Essentials Of Stochastic Processes

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Course: MATH 171, Stochastic Processes, Winter 2017Prerequisite: Math 33A and Math 170A (or Statistics 100A). It is helpful, though not required, to take Math 170B before this course or concurrently with this course. Course Content: A stochastic process is a collection of random variables. These random variables are often indexed by time, and the random variables are often related to each other by the evolution of some physical procedure. Stochastic processes can then model random phenomena that depend on time. We will study Markov chains, Martingales, Poisson Processes, Renewal Processes, and Brownian Motion Last update: 22 March 2017

Additional Recommended Course Texts (used in Spring 2016 and 2018): R. Durrett, Essentials of Stochastic processes, 1999, 2nd ed. 2010, 3rd ed. 2016, Springer. This is a good, readable book, with intuitive explanations and many interesting problems. The beta 2nd edition is available free online and also here.M. Lefebvre, Applied Stochastic Processes, 2007 Springer.This book is definitely easier than the others; it is radable at a more basic level and is a source for easier problems and more straightforward applications. The Serfozo and Lefebvre books can be freely downloaded as e-books (through the University of Maryland libraries) by currently registered University of Maryland College Park students, who can also purchase hard-copies for $25.00. (Inquire at the EPSL (STEM) Library for details.) Current HW Assignment Updated HW Solutions Info about Mid-term Test Info about Final Examination Sample Final-Exam Problems Course Coverage: The core material of the course consists of Chapters 1-4 in the Serfozo book, with considerable skipping of more advanced topics, and Chapters 3, 5, and 6 in the Lefebvre book: the primary topics are Discrete Time Markov Chains, Poisson Processes, Continuous Time Markov Chains, and Renewal Processes. See detailed Course Outline below. The primary applications are population and queueing processes, but many other special applied examples occur in both books.Overview: This course is about Stochastic Processes, or time-evolving collections of random variables, primarily about the discrete-statecontinuous time Markov chains which arise in applications in a variety of disciplines. For the first part of the course, both the random variables and the time index-set are discrete: in this setting, our object of study is discrete-time discrete-state Markov chains.Examples of "states" which arise in applications include the size of a population or a waiting-line, or the state ("in control" versus "out of control") of a manufacturing process, or other indicators such as "married" or "employed" etc. for an individual. "Markov chains" are time-evolving systems whose future trajectory does not depend on their past history, given their current state. But many of the most interesting applications involve the generalization of the same ideas to continuous time. Probability theory material needed throughout this course includes joint probability laws, probability mass functions and densities, conditional expectations, moment generating functions, and an understanding of the various kinds of probabilistic convergence, including the Law of Large Numbers. Various technical tools developed in the course and used in an essential way include: Laplace transforms and moment generating functions, methods of solving recursions and systems of difference equations, ergodic and renewal theorems for Markov chains, and (discrete-time) martingales.

Syllabus: This course is anintroduction to stochastic processes and Monte-Carlo methods.Prerequisite are a good working knowledge of calculus and elementary probability as in Stat 607, or Stat 605. And we will use from time to time some concepts from analysis and linear algebra. One of the main goal in the class is to develop a "probabilist intuition and way of thinking". We will present some proofs and we will skip some others in order to providea reasonably broad range of topics, concepts and techniques. We emphasize examples both in discrete and continuous time from a wide range of disciplines, for example branching processes, queueing systems, population models, chemical reaction networks and so on. We will also discuss the numerical implementation of Markov chains and discuss the basics of Monte-Carlo algorithms. Among the topicstreated in the class are

Non-Archimedean analogs of Markov quasimeasures and stochastic processes are investigated. They are used for the development of stochastic antiderivations. The non-Archimedean analog of the Itô formula isproved.

In previous posts I have often written about point processes, which are mathematical objects that seek to represent points scattered over some space. Arguably a more popular random object is something called a stochastic process. This type of mathematical object, also frequently called a random process, is studied in mathematics. But the origins of stochastic processes stem from various phenomena in the real world.

Stochastic processes find applications representing some type of seemingly random change of a system (usually with respect to time). Examples include the growth of some population, the emission of radioactive particles, or the movements of financial markets. There are many types of stochastic processes with applications in various fields outside of mathematics, including the physical sciences, social sciences, finance, and engineering.

Mathematically, a stochastic process is usually defined as a collection of random variables indexed by some set, often representing time. (Other interpretations exists such as a stochastic process being a random function.)

More formally, a stochastic process is defined as a collection of random variables defined on a common probability space \((\Omega,{\cal A}, \mathbb{P} )\), where \(\Omega\) is a sample space, \({\cal A}\) is a \(\sigma\)-algebra, and \(\mathbb{P}\) is a probability measure, and the random variables, indexed by some set \(T\), all take values in the same mathematical space \(S\), which must be measurable with respect to some \(\sigma\)-algebra \(\Sigma\).

Put another way, for a given probability space \(( \mathbb{P}, {\cal A}, \Omega)\) and a measurable space \((S, \Sigma)\), a stochastic process is a collection of \(S\)-valued random variables, which we can write as:

Often collection of random variables \(\{X(t):t\in T \}\) is denoted by simply a single letter such as \(X\). There are different notations for stochastic processes. For example, a stochastic process can also be written as \(\{X(t,\omega):t\in T \}\), reflecting that is function of the two variables, \(t\in T\) and \(\omega\in \Omega\).

The set \(T\) is called the index set or parameter set of the stochastic process. Typically this set is some subset of the real line, such as the natural numbers or an interval. If the set is countable, such as the natural numbers, then it is a discrete-time stochastic process. Conversely, an interval for the index set gives a continuous-time stochastic process.

The mathematical space \(S\) is called the state space of the stochastic process. The precise mathematical space can be any one of many different mathematical sets such as the integers, the real line, \(n\)-dimensional Euclidean space, the complex plane, or more abstract mathematical spaces. The different spaces reflects the different values that the stochastic process can take.

A single outcome of a stochastic process is called a sample function, a sample path, or, a realization. It is formed by taking a single value of each random variable of the stochastic process. More precisely, if \(\{X(t,\omega):t\in T \}\) is a stochastic process, then for any point \(\omega\in\Omega\), the mapping\[X(\cdot,\omega): T \rightarrow S,\]is a sample function of the stochastic process \(\{X(t,\omega):t\in T \}\). Other names exist such as trajectory, and path function.

The range of stochastic processes is limitless, as stochastic processes can be used to construct new ones. Broadly speaking, stochastic processes can be classified by their index set and their state space. For example, we can consider a discrete-time and continuous-time stochastic processes.

A very simple stochastic process is the Bernoulli process, which is a sequence of independent and identically distributed (iid) random variables. The value of each random variable can be one of two values, typically \(0\) and \(1\), but they could be also \(-1\) and \(+1\) or \(H\) and \(T\). To generate this stochastic process, each random variable takes one value, say, \(1\) with probability \(p\) or the other value, say, \(0\) with probability \(1-p\).

We can can liken this stochastic process to flipping a coin, where the probability of a head is \(p\) and its value is \(1\), while the value of a tail is \(0\). In other words, a Bernoulli process is a sequence of iid Bernoulli random variables. The Bernoulli process has the counting numbers (that is, the positive integers) as its index set, meaning \(T=1,\dots\), while in this example the state space is simply \(S=\{0,1\}\).

A random walk is a type of stochastic process that is usually defined as sum of a sequence of iid random variables or random vectors in Euclidean space. Given random walks are formed from a sum, they are stochastic processes that evolve in discrete time. (But some also use the term to refer to stochastic processes that change in continuous time.)

A classic example of this stochastic process is the simple random walk, which is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one. More specifically, the simple random walk increases by one with probability, say, \(p\), or decreases by one with probability \(1-p\). The index set of this stochastic process is the natural numbers, while its state space is the integers. 2b1af7f3a8