Reduction of Markov chains with two-time-scale state transitions

In this paper, we consider a general class of two-time-scale Markov chains whose transition rate matrix depends on a parameter $\lambda>0$. We assume that some transition rates of the Markov chain will tend to infinity as $\lambda\rightarrow\infty$. We divide the state space of the Markov chain $X$ into a fast state space and a slow state space and define a reduced chain $Y$ on the slow state space. Our main result is that the distribution of the original chain $X$ will converge in total variation distance to that of the reduced chain $Y$ uniformly in time $t$ as $\lambda\rightarrow\infty$.Comment: 30 pages, 3 figures; Stochastics: An International Journal of Probability and Stochastic Processes, 201

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.Comment: 83 page

Two-Time-Scale Systems In Continuous Time With Regime Switching And Their Applications

This dissertation is focuses on near-optimal controls for stochastic differential equation with regime switching. The random switching is presented by a continuous-time Markov chain. We use the idea of relaxed control and mean of martingale formulation to show a weak convergence result. The first chapter is devoted to the study of stochastic Li´enard equations with random switching. The motivation of our study stems from modeling of complex systems in which both continuous dynamics and discrete events are present. The continuous component is a solution of a stochastic Li´enard equation and the discrete component is a Markov chain with a finite state space that is large. A distinct feature is that the processes under consideration are time inhomogeneous. Based on the idea of nearly decomposability and aggregation, the state space of the switching process can be viewed as nearly decomposable into l subspaces that are connected with weak interactions among the subspaces. Using the idea of aggregation, we lump the states in each subspace into a single state. Considering the pair of process (continuous state, discrete state), under suitable conditions, we derive a weak convergence result by means of martingale problem formulation. The significance of the limit process is that it is substantially simpler than that of the original system. Thus, it can be used in the approximation and computation work to reduce the computational complexity. Finally, we investigate the system behavior of Van der Pol oscillator by introducing the noise. The system have been performed numerically and results are shown using Matlab. Simulations show that the proposed model gives limit cycles are more accurate as the noise decreased which the limit cycle is close to a sinusoidal oscillation and the shape of the signal becomes less sinusoidal as the noise increased

Large deviations of stochastic systems and applications

This dissertation focuses on large deviations of stochastic systems with applications to optimal control and system identification. It encompasses analysis of two-time-scale Markov processes and system identification with regular and quantized data. First, we develops large deviations principles for systems driven by continuous-time Markov chains with twotime scales and related optimal control problems. A distinct feature of our setup is that the Markov chain under consideration is time dependent or inhomogeneous. The use of two time-scale formulation stems from the effort of reducing computational complexity in a wide variety of applications in control, optimization, and systems theory. Starting with a rapidly fluctuating Markovian system, under irreducibility conditions, both large deviations upper and lower bounds are established first for a fixed terminal time and then for time-varying dynamic systems. Then the results are applied to certain dynamic systems and LQ control problems. Second, we study large deviations for identifications systems. Traditional system identification concentrates on convergence and convergence rates of estimates in mean squares, in distribution, or in a strong sense. For system diagnosis and complexity analysis, however, it is essential to understand the probabilities of identification errors over a finite data window. This paper investigates identification errors in a large deviations framework. By considering both space complexity in terms of quantization levels and time complexity with respect to data window sizes, this study provides a new perspective to understand the fundamental relationship between probabilistic errors and resources that represent data sizes in computer algorithms, sample sizes in statistical analysis, channel bandwidths in communications, etc. This relationship is derived by establishing the large deviations principle for quantized identification that links binary-valued data at one end and regular sensors at the other. Under some mild conditions, we obtain large deviations upper and lower bounds. Our results accommodate independent and identically distributed noise sequences, as well as more general classes of mixing-type noise sequences. Numerical examples are provided to illustrate the theoretical results

Sample Path Analysis of Integrate-and-Fire Neurons

Computational neuroscience is concerned with answering two intertwined questions that are based on the assumption that spatio-temporal patterns of spikes form the universal language of the nervous system. First, what function does a specific neural circuitry perform in the elaboration of a behavior? Second, how do neural circuits process behaviorally-relevant information? Non-linear system analysis has proven instrumental in understanding the coding strategies of early neural processing in various sensory modalities. Yet, at higher levels of integration, it fails to help in deciphering the response of assemblies of neurons to complex naturalistic stimuli. If neural activity can be assumed to be primarily driven by the stimulus at early stages of processing, the intrinsic activity of neural circuits interacts with their high-dimensional input to transform it in a stochastic non-linear fashion at the cortical level. As a consequence, any attempt to fully understand the brain through a system analysis approach becomes illusory. However, it is increasingly advocated that neural noise plays a constructive role in neural processing, facilitating information transmission. This prompts to gain insight into the neural code by studying the stochasticity of neuronal activity, which is viewed as biologically relevant. Such an endeavor requires the design of guiding theoretical principles to assess the potential benefits of neural noise. In this context, meeting the requirements of biological relevance and computational tractability, while providing a stochastic description of neural activity, prescribes the adoption of the integrate-and-fire model. In this thesis, founding ourselves on the path-wise description of neuronal activity, we propose to further the stochastic analysis of the integrate-and fire model through a combination of numerical and theoretical techniques. To begin, we expand upon the path-wise construction of linear diffusions, which offers a natural setting to describe leaky integrate-and-fire neurons, as inhomogeneous Markov chains. Based on the theoretical analysis of the first-passage problem, we then explore the interplay between the internal neuronal noise and the statistics of injected perturbations at the single unit level, and examine its implications on the neural coding. At the population level, we also develop an exact event-driven implementation of a Markov network of perfect integrate-and-fire neurons with both time delayed instantaneous interactions and arbitrary topology. We hope our approach will provide new paradigms to understand how sensory inputs perturb neural intrinsic activity and accomplish the goal of developing a new technique for identifying relevant patterns of population activity. From a perturbative perspective, our study shows how injecting frozen noise in different flavors can help characterize internal neuronal noise, which is presumably functionally relevant to information processing. From a simulation perspective, our event-driven framework is amenable to scrutinize the stochastic behavior of simple recurrent motifs as well as temporal dynamics of large scale networks under spike-timing-dependent plasticity