Estimation in Financial Models - RiskLab

First of all, as a simple though important example we approximate a Wiener
process using the central limit theorem. Let 1 ; 2 ;::: be a sequence of iid
random variables dened on a probability space (; B;P). Suppose E n =0,
Var n = 2 and S n = P n
i=1
i . Via the central limit theorem we have that the
distribution of S n =( p n) converges in distribution to the normal distribution
as n tends to innity. By means of the partial sums S n we dene piecewise
linear functions X n on the interval [0; 1]. For each n and each ! the function
X n (;!) is linear on each interval [(i , 1)=n; i=n], 1 i n and has the
value S i (!)=( p n) at the point i=n with starting point S 0 (!) = 0. Hence,
we construct a function X n of the form
X n (t; !) = 1
p n S i,1(!)+
t , (i , 1)=n
1=n
1
p n i(!);
for t 2 h i,1
n ; i ni
. For each !, Xn (;!) is a continuous function on [0; 1], i.e.
X n (;!) 2 C[0; 1]. Denote by P n the distribution of X n (;!)inC[0; 1]. Then
P n converges weakly to a Wiener measure W
P n ,! W
(see [10], x2). Note that this convergence result can be viewed in two dierent
ways: on the one hand X n can be seen as an approximation to a Wiener
process, and on the other hand the Wiener process as an approximation to
X n .We will come back to this point later.
The main statement of this chapter is a convergence theorem, where general
conditions are given for a sequence of discrete time Markov processes
to converge weakly to an It^o process. These conditions were developed by
Stroock and Varadhan (see [73], x11.2). As mentioned in the example above,
note that given a convergence result like this, we may use this fact to tackle
the approximation problem of continuous time by discrete models and vice
versa (see section 2.2). For a presentation of the convergence theorem and
the proof we also refer to Nelson [54].
The Convergence Theorem
For h > 0 arbitrarily, consider a discrete time Markov process X 0 ;X h ,
X 2h ;:::;X kh denoted by fX kh g, where X kh takes values in IR n for all k.
Assume that we know the transition probabilities of fX kh g and the distribution
of the initial random point X 0 .We construct the continuous time process
fX (h)
t g from the discrete time process fX kh g by making X (h)
t a step function
with jumps at times h; 2h; 3h::: and values X (h)
t = X kh almost surely for
kh t