Numerical approximation of invariant measures for hybrid diffusion systems

This work is concerned with numerical approximation of hybrid diffusions with regime switching modulated by continuous-time finite-state Markov chains. When using the Euler-Maruyama approximation algorithms, an important question is: Suppose the measure of the regime-switching diffusions converges to its invariant measure, will the sequence of measures of the approximation scheme converges to the same limit? This paper provides answers to this question. By appropriate interpolations and weak convergence methods, it shows that a suitably interpolated sequence resulted from the algorithm converges to the switching diffusion. The convergence to the invariant measure of the numerical algorithm is also studied.     

Consensus in Networks of Multiagents With Cooperation and Competition Via Stochastically Switching Topologies

In this brief, we provide some theoretical analysis of the consensus for networks of agents via stochastically switching topologies. We consider both discrete-time case and continuous-time case. The main contribution of this brief is that the underlying graph topology is more general in both cases than those appeared in previous papers. The weight matrix of the coupling graph is not assumed to be nonnegative or Metzler. That is, in the model discussed here, the off-diagonal entries of the weight matrix of the coupling graph may be negative. This means that sometimes, the coupling may not benefit, but may prevent the consensus of the coupled agents. In the continuous-time case, the switching time intervals also take a more general form of random variables than those appeared in previous works. We focus our study on such networks and give sufficient conditions that ensure almost sure consensus in both discrete-time case and continuous-time case. As applications, we give several corollaries under more specific assumptions, i.e., the switching can be some independent and identically distributed (i.i.d.) random variable series or a Markov chain. Numerical examples are also provided in both discrete-time and continuous-time cases to demonstrate the validity of our theoretical results.      

A stochastic approximation algorithm for option pricing model calibration with a switchable market

This paper is concerned with option pricing under a regime-switching model. The switching process takes two different modes, and the underlying stock price evolves in accordance with the two modes dictated by a continuous-time, finite-state Markov chain. At any given instance, the price follows either a geometric Brownian motion model or a mean-reversion model, depending on its market mode. Stochastic approximation/optimization algorithms are developed for model calibration. Convergence of the algorithm is proved; rate of convergence is also provided. Option market data are used to predict the future market mode.     

Predictive Consensus for Networked Multi‐Agent Systems with Switching Topology and Variable Delay

This paper investigates consensus problems of networked linear time invariant (LTI) multi‐agent systems, subject to variable network delays and switching topology. A new protocol is proposed for such systems with matrix B that has full row rank, based on stochastic, indecomposable, aperiodic (SIA) matrix and the predictive control scheme. With the predictive scheme the network delay is compensated. Consensus analysis based on the seminorm is provided. The conditions are obtained for such systems with periodic switching topology to reach consensus. The proposed protocol can deal with time‐varying delays, switching topology, and an unstable mode. The numerical examples demonstrate the effectiveness of the theoretical results.     

A numerical method for annuity-purchasing decision making to minimize the probability of financial ruin for regime-switching wealth models

This work develops an approximation procedure to find optimal annuity-purchasing strategies for minimizing the probability of lifetime ruin. The wealth is modelled as a regime-switching diffusion modulated by a continuous-time Markov chain. Based on Markov chain approximation techniques, a discrete-time controlled Markov chain with two components is constructed. Under simple conditions, the convergence of the approximation sequence to the wealth process is obtained. The convergence of the approximation to the value function is also established. Several examples are provided to demonstrate the performance of the algorithms.     

Distributed reliable H∞ consensus control for a class of multi‐agent systems under switching networks: A topology‐based average dwell time approach

This paper investigates the problem of distributed reliable H_∞ consensus control for high‐order networked agent systems with actuator faults and switching undirected topologies. The Lipschitz nonlinearities, several types of actuator faults, and exogenous disturbances are considered in subsystems. Suppose the communication network of the multi‐agent systems may switch among finite connected graphs. By utilizing the relative state information of neighbors, a new distributed adaptive reliable consensus protocol is presented for actuator failure compensations in individual nodes. Note that the Lyapunov function for error systems may not decrease as the communication network is time‐varying; as a result, the existing distributed adaptive control technique cannot be applied directly. To overcome this difficulty, the topology‐based average dwell time approach is introduced to deal with switching jumps. By applying topology‐based average dwell time approach and Lyapunov theory, the distributed controller design condition is given in terms of LMIs. It is shown that the proposed scheme can guarantee that the reliable H_∞ consensus problem is solvable in the presence actuator faults and external disturbance. Finally, two numerical examples are given the effectiveness of the proposed theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Sampled-data consensus of general linear multi-agent systems under switching topologies: averaging method

This paper investigates sampled-data consensus problems of general linear multi-agent systems under switching topologies. To perform the consensus analysis by the averaging method, the multi-agent system in the sampled-data setting is first converted into the continuous-time system with the input delay. Then, the approximate relationship between the states of this input delay system and the states of its averaged system is established by using the tool from non-standard analysis. Subsequently, a sufficient condition ensuring the sampled-data consensus is obtained by linear matrix inequalities. It is shown that the sampled-data consensus of general linear multi-agent systems, including exponential unstable agents, can be achieved, if the speed of topology switching is fast enough and the sampled-data consensus under the averaging topology can be achieved. Finally, numerical simulations are provided to demonstrate the effectiveness of theoretical analyses.     

Reaching consensus in unbalanced networks with coarse information communication

This paper is concerned with the problem of seeking consensus for a network of agents under a fixed or switching directed communication topology. Each agent is modeled as discrete‐time first‐order dynamics and interacts with its neighbors via logarithmically quantized information. We assume that the digraph is not necessarily balanced and, thus, avoiding the double stochasticity requirement for the adjacency matrix. For the case of a fixed topology that is strongly connected, it is shown that the proposed protocol is admissible for arbitrarily coarse logarithmic quantization and the β‐asymptotic weighted‐average consensus is achieved. For the case of a switching topology that is periodically strongly connected, it is shown that the proposed protocol is admissible for arbitrarily coarse quantization and the β‐asymptotic consensus is achieved. Furthermore, for both cases, not only are the convergence rates for consensus specified but also the bounds on the consensus error that highlight their dependence on the sector bound β of the logarithmic quantizer are also provided. Copyright © 2015 John Wiley & Sons, Ltd.     

Realization theory for two-time-scale distributions through approximation of Markov parameters

In this article, the problem of approximating the Markov parameters of a two-time-scale (TTS) distribution is studied. It is shown that the Markov parameters of a TTS distribution can be approximated in terms of the Markov parameters of its fast distribution only. This is an O(ε^1?i ) approximation, which deteriorates by a factor of 1/ε at each step as higher order Markov parameters are computed. In order to use the Markov parameters of the slow distribution in the approximation scheme, an inversion map is introduced by which a TTS distribution and its slow distribution are mapped into two new distributions. It is then shown that every Markov parameter of the inverted distribution approximates that of the inverted slow distribution with an O(ε) accuracy. An approximate expression for the Hankel matrix of the Markov parameters of a TTS distribution is also obtained. This expression is in the form of a telescopic series and involves the Markov parameters of the fast distribution only.     

An accessible implementation of interest rate models with Markov-switching

We examine the performance of interest rate models with regime-switching feature through a straightforward implementation. In particular, three short-rate models, the Vasicek, CIR and Black-Karasinski models, are extended to capture the switching of economic regimes using a finite-state Markov chain in discrete time. The Markov chain modulates the parameters of the model. We illustrate numerically that the resulting extended models are capable of reproducing various shapes of the yield curve. A quasi-maximum likelihood method based on James and Webber (2000) is employed to estimate the parameters of the regime-switching models. We demonstrate the implementation using actual financial datasets of Canadian yield rates. The numerical results show that under some model validation metrics, the two-state regime-switching models are more flexible, have better forecasting performance and provide better fit than the models without the regime-switching characteristic.