An elementary introduction to the Wiener process and stochastic integrals

By T. Szabados. 2010

46 citations recorded by the source corpus.

An elementary construction of the Wiener process is discussed, based on a proper sequence of simple symmetric random walks that uniformly converge on bounded intervals, with probability 1. This method is a simplification of F.B. Knight's and P. R\'ev\'esz's. The same sequence is applied to give elementary (Lebesgue-type) definitions of It\^o and Stratonovich sense stochastic integrals and to prove the basic It\^o formula. The resulting approximating sums converge with probability 1. As a by-product, new elementary proofs are given for some properties of the Wiener process, like the almost sure non-differentiability of the sample-functions. The purpose of using elementary methods almost exclusively is twofold: first, to provide an introduction to these topics for a wide audience; second, to create an approach well-suited for generalization and for attacking otherwise hard problems.

View Paper: https://www.semanticscholar.org/paper/6b5916cccab080fbef3976ff61755dcd6713238b

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Citations in the corpus

• Knowledge Tracing: A Survey (2022, background): “, 4PL) as a time-varying stochastic process such as a Wiener random process [105].”