Abstract
We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model
X
(
t
)
=
∫
0
t
a
(
t
-
s
)
d
Z
(
s
)
X(t)=\int_{0}^{t}a(t-s)\,dZ(s)
, where 𝑍 is a Lévy martingale and the kernel
a
(
.
)
a(\,{.}\,)
a deterministic function square integrable on
R
+
\mathbb{R}^{+}
.
Given 𝑁 i.i.d. continuous time observations
(
X
i
(
t
)
)
t
∈
[
0
,
T
]
(X_{i}(t))_{t\in[0,T]}
,
i
=
1
,
…
,
N
i=1,\dots,N
, distributed like
(
X
(
t
)
)
t
∈
[
0
,
T
]
(X(t))_{t\in[0,T]}
, we propose two types of nonparametric projection estimators of
a
2
a^{2}
under different sets of assumptions.
We bound the
L
2
\mathbb{L}^{2}
-risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.
Kernel estimation for Lévy driven stochastic convolutions
