Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay

We regard the stochastic functional differential equation with infinite delay as the result of the effects of stochastic perturbation to the deterministic functional differential equation , where is defined by x_t(θ)=x(t+θ),θ(−∞,0]. We assume that the deterministic system with infinite delay is exponentially stable. In this paper, we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable.     

On solutions to fuzzy stochastic differential equations with local martingales

This paper deals with a class of fuzzy stochastic differential equations (FSDEs) driven by a continuous local martingale under the Lipschitzian condition. Such equations can be useful in modeling hybrid systems, where the phenomena are simultaneously subjected to two kinds of uncertainties: randomness and fuzziness. The solutions of the FSDEs are the fuzzy stochastic processes, and their uniqueness is considered to be in a strong sense. Thus, the existence and uniqueness of solutions to the FSDEs under the Lipschitzian condition is first proven. Moreover, some asymptotic properties of the solutions to the FSDEs are investigated. Finally, an illustrating example on the interest term model is provided.     

Chaos expansion for the solutions of stochastic differential equations

In this note we generalize the Isobe–Sato formula for kernels of the Wiener–Ito chaos expansion to nonautonomous systems. Expansion of a transition density is obtained and some version of Wiener's famous “black-box” identification problem is solved.     

Efficient importance sampling maximum likelihood estimation of stochastic differential equations

Maximum likelihood estimation (MLE) of stochastic differential equations (SDEs) is difficult because in general the transition density function of these processes is not known in closed form, and has to be approximated somehow. An approximation based on efficient importance sampling (EIS) is detailed. Monte Carlo experiments, based on widely used diffusion processes, evaluate its performance against an alternative importance sampling (IS) strategy, showing that EIS is at least equivalent, if not superior, while allowing a greater flexibility needed when examining more complicated models.     

Split-step double balanced approximation methods for stiff stochastic differential equations

In the modelling of many important problems in science and engineering we face stiff stochastic differential equations (SDEs). In this paper, a new class of split-step double balanced (SSDB) approximation methods is constructed for numerically solving systems of stiff Itô SDEs with multi-dimensional noise. In these methods, an appropriate control function has been used twice to improve the stability properties. Under global Lipschitz conditions, convergence with order one in the mean-square sense is established. Also, the mean-square stability (MS-stability) properties of the SSDB methods have been analysed for a one-dimensional linear SDE with multiplicative noise. Therefore, the MS-stability functions of SSDB methods are determined and in some special cases, their regions of MS-stability have been compared to the stability region of the original equation. Finally, simulation results confirm that the proposed methods are efficient with respect to accuracy and computational cost.     

Stabilization and destabilization of hybrid systems of stochastic differential equations

This paper aims to determine whether or not a stochastic feedback control can stabilize or destabilize a given nonlinear hybrid system. New methods are developed and sufficient conditions on the stability and instability for hybrid stochastic differential equations are provided. These results are then used to examine stochastic stabilization and destabilization.     

On the exponential stability in mean square of neutral stochastic functional differential equations

In [1 and 2], some efforts have been devoted to the investigation of exponential stability in mean square of neutral stochastic functional differential equations. However, the results derived there are either difficult to demonstrate in a straightforward way for practical situations or somewhat too restricted to be applied to general neutral stochastic functional differential equations, for instance, nonautonomous cases. In this paper, we shall establish some results which are more effective and relatively easy to verify to obtain the required stability.     

Systematic physics constrained parameter estimation of stochastic differential equations

A systematic Bayesian framework is developed for physics constrained parameter inference of stochastic differential equations (SDE) from partial observations. Physical constraints are derived for stochastic climate models but are applicable for many fluid systems. A condition is derived for global stability of stochastic climate models based on energy conservation. Stochastic climate models are globally stable when a quadratic form, which is related to the cubic nonlinear operator, is negative definite. A new algorithm for the efficient sampling of such negative definite matrices is developed and also for imputing unobserved data which improve the accuracy of the parameter estimates. The performance of this framework is evaluated on two conceptual climate models.     

On stochastic Riccati equations for the stochastic LQR problem

This paper studies an approximation of stochastic Riccati equations for stochastic LQR problems some of which may be even with indefinite control weight costs.     

Infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems

This paper studies exponential convergence index assignment of stochastic control systems from the viewpoint of backward stochastic differential equation. Like deterministic control systems, it is shown that the exact controllability of an open-loop stochastic system is equivalent to the possibility of assigning an arbitrary exponential convergence index to the solution of the closed-loop stochastic system, formed by means of suitable linear feedback of the states. As an application, a sufficient and necessary condition for the existence and uniqueness of the solution of a class of infinite horizon forward-backward stochastic differential equations is provided.     

On stability in distribution of stochastic differential delay equations with Markovian switching

This paper provides a new sufficient condition for stability in distribution of stochastic differential delay equations with Markovian switching (SDDEs). It can be considered as an improvement to the result given by Yuan C. et al. in [6].     

On decomposition and integral representation of solutions for affine control systems

A decomposition formula for the trajectories of an affine control system with noncommuting control vector fields is presented.     

Functional differential equations driven by a fractional Brownian motion

In this paper, we prove a global existence and uniqueness result for the solution of a stochastic functional differential equation driven by a fractional Brownian motion with Hurst parameter H>1/2. We also study the dependence of the solution on the initial condition.     

Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems

A new approach to study the indefinite stochastic linear quadratic (LQ) optimal control problems, which we called the “equivalent cost functional method”, is introduced by Yu (2013) in the setup of Hamiltonian system. On the other hand, another important issue along this research direction, is the possible state feedback representation of optimal control and the solvability of associated indefinite stochastic Riccati equations. As the response, this paper continues to develop the equivalent cost functional method by extending it to the Riccati equation setup. Our analysis is featured by its introduction of some equivalent cost functionals which enable us to have the bridge between the indefinite and positive-definite stochastic LQ problems. With such bridge, some solvability relation between the indefinite and positive-definite Riccati equations is further characterized. It is remarkable the solvability of the former is rather complicated than the latter, hence our relation provides some alternative but useful viewpoint. Consequently, the corresponding indefinite linear quadratic problem is discussed for which the unique optimal control is derived in terms of state feedback via the solution of the Riccati equation. In addition, some example is studied using our theoretical results.     

Lie algebraic methods in optimal control of stochastic systems with exponential-of-integral cost

The purpose of this paper is to formulate and study the optimal control of partially observed stochastic systems with exponential-of-integral-sample cost, known as risk-sensitive problems, using Lie algebraic tools. This leads to the introduction of the sufficient statistic algebra, , through which one can determine á priori the maximum order of the controller. When , the construction of the control laws is addressed through extensions of the Wei–Norman method, as in nonlinear filtering problems. Aside from specific known finite-dimensional examples which are studied in order to delineate the application of the Lie algebraic tools, new classes of finite-dimensional controllers are identified as well. In addition, relations with minimax dynamic games are explored to best assess the importance and generality of the finite-dimensional control systems.     

Stability of weak numerical schemes for stochastic differential equations

This paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes, the regions of stability are also examined.