The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 219 Wednesday, February 4, 2004 12:50 PM

Stochastic Integrals 219

Here is where stochastic integration comes into play: It defines how the

stochastic rate process is transformed into the stochastic account pro-

cess. This is the direct stochastic integration approach.

It is possible to take a different approach. At any instant t, the

instantaneous interest rate and the cumulated bank account have two

probability distributions. We could use a partial differential equation to

describe how the probability distribution of the cumulated bank

account is linked to the interest rate probability distribution.

Similar reasoning applies to stock and derivative price processes. In

continuous-time finance, these processes are defined as stochastic pro-

cesses which are the solution of a stochastic differential equation.

Hence, the importance of stochastic integrals in continuous-time finance

theory should be clear.

Following some remarks on the informal intuition behind stochastic

integrals, this chapter proceeds to define Brownian motions and outlines

the formal mathematical process through which stochastic integrals are

defined. A number of properties of stochastic integrals are then estab-

lished. After introducing stochastic integrals informally, we go on to

define more rigorously the mathematical process for defining stochastic

integrals.

THE INTUITION BEHIND STOCHASTIC INTEGRALS

Let’s first contrast ordinary integration with stochastic integration. A

definite integral

b

A = ∫f x()xd

a

is a number A associated to each function f(x) while an indefinite inte-

gral

x

yx()= ∫ fs()sd

a

is a function y associated to another function f. The integral represents

the cumulation of the infinite terms f(s)ds over the integration interval.

A stochastic integral, that we will denote by