The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 277 Wednesday, February 4, 2004 12:51 PM

Stochastic Differential Equations 277

The above conditions state that the standard Brownian motion is a sto-

chastic process that starts at zero, has continuous paths, and has nor-

mally distributed increments whose variances grow linearly with time.

The next step is to extend the definition of the Itô integral in a

multi-dimensional environment. This is again a straightforward but

cumbersome extension of the 1-dimensional case. Suppose that the fol-

lowing r× d-dimensional matrix is given:

v

v 11 · v 1 d

· · ·

vr 1 · vrd

=

where each entry vij = vij(t,ω ) satisfies the following conditions:

1. vij are B
   d
   ℑ × measurable.


2. vij are ℑ t-adapted.
   t

3. P ∫()vij^2 sd < ∞ for all t ≥ 0 = 1.

0

Then, we define the multidimensional Itô integral

t t v 11 · v 1 d Bd 1

∫ v Bd = ∫ · · · ·

0 0 vr 1 · vrd Bd d

as the r-dimensional column vector whose components are the following

sums of 1-dimensional Itô integrals:

d t

∑∫ vij( sω , ) dBj( sω , )

i = (^10)
Note that the entries of the matrix are functions of time and state:
they form a vector of stochastic processes. Given the previous definition
of Itô integrals, we can now extend the definition of Itô processes to the
multidimensional case. Suppose that the functions u and v satisfy the
conditions established for the one-dimensional case. We can then form a
multidimensional Itô process as the following vector of Itô processes: