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The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. [60] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin, [63] [64] though the German term had been used earlier, for example, by Andrei Kolmogorov ...
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables.
In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process.
Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure that is absolutely continuous with respect to P then every P-semimartingale is a Q-semimartingale.
P.A. Meyer: " Martingales and Stochastic Integrals I," Springer Lecture Notes in Mathematics 284, 1972. Brelot's axiomatic theory of the Dirichlet problem and Hunt's theory, Annales de l'Institut Fourier, 13 no. 2 (1963), p. 357–372; Intégrales stochastiques I, Séminaire de probabilités de Strasbourg, 1 (1967), p. 72–94
Download as PDF; Printable version; ... If X(s) is a stochastic process, then for all functions t(s) ... Lecture Notes in Statistics 101. New York: Springer-Verlag.
In mathematics, the theory of stochastic processes is an important contribution to probability theory, [29] and continues to be an active topic of research for both theory and applications. [30] [31] [32] The word stochastic is used to describe other terms and objects in mathematics.
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...