When studying functionals of a Brownian motion, it may be useful to embed this functional into a larger dimensional Markov process.
Consider the case of the Levy area
where , , is a two dimensional Brownian motion started at 0. We can write
where . Since , we interpret as (two times) the algebraic area swept out in the plane by the Brownian curve up to time . The process is not a Markov process in its own natural filtration. However, if we consider the 3-dimensional process
then is solution of a stochastic differential equation
As a consequence is a Markov process with generator
where are the following vector fields
Observe that the Lie bracket
Thus, for every , is a basis of . From the celebrated Hormander’s theorem, this implies that for every the random variable has a smooth density with respect to the Lebesgue measure of . In particular also has smooth density whenever . We are interested in an expression for this density. The first idea is to reduce the complexity of the random variable by making use of symmetries.
Lemma: Let , . Then, the couple
is a Markov process with generator
From Ito’s formula, we have
Since the two processes
are two independent Brownian motions, the conclusion easily follows.
We are now ready to prove the celebrated Levy’s area formula.
Theorem: For and , and
First, we observe that by rotational symmetry of the Brownian motion , we have
Then, according to the previous lemma,
where is a Brownian motion independent from . We deduce
As we have seen, solves a stochastic differential equation
where is a one-dimensional Brownian motion.
One considers then the new probability
where is the natural filtration of . Observe that
In particular, one deduces that
which proves that is a martingale. By using this change of probability, if is a bounded and Borel function, we have
Putting things together, we are thus let with the computation of the distribution of under the probability . From Girsanov’s theorem, the process
is a Brownian motion under the probability . Thus
In law, this is the stochastic differential equation solved by where
We deduce that is distributed as , the norm of a two-dimensional Ornstein Uhlenbeck process with parameter . Since is a Gaussian random variable with mean 0 and variance , the conclusion follows from standard computations about the Gaussian distribution.
The Levy’s area formula has several interesting consequences. First, when , we deduce that
This gives a formula for the characteristic function of the algebraic stochastic area within the Brownian loop with length . Inverting this Fourier transform yields
Next, integrating the Levy’s area formula with respect to the distribution of yields the characteristic function of :
Unfortunately, this Fourier transform may not be easily inverted. However, one may deduce from it the following formula (due to Biane-Yor): For ,
where is the Dirichlet function. This provides an unexpected and fascinating connection with the Riemann zeta function.