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 $W=(W_{s}^{(1)},W_{s}^{(2)})_{s\geq 0}$ be a two-dimensional Brownian motion in $\mathbb {R} ^{2}$ then Lévy's stochastic area is the process

S(t,W)={\frac {1}{2}}\int _{0}^{t}\left(W_{s}^{(1)}dW_{s}^{(2)}-W_{s}^{(2)}dW_{s}^{(1)}\right),

where the Itō integral is used.[2]

Define the 1-Form $\vartheta ={\tfrac {1}{2}}(x^{1}dx^{2}-x^{2}dx^{1})$ then $S(t,W)$ is the stochastic integral of $\vartheta$ along the curve ${\displaystyle \varphi$

S(t,W)=\int _{W[0,t]}\vartheta .

[6]

Area formula

Let $x=(x_{1},x_{2})\in \mathbb {R} ^{2}$ , $a\in \mathbb {R}$ , $b=at/2$ and $S_{t}=S(t,W)$ then Lévy computed

\mathbb {E} [\exp(iaS_{t})]={\frac {1}{\cosh(b)}}

and

\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),

where $\|x\|_{2}$ 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.
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: 605–621.
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.
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). Azéma, J.; Yor, M. (eds.). Remarques sur une formule de paul levy. Séminaire de Probabilités XIV 1978/79. Lecture Notes in Mathematics. Vol. 784. Berlin, Heidelberg: Springer. doi:10.1007/BFb0089501.
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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[1] Lévy, Paul M. (1940). "Le Mouvement Brownien Plan". American Journal of Mathematics. 62 (1): 487–550. doi:10.2307/2371467.

[Levy-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.

[3] Ikeda, Nobuyuki; Taniguchi, Setsuo (2010). "The Itô–Nisio theorem, quadratic Wiener functionals, and 1-solitons". Stoch. Proc. Appl. 120: 605–621.

[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.

[5] 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.

[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. 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.