Estimation in Financial Models - RiskLab

(see [49], p.249f).
Asymptotic normality can be used to determine condence intervals for
and to determine a 'suitable' value of T .
Example 1. a) The regularity conditions are satised in the case of observations
coming from
dX t = ,X t dt + dW t ; X 0 =0; 0 t T; (3.10)
where 2 (; ), >0, that is , 0, the process has no
stationary distribution. Nevertheless we can show that the MLE ^ T is consistent.
Furthermore, e 0T (^ T , 0 ) is asymptotically normal with parameters
(0; 40) 2 (see [48], p.100).
c) In the case = 0 the MLE ^ T is consistent, but not asymptotically normal
(see [48], p.100).
We remark that in the discrete case (section 3.2) an analogous distinction
regarding parameter values is made, see p.43.
After having treated the special case of a drift term linear in we give a
short remark to the general case
dX t = b(; X t )dt + dW t ;
where b(; x) may be a nonlinear function. By Taylor expansion for we have
b(; x) =b( 0 ;x)+( , 0 ) d d b( 0;x)+O(( , 0 ) 2 );
where 0 again denotes the true value of parameter . Hence b(; x) can be
approximated by a term linear in . Since in practice a 'good guess' of 0
often is available, then with the above considerations it is sucient only to
consider the already treated case: a drift term linear in . This closes our
considerations about ML estimation.
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