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In this paper, we formulate a classical stochastic delayed chemostat model and verify that this model has a unique global positive solution. Furthermore, we investigate the dynamical behavior of this solution. We find that the solution of stochastic delayed system will oscillate around the equilibriums of the corresponding deterministic delayed model, moreover, analytical findings reveal that time delay has very significant effects on the extinction and persistence of the microorganism, that is to say, when the time delay is smaller, microorganism will be persistent; when the time delay is larger, microorganism will be extinct. Finally, computer simulations are carried out to illustrate the obtained results. In addition, we can also find by the computer simulation that larger noise may lead to microorganism become extinct, although microorganism is persistent in the deterministic delayed system when the time delay is smaller. [ABSTRACT FROM AUTHOR]

The pricing problem of European option is investigated under the generalized fractional jump-diffusion model. First of all, the generalized fractional jump-diffusion model is proposed, with the assumption that the underlying asset price follows this model, and the explicit solution is derived using the Itô formula. Then, the partial differential equation (PDE) of the European option price is obtained by using the delta-hedging strategy, and the analytical solutions of the European call and put option prices are obtained through the risk-neutral pricing principle. Moreover, the accuracy of closed-form formula for European option pricing is verified by the Monte Carlo simulation. Furthermore, the properties of the pricing formulas are discussed and the impact of main parameters on the option pricing model are analyzed via calculations of Greeks. Finally, the rationality and validity of the established option pricing model are verified by numerical analysis. [ABSTRACT FROM AUTHOR]

This work is on a general integration by parts formula generalizing the Ito formula. An application is given to stochastic inclusions which have a different form than usual. [ABSTRACT FROM AUTHOR]

Stochastic differential equation models are important and provide more valuable outputs to examine the dynamics of SARS-CoV-2 virus transmission than traditional models. SARS-CoV-2 virus transmission is a contagious respiratory disease that produces asymptomatically and symptomatically infected individuals who are susceptible to multiple infections. This work was purposed to introduce an epidemiological model to represent the temporal dynamics of SARS-CoV-2 virus transmission through the use of stochastic differential equations. First, we formulated the model and derived the well-posedness to show that the proposed epidemiological problem is biologically and mathematically feasible. We then calculated the stochastic reproductive parameters for the proposed stochastic epidemiological model and analyzed the model extinction and persistence. Using the stochastic reproductive parameters, we derived the condition for disease extinction and persistence. Applying these conditions, we have performed large-scale numerical simulations to visualize the asymptotic analysis of the model and show the effectiveness of the results derived.

In this research, a comprehensive stochastic model was developed to analyze the transmission dynamics of methicillin-resistant Staphylococcus aureus (MRSA) in both hospital-acquired and community-acquired contexts. The study explored the existence and uniqueness of solutions and established the presence of a global solution. The investigation focused on assessing the boundedness, extinction, and persistence of MRSA strains in terms of their means. The analysis of the model revealed a noteworthy phenomenon known as competitive exclusion, where the community-acquired MRSA strain gradually out-competes the hospital-acquired strain within hospital settings. In conclusion, effective interventions and preventive measures, such as improving hand hygiene compliance, implementing decolonization strategies, and enhancing disinfection rates, emerge as critical factors in controlling the transmission of these strains within hospital environments. [ABSTRACT FROM AUTHOR]

Stochastic differential equation models are important and provide more valuable outputs to examine the dynamics of SARS-CoV-2 virus transmission than traditional models. SARS-CoV-2 virus transmission is a contagious respiratory disease that produces asymptomatically and symptomatically infected individuals who are susceptible to multiple infections. This work was purposed to introduce an epidemiological model to represent the temporal dynamics of SARS-CoV-2 virus transmission through the use of stochastic differential equations. First, we formulated the model and derived the well-posedness to show that the proposed epidemiological problem is biologically and mathematically feasible. We then calculated the stochastic reproductive parameters for the proposed stochastic epidemiological model and analyzed the model extinction and persistence. Using the stochastic reproductive parameters, we derived the condition for disease extinction and persistence. Applying these conditions, we have performed large-scale numerical simulations to visualize the asymptotic analysis of the model and show the effectiveness of the results derived. [ABSTRACT FROM AUTHOR]

In the classical Itô formula, we propose replacing the second derivative (understood in the usual sense) by the second derivative in the sense of differentiation of distributions. In particular, we show that this can be done if the first derivative lies in the class L2,loc(R). Earlier, Föllmer, Protter, and Shiryayev [Bernoulli, 1 (1995), pp. 149--169] obtained a different form of the last term in the Itô formula under the same conditions. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]

This paper deals with a class of stochastic delay differential equations (SDDEs) of second order with multiple delays. Here, two main and novel results are proved on stochastic asymptotic stability and stochastic boundedness of solutions of the considered SDDEs. In the proofs of results, the Lyapunov-Krasovskii functional (LKF) method is used as the main tool. A comparison between our results and those are available in the literature shows that the main results of this paper have new contributions to the related ones in the current literature. Two numerical examples are given to show the applications of the given results. [ABSTRACT FROM AUTHOR]

This paper studies the 2nd-moment exponential stability of a class of infinite-dimensional linear stochastic switched systems comprising two unstable subsystems. We first construct an algebraic sufficient condition on the existence of multiple Lyapunov functions. Then, two switching strategies are designed to stabilize infinite-dimensional linear stochastic switched systems in terms of the multiple Lyapunov function method. Moreover, the system possesses good robust stability of the switching time with our switching strategies.

In this paper, we study a reducible method which is called linearization(Linear-transform) for some non-linear stochastic differential equations (SDEs) to linear by using the Ito-integrated formula. And then finding their analytic solution, we compare the obtained solution for the nonlinear SDEs with the approximate solution by using numerical (Euler -Maruyama and Milstein) Methods.

In this paper, we study the short-time behavior of the arithmetic average of Asian option price (AOP) derived from a general class of the stochastic volatility model with jumps. The AOP that rarely has explicit expression can reduce the volatility in the option price because of the average of the underlying asset price over the time interval. We consider the future average process in the model which is a non-adapted process. By using the Malliavin calculus operators, we get a non-adapted Itô formula, and also a decomposition formula of the option price in the model. We apply the decomposition formula to find the short-time limit of the arithmetic AOP and the implied volatility. [ABSTRACT FROM AUTHOR]

The exploration of stochastic partial differential equations in noisy perturbations of dynamical systems remains a major challenge at this stage. The study analyzes the effective dynamical system combining degenerate additive noise-driven stochastic partial differential equations, firstly in the first class of stochastic partial differential equations, the terms in the non-nuclear space formed by nonlinear interactions are overcome by effectively replacing the elements in the non-nuclear space through the ItÔ formulation, and thus the final effective dynamical system is obtained. The effective dynamical system is then obtained in the second type of stochastic partial differential equation using the O-U process similar to the terms in the non-nuclear space. At noise disturbance amplitudes of 5%, 10%, 15% and 20% AC voltage maxima in that order, the effective dynamical systems for the first type of stochastic partial differential equation and the second type of stochastic partial differential equation are more stable compared to the other types of partial differential equation dynamical systems, with the maximum range of error rate improvement for the sampling points 0–239 voltage rms and voltage initial phase value being 3.62% and 26.85% and 2.13% and 19.86% for sampling points 240–360, respectively. The effective dynamic system and stochastic partial differential equation obtained by the research have very high approximation effect, and can be applied to mechanical devices such as thermal power machines. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long time behavior of stochastic heat equations. The polynomial stability of stochastic heat equations are considered. Sufficient conditions are given to obtain the stability. All the results show the noise (time diffusion) will prevent the solutions to decay in $ p $-th moment, which coincides with the fact that the noise is a diffusion process but it will be different in the sense of almost surely, and the partial diffusion operator (spatial diffusion) will accelerate the decay of solutions. [ABSTRACT FROM AUTHOR]

This paper stands for the almost sure practical stability of nonlinear stochastic differential delay equations (SDDEs) with a general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functionals. Furthermore, we provide some numerical examples to validate the effectiveness of the abstract results of this paper.

We prove existence and uniqueness for the solution of a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with an absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

For a class of unstable stochastic differential equations (SDEs)with jumps, the problem of stabilization with linear time-delay feedback controller is studied. Such an equation will have a unique solution, provided that its coefficients satisfy polynomial growth condition and Khasminskii type condition. By constructing integral Lyapunov functional, the coefficient matrix and time delay of the controller are determined. [ABSTRACT FROM AUTHOR]

Considering that many prey populations in nature have group defense behavior, and the relationship between predator and prey is usually affected by environmental noise, a stochastic predator-prey model with group defense behavior is established in this paper. Some dynamical properties of the model, including the existence and uniqueness of global positive solution, sufficient conditions for extinction and unique ergodic stationary distribution, are investigated by using qualitative theory of stochastic differential equations, Lyapunov function analysis method, Itô formula, etc. Furthermore, the effects of group defense behavior and environmental noise on population stability are also discussed. Finally, numerical simulations are carried out to illustrate that the effects of environmental noise on both populations are negative, the appropriate group defense level of prey can maintain the stability of the relationship between two populations, and the survival threshold is strongly influenced by the intrinsic growth rate of prey population and the intensity of environmental noise.

The asymptotic properties of the solutions to a class of perturbed stochastic impulsive functional differential equations were investigated. Through comparison of the solution to the perturbed equation with the solution to the corresponding unperturbed one, the sufficient conditions for these solutions to be close in a finite time interval were derived. Then, when small perturbations approach zero and the length of the time interval approaches infinity, the 2 solutions will still be close to each other. Finally, an example illustrates the effectiveness of the results. [ABSTRACT FROM AUTHOR]

This paper investigates the polynomial stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS). Firstly, under the local Lipschitz condition and a more general nonlinear growth condition, the existence and uniqueness of the global solution to the addressed NSPDEsMS is considered. Secondly, by adopting the Razumikhin approach, one new criterion on the qth moment polynomial stability of NSPDEsMS is established. Moreover, combining with the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure polynomial stability of NSPDEsMS is examined. The results derived in this paper generalize the previous relevant ones. Finally, two examples are provided to illustrate the effectiveness of the theoretical work. [ABSTRACT FROM AUTHOR]

We consider a spread financial market defined by the multidimensional Ornstein–Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility functions using a stochastic dynamical programming method. We show a special verification theorem for this case. We find the solution to the Hamilton–Jacobi–Bellman (HJB) equation in explicit form and as a consequence we construct optimal financial strategies. Moreover, we study the constructed strategies with numerical simulations.

In this paper, we introduce a class of stochastic harvesting population system with Fractional Brownian Motion (FBM), which is still unclear when the stochastic noise has the character of memorability. Stochastic optimal control problems with FBM can not be studied using classical methods, because FBM is neither a Markov pocess nor a semi-martingale. When the external environment impact on the system of FBM, the necessary and sufficient conditions for the optimization are offered through the stochastic maximum principle, Hamilton function and Itô formula in our work. To illustrate our study, we provide an example to demonstrate the obtained theoretical results, which is the expansion of certainty population system.

The aim of this paper is to admit and stabilize T–S descriptive with time delays. We present some general conditions for the uniqueness and existence in terms of the impulse-free solution for delayed T–S fuzzy systems. Thus, we establish and obtain in terms of strict linear matrix inequalities new sufficient conditions for the stability of T–S fuzzy systems and delayed T–S fuzzy systems by constructing the Lyapunov–Krasovskii functional and applying the generalized Itô formula. Furthermore, a novel algorithm for real-time adjustment of controller membership functions is designed based on a gradient descent strategy to obtain superior H ∞ performance. By using non-PDC fuzzy control mechanism instead of conventional non-PDC control scheme, the performance of interference reduction can be effectively improved. [ABSTRACT FROM AUTHOR]

Clifford analysis has been the field of active research for several decades resulting in various methods to solve problems in pure and applied mathematics. However, the area of stochastic analysis has not been addressed in its full generality in the Clifford setting, since only a few contributions have been presented so far. Considering that the tools of stochastic analysis play an important role in the study of objects, such as positive definite functions, reproducing kernels and partial differential equations, it is important to develop tools for the study of these objects in the context of Clifford analysis. Therefore, in this work-in-progress paper, we present further steps towards stochastic Clifford analysis by studying random variables, martingales, Brownian motion, and Itô formula in the Clifford setting, as well as their applications in Clifford analysis. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to demonstrate the moment inequality theorems of the stochastic process. More specifically, we want to establish some stochastic moment inequalities in the stochastic process by applying the Ito formula and the Gronwall-type inequalities as well as introduce a new proofs of some parts of the Burkholder-Davis-Gundy inequality and induce inverse inequality. [ABSTRACT FROM AUTHOR]

Abstract Let u ( t , x ) $u(t,x)$ be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X ( t , x ) , t ≥ 0 , x ∈ R $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$ with initial condition u ( 0 , x ) ≡ 0 $u(0,x)\equiv 0$ , where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution u ( t , x ) $u(t,x)$ . In fact, it is well known that the solution is a Gaussian process such that the process t ↦ u ( t , x ) $t\mapsto u(t,x)$ is a bi-fractional Brownian motion with Hurst indices H = K = 1 2 $H=K=\frac{1}{2}$ for every real number x. However, the many properties of the process x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ are unknown. In this paper we consider the generalized quadratic covariations of the two processes x ↦ u ( ⋅ , x ) , t ↦ u ( t , ⋅ ) $x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$ . We show that x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with “Itô’s integral”, but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced.

The existent route convergence models are mainly deterministic ones, and various phenomena, such as packet losses, link noises, and sudden changes in interconnecting topology will occur in the route convergence process. Aimed at these random problems, a new stochastic dynamic system model was proposed by means of the Bernoulli white sequence distribution, the Wiener process and the Markov process. Based on the stochastic differential equation theory and the stochastic analysis methods, the sufficient conditions for the route convergence were given. The results prove that, the convergence of the routing state in a random environment is closely related to the Laplacian matrix of the router connection topology, the smooth distribution of the Markov switching, the successful transmission rate of the data packets, and the noise intensity in the network. Finally, a numerical example illustrates the effectiveness of the results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a stochastic chemostat model with impulsive perturbation on the nutrient. First, the existence and uniqueness of solutions are proved by constructing a pulseless equivalent system. Second, based on Khasminskii's Markov periodic process theory, we give sufficient conditions for the existence of positive T -periodic solutions. Third, under certain conditions, the existence and global attractivity of boundary periodic solutions are established by using comparison theorem. Finally, numerical simulations are provided to verify our main results. [ABSTRACT FROM AUTHOR]

The approach of Lyapunov functions is one of the most efficient ones for the investigation of the stability of stochastic systems, in particular, of singular stochastic systems. The main objective of the paper is the analysis of the stability of stochastic perturbed singular systems by using Lyapunov techniques under the assumption that the initial conditions are consistent. The uniform exponential stability in mean square and the practical uniform exponential stability in mean square of solutions of stochastic perturbed singular systems based on Lyapunov techniques are investigated. Moreover, we study the problem of stability and stabilization of some classes of stochastic singular systems. Finally, an illustrative example is given to illustrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

Effect of anti-predator defense due to fear of predator felt by prey and effect of toxic substance released by external sources on prey–predator system is a serious mater of concern in mathematical biology. In the proposed model we have discussed a prey–predator system in which both the species are infected by environmental toxicant. In our consideration prey species is directly infected by environmental toxicant and predator gets infected by consuming infected prey. Prey's growth rate is assumed to be affected by fear of predator. In this work the proposed predator–prey model is analyzed in presence of environmental fluctuation, i.e., stochastic analysis of this model is discussed. Using Itô formula: positivity, boundedness, uniform continuity criterion and global attractivity of solutions of this system have been established. Conditions for which the prey as well as the predator goes extinct have been derived. Conditions for persistence of the system have also been discussed. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different levels of toxicant and different levels of fear have been demonstrated by depicting figures in numerical simulation using MATLAB. [ABSTRACT FROM AUTHOR]

We consider a spread financial market defined by the multidimensional Ornstein--Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility functions using a stochastic dynamical programming method. We show a special verification theorem for this case. We find the solution to the Hamilton--Jacobi--Bellman (HJB) equation in explicit form and as a consequence we construct optimal financial strategies. Moreover, we study the constructed strategies with numerical simulations. [ABSTRACT FROM AUTHOR]

This paper studies the problem of principal-agent with moral hazard in continuous time. The firm's cash flow is described by geometric Brownian motion (hereafter GBM). The agent affects the drift of the firm's cash flow by her hidden effort. Meanwhile, the firm rewards the agent with corresponding compensation and equity which depend on the output. The model extends dynamic optimal contract theory to an inflation environment. Firstly, the authors obtain the dynamic equation of the firm's real cash flow under inflation by using the Itô formula. Then, the authors use the martingale representation theorem to obtain agent's continuation value process. Moreover, the authors derive the Hamilton-Jacobi-Bellman (HJB) equation of investor's value process, from which the authors derive the investors' scaled value function by solving the second-order ordinary differential equation. Comparing with He[1], the authors find that inflation risk affects the agent's optimal compensation depending on the firm's position in the market. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we are devoted to the numerical methods for mean-field stochastic differential equations with jumps (MSDEJs). By combining with the mean-field Itô formula (see Sun, Yang, and Zhao, Numer. Math. Theor. Meth. Appl., 10, pp. 798–828 (2017)), we first develop the Itô formula and further construct the Itô-Taylor expansion for MSDEJs. Then based on the Itô-Taylor expansions, we propose the strong order γ and the weak order η Itô-Taylor schemes for MSDEJs. We theoretically prove the strong convergence rate γ of the strong order γ Itô-Taylor scheme and the weak convergence rate η of the weak order η Itô-Taylor scheme, respectively. Some numerical tests are also presented to verify our theoretical conclusions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the almost sure partial practical stability of stochastic differential equations with general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functions. Finally, we provide a numerical example to demonstrate the efficiency of the obtained results. [ABSTRACT FROM AUTHOR]

For the asymplotic synchronization problem of neutral coupled complex networks with unknown control nodes, an adaptive pinning controller which is more suitable to engineering applications is designed, and the design basis of the adaptive update law is provided in the present study. According to the generalized Ito formula, linear matrix inequality and Lyapunov stability theory, the sufficient conditions for asymptotic synchronization of neutral coupled complex networks are given. Through a numerical simulation, the minimum number of pinned nodes required for synchronization is calculated, which further verifies the validity of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract Focusing on delay differential neoclassical growth model in random environments, we introduce the stochastic model to describe the dynamics of the long-run behavior of the economy with a parameter perturbed by white noises. We prove that the global positive solution exists uniquely and estimate its ultimate boundedness in mean and sample Lyapunov exponent. Finally, some numerical tests are given to illustrate theoretical results.

Abstract In this paper, the mean-square asymptotical heterogeneous synchronization of interdependent networks with stochastic disturbances, which is a zero-mean real Wiener process, is investigated. The network discussed consists of two sub-networks, which are one-by-one inter-coupled. The unknown but bounded nonlinear coupling functions not only exist in the intra-coupling but also in the inter-coupling between two sub-networks. Based on the stochastic Lyapunov stability theory, adaptive control, Itô formula and the linear matrix inequality, several sufficient conditions are proposed to guarantee adaptive mean-square heterogeneous asymptotical synchronization of the interdependent networks. In order to better illustrate the feasibility and effectiveness of the synchronization conditions derived in this brief, numerical simulations are provided finally.

A well-known Itô formula for finite-dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite-dimensional Itô formula for continuous semimartingales from Krylov (Probab Theory Relat Fields 147:583–605, 2010) to a class of L p -valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs. [ABSTRACT FROM AUTHOR]