Estimation in Financial Models - RiskLab

conditions for the continuity of the sample paths of X t and the denitions
for a h and b h . Later we will refer to these conditions as Nelson-conditions.
Now as an application of Theorem 1 Nelson [54] nds and analyzes the diffusion
limit of GARCH(1,1)-type processes.
The GARCH(1,1)-M process of Engle and Bollerslev (see [13]) is dened as
where f" t gN(0; 1) iid.
Y t = Y t,1 + c 2 t + t " t ; (2.1)
2 t+1
= + 2 t
h
+ "
2
t
i
; (2.2)
Now our purpose is to reduce the length of the time intervals more and more.
The parameters , and of the system may depend on h. The drift term
in (2.1) and the variance of " t are made proportional to h:
Y kh = Y (k,1)h + h ckh 2 + kh " kh ; (2.3)
(k+1)h 2 = h + kh
2 h +
h
h "2 kh ; (2.4)
where f" kh gN(0;h) iid and as for the initial distribution (k =0)wehave
P h (Y 0 ; 2 0) 2 A i = h (A)
and (h) 2
t are con-
for all A 2 B(IR 2 ). The continuous time processes Y (h)
t
structed by
Y (h)
t Y kh and (h) 2
t
2
kh
for kh t 0:
h#0
11