Estimation in Financial Models - RiskLab

continuous observations of X. X is a Markov process. P denotes the law
of the process X on the canonical space = C(IR + ; IR) with the canonical
ltration F t = (X s js t). Denote by P ;t := P jF t the restriction of P to
F t .
First consider the case where (; x) does not depend on . If does not
vanish, the measures P ;t and P 1 ;t for any ; 1 2 , t0, and the process (X t ) is observed in the
time interval [0;T]. Then the likelihood process (3.3) in T with 1 = 0 equals
"Z T b(X
L ;0
s )
T L T () = exp
0 2 (X s ) dX s , 1 Z T
#
2 b 2 (X s )
2 0 2 (X s ) ds ; (3.5)
see e.g. [51], x17, or [43], p.76, or [39]. In the following consider a constant
diusion term , say 1. For convenience we take the logarithm in (3.5)
and obtain the logarithmic likelihood (log-likelihood) function
ln L T () =
Z T
0
b(X s )dX s , 1 2
Z T
0
2 b 2 (X s )ds: (3.6)
Maximizing L T , or equivalently ln L T , with respect to we obtain the so
called Maximum Likelihood Estimator (MLE) ^ T based on continuous observations
of X in the interval 0 t T . We remark that by denition of the
likelihood function the measure P 1 , that dominates the measure P (that
is P