The Mathematics of Financial Modelingand Investment Management

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