The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 270 Wednesday, February 4, 2004 12:51 PM

270 The Mathematics of Financial Modeling and Investment Management

t t

Zt( ω , ) = ∫ φ( sω , ) sd + ∫ψ( sω , ) dBs ( sω , )

0 0

An Itô process is a process that is the result of the sum of two sum-

mands: the first is an ordinary integral, the second an Itô integral. Itô

processes are stable under smooth maps, that is, any smooth function

of an Itô process is an Itô process that can be determined through the

Itô formula (see Itô processes below).

■ Step 2: Definition of stochastic differential equations. As we have seen,

it is not possible to write a differential equation plus a white-noise term

which admits solutions in the domain of ordinary functions. However

we can meaningfully write an integral stochastic equation of the form

t t

Xt( ω , ) = ∫ φ( sX, ) sd + ∫ψ( sX, ) dBs

0 0

It can be demonstrated that this equation admits solutions in the

sense that, given two functions φ and ψ , there is a stochastic process X

that satisfies the above equation. We stipulate that the above integral

equation can be written in differential form as follows:

dX t ( ω , ) = φ( tX, ) td + ψ( tX, ) dBt

Note that this is a definition; a stochastic differential equation

acquires meaning only through its integral form. In particular, we can-

not divide both terms by dt and rewrite the equation as follows:

dX t ( ω , ) dBt

---------------------- = φ( tX) + ψ( tX) ---------

dt

, ,

td

The above equation would be meaningless because the Brownian

motion is not differentiable. This is the difficulty that precludes writ-

ing stochastic differential equations adding white noise pathwise. The

differential notation of a stochastic differential equation is just a

shorthand for the integral notation.

However we can consider a discrete approximation:

∆ X( t ω , ) = φ *( tX, )∆ t+ ψ *( tX, )∆ Bt