Finite Continuous Time Research Articles

Anticipated BSDEs driven by a single jump process

ABSTRACTIn this paper, we discuss a class of anticipated backward stochastic differential equations related to a finite continuous time single jump process. We prove the existence and uniqueness of the adapted solution. Moreover, a comparison theorem for the solutions is also established.

On Certain Average Characteristics of Finite Continuous-Time Markov Chains

We consider a nonhomogeneous continuous-time Markov chain with finite phase space and 1-periodic intensities. The behavior of two important characteristics of the process is studied, namely, we obtain the bounds on the rate of convergence to the limiting mean and to double mean. Some examples of introduced values analysis are considered.

Quasi-variational analysis of a dynamic Walrasian competitive exchange economy

In this paper we propose a new definition of competitive equilibrium for an exchange economy model over time. At each moment consumer is characterized by an initial endowment and by an utility function, which he adopts for choosing the better consumption that he can afford. The actions, the constrain budget and the preferences described by utility functions depend on the price vector, time and the instantaneous commodity holding. With reference to a finite continuous time period, assuming that in this marketplace the mean value of the aggregate excess demand is non-positive and measuring its principal activities only in terms of mean values, we treat this setting as a pure exchange economy and for it we give an existence result of such equilibria by means of variational methods.

Backward Stochastic Differential Equations for a Single Jump Process

We consider backward stochastic differential equations (BSDEs) related to a finite continuous time single jump process. We prove the existence and uniqueness of solutions when the coefficients satisfy Lipschitz continuity conditions. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are then investigated.

Mean-Variance Criteria for Finite Continuous-Time Markov Decision Processes

This technical note deals with the mean variance problem (known as the average variance (AV) minimization problem) for finite continuous time Markov decision processes. We first introduce a so called G-condition which is weaker than the well known ergodicity and unichain conditions and sufficient for the finiteness of the AV of a policy. Also, we present an example of a policy having infinite AV when the G-condition is not satisfied. Under the G-condition we prove that the AV criterion can be transformed into an equivalent mean (or expected) average criterion by using a martingale technique and an observation from the canonical form of a transition rate matrix, and thus the existence and calculation of an AV minimal policy over a class of mean optimal policies are obtained by a policy iteration algorithm in an finite number of iterations. As byproduct, we obtain some interesting new results about the mean average optimality.

Matrix‐analytic Models and their Analysis

We survey phase‐type distributions and Markovian point processes, aspects of how to use such models in applied probability calculations and how to fit them to observed data. A phase‐type distribution is defined as the time to absorption in a finite continuous time Markov process with one absorbing state. This class of distributions is dense and contains many standard examples like all combinations of exponential in series/parallel. A Markovian point process is governed by a finite continuous time Markov process (typically ergodic), such that points are generated at a Poisson intensity depending on the underlying state and at transitions; a main special case is a Markov‐modulated Poisson process. In both cases, the analytic formulas typically contain matrix‐exponentials, and the matrix formalism carried over when the models are used in applied probability calculations as in problems in renewal theory, random walks and queueing. The statistical analysis is typically based upon the EM algorithm, viewing the whole sample path of the background Markov process as the latent variable.

Some Bounds on the Computational Power of Piecewise Constant Derivative Systems

We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation time of these machines can be measured either as a discrete value, called discrete time, or as a continuous value, called continuous time. We relate the two notions of time for general PCD systems. We prove that general PCD systems are equivalent to Turing machines and linear machines in finite discrete time. We prove that the languages recognized by purely rational PCD systems in dimension d in finite continuous time are precisely the languages of the (d-2) th level of the arithmetical hierarchy. Hence the reachability problem of purely rational PCD systems of dimension d in finite continuous time is Σd-2 -complete.

The Distribution of the Maximum Length of a Poisson Queue During a Busy Period

In the design of waiting facilities for the customers in a queue, it is of interest to know probability distributions of extremal values of the queue length. In this paper we propose to calculate explicitly the probability that [Formula: see text] for all t in [0, t), in which Zt denotes the queue length at time t. This is the probability that in a Poisson queue the server remains busy throughout the time interval [0, t) without ever having as many as b customers in line. We express this probability in terms of the taboo-probabilities with taboo-states 0 and b for a finite continuous time random walk. As an incidental result we obtain new expressions for these taboo-probabilities. It is routine to determine the distribution of the maximum queue length before the next emptiness from our results.

Finite Continuous Time Markov Chains

In a recent book by the authors (see footnote on page 101) a systematic method was developed for computing the basic descriptive quantities of finite Markov chains. Matrix expressions were obtained for these quantities in terms of certain basic matrices, easily obtainable from the transition matrix. In this note, corresponding expressions are given for the basic quantities for finite continuous time chains.