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In this paper, a new probability distribution, referred to as the matrix Fisher-Gaussian distribution, is proposed on the product manifold of three-dimensional special orthogonal group and Euclidean space. It is constructed by conditioning a multivariate Gaussian distribution from the ambient Euclidean space into the manifold, while imposing a certain geometric constraint on the correlation term to avoid over-parameterization. The unique feature is that it may represent large uncertainties in attitudes, linear variables of an arbitrary dimension, and angular-linear correlations between them in a global fashion without singularities. Various stochastic properties and an approximate maximum likelihood estimator are developed. Furthermore, two methods are developed to propagate uncertainties through a stochastic differential equation representing attitude kinematics. Based on these, a Bayesian estimator is proposed to estimate the attitude and time-varying gyro bias concurrently. Numerical studies indicate that the proposed estimator provides more accurate estimates against the multiplicative extended Kalman filter and unscented Kalman filter for challenging cases.

Comparing with traditional stochastic differential equations (SDEs) involving white noise, random differential equations (RDEs) with colored noise are claimed to have more practical meaning. This paper considers the event-triggered adaptive tracking control for RDE systems with coexisting parametric uncertainties and severe nonlinearities. Combining a tracking error-based dynamic gain with a relative threshold event triggered control mechanism (ETCM), the tracking control problem for the random systems is solved without Zeno behavior. The tracking error can be rendered small enough by tuning design parameters. First, a series of adaptive control laws are designed by using backstepping technique. Then, two special cases are considered and the main results are extended to MIMO systems. Finally, a simulation example confirms the validity of the results. To the best of the authors knowledge, this paper serves as the first attempt of event-based control for RDE systems.

This paper is concerned with the design of a feedback control based on past states in order to make a given unstable hybrid stochastic differential equation (SDE) to be stable in distribution (stabilisation in distribution). This is the first paper in this direction. Under the global Lipschitz condition on the coefficients of the given unstable hybrid SDE, we will show that the stabilisation in distribution can be achieved by linear delay feedback controls. In particular, we discuss how to design the feedback controls in two structure cases: state feedback and output injection.

In this paper we study the problem of model reduction by moment matching for stochastic systems. We characterize the mathematical object which generalizes the notion of moment to stochastic differential equations and we find a class of models which achieve moment matching. However, differently from the deterministic case, these reduced-order models cannot be considered "simpler" because of the high computational cost paid to determine the moment. To overcome this difficulty, we relax the moment matching problem in two different ways and we present two classes of reduced-order models which, approximately matching the stochastic moment, are computationally tractable., This article has been accepted for publication by IEEE Transactions on Automatic Control. The manuscript included in this file is the open access accepted version. This open access version is released on arXiv in accordance with the IEEE copyright agreement

This paper presents a new asset pricing model incorporating fundamental uncertainties by choice of a probability measure. This approach is novel in that we incorporate uncertainties on Brownian motions describing risks into the existing asset pricing model. Particularly, we show extensions of interest rate models to the ones with uncertainties on the Brownian motions, which make the yield curve reflect not only economic factors but also views of the market participants on the Brownian motions. Such yield curve models are especially important in yield curve trading of hedge funds as well as monetary policy making of central banks under low interest rate environments observed after the global financial crisis, in which yield curves are less affected by economic factors since they are controlled by the central banks, but are driven mainly by sentiments of market participants. Firstly, to model aggressive (positive)/conservative (cautious) attitudes towards such fundamental uncertainties, we consider a sup-inf/inf-sup problem on the utility of a representative agent with respect to uncertainties over Brownian motions, i.e. fundamental market risks, by choice of a probability measure. Secondly, we show that the problem is solved via a backward-stochastic differential equations (BSDEs) approach. Then, under a probability measure determined by solving the sup-inf/inf-sup problem, we propose interest rate models with those uncertainties and explicitly obtain their term structures of interest rates. Particularly, we present two approaches to solving the relevant coupled forward-backward stochastic differential equations (FBSDEs) to obtain expressions of the equilibrium interest rate and the term structure of interest rates. In detail, the first approach is by comparison theorems, and the second approach is to predetermine the signs of the volatilities of the BSDE in the coupled system and confirm them by explicitly solving the separated BSDE. Finally, we present concrete examples with numerical experiments.

This article studies uniform stabilization and social optimality for linear quadratic (LQ) mean field control problems with multiplicative noise, where agents are coupled via dynamics and individual costs. The state and control weights in cost functionals are not limited to be positive semidefinite. This leads to an indefinite LQ mean field control problem, which may still be well-posed due to deep nature of multiplicative noise. We first obtain a set of forward–backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback control by decoupling the FBSDEs. By virtue of solutions to two Riccati equations, we design a set of decentralized control laws, which is further shown to be asymptotically social optimal. Some equivalent conditions are given for uniform stabilization of the systems with the help of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed control laws.

In this article, we consider the generalized risk-sensitive optimal control problem, where the objective functional is defined by the controlled backward stochastic differential equation (BSDE) with quadratic growth coefficient. We extend the earlier results of the risk-sensitive optimal control problem to the case of the objective functional given by the controlled BSDE. Note that the risk-neutral stochastic optimal control problem corresponds to the BSDE objective functional with linear growth coefficient, which can be viewed as a special case of the article. We obtain the generalized risk-sensitive dynamic programming principle for the value function via the backward semigroup associated with the BSDE. Then we show that the corresponding value function is a viscosity solution to the Hamilton–Jacobi–Bellman equation. Under an additional parameter condition, the viscosity solution is unique, which implies that the solution characterizes the value function. We apply the theoretical results to the risk-sensitive European option pricing problem.

We consider the indefinite linear-quadratic optimal control problem for stochastic systems with random coefficients, where the corresponding cost parameters need not be definite matrices. Although the solution to this problem was obtained previously via the stochastic maximum principle or by solving the stochastic Hamilton–Jacobi–Bellman (HJB) equation, our approach is simple and direct. Specifically, we develop a direct approach, also known as the completion of squares method, to characterize the explicit optimal solution and the optimal cost. The corresponding optimal solution is linear characterized in terms of the stochastic Riccati differential equation (SRDE) and the linear backward stochastic differential equation (BSDE). In our approach, the completion of squares method, together with the SRDE and the BSDE, allows us to construct an equivalent cost functional that is quadratic in the control $u$ , provided that an additional positive definiteness condition in terms of the SRDE holds. Then, the optimal control and the associated cost can be obtained by eliminating the quadratic term of $u$ in the equivalent cost functional. The additional positive definiteness condition is induced due to the indefiniteness of the cost parameters and the dependence of the control on the diffusion term. We verify the optimal solution by solving the stochastic HJB equation.

In this article, Stochastic port-Hamiltonian systems (SPHS) on infinite-dimensional spaces governed by Itô stochastic differential equations (SDEs) are introduced, and some properties of this new class of systems are studied. They are an extension of SPHSs defined on a finite-dimensional state space. The concept of well-posedness in the sense of Weiss and Salamon is generalized to the stochastic context. Under this extended definition, SPHSs are shown to be well posed. The theory is illustrated on an example of a vibrating string subject to a Hilbert space-valued Gaussian white noise process.

Given an unstable hybrid stochastic differential equation (SDE), can we design a feedback control, based on the discrete-time observations of the state at times $0, \tau, 2\tau, \ldots$ , so that the controlled hybrid SDE becomes asymptotically stable? It has been proved that this is possible if the drift and diffusion coefficients of the given hybrid SDE satisfy the linear growth condition. However, many hybrid SDEs in the real world do not satisfy this condition (namely, they are highly nonlinear) and there is no answer to the question, yet if the given SDE is highly nonlinear. The aim of this paper is to tackle the stabilization problem for a class of highly nonlinear hybrid SDEs. Under some reasonable conditions on the drift and diffusion coefficients, we show how to design the feedback control function and give an explicit bound on $\tau$ (the time duration between two consecutive state observations), whence the new theory established in this paper is implementable.

This technical note presents necessary and sufficient conditions for $2p$ th moment stability of several widely used linear stochastic systems. By the matrix derivative operator and vectorial Ito's formula, we first derive two extended systems: one is described by stochastic differential equation (SDE) and the other by ordinary differential equation (ODE). It is shown that the stochastic system is $2p$ th-moment exponentially stable if and only if the SDE is mean-square exponentially stable or the ODE is exponentially stable. Subsequently, combining the quasi-periodic homogeneous polynomial Lyapunov function methods with the proposed techniques, a series of nonconservative stability criteria are established for linear impulsive, sampled-data, and switched stochastic systems under dwell-time constraints. A numerical example illustrates the proposed theoretical results.

In this technical note, we consider linear exponential quadratic (LEQ) control for mean field stochastic differential equations (MFSDEs). The MFSDE includes the expectation value of state and control, and the objective functional is exponential of a quadratic functional in state, control, and their expectations. We obtain the explicit optimal solution as well as the optimal cost. The corresponding optimal solution is linear in state and its expectation, which is characterized by the Riccati differential equations (RDEs). The results are obtained by showing that after applying the completion of squares method, the remaining exponentiated stochastic integral and additional RDE terms can be eliminated together by taking expectation since they constitute the associated Radon–Nikodym derivative. As an extension of the problem, the LEQ zero-sum differential game is considered, for which we obtain the explicit optimal solution (saddle-point equilibrium) as well as the optimal cost.

We consider two-player risk-sensitive zero-sum differential games (RSZSDGs). In our problem setup, both the drift term and the diffusion term in the controlled stochastic differential equation are dependent on the state and controls of both players, and the objective functional is of the risk-sensitive type. First, a stochastic maximum principle type necessary condition for an open-loop saddle point of the RSZSDG is established via nonlinear transformations of the adjoint processes of the equivalent risk-neutral stochastic zero-sum differential game. In particular, we obtain two variational inequalities, namely, the pair of saddle-point inequalities of the RSZSDG. Next, we obtain the Hamilton–Jacobi–Isaacs partial differential equation for the RSZSDG, which provides a sufficient condition for a feedback saddle point of the RSZSDG, using a logarithmic transformation of the associated value function. Finally, we study the extended linear-quadratic RSZSDG (LQ-RSZSDG). We show intractability of the extended LQ-RSZSDG with the state and/or controls of both players appearing in the diffusion term. This unexpected intractability could lead to nonlinear open-loop and feedback saddle points even if the problem itself is essentially LQ and the Isaacs condition holds.

This note investigates the existence and uniqueness as well as the stability of the general decay rate of the global solution for neutral stochastic differential equations with time-varying delay under a locally Lipschitz condition, a contractive condition, and a monotonicity condition. The stability results are derived by using the Lyapunov function approach and some stochastic analysis techniques, which not only cover the exponential stability in the $p$ th $(p>0)$ -moment and the almost sure exponential stability, but also the polynomial stability in the $p$ th $(p>0)$ -moment and the almost sure polynomial stability. Two examples including one coupled system consisting of a mass–spring–damper connected to a pendulum and the nonlinear external random force are given to illustrate the effectiveness of the obtained results.

In this article, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of radicals (stubborn individuals). The distribution of radical opinions serves as an infinite-dimensional exogenous input to the FP equation, visibly influencing the steady opinion profile. We establish mathematical properties of the FP equation. In particular, we, first, show the well-posedness of the dynamic equation, second, provide existence result accompanied by a quantitative global estimate for the corresponding stationary solution, and, third, establish an explicit lower bound on the noise level that guarantees exponential convergence of the dynamics to stationary state. Combining the results in second and third readily yields the input-output stability of the system for sufficiently large noises. Next, using Fourier analysis, the structure of opinion clusters under the uniform initial distribution is examined. The results of analysis are validated through several numerical simulations of the continuum-agent model (partial differential equation) and the corresponding discrete-agent model (interacting stochastic differential equations) for a particular distribution of radicals.

A large number of significant applications involve numerical solution of stochastic differential equations (SDE's) evolving in Lie groups such as $SO(3)$ . In the engineering literature, the proper formulation of numerical schemes has largely been ignored so that many schemes are flawed, i.e., do not guarantee the solution stays in the Lie group. There is a small mathematics literature but it is not easily accessible. With this in mind, we give a directly accessible derivation of numerical schemes for solving SDE's that do not rely on differential geometry or advanced random process theory. In doing so, we develop some new results. We illustrate the numerical schemes with simulations.

This paper studies the linear–quadratic mean-field game (MFG) for a class of stochastic delayed systems. We consider a large-population system, where the dynamics of each agent is modeled by a stochastic differential delayed equation. The consistency condition is derived through an auxiliary system, which is an anticipated forward–backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such an AFBSDDE system can be obtained using a continuation method. Thus, the MFG strategies can be defined on an arbitrary time horizon, not necessary on a small time horizon by a commonly used contraction mapping method. Moreover, the decentralized strategies are verified to satisfy the $\epsilon$ -Nash equilibrium property. For illustration, three special cases of delayed systems are further explored, for which the closed-loop and open-loop MFG strategies are derived, respectively.

The behaviour of a standard super-twist controller under stochastic perturbations, when its dynamic is governed by the stochastic differential equation of Ito type with discontinuos right-hand side, is studied. The suggested analysis is based on the Lyapunov functions $V_{t}=V\left(x_{t},y_{t}\right)$ (Polyakov–Poznyak, Moreno–Osorio, Orlov and Utkin) designed for the stability analysis of the deterministic version of super-twist controllers. The major finding is that under stochastic (in fact, unbounded) perturbations, the special selection of a gain-parameter of such controller, making it depending on $V_{t}$ and its gradient $\frac{ \partial V}{\partial y}\left(x_{t},y_{t}\right)$ , provides the controller with an adaptivity property and guarantees the mean-square convergence of $V_{t}$ into the prespecified zone around the origin which depends on the diffusion parameter of stochastic noise, the upper estimate of the second derivative $\frac{\partial ^{2}V}{\partial y^{2}}$ as well as on the parameters of the controller.

This paper considers two classes of large population stochastic differential games connected to optimal and robust decentralized control of large-scale multiagent systems. The first problem ( P1 ) is one where each agent minimizes an exponentiated cost function, capturing risk-sensitive behavior, whereas in the second problem ( P2 ) each agent minimizes a worst-case risk-neutral cost function, where the “worst case” stems from the presence of an adversary entering each agent’s dynamics characterized by a stochastic differential equation. In both problems, the individual agents are coupled through the mean field term included in each agent’s cost function, which captures the average or mass behavior of the agents. We solve both P1 and P2 via mean field game theory. Specifically, we first solve a generic risk-sensitive optimal control problem and a generic stochastic zero-sum differential game, where the corresponding optimal controllers are applied by each agent to construct the mean field systems of P1 and P2 . We then characterize an approximated mass behavior effect on an individual agent via a fixed-point analysis of the mean field system. For each problem, P1 and P2 , we show that the approximated mass behavior is in fact the best estimate of the actual mass behavior in various senses as the population size, $N$ , goes to infinity. Moreover, we show that for finite $N$ , there exist $\epsilon$ - Nash equilibria for both P1 and P2 , where the corresponding individual Nash strategies are decentralized in terms of local state information and the approximated mass behavior. We also show that $\epsilon$ can be taken to be arbitrarily small when $N$ is sufficiently large. We show that the $\epsilon$ - Nash equilibria of P1 and P2 are partially equivalent in the sense that the individual Nash strategies share identical control laws, but the approximated mass behaviors for P1 and P2 are different, since in P2 , the mass behavior is also affected by the associated worst-case disturbance. Finally, we prove that the Nash equilibria for P1 and P2 both feature robustness, and as the parameter characterizing this robustness becomes infinite, the two Nash equilibria become identical and equivalent to that of the risk-neutral case, as in the one-agent risk-sensitive and robust control theory.

A team problem formulation of distributed stochastic differential systems with decentralized noisy information structures is considered, and a stochastic Pontryagin's maximum principle is applied to derive sufficient team and person-by-person optimality conditions. The sufficient conditions are given in terms of local convexity conditions of the Hamiltonian, and the corresponding conditional variational equations. These appear to be analogous to the convexity of the pay-off, in the action spaces, of static team problems. The stochastic maximum principle is applied to several examples in filtering and control. © 1963-2012 IEEE. 62 2 708 723

It is shown that the Sliding Mode Control (SMC) technique can be successfully applied to stochastic systems governed by the stochastic differential equations of the Ito type which contain additive stochastic unbounded white noise perturbations. The existence of a strong solution to the corresponding stochastic differential inclusion is discussed. To do this approach workable the gain control parameter is suggested to be done state-dependent on norms of system states. It is demonstrated that under such modification of the conventional SMC we can guarantee the exponential convergence of the averaged squared norm of the sliding variable to μ-zone (around the sliding surface) which is proportional to the diffusion parameter σ in the model description and inversely depending on the gain parameter k 0 .

This paper examines the long-standing problem of linear quadratic regulation and stabilization for Ito stochastic systems with input delay. This problem remains a primary challenge because the separation principle does not hold for Ito stochastic systems. This paper presents a complete solution to the problem: 1) The (sufficient and necessary) solvability condition of the optimal control and the analytical controller are given based on the modified Riccati differential equation defined herein. 2) The sufficient and necessary stabilization condition in mean square sense is explored. We show that the Ito stochastic system with input delay is stabilized if and only if the modified algebraic Riccati equation developed in this paper has a unique positive-definite solution. The essential obstacle encountered in this paper concerns a Delayed Forward-Backward Stochastic Differential Equation (D-FBSDE), which is mathematical challenging.

This paper introduces the backward mean-field (MF) linear-quadratic-Gaussian (LQG) games (for short, BMFLQG) of weakly coupled stochastic large-population system. In contrast to the well-studied forward mean-field LQG games, the individual state in our large-population system follows the backward stochastic differential equation (BSDE) whose terminal instead initial condition should be prescribed. Two classes of BMFLQG games are discussed here and their decentralized strategies are derived through the consistency condition. In the first class, the individual agents of large-population system are weakly coupled in their state dynamics and the full information can be accessible to all agents. In the second class, the coupling structure lies in the cost functional with only partial information structure. In both classes, the asymptotic near-optimality property (namely, $\epsilon$ - Nash equilibrium) of decentralized strategies are verified. To this end, some estimates to BSDE, are presented in the large-population setting.

In this technical note, we restage the positive effects of Gaussian white noises for continuous-time driftless input-affine nonholonomic systems. Compared with previous studies, we found that the positive effects are hidden in the shadows of nonuniqueness of stochastic integrals over multi-dimensional Wiener processes. Furthermore, we found that the positive effects are artificially restaged by numerical calculations with use of Euler scheme for ordinary differential equations. Our results demonstrate that the positive effects act as “hidden independent control inputs” for nonholonomic systems. This implies that our results provide a constructive procedure for designing control inputs such that the origins become $p$ -th moment exponentially or finite-time stable.

In this technical note, we formulate and investigate a class of mean-field linear-quadratic-Gaussian (LQG) games for stochastic integral systems. Unlike other literature on mean-field games where the individual states follow the controlled stochastic differential equations (SDEs), the individual states in our large-population system are characterized by a class of stochastic Volterra-type integral equations. We obtain the Nash certainty equivalence (NCE) equation and hence derive the set of associated decentralized strategies. The $\epsilon$ -Nash equilibrium properties are also verified. Due to the intrinsic integral structure, the techniques and estimates applied here are significantly different from those existing results in mean-field LQG games for stochastic differential systems. For example, some Fredholm equation in the mean-field setup is introduced for the first time. As for applications, two types of stochastic delayed systems are formulated as the special cases of our stochastic integral system, and relevant mean-field LQG games are discussed.

In this technical note, a nonzero sum differential game is studied, where the state dynamics follows a backward stochastic differential equation with time-delayed generator. An Arrow's sufficient condition for open-loop equilibrium point is proved. A linear-quadratic differential game and a pension fund management problem with time-delayed surplus are discussed as the applications of the sufficient optimality condition.

The active disturbance rejection control (ADRC) is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics, external disturbance, and unknown part in coefficient of the control. However, all theoretical works up to present are limited to deterministic uncertainty. In this technical note, we generalize the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise. We first design an extended state observer (ESO) that is used to estimate both state, and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance. It is shown that the resulting closed-loop system is practically stable in the mean-square topology. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.

This paper studies a linear-quadratic optimal control problem derived by forward-backward stochastic differential equations, where the drift coefficient of the observation equation is linear with respect to the state $x$ , and the observation noise is correlated with the state noise, in the sense that the cross-variation of the state and the observation is nonzero. A backward separation approach is introduced. Combining it with variational method and stochastic filtering, two optimality conditions and a feedback representation of optimal control are derived. Closed-form optimal solutions are obtained in some particular cases. As an application of the optimality conditions, a generalized recursive utility problem from financial markets is solved explicitly.

A consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest. This problem can be generalized in the context of convex metric spaces that extend the standard notion of convexity. In this paper we introduce and analyze a randomized gossip algorithm for solving the generalized consensus problem on convex metric spaces, where the communication between agents is controlled by a set of Poisson counters. We study the convergence properties of the algorithm using stochastic calculus. In particular, we show that the distances between the states of the agents converge to zero with probability one and in the $r^{th} $ mean sense. In the special case of complete connectivity and uniform Poisson counters, we give upper bounds on the dynamics of the first and second moments of the distances between the states of the agents. In addition, we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations.

In this technical note, we discuss a nonzero-sum stochastic differential game with delays. Not only the state variable, but also control variables of players involve delays. This kind of games are motivated by some interesting problems arising from economics and finance. Using anticipated backward stochastic differential equations, we establish a necessary condition and a sufficient condition of maximum principle for the delayed game problem. To explain theoretical results, we apply them to an economic problem.

Stochastic differential equations (SDEs) are widely adopted to describe systems with stochastic disturbances, while they are not necessarily the best models in some specific situations. This paper considers the nonlinear systems described by random differential equations (RDEs). The notions and the corresponding criteria of noise-to-state stability, asymptotic gain and asymptotic stability are proposed, in the $m$ -th moment or in probability. Several estimation methods of stochastic processes are presented to explain the reasonability of the assumptions used in theorems. As applications of stability criteria, some examples about stabilization, regulation and tracking are considered, respectively. A theoretical framework on stability of RDEs is finally constructed, which is distinguished from the existing framework of SDEs.

This technical note is concerned with a partially observed optimal control problem, whose novel feature is that the cost functional is of mean-field type. Hence determining the optimal control is time inconsistent in the sense that Bellman's dynamic programming principle does not hold. A maximum principle is established using Girsanov's theorem and convex variation. Some nonlinear filtering results for backward stochastic differential equations (BSDEs) are developed by expressing the solutions of the BSDEs as some Ito's processes. An illustrative example is demonstrated in terms of the maximum principle and the filtering.

This technical note focuses on optimal filtering for Ito stochastic continuous-time systems with multiple delayed measurements. Stochastic analysis and calculus of stochastic variables are the main tools employed for the analysis and design. For an Ito-stochastic system, its stochastic differential and integral have a significant place and are different from that for other stochastic systems owing to the Wiener or the Brownian process. In this technical note, an Ito stochastic continuous-time system with multiple delayed measurements is first reduced to an equivalent system with delay free measurements by solving stochastic equation via the non-singularity of the transition matrix instead of reorganizing the innovation. Then, based on the delay-free measurements the optimal filter is derived through calculation of the conditional expectation. It is should be stressed that the optimal filter follows directly from the manipulation of the performance. Finally, a short interest rate model in mathematical finance is chosen to demonstrate the design of the optimal filter via the approach proposed in the technical note.

This paper introduces Stochastic Port-Hamiltonian Systems (SPHS's), whose dynamics are described by Ito stochastic differential equations. SPHS's are extension of the deterministic port-Hamiltonian systems which are used to express various passive systems. First, we show a necessary and sufficient condition to preserve the stochastic port-Hamiltonian structure of the system under a class of coordinate transformations. Second, we derive a condition for the system to be stochastic passive. Third, we equip Stochastic Generalized Canonical Transformations (SGCT's), which are pairs of coordinate and feedback transformations preserving the stochastic port-Hamiltonian structure. Finally, we propose a stochastic stabilization framework based on stochastic passivity and SGCT's.

Due to the existence of various uncertainties, the design of distributed estimation algorithms with robustness and high accuracy is an urgent demand for sensor network applications. This paper is aimed at investigating the design of distributed parameter estimation algorithms and the analysis of their convergence properties in uncertain sensing and communication environments. Consensus-based distributed parameter estimation algorithms for both discrete-time and continuous-time cases are established, which are suitable for unreliable communication networks with stochastic communication noises, random link gains and Markovian signal losses. Under mild conditions on stochastic noises, gain function and topology-switching Markov chain, we establish both the mean square and almost sure convergence of the designed algorithms by use of probability limit theory, algebraic graph theory, stochastic differential equation theory and Markov chain theory. The effect of sensor-dependent gain functions on the convergence of the algorithm is also analyzed.

This paper is concerned with the problem of predicting the intensity process of an observed doubly stochastic multichannel Poisson process. Under the only hypothesis that the covariance function of the intensity process is separable, recursive algorithms for the computation of the optimal linear filter and predictor are designed. Approximate solutions to the nonlinear filtering and prediction problems are also given. The main advantage of the proposed solutions is that they can be applied to those situations where the intensity process does not satisfy a stochastic differential equation.

International audience; The Extended Kalman Filter (EKF) is a very popular tool dealing with state estimation. Its continuous-discrete version (CD-EKF) estimates the state trajectory of continuous-time nonlinear models, whose internal state is described by a stochastic differential equation and which is observed through a noisy nonlinear form of the sampled state. The prediction step of the CD-EKF leads to solve a differential equation that cannot be generally solved in a closed form. This technical note presents an overview of the numerical methods, including recent works, usually implemented to approximate this filter. Comparisons of theses methods on two different nonlinear models are finally presented. The first one is the Van der Pol oscillator which is widely used as a benchmark. The second one is a neuronal population model. This more original model is used to simulate EEG activity of the cortex. Experiments showed better stability properties of implementations for which the positivity of the prediction matrix is guaranteed.

An average consensus protocol is proposed for continuous-time double-integrator multi-agent systems with measurement noises under fixed topologies. The time-varying control gain is employed to attenuate noises. The closed-loop system is therefore a time-varying linear stochastic differential equation. By determining the state transition matrix of this closed-loop system, the dynamic characteristics of the multi-agent system can be fully described. It is proved that in the noisy communication environment the average consensus can be achieved if and only if the communication topology is a balanced and strongly connected graph, and the time-varying control gain satisfies the stochastic approximation-type conditions. Under the proposed protocol, the position of each agent is convergent in mean square to a common random variable whose mathematical expectation is the average of initial positions and initial velocities of all agents in the system, while each agent's velocity is convergent in mean square to a common random variable whose mathematical expectation and variance are both zero.

This note gives some new Razumikhin-type theorems on th moment stability of stochastic functional differential equations with Markovian switching (SFDEwMS) by using auxiliary ordinary differential equation. The main results of this note allow the diffusion operator associated with the underlying SFDEwMS of the Lyapunov function along a solution of the system to be not always negative. An example is provided to illustrate the effectiveness of the proposed results.

This technical note is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by backward stochastic differential equations (BSDEs). This kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities. We establish a necessary condition and a sufficient condition in the form of maximum principle for open-loop equilibrium point of the foregoing games respectively. To explain the theoretical results, we use them to study a financial problem.

A maximum principle for partially observed stochastic recursive optimal control problems is obtained under the assumption that control domains are not necessarily convex and forward diffusion coefficients do not contain control variables. Such kind of recursive optimal control problems have wide applications in finance and economics such as recursive utility optimization and principal-agent problems. By virtue of a classical spike variational approach and a filtering technique, the maximum principle is obtained, and the related adjoint processes are characterized by the solutions of forward-backward stochastic differential equations in finite-dimensional spaces. Then our theoretical results are applied to study a partially observed linear-quadratic recursive optimal control problem. In addition, for the case with initial and terminal state constraints, the corresponding maximum principle is also obtained by using Ekeland's variational principle.

This paper develops a notion of approximation for a class of stochastic hybrid systems that includes, as special cases, both jump linear stochastic systems and linear stochastic hybrid automata. Our approximation framework is based on the recently developed notion of the so-called stochastic simulation functions. These Lyapunov-like functions can be used to rigorously quantify the distance or error between a system and its approximate abstraction. For the class of jump linear stochastic systems and linear stochastic hybrid automata, we show that the computation of stochastic simulation functions can be cast as a tractable linear matrix inequality problem. This enables us to compute the modeling error incurred by abstracting some of the continuous dynamics, or by neglecting the influence of stochastic noise, or even the influence of stochastic discrete jumps.

In this note we study the problem of state estimation for a class of sampled-measurement stochastic hybrid systems, where the continuous state x satisfies a linear stochastic differential equation, and noisy measurements y are taken at assigned discrete-time instants. The parameters of both the state and measurement equation depend on the discrete state q of a continuous-time finite Markov chain. Even in the fault detection setting we consider-at most one transition for q is admissible-the switch may occur between two observations, whence it turns out that the optimal estimates cannot be expressed in parametric form and time integrations are unavoidable, so that the known estimation techniques cannot be applied. We derive and implement an algorithm for the estimation of the states x, q and of the discrete-state switching time that is convenient for both recursive update and the eventual numerical quadrature. Numerical simulations are illustrated

In this note, the problem of stochastic stability for linear systems with jump parameters being semi-Markovian rather than full Markovian is further investigated. In particular, the system under consideration is described by Ito type nonlinear stochastic differential equations with phase type semi-Markovian jump parameters. Stochastic stability conditions are presented

In this paper, we consider a dynamic M-ary detection problem when Markov chains are observed through a Wiener process. These systems are fully specified by a candidate set of parameters, whose elements are, a rate matrix for the Markov chain and a parameter for the observation model. Further, we suppose these parameter sets can switch according to the state of an unobserved Markov chain and thereby produce an observation process generated by time varying (jump stochastic) parameter sets. Given such an observation process and a specified collection of models, we estimate the probabilities of each model parameter set explaining the observation. By defining a new augmented state process, then applying the method of reference probability, we compute matrix-valued dynamics, whose solutions estimate joint probabilities for all combinations of candidate model parameter sets and values taken by the indirectly observed state process. These matrix-valued dynamics satisfy a stochastic integral equation with a Wiener process integrator. Using the gauge transformation techniques introduced by Clark and a pointwise matrix product, we compute robust matrix-valued dynamics for the joint probabilities on the augmented state space. In these new dynamics, the observation Wiener process appears as a parameter matrix in a linear ordinary differential equation, rather than an integrator in a stochastic integral equation. It is shown that these robust dynamics, when discretised, enjoy a deterministic upper bound which ensures nonnegative probabilities for any observation sample path. In contrast, no such upper bounds can be computed for Taylor expansion approximations, such as the Euler-Maryauana and Milstein schemes. Finally, by exploiting a duality between causal and anticausal robust detector dynamics, we develop an algorithm to compute smoothed mode probability estimates without stochastic integrations. A computer simulation demonstrating performance is included.

At important events or announcements, there can be large changes in the value of financial portfolios. Events and their corresponding jumps can occur at random or scheduled times. However, the amplitude of the response in either case can be unpredictable or random. While the volatility of portfolios is often modeled by continuous Brownian motion processes, discontinuous jump processes are more appropriate for modeling the response to important external events that significantly affect the prices of financial assets. Discontinuous jump processes are modeled by compound Poisson processes for random events or by quasi-deterministic jump processes for scheduled events. In both cases, the responses are randomly distributed and are modeled in a stochastic differential equation formulation. The objective is the maximal, expected total discounted utility of terminal wealth and instantaneous consumption. This paper was motivated by a paper by Rishel (1999) concerning portfolio optimization when prices are dependent on external events. However, the model has been significantly generalized for more realistic computational considerations with constraints and parameter values. The problem is illustrated for a canonical risk-adverse power utility model. However, the usual explicit canonical solution is not strictly valid. Fortunately, iterations about the canonical solution result in computationally feasible approximations.