The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 218 Wednesday, February 4, 2004 12:50 PM

218 The Mathematics of Financial Modeling and Investment Management

■ Moving from a deterministic to a stochastic environment does not

necessarily require leaving the realm of standard calculus. In fact, all

the stochastic laws of economics and finance theory could be

expressed as laws that govern the distribution of transition probabili-

ties. We will see an example of this mathematical strategy when we

introduce the Fokker-Planck differential equations (Chapter 20). The

latter are deterministic partial differential equations that govern the

probability distributions of prices. Nevertheless it is often convenient

to represent uncertainty directly through stochastic integration and

stochastic differential equations. This approach is not limited to eco-

nomics and finance theory: it is also used in the domain of the physi-

cal sciences. In economics and finance theory, stochastic differential

equations have the advantage of being intuitive: thinking in terms of

a deterministic path plus an uncertain term is easier than thinking in

terms of abstract probability distributions. There are other reasons

why stochastic calculus is the methodology of choice in economics

and finance but easy intuition plays a key role.

For example, a risk-free bank account, which earns a deterministic

instantaneous interest rate f(t), evolves according to the deterministic law:

y = Aexp( ∫ ft()td )

which is the general solution of the differential equation:

------dy = ft()td

y

The solution of this differential equation tells us how the bank account

cumulates over time.

However if the rate is not deterministic but is subject to volatility—

that is, at any instant the rate is f(t) plus a random disturbance—then

the bank account evolves as a stochastic process. That is to say, the

bank account might follow any of an infinite number of different paths:

each path cumulates the rate f(t) plus the random disturbance. In a sense

that will be made precise in this chapter and in Chapter 10 on stochastic

differential equations, we must solve the following equation:

------dy = ft()dt plus random disturbance

y