Switching Diffusion Processes Research Articles

General fluctuation-dissipation relations embedded in Markov switching diffusion processes

We present an integral theory for the fluctuation-dissipation relations embedded in Markov switching diffusion processes, which mimic the fluctuations induced simultaneously by noise originating from different time-scale levels. The formulas for the extended Einstein’s relation in terms of second-order moments, as well as the auto and cross spectra are derived. Several universal properties for two types of fluctuations are proved: The thermally induced one remains globally constant for isothermal systems and its spectrum is Lorentzian; the fluctuation induced by the switching noise has distinctive scaling laws that are embedded in the auto and cross spectra.

Least squares estimators for stochastic differential equations with Markovian switching

This work focuses on the parameter estimation for a class of switching diffusion processes which contains a continuous component and a discrete component. Under suitable conditions, we adopt the least square method to deal with the parameter estimation of stochastic differential equations with Markovian switching. More precisely, we first prove the consistency and the asymptotic distributions of the parameter estimator of least squares when $ \varepsilon\rightarrow0 $ and $ \Delta\rightarrow0 $. Then, we present an example of numerical simulation to illustrate the correctness of the results.

Optimal control of multiple Markov-switching stochastic systems with numerical applications

In this article the authors set up an optimal control framework for a hybrid stochastic system with dual or multiple Markov switching diffusion processes, while Markov chains governing these switching diffusions are not identical as assumed in the existing literature. For this purpose, the authors first combine two or more separate Markov chains into one compound Markov chain within continuous time context and give out concrete procedures about how to estimate corresponding generator matrix of the newly combined Markov chain via a modified EM algorithm, then the HJB equations for the optimal control problem with synthetic Markov switching diffusion are stated. As an application and illustration, the authors lay out an adapted numerical solution scheme for the problem about optimal consumption–investment decisions under multiple cyclical or switching markets. Relevant economic analysis and interpretations are provided as results.

Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics.

A novel approach to simulate simple protein-ligand systems at large time and length scales is to couple Markov state models (MSMs) of molecular kinetics with particle-based reaction-diffusion (RD) simulations, MSM/RD. Currently, MSM/RD lacks a mathematical framework to derive coupling schemes, is limited to isotropic ligands in a single conformational state, and lacks multiparticle extensions. In this work, we address these needs by developing a general MSM/RD framework by coarse-graining molecular dynamics into hybrid switching diffusion processes. Given enough data to parameterize the model, it is capable of modeling protein-protein interactions over large time and length scales, and it can be extended to handle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two protein-protein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules. To enable reproducibility, we have published our code in the MSM/RD software package.

An MCMC computational approach for a continuous time state-dependent regime switching diffusion process

ABSTRACT State-dependent regime switching diffusion processes or hybrid switching diffusion (HSD) processes are hard to simulate with classical methods which leads us to adopt a Markov chain Monte Carlo (MCMC) Bayesian approach very convenient to estimate complicated models such as the HSD one. In the HSD, the diffusion component is dependent on the switching discrete hidden regimes and the transition rates of the regime switching are dependent on the diffusion observations. Since in reality phenomena are only observed in discrete times, data imputation is called for to create more observations so as to have good approximations for the density of the diffusion process. Three categories of entities will be computed in a Bayesian context: The latent imputed observations, the regime switching states, and the parameters of the models. The latent imputed data is updated at random time intervals in block using a Metropolis Hastings algorithm. The switching states are computed by an adaptation of a forward filtering backward smoothing algorithm to the HSD model. The parameters are estimated after prior specifications and conditional posterior densities formulation using Gibbs sampler or Metropolis Hastings algorithm.

Bilinear Controllability of a Class of Advection–Diffusion–Reaction Systems

In this paper, we investigate the exact controllability properties of an advection–diffusion equation on a bounded domain, using time- and space-dependent velocity fields as the control parameters. This partial differential equation (PDE) is the Kolmogorov forward equation for a reflected diffusion process that models the spatiotemporal evolution of a swarm of agents. We prove that if a target probability density has bounded first-order weak derivatives and is uniformly bounded from below by a positive constant, then it can be reached in finite time using control inputs that are bounded in space and time. We then extend this controllability result to a class of advection–diffusion–reaction PDEs that corresponds to a hybrid switching diffusion process (HSDP), in which case the reaction parameters are additionally incorporated as the control inputs. For the HSDP, we first constructively prove controllability of the associated continuous-time Markov chain (CTMC) system in which the state space is finite. Then, we show that our controllability results for the advection–diffusion equation and the CTMC can be combined to establish controllability of the forward equation of the HSDP. Finally, we provide constructive solutions to the problem of asymptotically stabilizing an HSDP to a target nonnegative stationary distribution using time-independent state feedback laws, which correspond to spatially dependent coefficients of the associated system of PDEs.

Decomposition of the entropy production rate and nonequilibrium thermodynamics of switching diffusion processes.

A switching diffusion process (SDP) is a widely used stochastic model in physics and biology, especially for molecular motors that exhibit a discrete internal chemical kinetics as well as a continuous external mechanical motion. The nonequilibrium thermodynamics of switching diffusion processes has not been extensively studied yet. In the present paper, we propose the decomposition of the entropy production rate in one-dimensional SDPs, based on the flux decomposition. However, similar decompositions of the housekeeping heat dissipation rate and free energy dissipation rate cannot guarantee the non-negativity of each decomposed component. Hence, we modify this decomposition with the flow of exponential relative information under steady-state fluxes, resulting in another decomposition with all non-negative components. Furthermore, we also provide the nonequilibrium thermodynamics of one-dimensional SDPs under the perspectives of coarse -graining and exchange of information between the chemical kinetics and mechanical motion, resulting in several other decompositions of entropy production rate. Finally, we generalize all the results to high-dimensional SDPs with a more general mathematical treatment.

Decomposition of the entropy production rate and nonequilibrium thermodynamics of switching diffusion processes

A switching diffusion process (SDP) is a widely used stochastic model in physics and biology, especially for molecular motors that exhibit a discrete internal chemical kinetics as well as a continuous external mechanical motion. The nonequilibrium thermodynamics of switching diffusion processes has not been extensively studied yet. In the present paper, we propose the decomposition of the entropy production rate in one-dimensional SDPs, based on the flux decomposition. However, similar decompositions of the housekeeping heat dissipation rate and free energy dissipation rate cannot guarantee the non-negativity of each decomposed component. Hence, we modify this decomposition with the flow of exponential relative information under steady-state fluxes, resulting in another decomposition with all non-negative components. Furthermore, we also provide the nonequilibrium thermodynamics of one-dimensional SDPs under the perspectives of coarse -graining and exchange of information between the chemical kinetics and mechanical motion, resulting in several other decompositions of entropy production rate. Finally, we generalize all the results to high-dimensional SDPs with a more general mathematical treatment.

Some bivariate stochastic models arising from group representation theory

The aim of this paper is to study some continuous-time bivariate Markov processes arising from group representation theory. The first component (level) can be either discrete (quasi-birth-and-death processes) or continuous (switching diffusion processes), while the second component (phase) will always be discrete and finite. The infinitesimal operators of these processes will be now matrix-valued (either a block tridiagonal matrix or a matrix-valued second-order differential operator). The matrix-valued spherical functions associated to the compact symmetric pair (SU(2)×SU(2),diagSU(2)) will be eigenfunctions of these infinitesimal operators, so we can perform spectral analysis and study directly some probabilistic aspects of these processes. Among the models we study there will be rational extensions of the one-server queue and Wright–Fisher models involving only mutation effects.

Large deviations for empirical measures of switching diffusion processes with small parameters

We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small parameters.

Feynman-Kac formula for switching diffusions: connections of systems of partial differential equations and stochastic differential equations

Abstract This work develops Feynman-Kac formulae for switching diffusion processes. It first recalls the basic notion of a switching diffusion. Then the desired stochastic representations are obtained for boundary value problems, initial boundary value problems, and the initial value problems, respectively. Some examples are also provided.

On the goodness-of-fit testing for a switching diffusion process

We study the asymptotic behavior of the Cramér–von Mises type statistic in the goodness-of-fit hypotheses testing problem for ergodic diffusion processes. The basic (simple) hypothesis is defined by the stochastic differential equation with sign-type trend coefficient and known diffusion coefficient. It is shown that the limit distribution of the proposed test statistic (under hypothesis) is defined by the integral type functional of continuous Gaussian process. We provide the Karhunen–Loève expansion of the corresponding limiting process and show that the eigenfunctions in this expansion are expressed in terms of Bessel functions. This representation for the limit statistic allows us to approximate the threshold.

Comment on ‘Fault tolerance analysis for stochastic systems using switching diffusion processes’ by Yang, Jiang and Cocquempot

Results are given in Yang, Jiang and Cocquempot (Yang, H., Jiang, B., and Cocquempot, V. (2009), ‘Fault Tolerance Analysis for Stochastic Systems using Switching Diffusion Processes’, International Journal of Control, 82, 1516–1525) regarding the overall stability of switched diffusion processes based on stability properties of separate processes combined through stochastic switching. This article argues two main results to be empty, in that the presented hypotheses are logically inconsistent.

A Stochastic Approximation Algorithm for American Lookback Put Options

This work is concerned with pricing American fixed lookback put options. The underlying asset is modeled as a switching diffusion process, where the switching is represented by a continuous-time Markov chain. The switching diffusion delineates stochastic volatility effectively. Nevertheless, this formulation together with the lookback style put option makes it virtually impossible to find closed-form solutions. As a viable alternative, a stochastic approximation algorithm is suggested. The convergence and rates of convergence of the algorithm are established.

Weak convergence of Markov-modulated random sequences

This work is concerned with weak convergence of non-Markov random processes modulated by a Markov chain. The motivation of our study stems from a wide variety of applications in actuarial science, communication networks, production planning, manufacturing and financial engineering. Owing to various modelling considerations, the modulating Markov chain often has a large state space. Aiming at reduction of computational complexity, a two-time-scale formulation is used. Under this setup, the Markov chain belongs to the class of nearly completely decomposable class, where the state space is split into several subspaces. Within each subspace, the transitions of the Markov chain varies rapidly, and among different subspaces, the Markov chain moves relatively infrequently. Aggregating all the states of the Markov chain in each subspace to a single super state leads to a new process. It is shown that under such aggregation schemes, a suitably scaled random sequence converges to a switching diffusion process.

Fault tolerance analysis for stochastic systems using switching diffusion processes

This article models a class of stochastic systems with faults by switching diffusion processes (SDP), and analyses the fault tolerability of such stochastic systems in the sense of input-to-state stability of overall SDP. The fault tolerability relies on the trade-off among the fault occurrence transition rate, the frequency of switching, and the decreasing rate of Lyapunov functions along the solution of the system. Our results show that it may not be necessary to design the fault tolerant controller even though the stochastic system is not separately stable in the healthy and faulty situations, the stability of the overall system process is still guaranteed under some conditions. The proposed tools are illustrated through an example of a fault-prone manufacturing system.

Stability of Discrete-Time Regime-Switching Dynamic Systems with Delays

This work focuses on stability of regime-switching discrete-time systems with delays. Two-time-scale formulation is used for the purpose of reduction of complexity. It is demonstrated that associated with the original system, there is a limit system that is a switching diffusion process. An interesting problem is concerned with if the stability of the limit switching diffusion process can be carried over to the original system. This question is answered in the article. Furthermore, path excursion, mean recurrence time, and the associated error bounds are considered.

Discrete-time Markov chains with two-time scales and a countable state space: limit results and queueing applications

This work is concerned with discrete-time Markov chains having countable state spaces and two-time-scale structures. We examine two classes of Markov chains. In the first class, the state space of the Markov chain is nearly decomposable into a finite number of subspaces, each of which has countably many states, whereas in the second class, the state space is nearly decomposable into infinitely many subspaces each of which has finitely many states. Singular perturbation methods and two-time scales are used to alleviate the computational complexity. Under appropriate conditions, for the first class of models, we show that the formulation is ‘equivalent’ to a continuous-time Markov chain with a finite state space resulting in a substantial reduction of computational burden; for the second class of models, a similar ‘equivalence’ is established. These results are obtained using asymptotic expansions of probability vectors and transition matrices, and properties of aggregated processes. Moreover, we prove that suitably scaled sequences of occupation measures converge weakly to switching diffusion processes. An application to queueing networks is also presented.

Regularity and Recurrence of Switching Diffusions

This work is concerned with switching diffusion processes, also known as regime-switching diffusions. Our attention focuses on regularity, recurrence, and positive recurrence of the underlying stochastic processes. The main effort is devoted to obtaining easily verifiable conditions for the aforementioned properties. Continuous-state-dependent jump processes are considered. First general criteria on regularity and recurrence using Liapunov functions are obtained. Then we focus on a class of problems, in which both the drift and the diffusion coefficients are “linearizable” with respect to the continuous state, and suppose that the generator of the jump part of the process can be approximated by a generator of an ergodic Markov chain. Sufficient conditions for regularity, recurrence, and positive recurrence are derived, which are linear combination of the averaged coefficients (averaged with respect to the stationary measure of the Markov chain).

On the exponential stability of switching diffusion processes

In this note, we investigate almost sure exponential stability for a class of switching diffusion processes. Lyapunov type sufficient conditions to ensure this type of stability for nonlinear switching diffusions are derived. The conditions are an improvement over related results in the literature, since they do not rely on the moment stability of the system. These conditions are then specialized to case of linear switching diffusion processes, to provide easily checkable sufficient criteria for exponential stabilization and robust stability.