arXiv subjects

Jinghai Shao

Publications and source records attributed to Jinghai Shao.

At least 19 recordsLinked to original sources

arXiv2025

Averaging principle for two time-scale stochastic differential equations with fast component in noncompact space

The asymptotic behavior for fully coupled multiscale stochastic systems becomes much complicated when the fast processes do not locate in a compact space. An example is constructed to show that the averaged coefficients may become discontinuous even they are originally Lipschitz continuous when the fast process locate in a noncompact space. This work aims to reveal the impact of ergodicity of the fast process on the establishment of the averaging principle. The crucial point is to characterize the continuous dependence of the invariant probability measure on parameters related to the slow process with respect to various distances in the Wasserstein space.

math.PR↗

arXiv2023

Existence of optimal feedback controls for McKean-Vlasov SDEs

This work concerns the optimal control problem for McKean-Vlasov SDEs. We provide explicit conditions to ensure the existence of optimal Markovian feedback controls. Moreover, based on the flow property of the McKean-Vlasov SDE, the dynamic programming principle is established, which will enable to characterize the value function via the theory of Hamilton-Jacobi-Bellman equation on the Wasserstein space.

math.PR↗

arXiv2023

Viscosity solutions to HJB equations associated with optimal control problem for McKean-Vlasov SDEs

This work concerns the optimal control problem for McKean-Vlasov SDEs. In order to characterize the value function, we develop the viscosity solution theory for Hamilton-Jacobi-Bellman (HJB) equations on the Wasserstein space using Mortensen's derivative. In particular, a comparison principle for viscosity solution is established. Our approach is based on Borwein-Preiss variational principle to overcome the loss of compactness for bounded sets in the Wasserstein space.

math.PR↗

arXiv2023

Optimal control problem for reflected McKean-Vlasov SDEs

This work investigates the optimal control problem for reflected McKean-Vlasov SDEs and the viscosity solutions to Hamilton-Jacobi-Bellman(HJB) equations on the Wasserstein space in terms of intrinsic derivative. It follows from the flow property of reflected McKean-Vlasov SDEs that the dynamic programming principle holds. Applying the decoupling method and the heat kernel estimates for parabolic equations, we show that the value function is a viscosity solution to an appropriate HJB equation on the Wasserstein space, where the characterization of absolutely continuous curves on the Wasserstein space by the continuity equations plays an important role. To establish the uniqueness of viscosity solution, we generalize the construction of a distance like function initiated in Burzoni et al.(SICON, 2020) to the Wasserstein space over multidimensional space and show its effectiveness to cope with HJB equations in terms of intrinsic derivative on the Wasserstein space.

math.PR↗

arXiv2023

Viscosity solutions approach to finite-horizon continuous-time Markov decision process

This paper investigates the optimal control problems for the finite-horizon continuous-time Markov decision processes with delay-dependent control policies. We develop compactification methods in decision processes, and show that the existence of optimal policies. Subsequently, through the dynamic programming principle of the delay-dependent control policies, the differential-difference Hamilton-Jacobi-Bellman (HJB) equation in the setting of discrete space was established. Under certain conditions, we give the comparison principle and further prove that the value function is the unique viscosity solution to this HJB equation. Based on this, we show that among the class of delay-dependent control policies, there is an optimal one which is Markovian.

math.PR↗

arXiv2023

Ergodicity and stability of hybrid systems with piecewise constant type state-dependent switching

To deal with stochastic hybrid systems with general state-dependent switching, we propose an approximation method by a sequence of stochastic hybrid systems with piecewise constant type switching. The convergence rate in the Wasserstein distance is estimated in terms of the difference between transition rate matrices. Our method is based on an elaborate construction of coupling processes in terms of Skorokhod's representation theorem for jumping processes. Moreover, we establish explicit criteria on the ergodicity and stability for stochastic hybrid systems with piecewise constant type switching. Some examples are given to illustrate the sharpness of these criteria.

math.PR↗

arXiv2023

Conditional McKean-Vlasov SDEs with jumps and Markovian regime-switching: wellposedness, propagation of chaos, averaging principle

We investigate the conditional McKean-Vlasov stochastic differential equations with jumps and Markovian regime-switching. We establish the strong wellposedness using L2-Wasser-stein distance on the Wasserstein space. Also, we establish the propagation of chaos for the associated mean-field interaction particle system with common noise and provide an explicit bound on the convergence rate. Furthermore, an averaging principle is established for two time-scale conditional McKean-Vlasov equations, where much attention is paid to the convergence of the conditional distribution term.

math.PR↗

arXiv2022

Averaging principle for two time-scale regime-switching processes

This work studies the averaging principle for a fully coupled two time-scale system, whose slow process is a diffusion process and fast process is a purely jumping process on an infinitely countable state space. The ergodicity of the fast process has important impact on the limit system and the averaging principle. We showed that under strongly ergodic condition, the limit system admits a unique solution, and the slow process converges in the L1-norm to the limit system. However, under certain weaker ergodicity condition, the limit system admits a solution, but not necessarily unique, and the slow process can be proved to converge weakly to a solution of the limit system.

math.PR↗

arXiv2022

Comparison theorem and stability under perturbation of transition rate matrices for regime-switching processes

A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to control pathwisely the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on the elaborate constructions of switching processes in the spirit of Skorokhod's representation theorem varying according to the problem being dealt with. In particular, this method can cope with the switching processes in an infinite state space and not necessarily being of birth-death type. As an application, some known results on ergodicity and stability of state-dependent regime-switching processes can be improved.

math.PR↗

arXiv2020

The existence of optimal control for continuous-time Markov decision processes in random environments

In this work, we investigate the optimal control problem for continuous-time Markov decision processes with the random impact of the environment. We provide conditions to show the existence of optimal controls under finite-horizon criteria. Under appropriate conditions, the value function is continuous and satisfies the dynamic programming principle. These results are established by introducing some restriction on the regularity of the optimal controls and by developing a new compactification method for continuous-time Markov decision processes, which is originally used to solve the optimal control problem for jump-diffusion processes.

math.OC↗

arXiv2020

Permanence and extinction of regime-switching predator-prey models

In this work we study the permanence and extinction of a regime-switching predator-prey model with Beddington-DeAngelis functional response. The switching process is used to describe the random changing of corresponding parameters such as birth and death rates of a species in different environments. Our criteria can justify whether a prey die out or not when it will die out in some environments and will not in others. Our criteria are rather sharp, and they cover the known on-off type results on permanence of predator-prey models without switching. Our method relies on the recent study of ergodicity of regime-switching diffusion processes.

math.PR↗

arXiv2020

The existence of optimal feedback controls for stochastic dynamical systems with regime-switching

In this work we provide explicit conditions on the existence of optimal feedback controls for stochastic processes with regime-switching. We use the compactification method which needs less regularity conditions on the coefficients of the studied stochastic systems. Two different kinds of controls have been considered: one is the control on the coefficients of the diffusion processes, another is the control on the transition rate matrices of the continuous-time Markov chains. Moreover, the dynamic programming principle is established after showing the continuity of the value function.

math.OC↗

arXiv2019

Long time behavior of Levy-driven Ornstein-Uhlenbeck process with regime-switchin

In this work we investigate the long time behavior of the Ornstein-Uhlenbeck process driven by Levy noise with regime-switching. We provide explicit criteria on the transience and recurrence of this process. Contrasted with the Ornstein-Uhlenbeck process driven simply by Brownian motion, whose stationary distribution must be light-tailed, both the jumps caused by the Levy noise and regime-switching described by Markov chain can derive the heavy-tailed property of the stationary distribution. In this work, the different role played by Levy measure and regime-switching process is clearly characterized.

math.PR↗

arXiv2018

Stability of regime-switching processes under perturbation of transition rate matrices

This work is concerned with the stability of regime-switching processes under the perturbation of the transition rate matrices. From the viewpoint of application, two kinds of perturbations are studied: the size of the transition rate matrix is fixed, and only the values of entries are perturbed; the values of entries and the size of the transition matrix are all perturbed. Moreover, both regular and irregular coefficients of the underlying system are investigated, which clarifies the impact of the regularity of the coefficients on the stability of the underlying system.

math.PR↗

arXiv2018

Asymptotic Behavior of SIRS Models in State-dependent Random Environments

The extinction and persistence of infective individuals are closely related to the random change of the environment. In this paper, via the random/stochastic SIRS models, we analyze qualitatively and quantitatively the impact caused by the random change of the environment. Our contributions consist in (i) giving some sufficient conditions on extinction (persistence) of the infectious individuals even though they are persistent (resp. extinct) in certain fixed environments; (ii) revealing the influence of random switching of incidence functions on extinction for the infectious individuals, which has not been studied before; (iii) establishing a criterion to judge extinction of the infectious individuals for a range of random/stochastic SIRS models with state-dependent switching via a stochastic comparison for functionals of jump processes. Moreover, some examples are set to illustrate the applications of our theory.

math.PR↗

arXiv2018

Stabilization of regime-switching processes by feedback control based on discrete time observations II: state-dependent case

This work investigates the almost sure stabilization of a class of regime-switching systems based on discrete-time observations of both continuous and discrete components. It develops Shao's work [SIAM J. Control Optim., 55(2017), pp. 724--740] in two aspects: first, to provide sufficient conditions for almost sure stability in lieu of moment stability; second, to investigate a class of state-dependent regime-switching processes instead of state-independent ones. To realize these developments, we establish an estimation of the exponential functional of Markov chains based on the spectral theory of linear operator. Moreover, through constructing order-preserving coupling processes based on Skorokhod's representation of jumping process, we realize the control from up and below of the evolution of state-dependent switching process by state-independent Markov chains.

math.PR↗

arXiv2018

Weak convergence of path-dependent SDEs with irregular coefficients

In this paper we develop via Girsanov's transformation a perturbation argument to investigate weak convergence of Euler-Maruyama (EM) scheme for path-dependent SDEs with Hölder continuous drifts. This approach is available to other scenarios, e.g., truncated EM schemes for non-degenerate SDEs with finite memory or infinite memory. Also, such trick can be applied to study weak convergence of truncated EM scheme for a range of stochastic Hamiltonian systems with irregular coefficients and with memory, which are typical degenerate dynamical systems. Moreover, the weak convergence of path-dependent SDEs under integrability condition is investigated by establishing, via the dimension-free Harnack inequality, exponential integrability of irregular drifts w.r.t. the invariant probability measure constructed explicitly in advance.

math.PR↗

arXiv2018

Weak convergence of Euler-Maruyama's approximation for SDEs under integrability condition

This work establishes the weak convergence of Euler-Maruyama's approximation for stochastic differential equations (SDEs) with singular drifts under the integrability condition in lieu of the widely used growth condition. This method is based on a skillful application of the dimension-free Harnack inequality. Moreover, when the drifts satisfy certain regularity conditions, the convergence rate is estimated. This method is also applicable when the diffusion coefficients are degenerate. A stochastic damping Hamiltonian system is studied as an illustrative example.

math.PR↗