Estimation in Financial Models - RiskLab

The asymptotic properties of the estimator ^ n we obtain from the martingale
estimating functions (3.48), (3.50) and (3.52), or more generally from the
class of martingale estimating functions G n of the form (3.49), are discussed
by Bibby and Srensen [8]. Under natural regularity conditions (see Bibby
and Srensen [8], pp. 7{9) we have
Theorem 4 An estimator ^ n , which solves the equation
G n (^ n )=0;
exists with probability tending to one as n ,! 1 under P 0 . Moreover, as
n ,! 1,
^ n ,! 0
in probability under P 0 and ^ n is asymptotically normal in distribution under
P 0 .
For the proof we refer to [8].
As a rst example we consider the Ornstein-Uhlenbeck process where the
transition densities are well-known.
Example 1
The Ornstein-Uhlenbeck process is the solution of the stochastic dierential
equation
dX t = X t dt + dW t ; (3.53)
with X 0 = x 0 . In this case the drift coecientisb(x; ) =x and the diusion
coecient (x; ) is assumed to be known. The transition probability is
normal with mean F (x; ) =xe and variance () = 2
2 (e2 , 1). Hence
the estimating function ~G n has the form
~G n () = 1 2 nX
i=1
and we obtain ^ n as solution of ~G n () =0:
X (i,1) (X i , X (i,1) e ) ;
^ n = 1 P ni=1
log X (i,1) X i
Pni=1 :
X 2 (i,1)
The estimators ^ n we obtain from the martingale estimating functions G +
and G are the same because G + and G are proportional to ~G.
In the next example we consider a wider class of stochastic processes.
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