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:
- vij are B
d
ℑ × measurable. - 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: