Lévy's stochastic area

In probability theory, Lévy's stochastic area is a stochastic process that describes the enclosed area of a trajectory of a two-dimensional Brownian motion and its chord. The process was introduced by Paul Lévy in 1940,[1] and in 1950[2] he computed the characteristic function and conditional characteristic function.

The process has many unexpected connections to other objects in mathematics such as the soliton solutions of the Korteweg–De Vries equation[3] and the Riemann zeta function.[4] In the Malliavin calculus, the process can be used to construct a process that is smooth in the sense of Malliavin but that has no continuous modification with respect to the Banach norm.[5]

Lévy's stochastic area

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Let   be a two-dimensional Brownian motion in   then Lévy's stochastic area is the process

 

where the Itō integral is used.[2]

Define the 1-Form   then   is the stochastic integral of   along the curve  

 [6]

Area formula

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Let  ,  ,   and   then Lévy computed

 

and

 

where   is the Euclidean norm.[2]: 172–173 

Further topics

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  • In 1980 Yor found a short probabilistic proof.[7]
  • In 1983 Helmes and Schwane found a higher-dimensional formula.[8]

References

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  1. ^ Lévy, Paul M. (1940). "Le Mouvement Brownien Plan". American Journal of Mathematics. 62 (1): 487–550. doi:10.2307/2371467. JSTOR 2371467.
  2. ^ a b c Lévy, Paul M. (1950). "Wiener's random function, and other Laplacian random functions". Proc. 2nd Berkeley Symp. Math. Stat. Proba. II. Univ. California: 171–186.
  3. ^ Ikeda, Nobuyuki; Taniguchi, Setsuo (2010). "The Itô–Nisio theorem, quadratic Wiener functionals, and 1-solitons". Stoch. Proc. Appl. 120 (5): 605–621. doi:10.1016/j.spa.2010.01.009.
  4. ^ Biane, Philippe; Pitman, Jim; Yor, Marc (2001). "Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions". Bull. Amer. Math. Soc. (N.S.). 38 (4): 435–465. CiteSeerX 10.1.1.35.4158. doi:10.1090/S0273-0979-01-00912-0. S2CID 14710582.
  5. ^ Ikeda, Nobuyuki; Watanabe, Shinzō (1984). "An Introduction to Malliavin's Calculus". North-Holland Mathematical Library. 32. Elsevier: 1–52. doi:10.1016/S0924-6509(08)70387-8. ISBN 0-444-87588-3.
  6. ^ Ikeda, Nobuyuki; Taniguchi, Setsuo (2011). "Euler polynomials, Bernoulli polynomials, and Lévyʼs stochastic area formula". Bulletin des Sciences Mathématiques. 135 (6–7): 685. doi:10.1016/j.bulsci.2011.07.009.
  7. ^ Yor, Marc (1980). Azéma, J.; Yor, M. (eds.). Remarques sur une formule de paul levy (PDF). Séminaire de Probabilités XIV 1978/79. Lecture Notes in Mathematics. Vol. 784. Berlin, Heidelberg: Springer. doi:10.1007/BFb0089501.
  8. ^ Helmes, Kurt; Schwane, A (1983). "Levy's stochastic area formula in higher dimensions". Journal of Functional Analysis. 54 (2): 177–192. doi:10.1016/0022-1236(83)90053-8.