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This paper considers the problem of target tracking in complex manoeuvering scenarios with a lack of relevant prior knowledge. This is a challenge for classical model‐based manoeuvering target tracking algorithms because they rely heavily on accurate domain and prior knowledge of target motion. To address this problem, we propose a hybrid‐driven continuous‐time filter algorithm in this paper, which combines the advantages of the model‐driven and data‐driven. We use the stochastic differential equation (SDE) with the acceleration model as the basic framework of the proposed algorithm. In order to deal with unpredictable manoeuvres and unknown perturbations, we adopt neural networks as data‐driven to estimate target accelerations and compensations of composite perturbations in real time using historical and current measurements. And the direction of gradient descent of the neural network is constrained by the motion model, thus the learning efficiency of the network and the interpretability of the proposed algorithm is improved. The proposed algorithm can simultaneously utilise historical trajectory information and domain knowledge as hybrid‐driven to achieve complex manoeuvering target tracking with little prior information. Experimental results demonstrate the superiority of our algorithm in tracking accuracy, robustness and environmental adaptability. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for H>12$$ \mathcal{H}>\frac{1}{2} $$ these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for H(ξ)≠12$$ \mathcal{H}\left(\xi \right)\ne \frac{1}{2} $$. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm. [ABSTRACT FROM AUTHOR]

This paper is concerned with quantum stochastic differential equations driven by the fermion field in noncommutative space Lp(풞) for 2≤p<∞$$ 2\le p<\infty $$. First, we investigate the existence and uniqueness of Lp$$ {L}&#x0005E;p $$‐solutions of quantum stochastic differential equations in an infinite time horizon by using the noncommutative Burkholder–Gundy inequality given by Pisier and Xu and the noncommutative generalized Minkowski inequality. Then, we investigate the stability, self‐adjointness, and Markov properties of Lp$$ {L}&#x0005E;p $$‐solutions and analyze the error of numerical schemes of quantum stochastic differential equations. The results of this paper can be utilized for investigating the optimal control of quantum stochastic systems with infinite dimensions. [ABSTRACT FROM AUTHOR]

This paper deals with a Skorokhod's integral‐based least squares‐ (LS) type estimator of the drift parameter computed from multiple (possibly dependent) copies of the solution of a stochastic differential equation (SDE) driven by a fractional Brownian motion of Hurst index H∈(1/3,1)$$ H\in \left(1/3,1\right) $$. On the one hand, some convergence results are established on our LS estimator when H=1/2$$ H=1/2 $$. On the other hand, when H≠1/2$$ H\ne 1/2 $$, Skorokhod's integral‐based estimators cannot be computed from data, but in this paper some convergence results are established on a computable approximation of our LS estimator. [ABSTRACT FROM AUTHOR]

Many systems in the real world are periodic due to periodic phenomena in nature. Periodic hybrid stochastic differential equations are often used to model them. In many situations, it is inappropriate to study whether the solutions of periodic hybrid stochastic differential equations will converge to an equilibrium state (say, 0 or the trivial solution) but more appropriate to discuss whether the probability distributions of the solutions will converge to a stationary distribution, known as stability in distribution. This paper aims to determine whether or not a periodic stochastic state feedback control can make a given nonlinear periodic hybrid differential equation, which is not stable in distribution, to become stable in distribution. This problem will be referred to as stabilisation in distribution by periodic noise. There is little known on this problem so far. This paper initiates the study in this direction. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exact controllability of linear mean‐field stochastic systems with time‐variant random coefficients. We prove that the exact controllability, the validity of the observability inequality for the dual equation, the unique solvability of a family of optimal control problems, the unique solvability of a family of mean‐field forward‐backward stochastic differential equations (MF‐FBSDEs), and the unique solvability of a family of norm optimal control problems are all equivalent. Therefore, some approaches are provided to investigate the issue of exact controllability. [ABSTRACT FROM AUTHOR]

This paper investigates an optimal investment‐reinsurance problem for an insurer who possesses inside information regarding the future realizations of the claim process and risky asset process. The insurer sells insurance contracts, has access to proportional reinsurance business, and invests in a financial market consisting of three assets: one risk‐free asset, one bond, and one stock. Here, the nominal interest rate is characterized by the Vasicek model, and the stock price is driven by Heston's stochastic volatility model. Applying the enlargement of filtration techniques, we establish the optimal control problem in which an insurer maximizes the expected power utility of the terminal wealth. By using the dynamic programming principle, the problem can be changed to four‐dimensional Hamilton–Jacobi–Bellman equation. In addition, we adopt a deep neural network method by which the partial differential equation is converted to two backward stochastic differential equations and solved by a stochastic gradient descent‐type optimization procedure. Numerical results obtained using TensorFlow in Python and the economic behavior of the approximate optimal strategy and the approximate optimal utility of the insurer are analyzed. [ABSTRACT FROM AUTHOR]

Synchronization of hypernetworks has been extensively researched due to its potential applications in various contexts, such as secure communications and power grid engineering. However, the existing works thus far have primarily been devoted to inner synchronization or complete outer synchronization with identical node dynamics and topology structures. To investigate harmonious coexistence between two networks with nonidentical node dynamics and reduce control costs as well, this paper focuses on generalized outer synchronization (GOS) and parameters identification of uncertain delayed hypernetworks with stochastic noises. Regardless of whether the node parameters are known or not, based on Lyapunov stability and the LaSalle‐type invariance principle for stochastic differential equations with multiple delays, some sufficient conditions are exhibited to reach GOS almost surely via adaptive control schemes. Moreover, the unknown node parameter vectors are identified by the designed updating laws. In addition, some corollaries are deduced and indicate that our proposed methods contain single‐layer networks and complete synchronization methods as special cases. Furthermore, numerical simulations are provided to verify the effectiveness of our approaches. In particular, it is revealed that network parameters, including time delays, coupling strengths, and noise intensities, have an impact on the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper solves the consumption‐investment problem under Epstein‐Zin preferences on a random horizon. In an incomplete market, we take the random horizon to be a stopping time adapted to the market filtration, generated by all observable, but not necessarily tradable, state processes. Contrary to prior studies, we do not impose any fixed upper bound for the random horizon, allowing for truly unbounded ones. Focusing on the empirically relevant case where the risk aversion and the elasticity of intertemporal substitution are both larger than one, we characterize the optimal consumption and investment strategies using backward stochastic differential equations with superlinear growth on unbounded random horizons. This characterization, compared with the classical fixed‐horizon result, involves an additional stochastic process that serves to capture the randomness of the horizon. As demonstrated in two concrete examples, changing from a fixed horizon to a random one drastically alters the optimal strategies. [ABSTRACT FROM AUTHOR]

This paper studies the hybrid stochastic delay differential equations (SDDEs) with asynchronous switching and discrete observations. For SDDEs based on discrete observations, there are two methods: The discrete‐time approach and the input time delay method. For linear solvable equations, the discrete‐time approach is feasible but for unsolvable nonlinear hybrid SDSs, the results of the discrete‐time approach have not been discussed. So, it is natural to ask: Is the discrete‐time approach still workable for nonlinear hybrid SDSs? This paper focuses on this problem. By using tools of stochastic analysis, constructing Lyapunov functional and using the discrete‐time approach, the stability of hybrid SDSs by discrete‐time feedback control is obtained. Finally, a numerical example is presented to verify the theoretical result. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the averaging principle for a class of Hifer–Katugampola fractional stochastic differential equations driven by standard Brownian motion. Firstly, we establish the existence and uniqueness of mild solution for the considered system using Banach contraction principle. Then, under suitable assumptions, we demonstrate that the solution to the original differential equations converges to that of the averaged differential equations in the sense of mean square and probability as the time scale goes to zero. Finally, an illustrative example is provided to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exponential stability for a class of impulse stochastic differential equations. Different from the previous literature, the impulsive times considered in this paper are random times. By applying stochastic processes theory, stochastic analysis theory and Lyaponov method, we establish several novel exponential stability criteria of the suggested system. Finally, several simple examples are provided to show the validity and significance of the results. [ABSTRACT FROM AUTHOR]

In terms of system modeling, random differential equations (RDEs) with colored noise are demonstrated to be more appropriate than stochastic differential equations (SDEs) relating to white noise in most practical situations. This paper discusses a category of nonlinear random systems involving time‐varying delay and exogenous disturbances. The definitions for noise‐to‐state practically stable (NSPS) and noise‐to‐state practically exponentially mean‐square stable (NSPEMS) are first given. An event‐based controller is then constructed to render the closed‐loop system NSPEMS and circumvents the Zeno behavior at the same time. Since the introduction of event‐triggered mechanism endows the closed‐loop system with hybrid behaviors, event‐based control for RDE systems still remains difficult to deal with. Within our cognitive scope, this paper serves as a pioneering attempt for event‐based control of RDE systems. And the validity of the results is corroborated by two simulation examples. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

Small noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter H∈(1/4,1/2]$H\in (1/4, 1/2]$. We prove a moderate deviation principle for this equation as the scale parameter tends to zero. [ABSTRACT FROM AUTHOR]

This paper explores stochastic differential equation models to characterize the growth dynamics of a stock modeled by generalized logistic growth under a general harvesting function. The latter function adds diminishing marginal productivity to effort increases, while the former incorporates several well‐known growth functions as particular instances. In order to obtain optimal policies, a Crank–Nicolson discretization scheme is derived for the general models. We employ the optimal control theory to determine the optimal harvesting policy for a specific harvested stock and compare the expected net present value of the harvester's total profit earned across harvesting policies. The main conclusion is that models with a higher growth rate result in greater profit values. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a class of nonlinear stochastic differential equations driven by multifractional Gaussian noise. The presented method transforms problems under consideration into systems of nonlinear equations which can be simply solved by the Newton method. Moreover, the convergence analysis of the presented method is investigated. Finally, the efficiency of the new method is illustrated by solving the stochastic logistic equation and three test problems. [ABSTRACT FROM AUTHOR]

The focal point of this paper is to further enhance the existing stochastic epidemic models by incorporating several new disease characteristics, such as the validation time of the vaccination procedure, the stages of vaccine required to gain a long‐period immunity together with the time separating each stage, the deaths linked to the vaccine, and finally, the sudden environmental noise which is exhibited by sociocultural changes, such as antivaccination movements. To incorporate all the aforementioned characteristics, we extend the standard Susceptible‐Vaccinated‐Infected‐Recovered (SVIR) epidemic model to a new mathematical model, which is governed by a system of coupled stochastic delay differential equations, in which the disease transmission rates are driven by Gaussian noise and Lévy‐type jump stochastic process. First, under suitable conditions on the jump intensities, we address the mathematical well‐posedness and biological feasibility of the model, by virtue of the Lyapunov method and the stopping‐time technique. Then, by choosing an adequate positively invariant set for the considered model, we establish sufficient conditions guaranteeing the disease extinction and persistence. Lastly, to support the theoretical results, we provide the outcome of several numerical simulations which, together with our conducted analysis, indicate that the spread of the disease can be majorly altered by all the new considered characteristics. [ABSTRACT FROM AUTHOR]

We study the stochastic optimal control problem for fully coupled forward‐backward stochastic differential equations (FBSDEs) with jump diffusions. A major technical challenge of such problems arises from the dependence of the (forward) diffusion term on the backward SDE and the presence of jump diffusions. Previously, this class of problems has been solved via only the stochastic maximum principle, which guarantees only the necessary condition of optimality and requires identifying unknown parameters in the corresponding variational inequality. Our paper provides an alternative approach, which constitutes the sufficient condition for optimality. Specifically, the original fully coupled FBSDE control problem (referred to as (P)) is converted into the terminal state‐constrained forward stochastic control problem (referred to as (P′)$$ \left({\mathbf{P}}^{\prime}\right) $$) that includes additional (possibly unbounded) control variables. Then (P′)$$ \left({\mathbf{P}}^{\prime}\right) $$ is solved via the backward reachability analysis, by which the value function of (P′)$$ \left({\mathbf{P}}^{\prime}\right) $$ is expressed as the zero‐level set of the value function for the auxiliary unconstrained (forward) control problem (referred to as (P′′)$$ \left({\mathbf{P}}^{\prime \prime}\right) $$). Unlike (P′$$ \Big({\mathbf{P}}^{\prime } $$), (P′′)$$ \left({\mathbf{P}}^{\prime \prime}\right) $$ is an unconstrained problem, which includes additional control variables as a consequence of the martingale representation theorem. We show that the value function for (P′′)$$ \left({\mathbf{P}}^{\prime \prime}\right) $$ is the unique viscosity solution to the associated integro‐type Hamilton‐Jacobi‐Bellman (HJB) equation. The viscosity solution analysis presented in our paper requires a new technique due to additional control variables in the Hamiltonian maximization and the presence of the nonlocal integral operator in terms of the (singular) Lévy measure. To solve the original problem (P), we reverse our approach. Specifically, we first solve (P′′)$$ \left({\mathbf{P}}^{\prime \prime}\right) $$ to obtain the value function using the verification theorem and the viscosity solution of the HJB equation. Then (P′)$$ \left({\mathbf{P}}^{\prime}\right) $$ is solved by characterizing the zero‐level set of the value function of (P′′)$$ \left({\mathbf{P}}^{\prime \prime}\right) $$, from which the optimal solution of (P) can be constructed. To illustrate the theoretical results of this paper, applications to the linear‐quadratic problem for fully coupled FBSDEs with jumps are also presented. [ABSTRACT FROM AUTHOR]

This paper focuses on exponential stability of numerical solutions of neutral stochastic delay differential equations. A novel approach is introduced to study numerical approximation methods of neutral stochastic differential equations with time‐dependent delay. In contrast to the advances in the literature, this work provides a new method and new criteria for which the Euler‐Maruyama approximation method and the backward Euler‐Maruyama approximation method can reproduce exponential stability in mean square and almost sure exponential stability for sufficiently small step sizes. Two examples are provided to demonstrate the effectiveness of our criteria. [ABSTRACT FROM AUTHOR]

This paper addresses the existence of stability results for Ulam–Hyers (UHS) and Ulam–Hyers–Rassias (UHRS) in the setting of Caputo–Hadamard fractional functional stochastic differential equations with delay (FFSDEwD). We first prove existence and uniqueness using Banach fixed point theorem coupled with standard stochastic analysis techniques. Then, we deal with UHS and UHRS results of Caputo–Hadamard FFSDEwD through an application of Gronwall inequality. Our theoretical findings are corroborated with two numerical examples. [ABSTRACT FROM AUTHOR]

We construct a new family of random permutons, called skew Brownian permuton, which describes the limits of several models of random constrained permutations. This family is parameterized by two real parameters. For a specific choice of the parameters, the skew Brownian permuton coincides with the Baxter permuton, that is, the permuton limit of Baxter permutations. We prove that for another specific choice of the parameters, the skew Brownian permuton coincides with the biased Brownian separable permuton, a one‐parameter family of permutons previously studied in the literature as the limit of uniform permutations in substitution‐closed classes. This brings two different limiting objects under the same roof, identifying a new larger universality class. The skew Brownian permuton is constructed in terms of flows of solutions of certain stochastic differential equations (SDEs) driven by two‐dimensional correlated Brownian excursions in the nonnegative quadrant. We call these SDEs skew perturbed Tanaka equations because they are a mixture of the perturbed Tanaka equations and the equations encoding skew Brownian motions. We prove existence and uniqueness of (strong) solutions for these new SDEs. In addition, we show that some natural permutons arising from Liouville quantum gravity (LQG) spheres decorated with two Schramm–Loewner evolution (SLE) curves are skew Brownian permutons and such permutons cover almost the whole range of possible parameters. Some connections between constrained permutations and decorated planar maps have been investigated in the literature at the discrete level; this paper establishes this connection directly at the continuum level. Proving the latter result, we also give an SDE interpretation of some quantities related to SLE‐decorated LQG spheres. [ABSTRACT FROM AUTHOR]

In this paper, we study large‐scale linear fractional stochastic systems representing, e.g., spatially discretized stochastic partial differential equations (SPDEs) driven by fractional Brownian motion (fBm) with Hurst parameter H > ½. Such equations are more realistic in modeling real‐world phenomena in comparison to frameworks not capturing memory effects. To the best of our knowledge, dimension reduction schemes for such settings have not been studied so far. In this work, we investigate empirical reduced order methods that are either based on snapshots (e.g. proper orthogonal decomposition) or on approximated Gramians. In each case, dominant subspaces are learned from data. Such model reduction techniques are introduced and analyzed for stochastic systems with fractional noise and later applied to spatially discretized SPDEs driven by fBm in order to reduce the computational cost arising from both the high dimension of the considered stochastic system and a large number of required Monte Carlo runs. We validate our proposed techniques with numerical experiments for some large‐scale stochastic differential equations driven by fBm. [ABSTRACT FROM AUTHOR]

The solar wind ion transport across the magnetopause is one of the main sources of plasma for the Earth's magnetotail. Such a transport is supported by various dynamic processes at the flank magnetopause, where wave‐particle interactions play a crucial role in ion flow thermalization and diffusion across magnetic field surfaces of the magnetopause tangential discontinuity. In this paper we numerically model such ion thermalization by the most intense electromagnetic waves observed in the magnetosheath, kinetic Alfven waves. We aim to develop an approach for long‐term simulations of ion scattering by waves and ion dynamics around realistic magnetopause magnetic fields. This approach is based on a combination of test particle simulations and stochastic differential equations modeling ion diffusion in velocity space. We demonstrate that for realistic magnetopause configuration and wave characteristics, the magnetosheath ion flow can be substantially thermalized around the magnetopause. This result explains observations of ion energy conservation across the flank magnetopause: kinetic and thermal energies of flowing magnetosheath ions approximately equal to the thermal energy of stagnant magnetospheric ions. Key Points: We model the ion flow thermalization by kinetic Alfven waves around the flank magnetopauseThe model uses stochastic differential equations for particle trajectories to include effects of wave‐particle interactionsScattering and thermalization may explain ion energy conservation across the magnetopause [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness theorem (EUT) of Pantograph fractional stochastic differential equations (PFSDE) using the Banach fixed point theorem (BFPT). We show the Ulam–Hyers stability (UHS) of PFSDE by the generalized Gronwall inequalities (GGI). We illustrate our results by two examples. [ABSTRACT FROM AUTHOR]

This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully NPDEs are extremely difficult to solve because the computational cost of standard approximation methods grows exponentially with the number of dimensions. Therefore, we consider the following methods to overcome this difficulty. For the merged fully NPDEs and 2BSDEs system, combined with the time forward discretization and ReLU function, we use multiscale deep learning fusion and convolutional neural network (CNN) techniques to obtain two numerical approximation schemes, respectively. Finally, three practical high-dimensional test problems involving Allen-Cahn, Black-Scholes-Barenblatt, and Hamilton-Jacobi-Bellman equations are given so that the first proposed method exhibits higher efficiency and accuracy than the existing method, while the second proposed method can extend the dimensionality of the completely NPDEs-2BSDEs system over 400 dimensions, from which the numerical results highlight the effectiveness of proposed methods. [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]

In this paper, we study stability of hybrid stochastic differential equations (SDEs) with multiple delays by intermittent control based on time‐varying delay observations. By constructing an auxiliary non‐delayed hybrid stochastic differential equation and using M$$ M $$‐matrix theory, Lyapunov method, and the comparison principle, sufficient criterions to guarantee the q$$ q $$th moment exponential stability and almost sure exponential stability of hybrid SDEs are derived. Finally, a simulation example is shown to illustrate the validity of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the non-zero-sum differential game problem based on the delayed backward doubly stochastic differential equation. We discuss the case where the controlled system contains time-delayed variables, and the time-delayed variables are different. We construct a new kind of adjoint equation consisting of a doubly stochastic differential equation and three simple differential equations. Under appropriate premises for such equations, we deduce a stochastic maximum principle as a necessary condition and a verification theorem for the Nash equilibrium point by using the duality method and convex variation technique. As an application, we apply our result to a linear delayed backward doubly stochastic non-zero-sum game problem to verify the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a second-order stochastic maximum principle for optimal stochastic control of stochastic differential equations of general mean-field type. The coefficients of the system are nonlinear and depend on the state process as well as of its probability law. The control variable is allowed to enter into both drift and diffusion terms. We establish a set of second-order necessary conditions for the optimal control in integral form. The control domain is assumed to be convex. The proof of our main result is based on the first- and second-order derivatives with respect to the probability law and by using a convex perturbation with some appropriate estimates. [ABSTRACT FROM AUTHOR]

This paper investigates a linear quadratic mean field game with a leader and a large number of followers. The leader first gives its strategy, and then each follower optimizes its own cost. We first solve a mean field game problem, which gives the best response of followers to the leader's strategy. After applying the followers' strategies, the leader is faced to an optimal control problem driven by a high-dimensional forward-backward stochastic differential equations (FBS-DEs). By decoupling the high-dimensional Hamiltonian system with mean field approximations, we construct a set of decentralized strategies for all the players, which are further shown to be an ε-Stackelberg equilibrium.ε- [ABSTRACT FROM AUTHOR]

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms. [ABSTRACT FROM AUTHOR]

The evolution of a quantum system subject to continuous weak measurement is described by a stochastic differential equation (SDE). It has been shown that the conventional density matrix ρt$\rho _t$ characterized by such SDE, under experimentally relevant settings, remains within a deterministically evolving manifold of low dimension, whose associated equations can be computed explicitly. In this paper, as a complement, it is demonstrated that the effect matrix Et$E_t$, associated to a quantum system at the time t to express its measurement during the interval [t,T]$[t,T]$, also remains confined to a deterministically evolving manifold. In particular, by using the Bloch sphere representation together with the properties of Pauli matrices (as the quantum representation of quaternions), deterministic submanifolds for the effect matrix concerning a monitored qubit are given explicitly. The qubit stays on a deterministically evolving curve in the homodyne amplitude detection scenario, and on a deterministically moving surface under heterodyne amplitude detection. This deterministic characterization should be of interest for parameter estimation and towards obtaining computationally efficient quantum filters for real‐time quantum control. [ABSTRACT FROM AUTHOR]

The paper mainly studies globally pth moment exponentially ultimate boundedness and pth moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method of Razumikhin‐type condition and the principle of comparison, some sufficient conditions for globally pth moment exponentially ultimate boundedness and globally pth moment exponential stability are presented. Theorems require the linear coefficients of the upper bound of Lyapunov differential operators are time‐varying functions; this generalizes the previous results. When the time delay is not considered in the system, a unified criterion is given to achieve boundedness and stability when the system is disturbed by stabilizing impulse and destabilizing impulse. It shows that the stochastic differential equation may be unbounded or unstable, and it can be bounded or stable by adding appropriate impulsive perturbation. Finally, we use two examples to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

In this paper, we study partially observed optimal stochastic singular control problems of general Mckean–Vlasov type with correlated noises between the system and the observation. The control variable has two components, the first being absolutely continuous and the second is a bounded variation, nondecreasing continuous on the right with left limits. The dynamic system is governed by Itô‐type controlled stochastic differential equation. The coefficients of the dynamic depend on the state process and of its probability law and the continuous control variable. In terms of a classical convex variational techniques, we establish a set of necessary conditions of optimal singular control in the form of maximum principle. Our main result is proved by applying Girsanov's theorem and the derivatives with respect to probability law in Lions' sense. To illustrate our theoretical result, we study partially observed linear‐quadratic singular control problem of McKean–Vlasov type. [ABSTRACT FROM AUTHOR]

We consider the problem of existence of a (unique) weak solution to the SDE describing symmetric α‐stable process with a locally unbounded drift b:Rd→Rd$b:\mathbb {R}^d \rightarrow \mathbb {R}^d$, d≥3$d \ge 3$, 1<α<2$1<\alpha <2$. In this paper, b belongs to the class of weakly form‐bounded vector fields, the class providing the L2 theory of the non‐local operator behind the SDE, i.e. (−Δ)α2+b·∇$(-\Delta)^{\frac{\alpha }{2}} + b \cdot \nabla$. It contains as proper sub‐classes other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, the weak Ld/(α−1)$L^{\scriptscriptstyle d/(\alpha -1)}$ class and the Campanato–Morrey class (in general, such b makes invalid the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that the operator (−Δ)α2+b·∇$(-\Delta)^{\frac{\alpha }{2}} + b \cdot \nabla$ with weakly form‐bounded b admits a realization as (minus) Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of (−Δ)α2+b·∇$(-\Delta)^{\frac{\alpha }{2}} + b \cdot \nabla$ in Lp$L^p$, p>d−α+1$p>d-\alpha +1$. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and uniqueness result for a class of backward stochastic differential equations (BSDEs, in short) with non‐Lipschitz time delayed generators. In our point of view, this work is a non trivial extension of the pioneer result obtained by Delong and Imkeller under global Lipschitz condition. Indeed, for certain generators our result is valid while their result can not be hold. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the problem of optimal control where the system is given by a fully coupled forward‐backward stochastic differential equation with a risk‐sensitive performance functional. As a preliminary step, we use the risk neutral which is an extension of the initial control system where the admissible controls are convex, and an optimal solution exists.Then, we study the necessary as well as sufficient optimality conditions for risk sensitive performance. At the end of this work, we illustrate our main result by giving an example that deals with an optimal portfolio choice problem in financial market, specifically the model of control cash flow of a firm or project where, for instance, we can set the model of pricing and managing an insurance contract. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exponential stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs). The aim of this paper is to establish exponential stability criteria for a class of hybrid NPSDEs without the linear growth condition. The methods of Lyapunov functions and M‐matrix are used to study exponential stability and boundedness of the hybrid NPSDEs. [ABSTRACT FROM AUTHOR]

Understanding and predicting novel diseases have become very important owing to the huge global health burden. The organization and study of mathematical models are critical in predicting disease behavior of the disease. In this paper, a new stochastic Susceptible‐Infected‐Recovered‐Death (SIRD) model for spreading epidemic disease is investigated. First, the deterministic SIRD model is considered, and then, by allowing randomness in the recovery and death rates that are not deterministic, the system of nonlinear stochastic differential equations is derived. For the suggested model, the existence and uniqueness of a positive global solution are demonstrated. The parameter estimation is done with the conditional least square estimator for deterministic models and the maximum likelihood estimator for stochastic ones. After that, we investigate a nonadditive state‐space model for spreading epidemic disease by considering infected as the hidden process variable. The problem of the hidden process variable from noisy observations is filtered, predicted, and smoothed using a recursive Bayesian technique. For estimating the hidden number of infected variables, closed‐form solutions are obtained. Finally, numerical simulations with both simulated and real data are performed to demonstrate the efficiency and accuracy of the current work. [ABSTRACT FROM AUTHOR]

Energetic electron dynamics in the Earth's radiation belts and near‐Earth plasma sheet are controlled by multiple processes operating on very different time scales: from storm‐time magnetic field reconfiguration on a timescale of hours to individual resonant wave‐particle interactions on a timescale of milliseconds. The most advanced models for such dynamics either include test particle simulations in electromagnetic fields from global magnetospheric models, or those that solve the Fokker‐Plank equation for long‐term effects of wave‐particle resonant interactions. The most prospective method, however, would be to combine these two classes of models, to allow the inclusion of resonant electron scattering into simulations of electron motion in global magnetospheric fields. However, there are still significant outstanding challenges that remain regarding how to incorporate the long term effects of wave‐particle interactions in test‐particle simulations. In this paper, we describe in details two approaches that incorporate electron scattering in test particle simulations: stochastic differential equation (SDE) approach and the mapping technique. Both approaches assume that wave‐particle interactions can be described as a probabilistic process that changes electron energy, pitch‐angle, and thus modifies the test particle dynamics. To compare these approaches, we model electron resonant interactions with field‐aligned whistler‐mode waves in dipole magnetic fields. This comparison shows advantages of the mapping technique in simulating the nonlinear resonant effects, but also underlines that more significant computational resources are needed for this technique in comparison with the SDE approach. We further discuss applications of both approaches in improving existing models of energetic electron dynamics. Key Points: We discuss two approaches to incorporate resonant effects into test particle simulation modelsDetailed approaches have been shown for continuous stochastic differential equations (SDEs) and the mapping technique, respectivelyIn contrast to continuous SDE, the mapping technique allows one to simulate nonlinear resonant effects [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic optimal control problem for moving average control system. The corresponding moving average stochastic differential equation is a kind of integral differential equations. We prove the existence and uniqueness of the solution of the moving average stochastic differential equations. We obtain the stochastic maximum principle of the moving average optimal control system by introducing a kind of generalized anticipated backward stochastic differential equations. We prove the existence and uniqueness of the solution of this adjoint equation, which is singular at 0. As an application, the linear quadratic moving average control problem is investigated to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper studies delay effects on the mean‐square stabilization of stochastic systems with a single input delay. Both continuous‐ and discrete‐time systems are investigated. A continuous‐time system is described by a stochastic differential equation with high‐dimensional Brownian motion. Two conclusions are proved: (1) Once a system is unstabilizable for some value of delay, it will be unstabilizable for any larger delay; (2) If a system is stabilizable for some value of delay, it will be stabilizable in a small neighborhood of that delay. Hence, by excluding the trivial cases that the system is unstabilizable with zero delay and is stabilizable for any delay, there always exists a finite and positive number CM such that the system is stabilizable if and only if the delay is less than CM. Upper bounds of CM are presented. In a special case, an upper bound is given analytically in terms of eigenvalues of the matrix in the drift coefficient before the state. In general cases, upper bounds are given via LMI optimization problems. Two cases in which CM can be computed are studied. In the case that the system has no state‐dependent noise, CM is presented via an LMI optimization problem. In the case that both the state and the control of the system are scalars, CM is expressed analytically. All the above results have counterparts in discrete‐time system. [ABSTRACT FROM AUTHOR]

This paper is concerned with a n‐species stochastic food chain model with white noise and Lévy jumps. The global stochastic dynamics are established, including the existence of the global and positive solutions, stochastic ultimate boundedness, extinction with probability one, and the permanence and global stability in the mean by developing a new research method, using the auxiliary functions and theory of stochastic differential equations. Furthermore, the numerical simulations are presented to verify the theoretical results and illustrate the proposed open problems. [ABSTRACT FROM AUTHOR]

This paper deals with a partially observed risk‐sensitive optimal control problem described by mean‐field forward‐backward stochastic differential equations and the cost functional is a mean‐field exponential of integral type. By using Girsanov's theorem as well as classical convex variational techniques, we obtain two risk‐sensitive maximum principles, which are characterized in terms of the variational inequalities. Moreover, under certain concavity assumption, we give the sufficient condition for the optimality. Through the application of results, we consider the linear‐quadratic risk‐sensitive optimal control problem under partially observed information and fully observed information, respectively. [ABSTRACT FROM AUTHOR]

Optimizing aircraft collision avoidance and performing trajectory optimization are the key problems in an air transportation system. This paper is focused on solving these problems by using a stochastic optimal control approach. The major contribution of this paper is a proposed stochastic optimal control algorithm to dynamically adjust and optimize aircraft trajectory. In addition, this algorithm accounts for random wind dynamics and convective weather areas with changing size. Although the system is modeled by a stochastic differential equation, the optimal feedback control for this equation can be computed as a solution of a partial differential equation, namely, an elliptic Hamilton‐Jacobi‐Bellman equation. In this paper, we solve this equation numerically using a Markov Chain approximation approach, where a comparison of three different iterative methods and two different optimization search methods are presented. Simulations show that the proposed method provides better performance in reducing conflict probability in the system and that it is feasible for real applications. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we investigate the pth moment exponential stability for a class of impulsive stochastic functional differential equations with impulses at random times. The impulsive times considered in this paper are random times that are different from those investigated in the existing literature. By using the stochastic process theory, stochastic analysis theory, Razumikhin technique, and Lyapunov method, we obtain some new criteria of the pth moment exponential stability for the related system. Finally, some examples are provided to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Basis risk arises in a number of financial and insurance risk management problems when the hedging assets do not perfectly match the underlying asset in a hedging program. Notable examples in insurance include the hedging for longevity risks, weather index–based insurance products, variable annuities, etc. In the presence of basis risk, a perfect hedging is impossible, and in this paper, we adopt a mean‐variance criterion to strike a balance between the expected hedging error and its variability. Under a time‐dependent diffusion model setup, explicit optimal solutions are derived for the hedging target being either a European option or a forward contract. The solutions are obtained by a delicate application of the linear quadratic control theory, the method of backward stochastic differential equation, and Malliavin calculus. A numerical example is presented to illustrate our theoretical results and their interesting implications. [ABSTRACT FROM AUTHOR]

This paper concerns a class of stochastic differential equations driven by fractional Brownian motion. The existence and uniqueness of almost automorphic solutions in distribution are established provided the coefficients satisfy some suitable conditions. To illustrate the results obtained in the paper, a stochastic heat equation driven by fractional Brownian motion is considered. [ABSTRACT FROM AUTHOR]

Summary: Our recent paper (Fei W, etal. Delay dependent stability of highly nonlinear hybrid stochastic systems. Automatica. 2017;82:165‐170) is the first to establish delay‐dependent criteria for highly nonlinear hybrid stochastic differential delay equations (SDDEs) (by highly nonlinear, we mean that the coefficients of the SDDEs do not have to satisfy the linear growth condition). This is an important breakthrough in the stability study as all existing delay stability criteria before could only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions (namely, satisfy the linear growth condition). In this continuation, we will point out one restrictive condition imposed in our earlier paper. We will then develop our ideas and methods there to remove this restrictive condition so that our improved results cover a much wider class of hybrid SDDEs. [ABSTRACT FROM AUTHOR]