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In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.
Stochastic refers to a randomly determined process. The word first appeared in English to describe a mathematical object called a stochastic process, but now in mathematics the terms stochastic process and random process are considered interchangeable. The word, with its current definition meaning random, came from German, but it originally came from Greek στόχος (stókhos), meaning 'aim, guess'.
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In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. Progressively measurable processes are important in the theory of Itô integrals.
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The Engelbert–Schmidt zero–one law is a theorem that gives a mathematical criterion for an event associated with a continuous, non-decreasing additive functional of Brownian motion to have probability either 0 or 1, without the possibility of an intermediate value. This zero-one law is used in the study of questions of finiteness and asymptotic behavior for stochastic differential equations. This 0-1 law, published in 1981, is named after Hans-Jürgen Engelbert and the probabilist Wolfgang Schmidt.
Marc Oliver Rieger is a German mathematician and finance professor. His research covers the areas of behavioural finance, decision and game theory, financial derivatives, and calculus of variation. Together with Thorsten Hens and Mei Wang, he was one of the authors of the INTRA survey that collected risk and time preferences in 53 countries worldwide leading to various follow-up studies by other researchers and practitioners and media coverage by leading newspapers in Germany and elsewhere.
References
- ↑ "CV of Steven E. Shreve" (PDF). Retrieved 2020-01-03.
- ↑ [ permanent dead link ]
- ↑ "Interview with Steven Shreve". June 13, 2010. Retrieved 2020-01-03.
- ↑ Darrell Duffie (March 18, 2008). "A Review of Stochastic Calculus for Finance" (PDF).
- ↑ "Honored IMS Fellows". Institute of Mathematical Statistics. Retrieved 2020-01-03.
- ↑ "Steven Shreve". Carnegie Mellon University. Retrieved 2020-01-03.
