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1999 | Buch

Review of Probability Kapitel lesen Erstes Kapitel lesen

Basic Stochastic Processes

A Course Through Exercises

verfasst von: Zdzisław Brzeźniak, Tomasz Zastawniak

Verlag: Springer London

Buchreihe : Springer Undergraduate Mathematics Series

Enthalten in: Professional Book Archive

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Über dieses Buch

This book has been designed for a final year undergraduate course in stochastic processes. It will also be suitable for mathematics undergraduates and others with interest in probability and stochastic processes, who wish to study on their own. The main prerequisite is probability theory: probability measures, random variables, expectation, independence, conditional probability, and the laws of large numbers. The only other prerequisite is calculus. This covers limits, series, the notion of continuity, differentiation and the Riemann integral. Familiarity with the Lebesgue integral would be a bonus. A certain level of fundamental mathematical experience, such as elementary set theory, is assumed implicitly. Throughout the book the exposition is interlaced with numerous exercises, which form an integral part of the course. Complete solutions are provided at the end of each chapter. Also, each exercise is accompanied by a hint to guide the reader in an informal manner. This feature will be particularly useful for self-study and may be of help in tutorials. It also presents a challenge for the lecturer to involve the students as active participants in the course.

Inhaltsverzeichnis

Frontmatter
1. Review of Probability
Abstract
In this chapter we shall recall some basic notions and facts from probability theory. Here is a short list of what needs to be reviewed:
1)
Probability spaces, σ-fields and measures;
 
2)
Random variables and their distributions;
 
3)
Expectation and variance;
 
4)
The σ-field generated by a random variable;
 
5)
Independence, conditional probability.
 
The reader is advised to consult a book on probability for more information.
Zdzisław Brzeźniak, Tomasz Zastawniak
2. Conditional Expectation
Abstract
Conditional expectation is a crucial tool in the study of stochastic processes. It is therefore important to develop the necessary intuition behind this notion, the definition of which may appear somewhat abstract at first. This chapter is designed to help the beginner by leading him or her step by step through several special cases, which become increasingly involved, culminating at the general definition of conditional expectation. Many varied examples and exercises are provided to aid the reader’s understanding.
Zdzisław Brzeźniak, Tomasz Zastawniak
3. Martingales in Discrete Time
Abstract
A sequence ξ1, ξ2, … of random variables is typically used as a mathematical model of the outcomes of a series of random phenomena, such as coin tosses or the value of the FTSE All-Share Index at the London Stock Exchange on consecutive business days. The random variables in such a sequence are indexed by whole numbers, which are customarily referred to as discrete time. It is important to understand that these whole numbers are not necessarily related to the physical time when the events modelled by the sequence actually occur. Discrete time is used to keep track of the order of events, which may or may not be evenly spaced in physical time. For example, the share index is recorded only on business days, but not on Saturdays, Sundays or any other holidays. Rather than tossing a coin repeatedly, we may as well toss 100 coins at a time and count the outcomes.
Zdzisław Brzeźniak, Tomasz Zastawniak
4. Martingale Inequalities and Convergence
Abstract
Results on the convergence of martingales provide an insight into their structure and have a multitude of applications. They also provide an important interpretation of martingales. Namely, it turns out that a large class of martingales can be represented in the form
{fy(4.1)|67-1}
where ξ = limn ξn is an integrable random variable and F1,F2,… is the filtration generated by ξ1, ξ2 … , see Theorem 4.4 below. This makes it possible to think of ξ1, ξ1 … as the results of a series of imperfect observations of some random quantity ξ. As n increases, the accumulated knowledge Fn about ξ increases and ξn becomes a better approximation, approaching the observed quantity ξ in the limit.
Zdzisław Brzeźniak, Tomasz Zastawniak
5. Markov Chains
Abstract
This chapter is concerned with an interesting class of sequences of random variables taking values in a finite or countable set, called the state space, and satisfying the so-called Markov property. One of the simplest examples is provided by a symmetric random walk ξn with values in the set of integers ℤ. If ξn is equal to some i ∈ ℤ at time n, then in the next time instance n+1 it will jump either to i + 1, with probability 1/2, or to i-1, also with probability 1/2. What makes this model interesting is that the value of ξn+1at time n + 1 depends on the past only through the value at time n. This is the Markov property characterizing Markov chains. There are numerous examples of Markov chains, with a multitude of applications.
Zdzisław Brzeźniak, Tomasz Zastawniak
6. Stochastic Processes in Continuous Time
Abstract
The following definitions are straightforward extensions of those introduced earlier for sequences of random variables, the underlying idea being that of a family of random variables depending on time.
Zdzisław Brzeźniak, Tomasz Zastawniak
7. Itô Stochastic Calculus
Abstract
One of the first applications of the Wiener process was proposed by Bachelier, who around 1900 wrote a ground-breaking paper on the modelling of asset prices at the Paris Stock Exchange. Of course Bachelier could not have called it the Wiener process, but he used what in modern terminology amounts to W(t) as a description of the market fluctuations affecting the price X(t) of an asset. Namely, he assumed that infinitesimal price increments dX(t) are proportional to the increments dW(t) of the Wiener process, dX(t) = σdW(t), where σ is a positive constant. As a result, an asset with initial price X(0) = x would be worth X(t) = x + σW(t) at time t. This approach was ahead of Bachelier’s time, but it suffered from one serious flaw: for any t > 0 the price X(t) can be negative with non-zero probability. Nevertheless, for short times it works well enough, since the probability is negligible. But as t increases, so does the probability that X(t) < 0, and the model departs from reality.
Zdzisław Brzeźniak, Tomasz Zastawniak
Backmatter
Metadaten
Titel
Basic Stochastic Processes
verfasst von
Zdzisław Brzeźniak
Tomasz Zastawniak
Copyright-Jahr
1999
Verlag
Springer London
Electronic ISBN
978-1-4471-0533-6
Print ISBN
978-3-540-76175-4
DOI
https://doi.org/10.1007/978-1-4471-0533-6