Probability And Stochastic Processes Codexery

Wiener process

A continuous-time stochastic process with independent Gaussian increments.

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The Wiener process is a real-valued continuous-time stochastic process named after Norbert Wiener, also known as Brownian motion due to its historical connection with the physical process of the same name.[1] Among Lévy processes—stochastic processes with stationary independent increments that are càdlàg—the Wiener process is a particularly well-known example.[1]

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Key facts and history

It can also be constructed as the scaling limit of a random walk, a result known as Donsker's theorem.[2] Like the random walk, the Wiener process is recurrent in one or two dimensions but not in dimensions three and higher.[2] The expectation of the running maximum M_t is sqrt(2t/π).[3] The Wiener process is a Gaussian process with continuous paths, meaning any finite collection of its values forms a multivariate Gaussian random vector.[4]

Through the Feynman–Kac formula, the Wiener process provides a rigorous path integral formulation for quantum mechanics.[5] Within mathematical finance, the Wiener process plays a key role, especially in the Black–Scholes option pricing model.[6]

What the cited sources add

The same source notes that the hitting time T_a has the same distribution as a^2/W_1^2.[7][8] Lawler's 2005 book "Conformally invariant processes in the plane" explains that if f is a holomorphic function, then f(W_t) is a time-changed Wiener process in f(D).[9][10]

References

  1. Wikipedia, "Wiener process" (CC BY-SA 4.0), which cites: Dobrow, "Introduction to Stochastic Processes with R", Wiley, 2016, p. 321–322; Wiener, "Norbert Wiener: Collected Works with Commentaries", MIT Press, 1976 (original not read by us) en.wikipedia.org
  2. Wikipedia, "Wiener process" (CC BY-SA 4.0), which cites: "Pólya's Random Walk Constants", Wolfram Mathworld (original not read by us) en.wikipedia.org
  3. Wikipedia, "Wiener process" (CC BY-SA 4.0), which cites: Shreve, "Stochastic Calculus for Finance II: Continuous-Time Models", Springer, 2008, p. 114 (original not read by us) en.wikipedia.org
  4. Wikipedia, "Wiener process" (CC BY-SA 4.0), which cites: Klenke, "Probability Theory", Springer, 2020, p. 524, 526, 529, 531 (original not read by us) en.wikipedia.org
  5. Wikipedia, "Wiener process" (CC BY-SA 4.0), which cites: Kleinert, "Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets", World Scientific, 2004 (we also read that source) en.wikipedia.org
  6. Wikipedia, "Wiener process" (CC BY-SA 4.0), which cites: Karatsas, "Brownian Motion and Stochastic Calculus", Springer, 1998, p. 103, 108, 110 (original not read by us) en.wikipedia.org
  7. Wikipedia, "Wiener process" (CC BY-SA 4.0), which cites: Le Gall, "Brownian Motion, Martingales, and Stochastic Calculus", Springer, 2016, p. 11-19,20, 30, 32, 37, 39, 50, 51, 62, 63, 119, 121, 127, 155-161 (we also read that source) en.wikipedia.org
  8. Open Library record, ISBN 9783319310886 ("Brownian Motion, Martingales, and Stochastic Calculus"). CC0 (metadata); book content not read. openlibrary.org
  9. Wikipedia, "Wiener process" (CC BY-SA 4.0), which cites: Lawler, "Conformally invariant processes in the plane", American Mathematical Society, 2005 (we also read that source) en.wikipedia.org
  10. Open Library record, ISBN 9780821836774 ("Conformally invariant processes in the plane"). CC0 (metadata); book content not read. openlibrary.org

Each numbered sentence above was checked against the source it cites before publication. Sentences citing Wikipedia adapt text from the English Wikipedia article Wiener process (contributors), used under CC BY-SA 4.0; this entry is available under the same licence. Other sources were used for facts only; no text or images are copied from them.

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Sources

Compiled from Wikipedia and the sources listed below. Text from Wikipedia is available under CC BY-SA 4.0; this entry is adapted from it.

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