Estimation in Financial Models - RiskLab

time between observations is bounded away from zero, especially in the case
of time-equidistant observations with xed, Florens-Zmirou [25] shows that
the estimator ~ n of obtained by maximizing the approximate log-likelihood
function ~ l n () is inconsistent.
To overcome the diculties regarding parameter dependence and the dependence
of ~ l n ()onmax 1in jt i ,t i,1 j,we will propose three dierent estimation
approaches. The basic idea of the rst two approaches is to nd good approximations
to the transition densities. In the third approach we construct
martingale estimating functions.
Approximation to the transition densities of X by a sequence of
transition densities of approximating Markov processes
The following approach is proposed by Pedersen [58, 60].
We shall derive a sequence (l n;N ()) 1 N=1
of approximations to l n (), that
builds a connection between ~ l n () (see (3.17)) and l n () in the following sense:
the approximation l n;1 () is a generalization of ~ l n () with no restrictions
on (; ; ) regarding parameter dependence, each l n;N () for N 2 is an
improvement of l n;1 () and as N tends to innity l n;N () converges for each
in probability tol n ().
The crucial point of the approach is to approximate the transition densities
p(s; x; t; y; ) of X by a sequence of transition densities (p N (s; x; t; y; )) 1 N=1
of approximating Markov processes which converges to p(s; x; t; y; ) as N
tends to innity, and then to dene the approximate log-likelihood functions
l n;N () =
nX
i=1
log (p N (t i,1 ;X ti,1 ;t i ;X ti ; )): (3.18)
In the following we will derive the approximating densities p N (s; x; t; y; ).
We remark that we may relax the Lipschitz assumption made in (3.1) for b
and a little bit, namely we only assume b and to be locally Lipschitz
continuous as dened below.
Under the following conditions (A1), (A2) and (A3) which must hold for all
2 , the stochastic dierential equation (3.1) has a weak solution for all x 0
and and has the pathwise-uniqueness property which implies the uniqueness
in law (see [65], p.132 and p.151; in this context see also [73], x5-x8).
(A1) b and are continuous in t for all x 2 IR d .
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