Stochastic ProcessesF Baudoin, Purdue University, West Lafayette, IN, USA

ã 2010 Elsevier Ltd. All rights reserved.

Introduction and Historical Perspectives

A stochastic processes is a mathematical tool thatdescribes a random phenomenon evolving in time.

Let us for instance consider an economist who decidesto focus every day on the price of a barrel of oil over thenext year. The price at t = 1, . . . , 365, say Xt , can not bepredicted with absolute certainty, though some prices aremore likely than others. This price can then be mathe-matically modeled by the concept of a random variable: Xt

is random. The set, (Xt) t=1, . . . , 365 of these ramdom vari-ables in an example of a stochastic process.

The first stochastic process that has been extensivelystudied is the so-called Brownian motion, named in honorof the botanist Robert Brown (1773–1858), who observedand described in 1828 (see Brown, 1827) the randommovement of particles suspended in a liquid or gas. Oneof the first mathematical studies of this process goes backto the mathematician Louis Bachelier (1870–1946), whopresented in 1900, a stochastic modeling of the stock andoption markets. However, mainly because of the lack of arigorous foundation of probability theory at that time, theseminal work of Bachelier was ignored by mathematiciansfor a long time. However, in his 1905 paper (see Einstein,1905), Albert Einstein (1879–1955) brought this stochasticprocess to the attention of physicists by presenting it as away to indirectly confirm the existence of atoms and mole-cules. The rigorous mathematical study of stochastic pro-cesses really began with the mathematician AndreiKolmogorov (1903–1987). His monograph (Kolmogorov,1956), published in Russian in 1933, built up probabilitytheory rigorously from the fundamental axioms in a waycomparable to Euclid’s treatment of geometry. From thisaxiomatic approach, Kolmogorov gives a precise definitionof stochastic processes. His point of view stresses the factthat a stochastic process is nothing else but a randomvariable valued in a space of functions (or a space of curves).For instance, if an economist reads a financial newspaperbecause he/she is interested in the price of a barrel of oilevery day over the last year, then he/she will focus on thecurve of these prices. According to Kolmogorov’s point ofview, saying that these prices form a stochastic process isthen equivalent to saying that the curve that is seen is therealization of a random variable defined on a suitable prob-ability space. This point of view is mathematically quiteprofound and provides existence results for stochastic pro-cesses (Daniell-Kolmogorov existence result, 1933) as wellas pathwise regularity results (Kolmogorov, 1956).

Joseph Doob (1910–2004) writes in the introduction tohis famous book Stochastic Processes (see Doob, 1953).

[a stochastic process is] any process running along in time

and controlled by probability laws . . . [more precisely]

any family of random variables Xt [where] a random

variable . . . is simply a measurable function . . .

Doob’s point of view,which is consistentwithKolmogorov’sand built on the work by Paul Levy (1886–1971), is nowa-days commonly given as a definiation of a stochastic pro-cess. Relying on this point of view that emphasizes the roleof time, Doob’s work, developed during the 1940s and the1950s, has quickly become one of the most powerful toolsavailable to study stochastic processes. During the sameperiod in the 1940s, using Kolmogorov’s and Doob’sresults, Kiyosi Ito (1915-) developed the basic tools of thetheory of stochastic calculus (see ito, 1944). This theorymakes it possible to define the integrals of stochastic pro-cesses as against other stochastic processes and gives senseto the notion of differential equation driven by a randomsignal (stochastic differential equations).

These different seminal works contributed to give tothe theory of stochastic processes a rigorous basis that ledto many further developments within and outside mathe-matics. For further reading on the mathematical theory ofstochastic processes, we refer to the book by Revuz andYor (1999) or the book by Protter (2004). In the nextsection, we present some examples of stochastic processesand their applications.

Examples and Applications of StochasticProcesses

Markov Chains

Markov chains are among the most important stochasticprocesses. They are stochastic processes for which the des-cription of the present state fully captures all the informa-tion that could influence the future evolution of the process.

Predicting traffic flows, communications networks,genetic issues, and queues are examples where Markovchains can be used to model performance. Devising aphysical model for these chaotic systems would be impos-sibly complicated but doing so using Markov chains isquite simple. Modeling the to-and-fro communicationstraffic over the Internet is a prime example of whatMarkov chains can do. Nowadays, the Internet has becomeexceedingly complex, and traffic moves in a random

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way between network nodes. Markov chains are com-monly used to describe not only the traffic, which isinherently unpredictable, but also how the network willperform, despite the complexity of its structure. ThePageRank of a webpage as used by Google is defined bya Markov chain.

It is also worth mentioning that Markov chains havebeen recently used in education (see Duys and Headrick,2004). This study examined the efficacy of an infrequentlyused statistical analysis in counselor education research.A Markov chain analysis was used to examine hypothe-sized differences between students’ use of counselingskills in an introductory course.

Beyond Markov Chains, Fractional BrownianMotion

In engineering applications of probability, stochastic pro-cesses are often used to model the input of a system.Recent studies of communication networks show thatreal inputs exhibit long-range dependence: the behaviorof a real process after a given time t depends not only onthe situation at t but also on the whole history of theprocess up to time t. Moreover, it turns out that thisproperty is far from being negligible because of the effectsit induces on the expected behavior of the global system.Therefore, Markov chains cannot be used. Another prop-erty that the processes have encountered in communica-tion networks is the self-similarity: their behavior is indistribution of the same, up to a space-scaling, whateverthe time-scale is: this is to say that the process {Xat, t 2[0,1]} has the same distribution as the process {aH Xt, t 2[0,1]}, where H is called the Hurst parameter. Severalestimations on real data tend to show that H often liesbetween 0.7 and 0.8, whereas, for instance, the usualBrownian motion has a Hurst parameter equal to 0.5.Processes that satisfy these two properties, long-rangedependence and self-similarity, are nowadays being stud-ied extensively. Among them, the so-called fractionalBrownian motion introduced by Kolmogorov to modelturbulences phenomena is one of the simplest.

Brownian Motion and Financial Mathematics

As pointed out in the section titled ‘Introduction,’ theBrownian motion is one of the first stochastic processesthat has been studied. Since the seminal observation byL. Bachelier

It seems that the market, the aggregate of speculators, can

believe in neither a market rise nor a market fall, since, for

each quated price, there are as many buyers as sellers..

and the work by Black and Scholes (see 1973) and Merton(1973), Brownian motion is at the basis of the mathemati-cal modeling of financial markets. It can be used to modelthe fluctuations of stock prices. This modeling has inparticular led to a rational option: the pricing theory. Anoption is a contract written by a seller that extends to thebuyer the right but not the obligation to buy (in the case ofa call option) or to sell (in the case of a put option) aparticular asset at some time. In return for granting theoption, the seller collects a payment (the premium) fromthe buyer. This premium can be computed by usingmodels involving the Brownian motion (the so-calledBlack–Scholes formula). For further reading on this topic,we refer the interested reader to the book of Karatzas andShreve (1998).

Two-Dimensional Brownian Motion and RelatedProcesses

The study of the two-dimensional Brownian motion andof related stochastic processes (SLE) has made it possibleto solve many conjectures of physicists concerning criticalphenomena in dimension 2. The Fields medal wasawarded in 2006 to W. Werner for his work in this domain.For further reading on this topic, we refer readers to thebook of Lawler (2005).

Bibliography

Black, F. and Scholes, M. (1973). The pricing of options and corporateliabilities. Journal of Political Economy 81(3), 637–659.

Brown, R. (1827). A brief account of microscopical observations madein the months of June, July and August, on the particles contained inthe pollen of plants; and on the general existence of active moleculesin organic and inorganic bodies. Philosophical Magazine 4, 161–173.

Doob, J. L. (1953). Stochastic Processes. New York/London: Wiley/Chapman Hall.

Duys, D. and Headrick, T. (2004). Using Markov chain analyses incounselor education research. Counselor Education and Supervision44(2), 108–120.

Einstein, A. (1905). Uber die von der molekularkinetischen. Theorie derWarme geforderte Bewegung von in ruhenden Flussigkeitensuspendierten Teilchen. Annalen der Physik 17, 549–560.

Ito, K. (1944). Stochastic integral. Proceedings of the Imperial Academy,Tokyo 20, 519–524.

Karatzas, I. and Shreve, S. (1998). Methods of Mathematical Financial.New York: Springer.

Kolmogorov, A. N. (1956). Foundations of the Theory of Probability,N. Morrison (trans.), 2nd English edn. New York: Chelsea Publishing.

Lawler, G. (2005). Conformally Invariant Processes in the Plane,Mathematical Survey Monographs. Providence, RI: AMS.

Merton, R. C. (1973). Theory of rational option pricing. Bell Journal ofEconomics and Management Science 4, 141–183.

Protter, P. (2004). Stochastic Integration and Differential Equations, Vol.21 of Applications of Mathematics, 2nd edn. Springer.

Revuz, D. and Yor, M. (1999). Continuous Martingales and BrownianMotion, 3rd edn. Berlin: Springer.