e compute the
quantity
where the
are volatilities of the assets (positive real numbers),
is correlation (real number,
),
is an integrable function and
,
are independent standard Brownian motions.
According to the multidimensional version of the backward Kolmogorov's
equation (
Backward equation section
)
the function
is a solution of the
problem
Similarly to the previous section
(
one-dim case
) we introduce
the
process
and
observe
In
addition
and
Therefore, we define the
process
Conversely,
Hence, for the function
may be represented as
with
being the solution of
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