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
Let [math]\displaystyle{ W=(W_s^{(1)},W_s^{(2)})_{s\geq 0} }[/math] be a two-dimensional Brownian motion in [math]\displaystyle{ \mathbb{R}^2 }[/math] then Lévy's stochastic area is the process
- [math]\displaystyle{ S(t,W)=\frac{1}{2}\int_0^t \left(W_s^{(1)}dW_s^{(2)}-W_s^{(2)}dW_s^{(1)}\right), }[/math]
where the Itō integral is used.[2]
Define the 1-Form [math]\displaystyle{ \vartheta=\tfrac{1}{2}(x^1dx^2-x^2dx^1) }[/math] then [math]\displaystyle{ S(t,W) }[/math] is the stochastic integral of [math]\displaystyle{ \vartheta }[/math] along the curve [math]\displaystyle{ \varphi:[0,t]\to \R^2, s\mapsto (W_s^{(1)},W_s^{(2)}) }[/math]
- [math]\displaystyle{ S(t,W)=\int_{W[0,t]} \vartheta. }[/math][6]
Area formula
Let [math]\displaystyle{ x=(x_1,x_2)\in \R^2 }[/math], [math]\displaystyle{ a\in \R }[/math], [math]\displaystyle{ b=at/2 }[/math] and [math]\displaystyle{ S_t=S(t,W) }[/math] then Lévy computed
- [math]\displaystyle{ \mathbb{E}[\exp(iaS_t)]=\frac{1}{\cosh(b)} }[/math]
and
- [math]\displaystyle{ \mathbb{E}[\exp(iaS_t)\mid W_t=x]=\frac{b}{\sinh(b)}\exp\left(\frac{\|x\|_2}{2t}\left(1-b\coth\left(b\right)\right)\right), }[/math]
where [math]\displaystyle{ \|x\|_2 }[/math] is the Euclidean norm.[2]:172-173
Further topics
- In 1980 Yor found a short probabilistic proof.[7]
- In 1983 Helmes and Schwane found a higher-dimensional formula.[8]
References
- ↑ Lévy, Paul M. (1940). "Le Mouvement Brownien Plan". American Journal of Mathematics 62 (1): 487–550. doi:10.2307/2371467.
- ↑ 2.0 2.1 2.2 Lévy, Paul M. (1950). "Wiener's random function, and other Laplacian random functions". Proc. 2nd Berkeley Symp. Math. Stat. Proba. (Univ. California) II: 171–186.
- ↑ 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.
- ↑ 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. doi:10.1090/S0273-0979-01-00912-0.
- ↑ Ikeda, Nobuyuki; Watanabe, Shinzō (1984). "An Introduction to Malliavin's Calculus". North-Holland Mathematical Library (Elsevier) 32: 1–52. doi:10.1016/S0924-6509(08)70387-8. ISBN 0-444-87588-3.
- ↑ 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.
- ↑ Yor, Marc (1980). "Remarques sur une formule de paul levy". Séminaire de Probabilités XIV 1978/79.. 784. Berlin, Heidelberg: Springer. doi:10.1007/BFb0089501. http://www.numdam.org/item/SPS_1980__14__343_0.pdf.
- ↑ 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.
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