n
the formula (
Change of Brownian
motion
) the
and
are columns and the
are
scalar products. We intend to apply the result
(
Change of Brownian motion
) to a
situation of two assets given by one dimensional correlated diffusion
terms.
where the expression
stands for
's
Brownian motion with respect to
taken as numeraire. We assume
that
for some number
.
The increments
are jointly normal. We are seeking a matrix
such
that
where
is a column of independent standard Brownian motions.
Observe that the Brownian motions
given
by
are standard and satisfy the
condition
Hence, it suffices to set
We write
The formula (
Change of Brownian
motion
) takes the
form
Therefore, we write SDE for
,
in the
-measure
as
follows
where the Brownian motions
are
standard under the measure of numeraire
and satisfy the
relationship
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