Estimation in Financial Models - RiskLab

In the following we give some remarks to the case where the diusion term
depends on an unknown parameter . As mentioned before in this case we
cannot estimate by using Maximum Likelihood theory (see p.17).
As a rst estimating problem we deal with the process
dX t = ()dW t ; 0 t T: (3.11)
In order to estimate we discretize the process and derive its limit. Consider
partitions n = t (n)
0 ;:::;t m(n) of [0;T] constructed in such a way that
P 1n=1
sup k jt (n)
k+1
, t (n)
k j < 1 for n ,! 1. Then for the discretized Wiener
process W we have the convergence result
mX
k=1
jW t
(n)
k
, W (n) t
j 2 ,! T;
k,1
in probability for n ,! 1(see [15], p.262f). Hence for (3.11) we obtain
mX
k=1
jX t
(n)
k
, X (n) t
j 2 ,! T 2 ();
k,1
in probability for n ,! 1, thus we are able to give an arbitrarily precise
estimate of 2 (). Notice that we do not obtain a direct estimate of .
Next, if the unknown parameter splits into two parts ( 1 ; 2 ) in the following
way
dX t = b(t; 1 )dt + ( 2 )dW t ; (3.12)
wewant to estimate the part 2 in the diusion coecient. Denoting X (n)
k
X (n) t
, X (n)
k t
and W (n)
k W (n)
k,1
t
, W (n)
k t
we obtain by discretizing (3.12)
k,1
mX
k=1
X
(n)
k
2
, (2 )W (n) 2
k
=
mX
k=1
+2
b 2 (t k ; 1 ) t (n) 2
k
mX
k=1
b(t k ; 1 )( 2 )W (n)
k t (n)
k ;
which tends to 0 in probability as n ,! 1. Thus the drift term is insignificant
as n ,! 1 and we are able to estimate ( 2 ) just as in the previous
case. Then by treating 2 as known we can estimate 1 via the ML method.
Finally, for the general problem with a diusion coecient also depending
on X
dX t = (; X t )dW t ; (3.13)
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