Springer Finance

100 3 Brownian Motion

These will serve to determine the step size. We do not require the parti­

tion points to = 0, h, t 2 , ... , tn = T to be equally spaced, although they

are allowed to be. The maximum step size of the partition will be denoted

IIJIII = maxj=O, ... ,n-l(tj+l-tj)· We then define

n- 1

FVr(J) = lim L lf(ti+I)-f(tj)l.

111111 -+^0 j=O

(3.4.3)

The limit in (3.4.3) is taken as the number n of partition points goes to infinity

and the length of the longest subinterval ti+l -ti goes to zero.

Our first task is to verify that the definition (3.4.3) is consistent with the

formula (3.4.2) for the function shown in Figure 3.4.1. To do this, we use

the Mean Value Theorem, which applies to any function f(t) whose deriva­

tive f'(t) is defined everywhere. The Mean Value Theorem says that in each

subinterval [tj, ti+Il there is a p oint tj such that

/(tj+I)-f(tj) =

f'(t*).

tj+l - tj J

(3.4.4)

In other words, somewhere between ti and ti+l, the tangent line is parallel

to the chord connecting the points ( t i, f ( t i)) and ( t i +1 , f ( t i +1)) (see Figure

3.4.2).

f(t)

Fig. 3.4.2. Mean Value Theorem.

Multiplying (3.4.4) by ti+l- ti, we obtain

f(tj+I)-f(tj) = f'(tj)(tj+l -tj)·

The sum on the right-hand side of (3.4.3) may thus be written as

t