Affiliation:
1. Laboratory of Mathematical Engineering – EPT , Ecole Superieur des Sciences et de Technologie de Hammam Sousse , University of Sousse , Sousse , Tunisia
2. INSAT , Carthage University , Tunis , Tunisia
Abstract
Abstract
The local time (LT) of a given stochastic process
{
X
t
:
t
≥
0
}
\{X_{t}:t\geq 0\}
is defined informally as
L
X
(
t
,
x
)
=
∫
0
t
δ
x
(
X
s
)
d
s
,
L_{X}(t,x)=\int_{0}^{t}\delta_{x}(X_{s})\,ds,
where
δ
x
\delta_{x}
denotes the Dirac function; actually, it counts the duration of the process’s stay at 𝑥 up to time 𝑡.
Using an approximation approach, we study the existence and the regularity of the LT process for two kinds of stochastic processes.
The first type is the stochastic process defined by the indefinite Wiener integral
X
t
:=
∫
0
t
f
(
u
)
d
B
u
X_{t}:=\int_{0}^{t}f(u)\,dB_{u}
for a given deterministic function
f
∈
L
2
(
[
0
,
+
∞
[
)
f\in L^{2}([0,+\infty[)
, and secondly, for Lévy type processes, i.e. ones that are stationary and with independent increments.
Subject
Statistics and Probability,Analysis