![]() Write clear, complete and logical proofs for mathematical hypotheses that are new to the student. Explain definitions of Brownian and martingales Explain the length in the queue and solve simple waiting time problems Construct a continuous-time Markov chain and identify its generator in settings of practical problems in a diverse range of applications. Explain the basic properties of the Poisson process and use these to solve problems. Construct a Poisson process and identify its parameter from practical problem settings in a diverse range of applications. Explain Gambler's ruin problem and calculate extinction probability Explain and be able to apply limit theorems of discrete-time Markov chains and use those to identify and interpret their stationary distribution Construct a discrete-time Markov chain and identify its transition probability matrix from practical problem settings. Explain and apply the theoretical concepts of probability theory and stochastic processes. ![]() ![]() Students who undertake STAT3921/4021 will be expected to have a deeper, more sophisticated understanding of the theory and to be able to work with more complicated applications than students who complete the regular STAT3021 unit. By completing this unit, you will develop a solid mathematical foundation of stochastic processes for further studies in advanced areas such as stochastic analysis, stochastic differential equations, stochastic control, financial mathematics and statistical inference. Throughout the unit, various illustrative examples are provided in modelling and analysing problems of practical interest. This unit will also introduce basic concepts of Brownian motion and martingales. This unit will investigate simple queuing theory. This unit will derive key results of Poisson processes and simple continuous-time Markov chains. This unit will establish basic properties of discrete-time Markov chains including random walks and branching processes. They can be modelled as stochastic processes where the domain is a sufficiently large family of subsets of S, ordered by inclusion the range is the set of natural numbers and, if A is a subset of B, ƒ( A) ≤ ƒ( B) with probability 1.A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. Point processes: random arrangements of points in a space S.Martingales – processes with constraints on the expectation.Gauss–Markov processes: processes that are both Gaussian and Markov.Markov processes are those in which the future is conditionally independent of the past given the present.Special cases include stationary processes, also called time-homogeneous. Homogeneous processes: processes where the domain has some symmetry and the finite-dimensional probability distributions also have that symmetry.Gaussian process – a process where all linear combinations of coordinates are normally distributed random variables.Bernoulli schemes: discrete-time processes with N possible states every stationary process in N outcomes is a Bernoulli scheme, and vice versa.Bernoulli process: discrete-time processes with two possible states.Notice that the rows of P sum to 1: this is because P is a stochastic matrix. For an overview of Markov chains in general state space, see Markov chains on a measurable state space. All examples are in the countable state space. Stochastic processes topics This list is currently incomplete. This article contains examples of Markov chains and Markov processes in action. In practical applications, the domain over which the function is defined is a time interval ( time series) or a region of space ( random field).įamiliar examples of time series include stock market and exchange rate fluctuations, signals such as speech, audio and video medical data such as a patient's EKG, EEG, blood pressure or temperature and random movement such as Brownian motion or random walks.Įxamples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogeneous material. Markov processes, Poisson processes (such as radioactive decay), and time series are examples of basic stochastic processes, with the index variable referring. In the mathematics of probability, a stochastic process is a random function.
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