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# Stochastic Processes

A stochastic process is a random function; or more precisely, an indexed family of random variables.

"stochastic adjective randomly determined; having a random probability distribution or pattern that may be analysed statistically but may not be predicted precisely."
The New Oxford Dictionay of English, 1998

"stochastic process, n. a process that can be described by a RANDOM VARIABLE (stochastic variable) that depends on some parameter, which may be discrete or continuous, but is often taken to represent time; precisely, an indexed family of random variables, called states, on a probability space. A stochastic process is finite if the index familiy is countable and each state is a step function. A MARKOV CHAIN is a discrete-parameter stochastic process in which future probabilities are completely determined by the present state."
Borowski and Borwein, 1989

"In the mathematics of probability, a stochastic process is a function. In the most common applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series in applications) or a region of space (a stochastic process being called a random field)."
Wikipedia (2006)

## Papers

• CHANDRASEKHAR, S., Stochastic Problems in Physics and Astronomy, 1943. [Cited by 582]
• CLARK, Peter K., A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices, Econometrica, Vol. 41, No. 1. (Jan., 1973), pp. 135-155.
• COX, John C. and Stephen A. ROSS, The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics, 3(January/March 1976):145B166.
• DOOB, J. L., The Brownian Movement and Stochastic Equations, 1942. [Cited by 69]
• EINSTEIN, A. and R FUERTH, [BOOK] Investigations on the Theory of the Brownian Movement, 1956 - Dover Publications [Cited by 221]
• EINSTEIN, A., "Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-Kinetic Theory of Heat", Ann. Phys., Lpz, 1905. [Cited by 49]
• EINSTEIN, A., A New Determination of Molecular Dimensions", Annalen der Physik, 1906. [Cited by 19]
• EINSTEIN, A., The elementary theory of Brownian motion, Zeit. fuer Elektrochemie, 1908. [Cited by 4 ]
• HECKMAN, James J. "The Incidental Parameters Problem and the Problem of Initial Conditions in Estimating a Discrete Time-Discrete Data Stochastic Process." in Structural Analysis of Discrete Data with Econometric Applications, edited by C. Manski and D. McFadden (1981): 179-195. Massachusetts: The MIT Press.
• KAC, Mark, Random Walk and the Theory of Brownian Motion, 1947. [Cited by 61]
• MANTEGNA, Rosario N. and H. Eugene STANLEY, Stochastic Process with Ultraslow Convergence to a Gaussian: The Truncated Lévy Flight, Phys. Rev. Lett. 73, 2946–2949 (1994).
• RICE, S. O., Mathematical Analysis of Random Noise, 1945. [Cited by 465]
• UHLENBECK, G. E. and L. S. ORNSTEIN On the Theory of the Brownian Motion, 1930. [Cited by 280]
• WANG, Ming Chen and G. E. UHLENBECK, On the Theory of the Brownian Motion II, 1945. [Cited by 306]
• BACHELIER, Louis. �Theory of Speculation.� Reprinted in Paul H. Cootner, editor. The Random Character of Stock Market Prices. Cambridge: The MIT Press (1964), Ch. 2. Translated by A. James Boness.
• CHANCE, Don M. �The ABCs of Geometric Brownian Motion.� Derivatives Quarterly (Winter 1994), 41-47.
• DIMAND, Robert W. �The Case of Brownian Motion: A Note on Bache1ier�s Contribution.� British Journal of the History of Science (1993), 233-234.
• MAIOCCHI, Robert. �The Case of Brownian Motion.� British Journal of the History of Science (1990), 257-283.
• MERTON, Robert C. �On the Role of the Wiener Process in Finance Theory and Practice: The Case of Replicating Portfolios.� Proceedings of Symposia in Pure Mathematics 60 (1997), 209-221.