential equation and prove an upper bound for its weak rate error. In addition, we prove the central limit theorem for the proposed method in order to obtain confidence intervals for it. As an application, the Monte Carlo scheme applies to the stochastic heat equation with various boundary conditions, and we provide the result of numerical experiments which confirm the theoretical results in this paper. [ABSTRACT FROM AUTHOR]

This paper focuses on a nonlinear second-order stochastic evolution equations driven by a fractional Brownian motion (fBm) with Poisson jumps and which is dependent upon a family of probability measures. The global existence of mild solutions is established under various growth conditions, and a related stability result is discussed. An example is presented to illustrate the applicability of the theory.

In this paper, we study two inverse problems for stochastic parabolic equations with additive noise. One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇. For this inverse problem, we obtain a conditional stability result. The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary. The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation. [ABSTRACT FROM AUTHOR]

We consider 3D Navier–Stokes–Voigt equations in smooth bounded domains with homogeneous Dirichlet boundary conditions. First, we study the existence and exponential stability of strong stationary solutions to the problem. Then we show that any unstable steady state can be exponentially stabilized by using either an internal feedback control with a support large enough or a multiplicative Itô noise of sufficient intensity. [ABSTRACT FROM AUTHOR]

In this article we show that a finite-dimensional stochastic differential equation driven by a Lévy noise can be formulated as a stochastic partial differential equation (SPDE) driven by the same Lévy noise. We prove the existence result for such an SPDE by Itô's formula for translation operators, and the uniqueness by an adapted form of "Monotonicity inequality", proved earlier in the diffusion case. As a consequence, the solutions that we construct have the "translation invariance" property. [ABSTRACT FROM AUTHOR]

In the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai's equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai's equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method. [ABSTRACT FROM AUTHOR]

This paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations. [ABSTRACT FROM AUTHOR]

In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time τ. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when τ is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided. [ABSTRACT FROM AUTHOR]

We investigate the effect of Ito noise on the stability of stationary solutions to a class of semilinear degenerate parabolic equations with the nonlinearity satisfying an arbitrary polynomial growth condition. We will show that an Ito noise of sufficient intensity will stabilize the unstable stationary solution. [ABSTRACT FROM AUTHOR]

In this paper we present an Lp-theory for a class of stochastic partial differential equations (SPDEs in abbreviation) driven by Lévy processes. The SPDEs under consideration can have random coefficients that depend both on the time and space variable. Existence and uniqueness of solutions in various Sobolev spaces are obtained. These Sobolev spaces describe the regularity of the solutions of the SPDEs. [ABSTRACT FROM AUTHOR]

In this paper we consider a white noise calculus for fractional Brownian motion with values in a separable Hilbert space, whereby the covariance operator Q is a kernel operator ( Q-fractional Brownian motion). We prove a Q-fractional version of the Itô's formula. Furthermore, we introduce Malliavin derivative for Q-fractional motion, prove a Q-fractional integration-by-parts formula and a Q-fractional version of the Itô isometry. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the existence, uniqueness and exponential stability of mild solutions for some stochastic neutral partial functional integrodifferential equations. We suppose that the linear part has a resolvent operator in the sense given in Grimmer [Transactions of the American Mathematical Society 273 (1982), 333-349]. The nonlinear part is assumed to be continuous and lipschitzian with respect to the second argument. Firstly, we study the existence of mild solutions. Secondly we give some results on the exponential stability in mean square sense. An example is provided to illustrate the results of this work. [ABSTRACT FROM AUTHOR]

Our work is concerned with a neural network with n nodes, where the activity of the k-th cell depends on external, stochastic inputs as well as the coupling generated by the activity of the adjacent cells, transmitted through a diffusion process in the network. This paper aims to throw some light on time-varying, stochastically perturbed, neuronal networks. We show that when the coefficients oscillate around a reference value, with oscillations that are almost periodic and suitably small in percentage, then there exists a unique solution for the system, that is almost periodic and uniformly bounded in the square-mean norm for all times. [ABSTRACT FROM AUTHOR]