Estimation in Financial Models - RiskLab

a convergence in probability result is given by
mX
k=1
jX t
(n)
k
, X (n) t
j 2 ,!
k,1
Z T
0
2 (; X s )ds; (3.14)
for n ,! 1 (see [24], [51], x4). A hint towards the correctness of (3.14) is
given by the well-known equality
E(X 2 t )=
Z t
0
E( 2 (; X s ))ds:
Again
R
note that we cannot estimate directly but are only able to estimate
T
0
2 (; X t )dt.
3.1.2 Discrete observations
In this section we deal with the (more realistic) situation where the diusion
process X (3.1) has only been observed at not necessarily equally spaced
discrete time points 0 = t 0 < t 1 < ::: < t n . In our discussion below, we
basicly follow Kloeden, Schurz, Platen and Srensen [46], Bibby [4, 5, 6, 7],
Bibby and Srensen [8] and Pedersen [58, 59, 60, 61, 62, 63].
For the estimation of from discrete observations of X wehave to distinguish
between two cases: the transition densities of X are known or unknown.
If the transition densities p(s; x; t; y; ) of X are known, e.g. in the case of
an Ornstein-Uhlenbeck process (see p.36, Example 1), an obvious choice of
an estimator for is the Maximum Likelihood Estimator (MLE) ^ n which
maximizes the likelihood function
L n () =
nY
i=1
or equivalently the log-likelihood function
l n () ln L n () =
p(t i,1 ;X ti,1 ;t i ;X ti ; );
nX
i=1
log (p(t i,1 ;X ti,1 ;t i ;X ti ; )) (3.15)
for , see e.g. [2], p.14. In the case of time-equidistant observations (t i =
i; i = 0; 1 :::;n for some xed > 0) Dacunha-Castelle and Florens-
Zmirou [19] prove consistency and asymptotic normality of ^ n as n ! 1,
independent of the value of . Unfortunately in general the transition densities
of X are unknown.
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