Estimation in Financial Models - RiskLab

In the case jj 1 we have Var X n ,!1as n ,! 1, that is the process
explodes.
In both cases = 1, fX n g reduces to a random walk.
From these considerations we suppose that as for probabilistic properties of
estimators of , e.g. asymptotic behaviour, we have to distinguish between
the cases jj > 1, jj < 1 and jj =1.
In the following we assume for the AR(1) model (3.62) X 0 = 0 and 2 = 1
for convenience.
Denoting by p (X 1 ;:::;X n ) the joint density of X 1 ;:::;X n , the Maximum
Likelihood Estimator (MLE) ^ n is dened to be a value such that the joint
density p reaches a maximum that is
^ n = argmax p (X 1 ;:::;X n );
(for the denition of the MLE in continuous time see x3.1.1, p.17). We know
the joint density p of X 1 ;:::;X n
"
p (X 1 ;:::;X n )=(2) , n 2 exp , 1 #
nX
(X i , X i,1 ) 2 ;
2
and hence are able to calculate ^ n by solving d d p =0
Inserting (3.62) for X i we obtain
i=1
P ni=1
X i,1 X i
^ n = Pni=1 :
Xi,1
2
^ n = +
P ni=1
X i,1 " i
Pni=1
X 2 i,1
P a:s: (3.65)
Denoting
M n =
nX
i=1
X i,1 " i ;
we see immediately that the process M n is a martingale with respect to the
ltration F n = (X 1 ;:::;X n ) under P for any . The martingale M n is
square integrable, i.e. EMn
2 < 1, n 0, and we know that the stochastic
sequence Mn 2 is a submartingale (see [66] x7.1, p.455). By means of the
Doob decomposition (see e.g. [66], p.454) there is a martingale m n and a
predictable 2 increasing 3 sequence hMi n such that
M 2 n = m n + hMi n :
2 A process X n is predictable if X n is F n,1 measurable.
3 A process X n is increasing if X 0 =0,X n X n+1 P a.s.
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