Estimation in Financial Models - RiskLab

we can write the Euler scheme (B.1) in the impressive form
Y n+1 = Y n + a n + bW n
(B.2)
for n =0; 1;:::;N , 1.
In order to compute the sequence fY n ;n = 0; 1;:::;N , 1g of values of
the Euler approximation we have to generate the random increments W n
for n = 0; 1;:::;N , 1 of the Wiener process W = fW t ;t 0g. These
increments are independent Gaussian random variables with E(W n ) = 0
and Var(W n )= n and can be generated by a random number generator
(see e.g. [45], x1.3).
For the multi-dimensional case of the Euler scheme see e.g. [45], x10.2.
Note that when the diusion coecient b is identically zero the stochastic
iterative scheme (B.2) reduces to the well-known deterministic Euler scheme
for the ordinary dierential equation x 0 = a(t; x).
We introduce the notion of strong convergence.
Denition 1 A time discrete approximation Y with maximum step size
converges strongly to X at time T if
lim E(jX T , Y (T )j) =0:
#0
The rate of strong convergence is crucial if we want to compare dierent time
discrete approximation methods.
Denition 2 A time discrete approximation Y converges strongly with
order > 0 at time T , if there exists a constant C > 0, independent of ,
and a 0 > 0 such that
for each 2 (0; 0 ).
E(jX T , Y (T )j) C
Under some regularity assumptions the Euler scheme converges strongly
with order =0:5 :
Theorem 8 Suppose
E(jX 0 j 2 ) < 1;
E jX 0 + Y
0
j 2 1 2
C 1 1 2 ;
ja(t; x) , a(t; y)j + jb(t; x) , b(t; y)j C 2 jx , yj;
ja(t; x)j + jb(t; x)j C 3 (1 + jxj)
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