Variations sur une formule de Paul Lévy. (Variations on a formula of Paul Levý). (French) Zbl 0623.60099

Biane, Ph.; Yor, M.

Ann. Inst. Henri Poincaré, Probab. Stat. 23, Suppl., 359-377 (1987).

Let \(Z=X+iY\) be the complex Brownian motion, \(S=\int^{1}_{0}X_ sdY_ s-Y_ sdX_ s\) be Lévy’s stochastic area. Extending the famous Lévy’s formula it is proved that \(S=\sum^{\infty}_{p=0}\beta_ p\times \beta_{p+1}\) and for each \(n=0,1,..\). \[ (*)\quad E[\exp (i\lambda \sum^{\infty}_{p=n}\beta_ p\times \beta_{p+1}| \beta_ n=z)]=\frac{\lambda^{\nu}}{2^{\nu}\Gamma (\nu +1)I_{\nu}(\lambda)}e^{-(| z|^ 2/2)\lambda I_{\nu +1}(\lambda)/I_{\nu}(\lambda)}, \] where \(\beta_ p=\int^{1}_{0}P_ p(2s-1)dz(s)\), \(P_ n(t)=(1/2^ nn!)(d^ n/dt^ n)((t^ 2-1)^ n)\) is a Legendre polynomial, \(\xi \times \eta =Im({\bar \xi}\eta)\), \(\nu =n+1/2\) and \(I_{\nu}\) is the modified Bessel function. Using some Gaussian enlargements of the Brownian filtrations the relationship of the formula (*) with the known formula: \[ E[\exp (- (\lambda^ 2/2)\int^{\infty}_{0}1_{\{| B_ s| \leq 1\}}ds)| L=t]=\frac{\lambda^{\nu}}{2^{\nu}\Gamma (\nu +1)I_{\nu}(\lambda)}e^{(-t/2)\lambda I_{\nu +1}(\lambda)/I_{\nu}(\lambda)} \] is explained, where B is the d- dimensional Brownian motion starting from 0, L is the local time of \(| B|\) at level 1, \(\nu =d/2-1\).

Reviewer: B.Grigelionis

MSC:

60J55	Local time and additive functionals
60G48	Generalizations of martingales
60J65	Brownian motion
60H05	Stochastic integrals

Keywords:

complex Brownian motion; Lévy’s stochastic area; Lévy’s formula; Bessel function; local time