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2013 | Buch

Stochastic World

verfasst von: Sergey S. Stepanov

Verlag: Springer International Publishing

Buchreihe : Mathematical Engineering

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SUCHEN

Über dieses Buch

This book is an introduction into stochastic processes for physicists, biologists and financial analysts. Using an informal approach, all the necessary mathematical tools and techniques are covered, including the stochastic differential equations, mean values, probability distribution functions, stochastic integration and numerical modeling. Numerous examples of practical applications of the stochastic mathematics are considered in detail, ranging from physics to the financial theory. A reader with basic knowledge of the probability theory should have no difficulty in accessing the book content.

Inhaltsverzeichnis

Frontmatter
Random Events
Abstract
Absolutely deterministic events and processes do not exist. The Universe speaks to us in the language of probability theory. We assume that the Reader is familiar with the basics of probability, therefore, only those terms and concepts that are necessary for the understanding of further material are introduced.
Sergey S. Stepanov
Stochastic Equations
Abstract
This is the key chapter of the book. Stochastic differential equations, which are the main mathematical object we consider, are introduced here. We choose an informal, intuitive way believing that obtaining specific practical results is more important than their rigorous mathematical proof.
Sergey S. Stepanov
Mean Values
Abstract
The differential equation for the random function x(t) is only one of many possible languages to describe a stochastic process. In the situation when the system changes with time, the mean values also change and comply with certain differential equations. In fact, their solution is the most direct way of obtaining practically useful results.
Sergey S. Stepanov
Probabilities
Abstract
There is one more way to obtain information about the behavior of a stochastic process: it is the solution of equations for the conditional probability density P(x 0, t 0, ⇒ x, t). This chapter is devoted to such equations.
Sergey S. Stepanov
Stochastic Integrals
Abstract
Similarly to the standard analysis, when the stochastic differentiation is defined, it is natural to introduce the stochastic integration. The corresponding approach will give us one more instrument of obtaining the expressions for some general random processes. This is an elegant domain of stochastic mathematics; moreover, it is actively used in the textbooks and academic literature.
Sergey S. Stepanov
Systems of Equations
Abstract
One-dimensional stochastic equations can describe only relatively simple systems. Even for the ordinary physical oscillator it is necessary to solve the system of two equations of the first order. In general case, the reality is multidimensional, and supplies us with a lot of examples of very complex but surprisingly interesting random processes.
Sergey S. Stepanov
Stochastic Nature
Abstract
In this chapter we consider the examples of natural systems which can be described with stochastic differential equations. These systems cover the wide range of applications from physics to biology. However, they don’t require the deep knowledge of the corresponding fields of science. The majority of sections are not closely connected with each other, and one can read them in any sequence, independently. The first stochastic differential equation was written by Paul Langevin in 1908. It is this very equation that opens the present chapter.
Sergey S. Stepanov
Stochastic Society
Abstract
In this chapter we give some examples of applying the stochastic methods to financial markets and economy. The volatile character of prices and economic indicators is a manifestation of significantly stochastic dynamics of the corresponding systems, and the δW term plays the leading role in the Ito equations. First we make a small diversion into the financial markets and the empirical properties of financial instruments’ prices. Then we consider the theory of diversification and beta-coefficients. Stochastic methods appear to be very useful when studying the complex financial instruments. Options are one example of such instrument. We will consider the main properties of the options and derive the Black-Scholes formula in two different ways. After that a simple one-factor model of the yield curve will be considered.
Sergey S. Stepanov
Computer Modeling
Abstract
Sometimes computer modeling of the behavior of complicated stochastic systems is the only way to investigate them. This chapter is for the readers who like not only formulas but also algorithms. Programming is merely a process of recording a strictly fixed sequence of actions in restricted and formalized English. This sequence can be executed by a human, a computer or an alien. Any program could be easily translated into a human language, however the result will take more space and it can be easily misunderstood because of the ambiguousness of the natural language. This chapter is not meant to teach the reader C++ programming; any manual would serve this purpose much better. At the same time we will discuss some features of C++, which will help to understand how the code works in the examples.
Sergey S. Stepanov
Backmatter
Metadaten
Titel
Stochastic World
verfasst von
Sergey S. Stepanov
Copyright-Jahr
2013
Verlag
Springer International Publishing
Electronic ISBN
978-3-319-00071-8
Print ISBN
978-3-319-00070-1
DOI
https://doi.org/10.1007/978-3-319-00071-8

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