Stochastic processes

A stochastic process, or often random process, is a collection of random variables representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic, or random process, there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.

In the simple case of discrete time, as opposed to continuous time, a stochastic process is a sequence of random variables. The random variables corresponding to various times may be completely different, the only requirement being that these different random quantities all take values in the same space (the codomain of the function). One approach may be to model these random variables as random functions of one or several deterministic arguments (in most cases, the time parameter). Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical dependence.

Examples of stochastic processes:

• Stock market and exchange rate fluctuations
• Signals such as speech; audio and video.
• Medical data such as a patient’s EKG, EEG, blood pressure or temperature.

A generalization, the random field, is defined by letting the variables be parametrized by members of a topological space instead of time. Examples of random fields include static images, random terrain (landscapes), wind waves and composition variations of a heterogeneous material.

See also

Material

• Popular Stochastic Processes used in Quantitative Finance
• Interactive Web Application: Stochastic Processes used in Quantitative Finance. TuringFinance.com.
• StochPy. Sourgeforce

Books

• Øksendal, B. (2003). Stochastic differential equations. An introduction with applications. Springer Berlin Heidelberg.
• Evans, L. C. (2012). An introduction to stochastic differential equations (Vol. 82). American Mathematical Soc.
• Daley, Daryl J.; Vere-Jones, David (2003). An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. Springer.
• Daley, Daryl J.; Vere-Jones, David (2007). An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure. Springer.
• Cox, D. R.; Isham, V. I. (1980). Point Processes. Chapman & Hall.
• Bhat, U. N., & Miller, G. K. (1972). Elements of applied stochastic processes. J. Wiley.
• Gardiner, C. W. (1985). Handbook of stochastic methods (Vol. 3). Berlin: Springer.
• Itô, K. (1974). Diffusion Processes. John Wiley & Sons, Inc.
• Allen, Linda J. S. (2010). An Introduction to Stochastic Processes with Applications to Biology, 2nd Edition, Chapman and Hall.
• Klebaner, Fima C. (2011). Introduction to Stochastic Calculus With Applications. Imperial College Press.