The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 268 Wednesday, February 4, 2004 12:51 PM

268 The Mathematics of Financial Modeling and Investment Management

will first provide the basic intuition behind stochastic differential equa-

tions and then proceed to formally define the concept and the properties.

THE INTUITION BEHIND STOCHASTIC DIFFERENTIAL EQUATIONS

Let’s go back to the equation

------ = [ft+

dy

() ε]y

dt

where εis a continuous-time noise process. It would seem reasonable to

define a continuous-time noise process informally as the continuous-

time limit of a zero-mean, IID sequence, that is, a sequence of indepen-

dent and identically distributed variables with zero mean (see Chapter

6). In a discrete time setting, a zero-mean, IID sequence is called a white

noise. We could envisage defining a continuous-time white noise as the

continuous-time limit of a discrete-time white noise. Each path of εis a

function of time ε(⋅,ω). It would therefore seem reasonable to define the

solution of the equation pathwise, as the family of functions that are

solutions of the equations,

------ = [ft+

dy

() ε(t ω , )]y

dt

where each equation corresponds to a specific white noise path.

However this definition would be meaningless in the domain of

ordinary functions. In other words, it would generally not be possible to

find a family of functions y(⋅,ω) that satisfy the above equations for each

white-noise path and that form a reasonable stochastic process.

The key problem is that it is not possible to define a white noise pro-

cess as a zero-mean stationary stochastic process with independent

increments and continuous paths. Such a process does not exist in the

domain of ordinary functions.^1 In discrete time the white noise process

is obtained as the first-difference process of a random walk. Anticipat-

ing concepts that will be developed in Chapter 12 on time series analy-

sis, the random walk is an integrated nonstationary process, while its

first-difference process is a stationary IID sequence.

(^1) It is possible to define a “generalized white noise process” in the domain of “tem-
pered distributions.” See Bernd Oksendal, Stochastic Differential Equations: Third
Edition (Berlin: Springer, 1992).