e have two instruments with
prices
and
,
default intensities
and
and default times
.
The default times are independent random variables (or conditionally
independent, see (
Copula calculation
for CDO
)). The default times generate filtration
.
The
generate filtration
.
The composition of the two filtrations
and
is
.
Let
be the time of the first default of any asset. We aim to calculate
According to the
transformation
(see the (
Total probability rule
)) it
suffices to compute
as if the
are deterministic functions of time. We proceed according to such recipe and
use the techniques of the section
(
Distribution of Poisson
process
section
),
We perform evaluation of the summation
terms:
and continue the main
calculation,
Suppose that we have three instruments and we are interested in calculation of
the
defined as the time of second
default.
