The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 274 Wednesday, February 4, 2004 12:51 PM

274 The Mathematics of Financial Modeling and Investment Management

where μ ,σ are real constants, and consider the map g(t,x) = log x. In this

case, we can write

∂ g ∂ g 1 ∂^2 g 1

------= 0 , ------ = ---, --------- = ------

∂ t ∂ x x ∂ x^2 x^2

and Itô’s formula yields

1

dYt = dlog Xt = μ – ---σ^2  dt+ σ dBt

2 

STOCHASTIC DIFFERENTIAL EQUATIONS

An Itô process defines a process Z(t,ω ) as the sum of the time integral of

the process a(t,ω ) plus the Itô integral of the process b(t,ω ). Suppose

that two functions φ (t,x), ψ (t,x) that satisfy conditions established

below are given. Given an Itô process X(t,ω ), the two processes φ (t,X),

ψ (t,X) admit respectively a time integral and an Itô integral. It therefore

makes sense to consider the following Itô process:

t t

Zt( ω, ) = ∫ φ[ sXs, ( ω, )] ds+ ∫ψ[ sXs, ( ω, )] dBs

0 0

The term on the right side transforms the process Xinto a new process

Z. We can now ask if there are stochastic processes Xthat are mapped

into themselves such that the following stochastic equation is satisfied:

t t

Xt( ω, ) = ∫ φ[ sXs, ( ω, )] ds+ ∫ψ[ sXs, ( ω, )] dBs

0 0

The answer is positive under appropriate conditions. It is possible

to prove the following theorem of existence and uniqueness. Suppose

that a 1-dimensional Brownian motion Bt is defined on a probability

space (Ωℑ ,, P) equipped with a filtration ℑ tand that Btis adapted to

the filtration ℑ t. Suppose also that the two measurable functions φ (t,x),

ψ (t,x) map [0,T] × R→ Rand that they satisfy the following conditions:

2 2

)

2

φ( tx, ) + ψ( tx, ) ≤ C( 1 + x , t∈ [ 0 , T] , x∈ R