probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

, Lévy's stochastic area is a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...

that describes the enclosed area of a

trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tra ...

of a two-dimensional

Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

and its chord. The process was introduced by Paul Lévy in 1940, and in 1950 he computed the

characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function \mathbf_A\colon X \to \, which for a given subset ''A'' of ''X'', has value 1 at points ...

and conditional characteristic function. The process has many unexpected connections to other objects in mathematics such as the

soliton In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...

solutions of the Korteweg–De Vries equation and the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

. In the

Malliavin calculus In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...

, the process can be used to construct a process that is smooth in the sense of Malliavin but that has no

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

modification with respect to the

Banach norm In mathematics, more specifically in functional analysis, a Banach space (, ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation of Norm ( ...

Lévy's stochastic area

Let

W=(W_s^,W_s^)_

be a two-dimensional Brownian motion in

\mathbb^2

then Lévy's stochastic area is the process :

S(t,W)=\frac\int_0^t \left(W_s^dW_s^-W_s^dW_s^\right),

where the Itō integral is used. Define the

1-Form In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the t ...

\vartheta=\tfrac(x^1dx^2-x^2dx^1)

then

S(t,W)

is the stochastic integral of

\vartheta

along the curve

to \R^2, s\mapsto (W_s^,W_s^)

S(t,W)=\int_ \vartheta.

Area formula

Let

x=(x_1,x_2)\in \R^2

a\in \R

b=at/2

and

S_t=S(t,W)

then Lévy computed :

\frac

and :

\frac\exp\left(\frac\left(1-b\coth\left(b\right)\right)\right),

where

\, x\, _2

is the

Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...

Further topics

* In 1980 Yor found a short probabilistic proof. * In 1983 Helmes and Schwane found a higher-dimensional formula.{{cite journal, first1=Kurt, last1=Helmes, first2=A, last2=Schwane, title=Levy's stochastic area formula in higher dimensions, journal=Journal of Functional Analysis, volume=54, number=2, date=1983, pages=177–192, doi=10.1016/0022-1236(83)90053-8

References

Stochastic processes Paul Lévy (mathematician)