Dynkin Formula Research Articles

Event-triggered interval type-2 fuzzy control of networked systems with extended dissipative under Markovian switching

In this technical note, the problem of the extended dissipative control for interval type-2 fuzzy model with nonlinear network systems in the event-triggered framework is investigated. A novel dynamic event-triggered strategy is developed to ascertain the instants of activation. It has the advantage of reducing data transmission and optimising the utilisation of communication resources. On the basis of the proposed mechanism, the stochastic Lyapunov function including internal dynamic variables is constructed. Based on this function, coupled with the inequality scaling technique and the application of Dynkin's formula, we derive a sufficient condition for the system's stochastic asymptotic stability and extended dissipativity. The design of the dynamic event-triggered extended dissipative controller becomes challenging because the new sufficient condition is nonlinear for some matrix variables. However, utilising the Projection Lemma and the co-design approach, such a controller can be developed based on the obtained analysis result. The methodology is demonstrated through a rigorous numerical simulation case study.

Output-Feedback Control for Stochastic Feedforward Nonlinear Systems With Markovian Switching

This paper studies the output-feedback control for stochastic feedforward nonlinear systems with Markovian switching and unknown measurement sensitivity. By developing a stochastic dual-domination design approach, a state observer and an output feedback control law are designed. By using the generalized It<i>&#x00F4;</i> formula and Dynkin formula for Markovian switching systems, it is shown that the closed-loop system has a unique solution and the solution of the closed-loop systems is almost surely asymptotically stable. Finally, a simulation example is given to illustrate the effectiveness of the control scheme.

Development of Task-Space Nonholonomic Motion Planning Algorithm Based on Lie-Algebraic Method

In this paper, a Lie-algebraic nonholonomic motion planning technique, originally designed to work in a configuration space, was extended to plan a motion within a task-space resulting from an output function considered. In both planning spaces, a generalized Campbell–Baker–Hausdorff–Dynkin formula was utilized to transform a motion planning into an inverse kinematic task known for serial manipulators. A complete, general-purpose Lie-algebraic algorithm is provided for a local motion planning of nonholonomic systems with or without output functions. Similarities and differences in motion planning within configuration and task spaces were highlighted. It appears that motion planning in a task-space can simplify a planning task and also gives an opportunity to optimize a motion of nonholonomic systems. Unfortunately, in this planning there is no way to avoid working in a configuration space. The auxiliary objective of the paper is to verify, through simulations, an impact of initial parameters on the efficiency of the planning algorithm, and to provide some hints on how to set the parameters correctly.

Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic [Formula: see text]-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

On infinitesimal generators of sublinear Markov semigroups

We establish a Dynkin formula and a Courrege-von Waldenfels theorem for sublinear Markov semigroups. In particular, we show that any sublinear operator $A$ on $C_c^{\infty}(\mathds{R}^d)$ satisfying the positive maximum principle can be represented as supremum of a family of pseudo-differential operators: \begin{equation*} Af(x) = \sup_{\alpha \in I} (-q_{\alpha}(x,D) f)(x). \end{equation*} As an immediate consequence, we obtain a representation formula for infinitesimal generators of sublinear Markov semigroups with a sufficiently rich domain. We give applications in the theory of non-linear Hamilton-Jacobi-Bellman equations and Levy processes for sublinear expectations.

Baker–Campbell–Hausdorff–Dynkin Formula for the Lie Algebra of Rigid Body Displacements

The paper proposes, for the first time, a closed form of the Baker–Campbell–Hausdorff–Dynkin (BCHD) formula in the particular case of the Lie algebra of rigid body displacements. For this purpose, the structure of the Lie group of the rigid body displacements S E ( 3 ) and the properties of its Lie algebra s e ( 3 ) are used. In addition, a new solution to this problem in dual Lie algebra of dual vectors is delivered using the isomorphism between the Lie group S E ( 3 ) and the Lie group of the orthogonal dual tensors.

Guaranteed cost control for impulsive nonlinear Itô stochastic systems with mixed delays

This paper mainly investigates the problem of guaranteed cost control for a class of impulsive nonlinear Itô stochastic systems with mixed delays. Several sufficient conditions are proposed to ensure the closed-loop systems are asymptotically stable in probability and the given cost function is guaranteed to have an upper bound by using Lyapunov–Krasovskii functional, Hamilton–Jacobi inequalities, Dynkin formula and Razumikhin technique. Finally, a simulation result shows the effectiveness of the proposed method. Different the previous works, the employed method in this paper is novel and the obtained results are less conservative.

Multiconstrained Finite-Horizon Piecewise Deterministic Markov Decision Processes with Unbounded Transition Rates

This paper studies a multiconstrained problem for piecewise deterministic Markov decision processes (PDMDPs) with unbounded cost and transition rates. The goal is to minimize one type of expected finite-horizon cost over history-dependent policies while keeping some other types of expected finite-horizon costs lower than some tolerable bounds. Using the Dynkin formula for the PDMDPs, we obtain an equivalent characterization of occupancy measures and express the expected finite-horizon costs in terms of occupancy measures. Under suitable assumptions, the existence of constrained-optimal policies is shown, the linear programming formulation and its dual program for the constrained problem are derived, and the strong duality between the two programs is established. An example is provided to demonstrate our results.

An elementary derivation of moments of Hawkes processes

AbstractHawkes processes have been widely used in many areas, but their probability properties can be quite difficult. In this paper an elementary approach is presented to obtain moments of Hawkes processes and/or the intensity of a number of marked Hawkes processes, in which the detailed outline is given step by step; it works not only for all Markovian Hawkes processes but also for some non-Markovian Hawkes processes. The approach is simpler and more convenient than usual methods such as the Dynkin formula and martingale methods. The method is applied to one-dimensional Hawkes processes and other related processes such as Cox processes, dynamic contagion processes, inhomogeneous Poisson processes, and non-Markovian cases. Several results are obtained which may be useful in studying Hawkes processes and other counting processes. Our proposed method is an extension of the Dynkin formula, which is simple and easy to use.

Mean-Square Exponential Input-to-State Stability of Stochastic Gene Regulatory Networks with Multiple Time Delays

This paper is concerned with the input-to-state stability of stochastic gene regulatory networks with multiple time delays. It is well acknowledged that stochastic systems can accurately describe some complex systems with random disturbances. So it is significant that stochastic systems are applied to model gene regulatory networks because of the complex relationship between genes and proteins from a micro perspective. Considering the differences between stochastic differential equations and ordinary differential equations, we introduce the new stability criterion which is different from the general stability criteria. Making use of Lyapunov functionals, It$$\hat{o}$$ formula and Dynkin formula, we present sufficient conditions to guarantee that the proposed system is mean-square exponentially input-to-state stable. Moreover, numerical examples are given to illustrate validity and feasibility of the obtained results.

Analysis of Exponential Stability for Neutral Stochastic Cohen-Grossberg Neural Networks with Mixed Delays

This paper is concerned with the mean-square exponential input-to-state stability problem for a class of stochastic Cohen-Grossberg neural networks. Different from prior works, neutral terms and mixed delays are discussed in our system. By employing the Lyapunov-Krasovskii functional method, Itô formula, Dynkin formula, and stochastic analysis theory, we obtain some novel sufficient conditions to ensure that the addressed system is mean-square exponentially input-to-state stable. Moreover, two numerical examples and their simulations are given to illustrate the correctness of the theoretical results.

Stability of reaction–diffusion systems with stochastic switching

In this paper, we investigate the stability for reaction systems with stochastic switching. Two types of switched models are considered: (i) Markov switching and (ii) independent and identically distributed switching. By means of the ergodic property of Markov chain, Dynkin formula and Fubini theorem, together with the Lyapunov direct method, some sufficient conditions are obtained to ensure that the zero solution of reaction–diffusion systems with Markov switching is almost surely exponential stable or exponentially stable in the mean square. By using Theorem 7.3 in [R. Durrett, Probability: Theory and Examples, Duxbury Press, Belmont, CA, 2005], we also investigate the stability of reaction–diffusion systems with independent and identically distributed switching. Meanwhile, an example with simulations is provided to certify that the stochastic switching plays an essential role in the stability of systems.

Partial self-exciting point processes and their parameter estimations

A partial self-exciting point process is proposed based on some phenomena in real world, which is a new kind of Hawkes process with partial self-exciting effect. The features, properties and moments of the partial self-exciting point process are investigated in terms of Dynkin formula and the method proposed by Cui et al. (2017), and the relationship between the new and conventional Hawkes processes is given. The maximum-likelihood estimations (MLEs) for the parameters in the partial self-exciting point process for two data situations are given via some sets of equations that can be solved numerically by suing the commend fslove in Matlab. Some numerical examples are presented based on simulation data to illustrate the MLE method, and an algorithm to generate the random event times for the partial self-exciting point process is presented as well. A primary goal of the work is to extend the applicability of Hawkes process and to present the MLE of parameters for partial self-exciting point process.

Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula

Verification theorems are key results to successfully employ the dynamic programming approach to optimal control problems. In this paper, we introduce a new method to prove verification theorems for infinite dimensional stochastic optimal control problems. The method applies in the case of additively controlled Ornstein–Uhlenbeck processes, when the associated Hamilton–Jacobi–Bellman (HJB) equation admits a mild solution (in the sense of [J. Differential Equations 262 (2017) 3343–3389]). The main methodological novelty of our result relies on the fact that it is not needed to prove, as in previous literature (see, e.g., [Comm. Partial Differential Equations 20 (1995) 775–826]), that the mild solution is a strong solution, that is, a suitable limit of classical solutions of approximating HJB equations. To achieve the goal, we prove a new type of Dynkin formula, which is the key tool for the proof of our main result.

Input-to-state stability of stochastic nonlinear fuzzy Cohen–Grossberg neural networks with the event-triggered control

ABSTRACT In this paper, we investigate a class of stochastic nonlinear fuzzy Cohen–Grossberg neural networks with feedback control and an unknown exogenous disturbance. By using the Lyapunov function, Itô's formula, Dynkin's formula, Comparison principle and stochastic analysis theory, we show that the considered system is input-to-state stable with the help of the designed event-triggered mechanism. Moreover, we also guarantee that the internal execution time intervals of control task will not be arbitrarily small. Finally, some remarks and discussions have been provided to show that our results are meaningful.

The Estimates of the Mean First Exit Time of a Bistable System Excited by Poisson White Noise

We propose a method to find an approximate theoretical solution to the mean first exit time (MFET) of a one-dimensional bistable kinetic system subjected to additive Poisson white noise, by extending an earlier method used to solve stationary probability density function. Based on the Dynkin formula and the properties of Markov processes, the equation of the mean first exit time is obtained. It is an infinite-order partial differential equation that is rather difficult to solve theoretically. Hence, using the non-Gaussian property of Poisson white noise to truncate the infinite-order equation for the mean first exit time, the analytical solution to the mean first exit time is derived by combining perturbation techniques with Laplace integral method. Monte Carlo simulations for the bistable system are applied to verify the validity of our approximate theoretical solution, which shows a good agreement with the analytical results.

Unifying the Dynkin and Lebesgue–Stieltjes formulae

AbstractWe establish a local martingaleMassociate withf(X,Y) under some restrictions onf, whereYis a process of bounded variation (on compact intervals) and eitherXis a jump diffusion (a special case being a Lévy process) orXis some general (càdlàg metric-space valued) Markov process. In the latter case,fis restricted to the formf(x,y)=∑k=1Kξk(x)ηk(y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes anL2-martingale. Convergence of the product of this Martingale with some deterministic function ( of time ) to 0 both inL2and almost sure is also considered and sufficient conditions for functions for which this happens are identified.

Synchronization of master-slave markovian switching complex dynamical networks with time-varying delays in nonlinear function via sliding mode control

In this article, a synchronization problem for master-slave Markovian switching complex dynamical networks with time-varying delays in nonlinear function via sliding mode control is investigated. On the basis of the appropriate Lyapunov-Krasovskii functional, introducing some free weighting matrices, new synchronization criteria are derived in terms of linear matrix inequalities (LMIs). Then, an integral sliding surface is designed to guarantee synchronization of master-slave Markovian switching complex dynamical networks, and the suitable controller is synthesized to ensure that the trajectory of the closed-loop error system can be driven onto the prescribed sliding mode surface. By using Dynkin's formula, we established the stochastic stablity of master-slave system. Finally, numerical example is provided to demonstrate the effectiveness of the obtained theoretical results.

Quadratic Optimal Control of Fractional Stochastic Differential Equation with Application

The paper is devoted to the study of optimal control of Quadratic Optimal Control of Fractional stochastic differential Equation with application of Economy Mode with different types of fractional stochastic formula (ITO, Stratonovich), By using the Dynkin formula, Hamilton-Jacobi-Bellman (HJB) equation and the inverse HJB equation are derived. Application is given to a stochastic model in economics.

A closed-form expansion approach for pricing discretely monitored variance swaps

Variance swaps are among the most useful tools for derivatives trading and risk management. For pricing discretely monitored variance swaps under a general class of jump–diffusion models, we propose a closed-form expansion based on the length of monitoring interval. Our method relies on an iterative application of the Dynkin formula, which is usually called the operator method in financial econometrics. Numerical examples are given for demonstrating the efficiency of the method.