Non-linear functionals of the Brownian bridge and some applications

Corinne Berzin-Joseph (1) , José R. León (2) , Joaquín Ortega (3, 2)

1 UPMF - Université Pierre Mendès France - Grenoble 2
2 UCV - Universidad Central de Venezuela
3 IVIC - Instituto Venezolano de Investigaciones Cientificas = Venezuelan Institute for Scientific Research

Let $b^F(t)$; $t \in [0,1]$ be an $F$-Brownian bridge process. We study the asymptotic behaviour of non-linear functionals of regularizations by convolution of this process and apply these results to the estimation of the variance of a non-homogeneous diffusion and to the convergence of the number of crossings of a level by the regularized process to a modification of the local time of the Brownian bridge as the regularization parameter goes to 0.

Mots clés: Crossings; Local time; Regularization by convolution; Brownian bridge; Non-linear functionals; Non-homogeneous difusion

Domaines: Probabilités [math.PR]

Corinne Berzin-Joseph, José R. León, Joaquín Ortega. Non-linear functionals of the Brownian bridge and some applications. Stochastic Processes and their Applications, 2001, 92 (1), pp.11-30. ⟨10.1016/S0304-4149(00)00068-5⟩. ⟨hal-00319148⟩