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The two- and three-dimensional incompressible backward stochastic convective Brinkman–Forchheimer (BSCBF) equations on a torus driven by Lévy noise are considered in this paper. A-priori estimates for adapted solutions of the finite-dimensional approximation of 2D and 3D BSCBF equations are obtained. For a given terminal data, the existence and uniqueness of pathwise adapted strong solutions is proved by using a standard Galerkin (or spectral) approximation technique and exploiting the monotonicity arguments. We also establish the continuity of the adapted solutions with respect to the terminal data. The above results are obtained for the absorption exponent r ∈ [ 1 , ∞) for d = 2 and r ∈ [ 3 , ∞) for d = 3 , and any Brinkman coefficient μ > 0 , Forchheimer coefficient β > 0 , and hence the 3D critical case ( r = 3 ) is also handled successfully. We deduce analogous results for 2D backward stochastic Navier–Stokes equations perturbed by Lévy noise also. [ABSTRACT FROM AUTHOR]

Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study a stochastic process from a macro point of view, and thus, periodic solution of a stochastic process in distributional sense is introduced. We first give the definition and then establish the existence of periodic solution on bounded domain. Lastly, for the case that probability density function exists, we obtain the existence periodic solutions of the probability density function corresponding to the stochastic process by using the technique of deterministic partial differential equations. [ABSTRACT FROM AUTHOR]

By introducing some parameters perturbed by white noises, we propose a class of stochastic inertial neural networks in random environments. Constructing two Lyapunov–Krasovskii functionals, we establish the mean-square exponential input-to-state stability on the addressed model, which generalizes and refines the recent results. In addition, an example with numerical simulation is carried out to support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a two-group SVIR epidemic model with random effect. Firstly, the two-group SVIR epidemic model with random perturbation of natural death rate is established. The existence and uniqueness of positive solution are proved by using stopping time theory and the Lyapunov analysis method. Secondly, a property of the system solution is obtained by using the law of strong numbers and the continuous local martingale. Finally, a new combination of Lyapunov functions is applied. The solution of the model we obtained is oscillating around a steady state if the basic reproduction number is less than one, which is the disease-free equilibrium of the corresponding deterministic model. A numerical simulation is presented to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics property of a stochastic HIV model with Beddington–DeAngelis functional response. It has a unique uninfected steady state. We prove that the model has a unique global positive solution. Furthermore, if the basic reproductive number is not larger than 1, the asymptotic behavior of the solution is stochastically stable. Otherwise, it fluctuates randomly around the infected steady state of the corresponding deterministic HIV model. Finally, some numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we use an Ornstein–Uhlenbeck process to describe the environmental stochasticity and propose a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. For each species, sharp sufficient conditions for persistence in the mean and extinction are respectively obtained. The results demonstrate that the persistence and extinction of the species have close relationships with the environmental stochasticity. In addition, the theoretical results are numerically illustrated by some simulations. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a version of Itô's formula for the backwards Itô-Henstock integral of an operator-valued stochastic process. Itô's formula is the stochastic analogue of the change of variable for deterministic integrals. [ABSTRACT FROM AUTHOR]

In this paper, a two prey and one predator model with logistic growth in prey species have been considered in presence of stochastic perturbation. The cooperation among the preys against predation is assumed. Necessary conditions for asymptotic behavior around the equilibrium points of its deterministic system with and without stochasticity have been discussed. Finally, to illustrate the analytical results, numerical simulations have been performed. [ABSTRACT FROM AUTHOR]

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512-5524 2011), and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in Lp) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370-384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exponential input-to-state stability for a class of stochastic neural networks with mixed delays. Based on a new Lyapunov-Krasovskii functional, Itô’s formula, Dynkin’s formula and some inequality techniques, some novel sufficient conditions ensuring the exponential input-to-state stability in the mean square for the given stochastic neural networks are derived. Some existing results are extended. Two numerical examples are provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The authors consider the problem of the existence, with probability 1, of Green’s function of a stochastic m-point Cauchy problem for a higher-order parabolic equation with white noise perturbations taken with negative values only. Estimates are obtained in spaces of functions whose norm contains the mathematical expectation. [ABSTRACT FROM AUTHOR]

With the methods of H2/H∞-control theory, in the presence of noise we solve the optimization problem for a multiplicative stochastic system with several external disturbances (the multiperturbation problem) and vector Wiener processes with arbitrary intensity matrices. We obtain matrix differential equations of Riccati type, reducing the original optimization problem to solving these equations. [ABSTRACT FROM AUTHOR]

We propose a new stochastic competition chemostat system with saturated growth rate and impulsive toxicant input. The main purpose of this paper is to study the stochastic dynamics of a high-dimensional impulsive stochastic chemostat model and find the threshold between persistence and extinction for the impulsive stochastic chemostat system. First, we investigate the stability of the periodic solution of a deterministic impulsive chemostat model and obtain the threshold between persistence and extinction for the system. Second, by using qualitative analysis method of impulsive stochastic differential equations, we obtain conditions for the extinction and persistence in mean of two microorganisms in the stochastic chemostat model. The results show that a stochastic disturbance or the impulsive effect can cause the microorganisms to go to extinction. Finally, we provide some examples together with numerical simulations to illustrate the analytical results and explain the biological implications. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic SIS model with saturated incidence rate. We investigate the existence and uniqueness of the positive solution to the system, and we show the condition for the infectious individuals to be extinct. Moreover, we prove that the system has the ergodic property and derive the expression for its invariant density. The simulation results are illustrated finally. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of stationary distribution and extinction for a stochastic generalized logistic system. Sufficient and necessary conditions for the existence of a stationary distribution and extinction are obtained. (a) The system has a unique stationary distribution if and only if the noise intensity is less than twice the intrinsic growth rate. The probability density function has been solved by the stationary Fokker-Planck equation. (b) The system will become extinct when and only when the noise intensity is no less than twice the intrinsic growth rate, and the exponential extinction rate is estimated precisely by two parameters of the systems. A new perspective is provided to explain the recurrence phenomenon in practice. Nontrivial examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we develop and study a stochastic model for the competition of three species with a generalized dose-response function in a polluted environment. We first carry out the survival analysis and obtain sufficient conditions for the extinction, non-persistence, weak persistence in the mean, strong persistence in the mean and stochastic permanence. The threshold between weak persistence in the mean and extinction is established for each species. Then, using Hasminskii's methods and a Lyapunov function, we derive sufficient conditions for the existence of stationary distribution for each population. Numerical simulations are carried out to support our theoretical results, and some biological significance is presented. [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic two-species predator-prey model which is ratio-dependent and a specific functional response is considered in, is proposed. The existence of a global positive solution to the model for any positive initial value is shown. Stochastically ultimate boundedness and uniform continuity are derived. Moreover, under some sufficient conditions, the stochastic permanence and extinction are established for the model. At last, numerical simulations are carried out to support our results. [ABSTRACT FROM AUTHOR]

In this work we deal with a Backward doubly stochastic differential equation associated to a Poisson random measure. We establish a comparison theorem and prove existence of a minimal solution under weaker conditions on the coefficients. [ABSTRACT FROM AUTHOR]

A comparison principle for stochastic integro-differential equations driven by Lévy processes is proved. This result is obtained via an extension of an Itô formula, proved by N.V. Krylov, for the square of the norm of the positive part of L − valued, continuous semimartingales, to the case of discontinuous semimartingales. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to discuss the Lp-estimate of solution of stochastic differential equations under nonlinear growth condition. Furthermore, almost surely asymptotic estimates for the solutions of these equations are given. More accurately, we shall estimate sample Lyapunov exponent almost surely. [ABSTRACT FROM AUTHOR]

Let SH be a sub-fractional Brownian motion with index 0 < H < 1/2. In this paper we study the existence of the generalized quadratic covariation [ f( SH), SH] (W) defined by $[f(S^H ),S^H ]_t^{(W)} = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \tfrac{1} {{\varepsilon ^{2H} }}\int_0^t {\{ f(S_{s + \varepsilon }^H ) - f(S_s^H )\} (S_{s + \varepsilon }^H - S_s^H )ds^{2H} } , $ provided the limit exists in probability, where x ↦ f( x) is a measurable function. We construct a Banach space ℋ of measurable functions such that the generalized quadratic covariation exists in L2 provided f ∈ ℋ. Moreover, the generalized Bouleau-Yor identity takes the form [Figure not available: see fulltext.] for all f ∈ ℋ, where ℒ H( x, t) is the weighted local time of SH. This allows us to write the generalized Itô’s formula for absolutely continuous functions with derivative belonging to ℋ. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the problem of pricing and hedging of certain catastrophe (CAT) options. In particular, a self-financing hedging strategy that minimizes the risk in the loss period and replicates the option in the development period is proposed. [Copyright &y& Elsevier]

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka-Rosen-Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones. [ABSTRACT FROM AUTHOR]

Nakao's stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Itô's formula in terms of the extended stochastic integrals is obtained. [ABSTRACT FROM AUTHOR]

This paper develops variational inequality approach to establish optimal stopping problems arising from the pricing of stock loan and capped stock loan models. The valuations of the stock loans and ranges of reasonable values of key parameters such as loan sizes, interest rates, fees for providing such a service and quantity of this automatic termination clause and relationships among these parameters as well as the optimal terminable stopping times are explicitly derived. The paper also gives two numerical samples to portray the results and the difference between stock loan model and capped stock loan model. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay. We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory. [ABSTRACT FROM AUTHOR]

We prove Itô’s formula for the L p-norm of a stochastic $${W^{1}_{p}}$$ -valued processes appearing in the theory of SPDEs in divergence form. [ABSTRACT FROM AUTHOR]

We will concentrate on a 1-dimensional nonlinear filtering problem, which allows an explicit solution in terms of a stochastic differential equation for $$ \hat X_t $$. [ABSTRACT FROM AUTHOR]

This paper provides new insights into the solution of optimal stochastic control problems by means of a system of partial differential equations, which characterize directly the optimal control. This new system is obtained by the application of the stochastic maximum principle at every initial condition, assuming that the optimal controls are smooth enough. The type of problems considered are those where the diffusion coefficient is independent of the control variables, which are supposed to be interior to the control region. [ABSTRACT FROM AUTHOR]

This paper studies the existence, uniqueness, and stability of solutions for stochastic impulsive systems. By employing Lyapunov-like functions, some sufficient conditions of the global existence, uniqueness, and stability of solutions for stochastic impulsive systems are established. Furthermore, the results are specialized to the case of linear stochastic impulsive systems. Finally, some examples are given to illustrate the applications of our theory. [ABSTRACT FROM AUTHOR]

Let Z=( Z t ) t≥0 be a Bessel process of dimension δ( δ>0) starting at zero and let K( t) be a differentiable function on [0, ∞) with K( t)>0 (∀ t≥0). Then we establish the relationship between L p -norm of log1/2(1+ δJ τ) and L p -norm of sup Z t [ t+ k( t)]−1/2 (0≤ t≤ τ) for all stopping times τ and all 0< p<+∞. As an interesting example, we show that |log1/2(1+ δL m+1( τ))| p and |sup Z t Π[1+ L j ( t]−1/2| p (0≤ j≤ m,j∈ —; 0≤ t≤ τ) are equivalent with 0< p<+∞ for all stopping times τ and all integer numbers m, where the function L m ( t) ( t≥0) is inductively defined by L m+1( t)=log[1+ L m ( t)] with L 0( t)=1. [ABSTRACT FROM AUTHOR]

In this paper we prove L p estimates ( p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE. [ABSTRACT FROM AUTHOR]

Let $$X = (X_t)_{t \geq 0}$$ be a continuous semimartingale and let $$b: \mathbb{R}_+ \rightarrow \mathbb{R}$$ be a continuous function of bounded variation. Setting $$C = \{(t, x) \in \mathbb{R} + \times \mathbb{R} | x < b(t)\}$$ and $$D = \{(t,x) \in \mathbb{R}_+ \times \mathbb{R} | x > b(t)\}$$ suppose that a continuous function $$F: \mathbb{R}_+ \times \mathbb{R} \rightarrow \mathbb{R}$$ is given such that F is C1,2 on $$\bar{C}$$ and F is $$C^{1,2}$$ on $$\bar{D}$$. Then the following change-of-variable formula holds: $$\eqalign{ F(t,X_t) = F(0,X_0)+\int_0^{t} {1 \over 2} (F_t(s, X_s+) + F_t(s,X_s-)) ds\cr + \int_0^t {1 \over 2} (F_x(s,X_s+) + F_x(s,X_s-))dX_s\cr + {1 \over 2} \int_0^t F_{xx} (s,X_s)I (X_s \neq b(s)) d \langle X, X \rangle_s\cr + {1 \over 2} \int_0^t (F_x(s,X_s+)-F_x(s,X_s-)) I(X_s = b(s)) d\ell_{s}^{b} (X),\cr} $$ where $$\ell_{s}^{b}(X)$$ is the local time of X at the curve b given by $$\ell_{s}^{b}(X) = \mathbb{P} - \lim_{\varepsilon \downarrow 0} {1 \over 2 \varepsilon} \int_0^s I(b(r)- \varepsilon < X_r < b(r) + \varepsilon) d \langle X, X \rangle_{r} $$ and $$d\ell_{s}^{b}(X)$$ refers to the integration with respect to $$s \mapsto \ell_{s}^{b}(X)$$. A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping. [ABSTRACT FROM AUTHOR]

Using the products of exponential functions, a family of unitary operators will be defined for all square integrable continuous martingale processes having a deterministic quadratic variation. A Heisenberg inequality for stochastic integrals will be proved using this family of operators. [ABSTRACT FROM AUTHOR]

Let X be a Cox–Ingersoll–Ross (CIR) process given by$\{\backslash rm\; d\}X\_t=(a+bX\_t)\backslash ,\{\backslash rm\; d\}t+c\backslash sqrt\{\backslash vert\; X\_t\backslash vert\}\backslash ,\{\backslash rm\; d\}B\_t$with X0=0, where a, c>0, $b\backslash in\{\backslash open\; R\}$ and B a standard Brownian motion starting at zero. We obtain some inequalities between the integral functional $J\_\{\backslash varphi\}(t)=\backslash int\_0^t\; \backslash varphi(X\_s)\; \backslash ,\; \{\backslash rm\; d\}s$, t≥0 and the maximal process sup0≤s≤tXs, t≥0, where x↦ϕ(x) a nonnegative continuous function with some suitable conditions. [ABSTRACT FROM AUTHOR]

We prove a uniform bound for the density, p t( x), of the solution at time t∈(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p t( x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al.(5) [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of the quadratic covariation [f( X), X], where f is a locally square integrable function and Xt = ∫ t 0 us dWs is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. [ABSTRACT FROM AUTHOR]

Let M={M z, z ∈ R} be a continuous square integrable martingale and A={A z, z ∈ R be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX z=α(z, Xz)dMz+β(z, Xz)dAz, z∈R, Xz=Zz, z∈∂R, where R =[0, +∞)×[0,+∞) and ∂ R is its boundary, Z is a continuous stochastic process on ∂ R . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. [ABSTRACT FROM AUTHOR]

We develop analogues of Itô's, Tanaka's and Stratonovich's formulae for the case in which one integrates with respect to a deterministic function taken from a class including self-affine functions of order 1/2. Föllmer and Bertoin have also defined types of deterministic 'stochastic' integrals, but the values of their integrals are dependent on choices of partitions made in the definition of the integral. We avoid this problem by the use of a second-order averaging method to define a sort of derivative for a self-affine function. [ABSTRACT FROM AUTHOR]

Acknowledging many effects on humans, which are ignored in deterministic models for COVID-19, in this paper, we consider stochastic mathematical model for COVID-19. Firstly, the formulation of a stochastic susceptible–infected–recovered model is presented. Secondly, we devote with full strength our concentrated attention to sufficient conditions for extinction and persistence. Thirdly, we examine the threshold of the proposed stochastic COVID-19 model, when noise is small or large. Finally, we show the numerical simulations graphically using MATLAB. [ABSTRACT FROM AUTHOR]

This paper is concerned with almost sure exponential stabilization of neural networks by intermittent control based on delay observations. By the stochastic comparison principle and Itô's formula, a sufficient criterion is derived, under which unstable neural networks can be stabilized by stochastic intermittent control based on delay observations. The range of intermittent rate is given, and the upper bound of time delay can be solved from a transcend equation. Finally, two examples are provided to demonstrate the feasibility and validity of our proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, stochastic effect on the spread of the infectious disease with saturated incidence rate and the special transfer from infectious is discussed. The threshold dynamics is explored for the case of relatively small noise. Our results show that large noise will cause the elimination of the disease, which will help suppress the spread of the disease. [ABSTRACT FROM AUTHOR]