Applications of stochastic processes. Stochastic processes, which started from research on physical phenomena modelled as a class of random variables evolving over time, have been intensively studied over decades. This has led to the emergence of new and fascinating mathematical models and probabilistic techniques. The theory and methodology of stochastic processes have been widely applied to various fields, such as physics, mechanics, computer science, image and signal processing, biology, finance, among others. We next illustrate the general topic with a few examples. In computer science, stochastic models are often used to describe dynamic networks, and investigate long- term behaviour in order to compute quantities of interest such as probabilities, ex- pectations, and other distributional characteristics. Markov Chain Xxxxx-Xxxxx (MCMC) is a sampling method for obtaining infor- mation about probability distributions (see [58] for a more technical exposition). The MCMC method generates random variables based on a random number generator for systematic random sampling from complicated and often intractable probability dis- tributions, often where the next sample is dependent on the current existing sample. Often direct computation is not feasible as it may be complex, time-consuming, or too expensive. No matter what the distribution, a Markov chain can be set up so that the steady-state distribution of the random variables produced by MCMC has the desired unconditional distribution from which it is sampled. Therefore, one can char- acterise a probability distribution using MCMC without knowing its mathematical properties. Queueing systems, which originated from modelling the number of phone calls at the Copenhagen Telephone Exchange [99], consist of several servers to perform tasks, and are used to describe dynamics that involve spontaneous arrivals and departures, random waiting and service times, which occur in systems such as computers execut- ing tasks, or customers waiting to be served. Gaussian processes have received increased attention in the field of machine learn- ing as they provide a practical approach in challenging learning problems. They are commonly used for regression and classification tasks with reliable estimation of un- certainty [93]. In biology, a model of great interest is the Xxxxx process [87], which models the spread of genetic mutations in a population of a community. It is a discrete-time stochastic process on graphs, as adapted by [78]. In a Xxxxx process, th...
