Parameter estimation for stochastic equations with ... - samos-matisse

of (3.1). We get
(5.2)
Z t,s :=
=
∫ t ∫ s
0 0
∫ t ∫ s
0
0
k α (t, v)k β (s, u) dX v,u
k α (t, v)k β (s, u)b(X v,u ) dudv + M α,β
t,s .
5.3 Remark. Moreover, for α, β > 1 2
, it follows from [6] and [16] that if we denote
then it holds that
(5.4) X t,s =
Denote
(5.5) R t,s = d
dω α t
Then we have the following:
K α (t, v) = α(2α − 1)
d
dω β s
∫ t ∫ s
0
0
∫ t ∫ s
0
∫ t
v
r 2α−1 (r − v) α− 3 2 dr
K α (t, v)K β (s, u) dZ u,v .
0
k α (t, v)k β (s, u)b(X v,u ) dudv.
• For every (α, β) ∈ (0, 1) 2 and if b is Lipschitz, the sample paths of the process R
given by (5.5) belong to L 2 ([0, 1] 2 , ω α ⊗ ω β ). This can be viewed in the same
way as in Theorem 2.
• Clearly, the process R is related to the process Q (4.9) by
(5.6) R t,s = c α,β t α− 1 2 s
β− 1 2 Qt,s .
¿From (5.2) and (5.5) we obtain that
(5.7) Z t,s = θ
∫ t ∫ s
0
0
R v,u dω β udω α v + M α,β
t,s .
and then the MLE for the parameter θ in (3.1) can be written as
(5.8) θ t = −
∫ t ∫ t
0
∫ t
0
0 R v,u dMu,v
α,β
∫ t
0 R2 v,u dωudω β v
α
As a final remark, we derive an easier expression for the process R (or, equivalently,
for the process Q) appearing in the expression of the MLE in the linear case.
In this case we have
(5.9) R t,s = d
dω α t
d
dω β s
∫ t ∫ s
0
0
k α (t, v)k β (s, u)X v,u dudv.
We need first a more suitable expression of the process R given by (5.9).
.
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