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

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
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