Thanks to the work by M. Yor and colleagues, much is known about the following exponential of Brownian motion:

$X= \int_0^{\infty}{\rm d}t \ e^{-t + g \ B(t)}$

where $g$ is a ** real** scale parameter.

A nice probabilistic way to obtain results is to use Lamperti representation to recognize $X$ as a time-changed Bessel process. Among other things, one can show that for $D\equiv g^2/2 <1$, $D X$ is a Gamma distribution of parameter $-1/D$.

For a physicist (which I am), a cheap way out to recover this would be to determine the stationary solution to the forward Chapman-Kolmogorov equation associated to the stochastic differential equation:

$dX(t) = [1+(D-1)X(t)]dt + g X(t) dB(t)$

The latter is readily integrated, and taking the $t \to \infty$ limit, one encounters the definition above for the functional $X$.

Motivated by various questions, with colleagues we have bee working hard over the last year on the following problem: what happens if $g \to \imath \ g, \ \imath \equiv \sqrt{-1}$ (in either the s.d.e. or the explicit exponential functional of BM)?

Now, of course the variable:

$Z= \int_0^{\infty}{\rm d}t \ e^{-t + \imath g \ B(t)}.$

do not live anymore on the real axis, but in the complex plane, and actually in the unit disk ($\forall g$), because of $|Z| \le \int_0^{\infty}{\rm d}t \ e^{-t}=1$.

But by whatever means, the distribution of $Z$, i.e. the joint distribution of its real and imaginary part, or of its modulus and of its angle (writing $Z=X+ \imath \ Y= R e^{\imath \theta}$), seems very hard to determine reasonably explicitely.

For instance, one can compute the integer moments:

$ E[(D Z)^m] = \frac{\Gamma(1+1+1/D)}{\Gamma(m+1+1/D)}$

but of course this is not enough to reconstruct the joint law of $(X,Y)$: one would need to solve the quadratic recurrence for the joint moments $a_{m,n} \equiv E[Z^m {\overline{Z}}^n]$ of $Z$ and of its complex conjugate $\overline{Z}$:

$ (m+n+D(m-n)^2) a_{m,n} = m a_{m-1,n}+ n a_{m,n-1}.$

The most compact characterization I have found is through the following (double) exponential generating function: $\phi(p,q) \equiv E[\exp{(p Z + q \overline{Z})}]$. The latter obeys a certain second order linear p.d.e., which can be brought to canonical form by setting $p=(\mu/2)e^{-\imath \nu}, \ q= (\mu/2)e^{\imath \nu}$, so that, with $Z=R e^{\imath \theta}$, one finds that

$\phi(p,q)=\Omega(\mu,\nu)= E[e^{\mu R \cos{(\theta-\nu)}}]$

where $\Omega$ is the unique solution with $\Omega(0,\nu)=1$ to:

$[\mu \partial_{\mu} -D \partial^2_{\nu} - \mu \cos{\nu}]\Omega(\mu,\nu)=0$

I shall post soon a paper on the arXiv on this, but I thinks it's worth drawing the attention of mathematicians to this kind of problem, which is well-posed, and for which no solution seems to have been given.

If anybody has an idea, she/he would be most welcome...