In this paper, we will establish the existence and the Hölder regularity of the local time of the Brownian bridge.
Our results are obtained by using a result on Malliavin calculus in
[K. Es-Sebaiy, D. Nualart, Y. Ouknine and C. A. Tudor,
Occupation densities for certain processes related to fractional Brownian motion,
Stochastics 82 2010, 1–3, 133–147]
for a Gaussian process with an absolutely continuous random drift, jointly with the classical approach
based on the concept of local nondeterminism for Gaussian processes introduced by Berman
[S. M. Berman,
Local nondeterminism and local times of Gaussian processes,
Indiana Univ. Math. J. 23 1973/74, 69–94].
The generalized Brownian bridge
X
a
,
b
,
T
{X^{a,b,T}}
from a to b of length T was used in several fields such as in mathematical finance, biology and statistics. In this paper, we study the following stochastic properties and characteristics of this process: The Hölder continuity, the self-similarity, the quadratic variation, the Markov property, the stationarity of the increments,
and the α-differentiability of the trajectories.
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