Patterns for the waiting time in the context of discrete-time stochastic processes.

A stochastic process or random process consists of chronologically ordered random variables {X t ; t ≥ 0}. For simplicity we assume that the process starts at time t = 0 in X 0 = 0. In this chapter, we consider exclusively processes in discrete time, i.e. processes which are observed at equally spaced points of time t = 0, 1, 2,... . Typical examples are daily, monthly or yearly observed economic data as stock prices, rates of unemployment or sales amount.

This paper present a development of new multivariate optimal stopping problems which associated the discrete time or continuous time stochastic processes with a fractional reward criterion. By applying the established reasoning approach in mathematical programming theory, the existence of an optimal stopping time for these problems will be established by assuming certain regularity conditions on the stochastic processes.

If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting “MRCA age” process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright–Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on ℝ++×ℝ++ with intensity measures of the form λ⊗μ where λ is Lebesgue measure, and μ (the “family lifetime measure”) is an arbitrary, absolutely continuous measure satisfying μ((0, ∞))=∞ and μ((x, ∞))<∞ for all x>0. Special cases of this construction describe the time evolution of the MRCA age in (1+β)-stable continuous state branching processes conditioned on nonextinction—a particular case of which, β=1, is Feller’s continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist. Moreover, we introduce a new family of Markov processes that stands in a relation with respect to the general (1+β)-stable continuous state branching process and its conditioned version that is similar to the one between the family of Bessel-squared diffusions and the unconditioned and conditioned Feller continuous state branching process.

This chapter was a very brief introduction to the basic concepts of the stochastic processes including the definition of a stochastic process, a discrete-time and continuous-time stochastic process, state space, i.i.d. stochastic process, stopping time, and the hitting time of a process to a state. Some typical examples and problems including the daily stock prices of a company, and the evolution of the population of a country have been provided to clarify the basic concepts.

Relative branch size in branch clusters modelled through a Markovian process

Chapter 4 - Stochastic Processes in Discrete Time

Models of maternity histories in female birth cohorts, although interesting in their own right, are only one facet of the dynamics of human populations. In order to gain insights into the implication of these models from the standpoint of the evolutionary development of human populations, they need to be linked, primarily through cohort fertility density functions, to a methodology for making population projections. The primary purpose of this chapter is to provide such a linkage to a population projection methodology based on a class of stochastic processes in discrete time with either time homogeneous or time inhomogeneous laws of evolution.KeywordsCovariance FunctionRandom FunctionPopulation ProjectionCrude Birth RatePopulation ProcessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity conditions, also obey the Kirchhoff conditions, expressing the balance of tensions. In this paper, we propose a new look at quantum graphs as temporal networks, which means that the variable parametrizing the edges of a graph is interpreted as time, while each internal vertex is a branching point giving several different scenarios for the further trajectory of a process. Then Kirchhoff‐type conditions may also arise. Namely, they will be satisfied by such a trajectory of the process that is optimal with account of all the scenarios simultaneously. By employing the recent concept of global delay, we extend the problem of damping a first‐order control system with aftereffect, considered earlier only on an interval, to an arbitrary tree graph. The first means that the delay, imposed starting from the initial moment of time, associated with the root of the tree, propagates through all internal vertices. Bringing the system into the equilibrium and minimizing the energy functional with account of the anticipated probability of each scenario, we come to a variational problem. Then, we establish its equivalence to a self‐adjoint boundary value problem on the tree for some second‐order equations involving both the global delay and the global advance. The unique solvability of both problems is proved. We also illustrate that the interval case when the coefficients of the equation are discrete stochastic processes in discrete time can be viewed as the extension to a tree.

On exponential families of Markov processes

This paper establishes several properties of sequential plans proposed earlier by the same authors for estimation of parameters of random processes described by stochastic difference equations. Upper and lower bounds for mean time of sequential estimation are obtained. The asymptotic normality of the stopping time is shown in the case of autoregressive models.

Quantile Diffusions

Evaluating the uncertainty in mean residual times: Estimators based on residence times from discrete time processes

A general class of stochastic models for choice behaviour is described, which can be regarded as finite Markov chains with absorbing states and which subsumes some particular models discussed by Audley and Pike (1965). The equations for response probabilities and moments of response latency distributions are obtained from the properties of such chains and a very general approach is given in terms of ‘transition‐similar’ systems, which includes discrete time and continuous time processes. Generating functions for the systems are obtained and variations in parameters controlling the rate of occurrence of events in time are also considered. A brief discussion centres around some of the problems which arise in such a general treatment and the reasons for its usefulness.

The dynamic of a financial asset's returns and prices can be expressed using a deterministic process if there is no uncertainty about its future behavior, or with a stochastic process in the more likely case when the value is uncertain. Stochastic processes in continuous time are the most used tool to explain the dynamics of a financial asset's returns and prices. They are the building blocks with which to construct financial models for portfolio optimization, derivatives pricing, and risk management. Continuous-time processes allow for more elegant theoretical modeling compared to discrete-time models and many results proven in probability theory can be applied to obtain a simple evaluation method. Keywords: Brownian motion; Poisson process; jump processes; Levy processes; stochastic process X =; gamma process; exponential process; variance gamma; process; stable process; standard CTS process; generalized tempered stable; process; modified tempered stable; process; standard MTS process; normal tempered stable; process; standard NTS process; normal inverse Gaussian; standard NIG process; Kim-Rachev tempered stable; standard KRTS process; rapidly decreasing tempered stable; standard RDTS process; Brownian motion; Independent increments; Stationary increments; time-changed Brownian motion; The Modified Tempered Stable Processes with Application to Finance; conditional expectation; change of measure; Esscher transform; Brownian Motion and Stochastic Calculus; Risk Assessment Decisions in Banking and Finance

Markov Models and Social Analysis