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This paper focuses on the optimal output‐feedback control and stabilization of a networked control system, where both time delay and packet loss are involved. Different from most previous work, the packet loss is Markovian, and the time delay occurs between the actuator and controller. The difficulty is rooted in the failure of the separation principle. The main tools are solving the delayed forward‐backward stochastic difference equations (D‐FBSDEs) and convergent analysis. Resorting to solving D‐FBSDEs and completing the square, we acquire the analytical solution, including the necessary and sufficient solvability condition and the explicit controller, of the finite‐horizon optimal control problem. Based on the finite‐horizon result and convergent analysis, we then obtain the analytical solution of the infinite‐horizon optimal control and stabilization problems. In addition, by analyzing the convergence of the optimal estimator, we establish a constraint relationship between the packet loss probability q$$ q $$ and the system matrix A$$ A $$. The results are evaluated by numerical examples. [ABSTRACT FROM AUTHOR]

An uncertain stochastic noncausal system (USNS) is an uncertain stochastic singular system that is expected to be regular but not impulse-free. This study investigates zero-sum games (ZSGs) under the constraint of nonlinear uncertain stochastic noncausal systems (USNSs) considering polynomial control functions. Firstly, a method is introduced to convert USNSs into subsystems, including forward uncertain stochastic difference equations as well as backward uncertain stochastic difference equations. Equilibrium equations are then derived to determine the saddle-point equilibrium solution of the zero-sum game (ZSG) for USNSs. Subsequently, a unified framework is developed to find the saddle-point equilibrium and equilibrium value of the ZSG. Additionally, a numerical example and an analysis of the treatment of industrial wastewater are provided to demonstrate the effectiveness of the proposed framework. [ABSTRACT FROM AUTHOR]

Based on the Feynman-Kac formula, we develop a new approach to show the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations, which are associated with a class of time-dependent stochastic differential equations driven by symmetric (or asymmetric) non-Gaussian Lévy process. First, after deriving the forward time-dependent multidimensional nonlocal Fokker-Planck equation, the Feynman-Kac formula for the forward nonlocal Fokker-Planck equation is established by using Itô's formula and techniques for the backward nonlocal Fokker-Planck equation corresponding to the backward stochastic differential equation driven by Lévy process. Then, from the Feynman-Kac formula, we prove the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations by using techniques from stochastic analysis. Finally, the dynamic evolution of the probability density function corresponding to the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, comK, comS) is investigated over an extended period by Feynman-Kac formula and Monte Carlo simulations. In particular, we reveal the influence of symmetric and asymmetric stable noises as well as tempered stable noise on the probability density function corresponding to the stochastic MeKS network model. [ABSTRACT FROM AUTHOR]

This article examines the stability properties of linear stochastic difference equations with delays. For this purpose, a novel approach is used that combines the theory of inverse-positive matrices and the asymptotic methods developed by N.V. Azbelev and his students for deterministic functional differential equations. Several efficient conditions for p-stability and exponential p-stability (2 ≤ p < ∞) of systems of linear Itô-type difference equations with delays and random coefficients are found. All results are conveniently formulated in terms of the coefficients of the equations. The suggested examples illustrate the feasibility of the approach. [ABSTRACT FROM AUTHOR]

Stability estimates are proposed for two variants of Markov processes defined by stochastic difference equations: uncontrolled and controlled. Processes of this type are widely used in applications where their “governing distributions” are known only approximately, for example, as statistical estimates obtained from real data. Therefore, the problem of estimating deviations of output characteristics arises. The Kantorovich metric is used to measure the variations of probability distributions that govern the processes. In the uncontrolled case, the Kantorovich distance between the stationary distributions of the initial process and its perturbation is evaluated. On the other hand, the control processes being compared are endowed with an expected total discounted cost, and the inequality for the corresponding stability index is obtained. The stability index measures the increase in costs when using the control policy optimal for the “approximating process.”

In this article, we consider the solvability problems for the fully coupled forward-backward stochastic difference equations (FBSΔEs). On one hand, we provide the necessary and sufficient condition for the solvability of the linear FBSΔEs. On the other hand, under the assumption that the coefficients satisfy the monotonicity condition, we investigate the existence and uniqueness theorems for the general non linear FBSΔEs. [ABSTRACT FROM AUTHOR]

Despite the fact that the theory of stability of continuous-time difference equations has a long history, is well developed and very popular in research, there is a simple and clearly formulated problem about the stability of stochastic difference equations with continuous time and distributed delay, which has not been solved for more than 13 years. This paper offers to readers some generalization on this unsolved problem in the hope that it will help move closer to its solution. [ABSTRACT FROM AUTHOR]

This article investigates the finite horizon linear quadratic gaussian (LQG) control problem for networked control systems with two‐way Markovian packet dropouts and input delay, where packet dropouts occur from the sensor to the estimator and from the controller to the actuator, and delay occurs from the controller to the actuator. The novelty of this work lies in providing a complete solution for the optimal control problem subject to two‐way Markovian packet dropouts and input delay. The contributions of this article are as follows: first, by applying the maximum principle to the discrete‐time linear systems and the quadratic cost function involving input delay and Markovian packet dropouts, a solution to the forward and backward stochastic difference equations (FBSDEs) is derived. Second, a sufficient and necessary condition for the optimal control problem is obtained by decoupling the coupled Riccati equations. Finally, the explicit solution of the optimal controller is presented based on the complete square method. Numerical examples are provided to demonstrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of Lp space. This exploration involved the utilization of Jensen’s, Gröenwall-Bellman’s, Hölder’s, Burkholder-Davis-Gundy’s inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example. [ABSTRACT FROM AUTHOR]

Based on the Girsanov theorem for a kind of Volterra-Gaussian process, which are the generalization of fractional Brownian motion, Liouville fractional Brownian motion, and fractional Ornstein-Uhlenbeck process, we establish the Harnack inequalities for a class of stochastic functional differential equations driven by a kind of Volterra-Gaussian processes with a subordinator by an approximation technique. Some known results have been generalized and improved. [ABSTRACT FROM AUTHOR]

This article studies the optimal control for networked control systems (NCSs) subject to Markovian packet dropouts and input delay. In this scenario, Markovian packet dropouts occur in two channels, that is, from sensor to controller and from controller to actuator. Meanwhile, the input delay arises in the channel from controller to actuator. The contributions of this paper are as the following two aspects: Firstly, based on Pontryagin's maximum principle, the solution to the forward and backward stochastic difference equations (FBSDEs) is obtained. Secondly, in virtue of the obtained solution to the FBSDEs, sufficient and necessary optimal control conditions are derived and the explicit expression of the optimal controller is presented. The validity of the results is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses the stochastic linear quadratic (LQ) control problem with first‐ and second‐order moment constraints on the terminal state. The problem is a modified version of the optimal covariance control problem, where the terminal state is steered to a given probability distribution. Studying a multiplicative‐noise stochastic system rather than an additive‐noise system is a salient feature. Unlike the existing ideas in the optimal steering, by using the Lagrange multipliers method and establishing the stochastic maximum principle, our problem is converted into solving forward–backward stochastic difference equations (FBSDEs), which is a special stochastic two‐point boundary‐value problem (TPBVP). We provide the optimal closed‐loop controller and necessary and sufficient solvability conditions by developing a nonhomogeneous relationship between the optimal state and costate in FBSDEs. Finally, numerical examples are given to demonstrate our results. [ABSTRACT FROM AUTHOR]

This paper investigates the discrete‐time linear quadratic (LQ) stochastic Stackelberg game, which has not been thoroughly addressed in previous literature. Firstly, we derive the maximum principle for the stochastic Stackelberg difference game using the variational method, and obtain the necessary and sufficient solvability conditions. However, due to the coupling between the two players and the presence of stochastic noise, obtaining explicit optimal leader and follower's strategies becomes challenging. Therefore, we present a feasible suboptimal control strategy instead. As a result, we derive a feasible suboptimal control strategy. To achieve this, we assume a linear homogeneous relationship to decouple the group of stochastic game forward‐backward stochastic differential equations (SG‐FBSDEs), which serves as a compromise for obtaining the optimal solution. With this approach, we derive a feasible solution to the stochastic Stackelberg difference game based on the solution to symmetric Riccati equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with approximations of invariant probability measures for stochastic functional differential equations (SFDEs) using a backward Euler-Maruyama (BEM) scheme under one-sided Lipschitz condition on the drift coefficient. Firstly, the strong convergence of the numerical "segment sequence" from the BEM scheme on finite time interval [ 0 , T ] is established. In addition, it is also demonstrated that the numerical segment sequence from the BEM scheme is a Markov process, and the corresponding discrete-time semigroup generated by this BEM scheme admits a unique numerical invariant probability measure. Finally, it is revealed that the numerical invariant probability measure converges to the underlying one in a Wasserstein distance. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behaviors of the solutions of an impulsive stochastic neural field lattice model driven by nonlinear noise. We first show the existence and uniqueness of weak pullback mean random attractors for the impulsive stochastic systems. Then by the properties of Markov processes, the existence of evolution system of measures for the impulsive stochastic systems is established. To this end, we employ the idea of uniform estimates on the tails of the solutions to show the tightness of a family of distributions of the solutions of the lattice systems. [ABSTRACT FROM AUTHOR]

In this paper, the finite horizon asymmetric information linear quadratic (LQ) control problem is investigated for a discrete-time mean field system. Different from previous works, multiple controllers with different information sets are involved in the mean field system dynamics. The coupling of different controllers makes it quite difficult in finding the optimal control strategy. Fortunately, by applying the Pontryagin's maximum principle, the corresponding decentralized control problem of the finite horizon is investigated. The contributions of this paper can be concluded as follows: For the first time, based on the solution of a group of mean-field forward and backward stochastic difference equations (MF-FBSDEs), the necessary and sufficient solvability conditions are derived for the asymmetric information LQ control for the mean field system with multiple controllers. Furthermore, by the use of an innovative orthogonal decomposition approach, the optimal decentralized control strategy is derived, which is based on the solution to a non-symmetric Riccati-type equation. [ABSTRACT FROM AUTHOR]

In the present note, we state and solve the linear quadratic (LQ) control problem for a class of McKean--Vlasov stochastic differential equations for which the control actions are of impulsive nature. After reformulating the original optimization problem using an orthogonal decomposition of the state variables, we introduce an adequately defined system of two coupled matrix Lyapunov-type differential equations with jumps. Such equations are central for the definition of the optimal feedback gains corresponding to the closed-loop representation of the optimal LQ control. [ABSTRACT FROM AUTHOR]

This paper focuses on the existence and uniqueness, the pth moment and almost sure stability with a general decay rate and the practical stability with general decay rate of a class of hybrid neutral stochastic functional differential equations driven by Lévy noise. Our crucial techniques include Lyapunov functions and the nonnegative semi-martingale convergence theorem. The conditions on the diffusion operator and neutral term are weaker than those in the related existing works. Examples are given to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the effect of "stabilization by noise". The essence of this effect is that an unstable deterministic system is stabilized by stochastic perturbations of sufficiently high intensity. The problem is that the effect of "stabilization by noise", well-known already for more than 50 years for stochastic differential equations, still has no analogue for stochastic difference equations. Here, a corresponding hypothesis is formulated and discussed, the truth of which is illustrated and confirmed by numerical simulation of solutions of stochastic linear and nonlinear difference equations. However, a problem of a formal proof of this hypothesis remains open. [ABSTRACT FROM AUTHOR]

In this article, the local and remote stochastic nonzero‐sum game for a multiplicative noise system with inconsistent information is investigated, in which the multiplicative noise can cause nonlinear characteristics of linear systems, making it difficult to solve the optimal linear feedback Nash equilibrium. For the considered local and remote stochastic nonzero‐sum game, the local player and the remote player obtain different information sets, which leads to inconsistent information between the two players. The goal is that each player is desired to minimize their own cost function. Our approach is based on a combination of orthogonal decomposition and completing square techniques, which allow us to derive a set of coupled Riccati equations that characterize the optimal feedback explicit (closed‐loop) Nash equilibrium. The contributions of this article are summarized as follows. First, the optimal open‐loop Nash equilibrium is obtained in terms of the forward and backward stochastic difference equations (FBSDEs) by adopting the Pontryagin maximum principle. Second, the closed‐loop Nash equilibrium of this local and remote stochastic nonzero‐sum game for a multiplicative noise system with inconsistent information is obtained by using the orthogonal decomposition methods. Finally, a simulation example is given to illustrate the validity of theoretical results and discuss potential extensions to more complex systems. [ABSTRACT FROM AUTHOR]

This article is concerned with the infinite‐horizon optimal control for networked control systems with Markovian packet losses. We focus on the case that packet losses may occur in an unreliable communication channel between the sensor and the controller. This sort of packet loss enables the optimal control problem to fall into the category of optimal output‐feedback control. Also, the output involves multiplicative noise. As well known, the separation principle fails to hold in a multiplicative‐noise setting. Thus the problem is very involved. By solving forward and backward stochastic difference equations and making convergence analysis, we acquire the necessary and sufficient solvability condition and the explicit optimal controller. The achieved results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we will consider periodic EGARCH ⁡ (p , p) (exponential generalized autoregressive conditional heteroscedastic) processes denoted by PEGARCH ⁡ (p , p) . These processes are similar to the standard EGARCH processes, but include seasonally varying coefficients. We examine the probabilistic structure of an EGARCH-type stochastic difference equation with periodically-varying parameters. We propose necessary and sufficient conditions ensuring the existence of stationary solutions (in a periodic sense) based on a Markovian representation. The closed forms of higher moments are, under these conditions, established. Furthermore, the expressions for the Kurtosis coefficient and the autocorrelations of squared observations are derived. The general theory is illustrated by considering special cases such as the symmetric and the asymmetric cases of the second order PEGARCH model. [ABSTRACT FROM AUTHOR]

This article examines the stability properties of linear stochastic difference equations with delays. For this purpose, a novel approach is used that combines the theory of inverse-positive matrices and the asymptotic methods developed by N.V. Azbelev and his students for deterministic functional differential equations. Several efficient conditions for p-stability and exponential p-stability (2≤p<∞) of systems of linear Itô-type difference equations with delays and random coefficients are found. All results are conveniently formulated in terms of the coefficients of the equations. The suggested examples illustrate the feasibility of the approach.

The results of the application of the methods of the synergetic control theory to a high-dimensional immunology object with uncertainty in its descriptions are reported. The control here is the therapy treated as a problem for constructing an optimal cure program. The control object is presented in continuous and discrete forms, i.e., mathematical models given by a system of ordinary differential equations with a bounded disturbance and a system of stochastic difference equations, respectively. Two algorithms for deriving robust regulators applicable to a 10-dimensional nonlinear multi-loop system with unstable limit states, which models an immune response to the hepatitis B infection, are obtained. Analytical control design for a continuous model relies on the method of nonlinear adaptation on the target manifold. The second algorithm represents a stochastic extension of the method of analytical design of aggregated discrete regulators minimizing the variance of the target macro variable. The numerical simulation of the developed control systems indicates the performance of the designed control algorithms. The results of this study can be used as a component part of the mathematical tools of expert systems and decision support systems. [ABSTRACT FROM AUTHOR]

A class of fully coupled forward–backward stochastic difference equations with delay (FBSDDEs) over infinite horizon are considered in this article. By establishing a non‐homogeneous explicit relation between the forward and backward equations in terms of Riccati‐like difference equations, we derive the unique solution to the FBSDDEs under certain conditions. Then, we deduce that the FBSDDEs are solvable if and only if the corresponding stochastic delayed system is β$$ \beta $$‐degree open‐loop mean‐square exponentially stabilizable. Finally, as an application, the FBSDDEs are employed to demonstrate the maximum principle of the stochastic LQ optimal control problem. [ABSTRACT FROM AUTHOR]

The method of studying the stability in the probability for nonlinear systems of stochastic difference equations is demonstrated on two systems with exponential and fractional nonlinearities. The proposed method can be applied to nonlinear systems of higher dimensions and with other types of nonlinearity, both for difference equations and for differential equations with delay. [ABSTRACT FROM AUTHOR]

We explore stabilization for nonlinear systems of difference equations with modified Target-Oriented Control and a chosen equilibrium as a target, both in deterministic and stochastic settings. The influence of stochastic components in the control parameters is explored. The results are tested on the Hénon and the Lozi maps. [ABSTRACT FROM AUTHOR]

This paper considers a linear‐quadratic mean field control problem involving a major agent and N minor agents. We aim to optimize a social cost as a weighted sum of the individual costs under decentralized information. Firstly, the forward‐backward stochastic difference equations (FBSDEs) are obtained for this problem by variational analysis. Then, by decoupling the FBSDEs, we design the decentralized control laws, which are further shown to be asymptotically optimal. [ABSTRACT FROM AUTHOR]

In this paper, we consider the indefinite stochastic optimal control problems of discrete‐time Markov jump linear systems. Firstly, we establish the new stochastic maximum principle, and by solving the forward‐backward stochastic difference equations with Markov jump (FBSDEs‐MJ), we derive the necessary and sufficient solvability condition of the indefinite control problem with non‐discounted cost, which is in an explicit analytical expression. Then, the optimal control is designed by a series of coupled generalized Riccati difference equations with Markov jump (GRDEs‐MJ) and linear recursive equations with Markov jump (LREs‐MJ). Moreover, based on the non‐discounted cost case, we deduce the optimal control problem with discounted cost. Finally, a numerical example for defined‐benefit (DB) pension fund with regime switching is exploited to illustrate the validity of the obtained results. [ABSTRACT FROM AUTHOR]

The decentralized linear–quadratic–Gaussian (LQG) control problem for networked control systems (NCSs) with asymmetric information is investigated, where controller 1 shares its historical information with controller 2, and not vice versa. The asymmetry of the information structure leads to the coupling between controller 2 and estimator 1, and hence the classical separation principle fails. Through the assumption of linear control strategy, the coupling between controller 2 and estimator 1 (CCE) is decoupled, but the estimation gain is still coupled with the control gain. It is noted that the control gain conforms to the backward Riccati equation while estimation gain abides by the forward equation, which is computationally challenging. Applying the stochastic maximum principle, the solvability of the decentralized LQG control problem is reduced to that of corresponding forward and backward stochastic difference equations (FBSDEs). Further, necessary and sufficient conditions for the solvability of optimal control problem are presented by two Riccati equations, one of which is nonsymmetric. Moreover, a novel iterative forward method is proposed to calculate the coupled backward control gain and forward estimation gain. [ABSTRACT FROM AUTHOR]

This paper is concerned with the linear quadratic optimal control problem for the time-delay stochastic system with partial information. The main contribution is to derive the equivalent solvability condition and give the explicitly optimal controller under partial information in terms of the Riccati equations. The key is to explicitly solve the forward and backward stochastic difference equations with partial information, which is derived from the stochastic maximum principle. [ABSTRACT FROM AUTHOR]

This article is concerned with the optimal control and stabilization for the discrete‐time networked control systems with Markovian fading channels in a diagonal form, which indicates that different channel possesses different correlated fading property. To overcome the essential difficulty resulting from the multi communication channel packet losses and the correlation of the Markov chain, a new multistate Markov chain is introduced, by which the optimal control and stabilization for the NCSs with the diagonal form of Markovian fading channels are transformed into the ones for the general Markovian jump linear system. According to the converted system, a new modified maximum principle is presented, that is, a forward‐backward stochastic difference equation (FBSDE) is obtained. In the finite horizon design, the key step is to seek for the relationship between the system state and optimal costate. By the obtained relation and based on a new coupled difference Riccati equation (CDRE), a necessary and sufficient solvability condition for the optimal control is obtained as well as an explicit expression for the optimal controller. For the infinite horizon case, a new type of coupled Lyapunov function is defined, which keeps consistent with the finite horizon optimal performance index. By introducing a new coupled algebraic Riccati equation and using the defined Lyapunov function, a necessary and sufficient stabilization condition in the mean‐square sense and an infinite horizon optimal controller are presented. Finally, a numerical example is supplied to illustrate the efficiency of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we shall deal with stochastic singular difference equations (SSDEs) with delay. The aim of the paper is to study the solvability and stability of SSDEs with delay. It is difficult to investigate these properties for SSDEs with a delay because of the singularity of the leading coefficient matrix. An index-ν concept is derived for the solvability of these equations. Stability of SSDEs with delay is studied by using the method of Lyapunov functions and comparison principle. An example is given to illustrate the results. To the best of our knowledge, these results are novel for SSDEs with delay. [ABSTRACT FROM AUTHOR]

This paper deals with finite nonzero‐sum Markov games under a discounted optimality criterion and infinite horizon. The state process evolves according to a stochastic difference equation and depends on players' actions as well as a random disturbance whose distribution is unknown to the players. The actions, the states, and the values of the disturbance are observed by the players, then they use the empirical distribution of the disturbances to estimate the true distribution and make choices based on the available information. In this context, we propose an almost surely convergent procedure—possibly after passing to a subsequence—to approximate Nash equilibria of the Markov game with the true distribution of the random disturbance. [ABSTRACT FROM AUTHOR]

Based on the perspective of government and enterprises, we explore the cooperative strategy and cost-sharing problem of cooperative open sharing of data between government and enterprises. In order to accurately analyze the data-opening strategies of government and enterprises, stochastic differential game theory is applied to construct the Nash non-cooperative game, Stackelberg master-slave game and cooperative game models with government and enterprises as game subjects to obtain the optimal open data effort, the optimal trajectory of social data open sharing level and the optimal benefit function of government and enterprises in three scenarios. Combined with numerical simulations to analyze the sensitivity of the relevant parameters affecting the level of social data openness, the results of the study revealed the following: 1 When the government's income distribution ratio is greater than 1/3, the benefits of the government and the enterprises under the Stackelberg master-slave game and the effort to open and share data are greater than in the Nash non-cooperative situation; in the case of a cooperative game, the degree of effort and total revenue of both parties reach the Pareto optimal state. 2 When the government's income distribution ratio is greater than 1/3, the expectation and variance of the open data and shared stock under the cost-sharing situation and the corresponding limit value are all greater than the value in the Nash non-cooperative situation, and in the cooperative game, the expectation and variance of open data and shared stock and its corresponding limit value are the greatest. 3 The government and enterprises coexist with profit and risk under the influence of random interference factors, and high profit means high risk. This research provides a theoretical basis and practical guidance for promoting the open sharing of government and enterprise data. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of p-mean θ-pseudo almost periodic stochastic processes, which is slightly weaker than p-mean pseudo almost periodic stochastic processes. Using the operator semigroup theory and stochastic analysis theory, we obtain the existence and uniqueness of squaremean θ-pseudo almost periodic mild solutions for a semilinear stochastic differential equation in infinite dimensions. Moreover, we prove that the obtained solution is also pseudo almost periodic in path distribution. It is noteworthy that the ergodic part of the obtained solution is not only ergodic in squaremean but also ergodic in path distribution. Our main results are even new for the corresponding stochastic differential equations (SDEs) in finite dimensions. [ABSTRACT FROM AUTHOR]

We study the behavior in a large time regime of an explicit tamed Euler-Maruyama scheme applied to a class of ergodic Ito stochastic differential equations with one-sided Lipschitz continuous drift coefficient and bounded globally Lipschitz diffusion coefficient. Our first main contribution is to prove moments for the numerical scheme, which, on the one hand, are uniform with respect to the time-step size, and which, on the other hand, may not be uniform but have at most polynomial growth with respect to time. Our second main contribution is to apply this result to obtain weak error estimates to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process, as a function of the time-step size and of the time horizon. The explicit tamed Euler scheme is shown to be computationally effective for the approximation of the invariant distribution: even if the moment bounds and error estimates are not proved to be uniform with respect to time, the obtained polynomial growth results in a marginal increase in the upper bound of the computational cost. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for stochastic differential equations with non-globally Lipschitz coefficients using an explicit tamed Euler-Maruyama scheme. [ABSTRACT FROM AUTHOR]

This paper is concerned with the linear quadratic decentralized control of forward and backward stochastic difference system (FBSDS) with multiplicative noises and nested asymmetric information. The main contribution is to give the equivalent condition for the solvability of the problem and the explicit decentralized optimal controllers in terms of Riccati equations. The key technology is solving the forward and backward stochastic difference equations with partial information derived from stochastic maximum principle. • LQ decentralized control of forward and backward stochastic difference system with nested asymmetric information. • Forward and backward stochastic difference equations with partial information. • Present solvability condition and explicitly decentralized optimal controllers of the problem. • Propose novel Riccati equations to solve forward and backward stochastic difference equations. [ABSTRACT FROM AUTHOR]

We investigate the interplay between forward uncertain difference equations and backward stochastic difference equations, which belong to distinct mathematical frameworks: uncertainty theory and probability theory, respectively. By establishing the connections between uncertain expectation, probabilistic expectation, and chance expectation, we develop the chance theory framework for analyzing uncertain stochastic hybrid zero-sum games constrained by both forward uncertain difference equations and backward stochastic difference equations. Through an equivalent transformation, we convert these games into deterministic difference equation solving problems. By backward-solving the difference equations and applying mathematical induction, analytical expressions for the equilibrium solutions of two types of uncertain stochastic hybrid zero-sum games are obtained. Furthermore, we explore solution methods for zero-sum games with different operational orders of uncertain and probabilistic expectations and highlight the fundamental differences through numerical examples. • Present uncertain stochastic hybrid zero-sum games of forward and backward systems. • Chance theory framework aids analysis of uncertain stochastic hybrid zero-sum games. • Transforms these games into deterministic difference equation solving problems. • Explores solution methods for games with different expectation operational orders. • Provides numerical examples to illustrate effectiveness of solution methods. [ABSTRACT FROM AUTHOR]

We derive an asymptotic expansion for the quadratic variation of a stochastic process satisfying a stochastic differential equation driven by a fractional Brownian motion, based on the theory of asymptotic expansion of Skorohod integrals converging to a mixed normal limit. In order to apply the general theory, it is necessary to estimate functionals that are a randomly weighted sum of products of multiple integrals of the fractional Brownian motion, in expanding the quadratic variation and identifying the limit random symbols. To overcome the difficulty, we utilized the theory of exponents of functionals, which was introduced by the authors in Yamagishi and Yoshida (2023). [ABSTRACT FROM AUTHOR]

This paper aims to test the structure of interest rates during the period from 1 September 1981 to 28 December 2020 by using Lie algebras and groups. The selected period experienced substantial events impacting interest rates, such as the economic crisis, the military intervention of the USA in Iraq, and the COVID-19 pandemic, in which economies were in lockdown. These conditions caused the interest rate to have a nonlinear structure, chaotic behavior, and outliers. Under these conditions, an alternative method is proposed to test the random and nonlinear structure of interest rates to be evolved by a stochastic differential equation captured on a curved state space based on Lie algebras and group. Then, parameter estimates of this equation were obtained by OLS, NLS, and GMM estimators (hereafter, LieNLS, LieOLS, and LieGMM, respectively). Therefore, the interest rates that possess nonlinear structures and/or chaotic behaviors or outliers were tested with LieNLS, LieOLS, and LieGMM. We compared our LieNLS, LieOLS, and LieGMM results with the traditional OLS, NLS, and GMM methods, and the results favor the improvement achieved by the proposed LieNLS, LieOLS, and LieGMM in terms of the RMSE and MAE in the out-of-sample forecasts. Lastly, the Lie algebras with NLS estimators exhibited the lowest RMSE and MAE followed by the Lie algebras with GMM, and the Lie algebras with OLS, respectively. [ABSTRACT FROM AUTHOR]

Pulse stabilization of cycles with Prediction-Based Control including noise and stochastic stabilization of maps with multiple equilibrium points is analyzed for continuous but, generally, non-smooth maps. Sufficient conditions of global stabilization are obtained. Introduction of noise can relax restrictions on the control intensity. We estimate how the control can be decreased with noise and verify it numerically. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability problems for the fully coupled forward–backward stochastic difference equations (FBSΔEs) on spaces related to discrete time, finite state processes. On one hand, we provide the necessary and sufficient condition for the solvability of the linear FBSΔEs. On the other hand, under the assumption that the coefficients satisfy the monotonicity condition, we investigate the existence and uniqueness theorems for the general nonlinear FBSΔEs. [ABSTRACT FROM AUTHOR]

In this paper, we shall deal with stochastic singular difference equations (SSDEs) with constant coefficient matrices and nonlinear stochastic perturbations. The solvability and stability of SSDEs are difficult to study because of the singularity of the leading coefficient matrix. An index-ν concept is derived and formulas of solution are established for these equations. The continuous dependence of solution on initial condition is also considered. Finally, the stability of SSDEs is studied by using the method of Lyapunov functions. Some examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]