The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 278 Wednesday, February 4, 2004 12:51 PM

278 The Mathematics of Financial Modeling and Investment Management

dX 1 = u 1 dt + v 11 dB 1 + ... + v 1 ddBd

...

dX 1 r = u dt r + vr 1 dB 1 + ... + vrddBd

or, in matrix notation

dX = udt + vdB

After defining the multidimensional Itô process, multidimensional sto-

chastic equations are defined in differential form in matrix notation as

follows:

dX( t ω, ) = u[ tX, 1 ( t ω, ), ..., Xd( t ω, )] dt

+ v[ tX, 1 ( t ω, ), ..., Xd( t ω, )] dB

Consider now the multidimensional map: g(t,x) ≡ [g 1 (t,x), ...,

gd(t,x)], which maps the process X into another process Y = g(t,X). It

can be demonstrated that Y is a multidimensional Itô process whose

components are defined according to the following rules:

∂ gk( tX, ) ,

∑

∂ gk( tX) 1 ∂

2

gk( tX)

dYk = -----------------------dt + -----------------------dXi + --- --------------------------dXidXj

∂ t ∂ Xi 2 ∑ ∂ Xi∂

,

i ij, Xj

dBidBj = 1if i = j, 0if ij≠ , dBidt = dtdBi = 0

SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS

It is possible to determine an explicit solution of stochastic differential

equations in the linear case and in a number of other cases that can be

reduced to linear equations through functional transformations. Let’s

first consider linear stochastic equations of the form:

dXt = [ At() Xt + at()] dt + σ() t dBt , 0 ≤ t < ∞

X 0 = ξ

where B is an r-dimensional Brownian motion independent of the d-

dimensional initial random vector ξ and the (d× d), (d× d), (d× r) matrices

A(t), a(t), σ (t) are nonrandom and time dependent.