Preface

This book contains four parts:

• Basics of Probability

• Discrete Time Processes

• Continuous Time Processes

• Applications.

The first part concerns the basics of Probability Theory which, in fact, is the probability space. The key idea behind probability space is the stabilization of the relative frequencies when one performs ‘independent’ repetition of a random experiment and records whether each time ‘event’, say A, occurs ornot. Define the characteristicfunction ofevent A during trial t = 1, 2,… by χ (At), namely,

At:=1ifAoccurs,i.e.,At=A0ifnot

Denoted by

nA:=1n∑t=1nχAt

the relative frequency of event A after the first n trials, because of the dawn of history one can observe the stabilization of the relative frequencies; that is, it seems natural that as n → ∞

rn (A) converges to some real number called the probability of A.

Although games of chance have been performed for thousands of years, probability theory, as a science, originated in the middle of the 17th century with Pascal (1623‒1662), Fermat (1601‒1655) and Huygens (1629‒1695). The real history of probability theory began with the works of James Bernoulli (1654‒1705) and De Moivre (1667‒1754). Bernoulli was probably the first to realize the importance of consideration of infinite sequences of random trials and made a clear distinction between the probability of an event and the frequency of its realization. In 1812 there appeared Laplace’s (1749‒1827) great treatise containing the analytical theory of probability with application to the analysis of observation errors. Then limit theorems were studied by Poisson (1781‒1840) and Gauss (1777‒1855).

The next important period in the development of probability theory is associated with the names of P.L. Chebyshev (1857‒1894), A.A. Markov (1856‒1922) and A.M. Lyapunov (1857‒1918), who developed effective methods for proving limit theorems for sums of independent but arbitrarily distributed random variables. Before Chebyshev the main interest had been in the calculation of the probabilities of random events. He, probably, was the first to understand clearly and exploit the full strength of the concepts of random variables. The number of Chebyshev’s publications on probability theory is not large ‒ four in all ‒ but it would be hard to overestimate their role and in the development of the classical Russian school of that subject.

The modern period in the development of probability theory began with its axiomatization due to the publications of S.N. Bernstein (1880‒1968), R. von Mises (1883‒1953) and E. Borel (1871‒1956). But the first mathematically rigorous treatment of probability theory came only in the 1930s by the Russian mathematician A.N. Kolmogorov (1903‒1987) in his seminal paper (Kolmogorov, 1933). His first observation was that a number of rules that hold for relative frequencies rn (A) should also hold for probabilities. This immediately raises the question: which is the minimal set of such rules? According to Kolmogorov, the answer is based on several axiomatic concepts. These fundamental concepts are:

(1) α-algebra

(2) probability measure

(3) probability space

(4) distribution function.

In this volume we will discuss each of these principal notions in details.

The second part deals with discrete time processes, or, more exactly, random sequences where the main role is played by Martingale Theory, which takes a central place in Discrete-Time Stochastic Process Theory because of the asymptotic properties of martingales providing a key prototype of probabilistic behavior which is of wide applicability. The first appearance of a martingale as a mathematical term was due to J. Ville (1939). The major breakthrough was associated with the classic book Stochastic Processes by J. Doob (1953). Other recent books are J. Neveu (1975), R. Liptser and A. Shiryaev (1989) and D. Williams (1991). The martingale is a sequence {ξn, n}n ≥ 1 of random variables ξn associated with a corresponding prehistory (α-algebra) n − 1 such that the conditional mathematical expectation of ξn under fixed n − 1 is equal to ξn–1 with probability 1, that is,

ξn/Fn−1a.s.¯¯ξn−1

Martingales are probably the most inventive and generalized of sums of independent random variables with zero-mean. Indeed, any random variable ξn (maybe, dependent) can be expressed as a sum of ‘martingale-differences’

n:=E{ξn/Fk}−E{ξn/Fk−1},E{ξn/F0}=0,k=1,…,n

because of the representation

n=ξn−Eξn/Fn−1+Eξn/Fn−1−Eξn/Fn−2+…+Eξn/F1−Eξn/F0=∑k=1nΔk

In some sense martingales occupy the intermediate place between independent and dependent sequences. The independence assumption has proved inadequate for handling contemporary developments in many fields.

Then, based on such considerations, there are presented and discussed in detail the three most important probabilistic laws: the Weak Law of Large Numbers (LLN) and its strong version known as the Strong Law of Large Numbers (SLLN), the Central Limit Theorem (CLT), and, finally, the Law ofthe Iterated Logarithm (LIL). All of them may be interpreted as invariant principles or invariant laws because of the independence of the formulated results of the distribution of random variables forming considered random sequences.

The third part discusses Continuous-Time Processes basically governed by stochastic differential equations. The notion of the mean-square continuity property is introduced along with its relation with some properties of the corresponding auto-covariance matrix function. Then processes with orthogonal and independent Increments are introduced, and, as a particular case, the Wiener process or Brownian motion is considered. A detailed analysis and discussion of an invariance principle and LIL for Brownian motion are presented.

The so-called Markov Processes are then introduced. A stochastic dynamic system satisfies the Markov property if the probable (future) state of the system is independent of the (past) behavior of the system. The relation of such systems to diffusion processes is deeply analyzed. The ergodicity property of such systems is also discussed.

Next, the most important constructions of stochastic integrals are studied: namely

• a time-integral of a sample path of a second order (s.o.) stochastic process;

• the so-called Wiener integral of a deterministic function with respect to a stationary orthogonal increment random process such that this integral is associated with the Lebesgue integral, it is usually referred to as a stochastic integral with respect to an ‘orthogonal random measure’ ;

• the so-called Itô integral of a random function with respect to a stationary orthogonal increment random process;

• and, finally, the so-called Stratonovich integral of a random function with respect to an s.o. stationary orthogonal increment random process where the ‘summation’ on the right-hand side is taken in a special sense.

All of these different types of stochastic integral are required for the mathematically rigorous definition of a solution of a stochastic differential equation. We discuss the class of the so-called Stochastic Differential Equation, introduced by K. Itô, whose basic theory was developed independently by Itô and I. Gihman during 1940s. There the Itô-type integral calculus is applied. The principal motivation for choosing the Itô approach (as opposed to the Stratonovich calculus as another very popular interpretation of stochastic integration) is that the Itô method extends to a broader class of equations and transformation the probability law of the Wiener process in a more natural way. This approach implements the so-called diffusion approximation which arises from random difference equation models and has wide application to control problems in engineering sciences, motivated by the need for more sophisticated models which spurred further work on...