8-Stochastic Integrals Page 222 Wednesday, February 4, 2004 12:50 PM
222 The Mathematics of Financial Modeling and Investment Management
Recall that the total variation of a function f(x) is the limit of the
sums
∑ fx()i – fx( i– 1 )
while the quadratic variation is defined as the limit of the sums
∑
()– fx^2
fxi ( i– 1 )
Quadratic variation can be interpreted as the absolute volatility of a
process. Thanks to this property, the ∆Bi of the Brownian motion
provides the basic increments of the stochastic integral, replacing the
∆xi of the Rieman-Stieltjes integral.
■ Step 2. The second step consists in defining the stochastic integral for a
class of simple functions called elementary functions. Consider the time
interval [S,T] and any partition of the interval [S,T] in N subintervals:
St≡ 0 < <t 1 ...ti < ...tN ≡T. An elementary function φis a function
defined on the time t and the outcome ωsuch that it assumes a constant
value on the i-th subinterval. Call I[ti+1,ti) the indicator function of the
interval [ti+1,ti). The indicator function of a given set is a function that
assumes value 1 on the points of the set and 0 elsewhere. We can then
write an elementary function φas follows:
φ(t, ω)= ∑εi ()ω [Iti + 1 ,)ti
i
In other words, the constants εi(ω) are random variables and the
function φ(t,ω) is a stochastic process made up of paths that are con-
stant on each i-th interval.
We can now define the stochastic integral, in the sense of Itô, of
elementary functions φ(t,ω) as follows:
T
W = ∫φ(t, ω)dBt()ω = ∑εi ()ω[Bi + 1 ()ω – Bi ()ω]
S i
where B is a Brownian motion.
It is clear from this definition that W is a random variable ω→
W(ω). Note that the Itô integral thus defined for elementary functions