Assume now that E is a complete separable metric space, and that the
sample paths
of
X are right-continuous with left limits; in particular (21) is
valid. In this
situation all of the previously displayed formulae have versions involving a
conditioning on
, as indicated in various settings by
Kac [33], Ray [55] and Dynkin [18, 19].
Such conditioning can be achieved in great generality using h-processes.
To illustrate, the h-process version of (27), as formulated
in [19], takes a simple form due to cancellation of the
h-factors in the product of Green's kernels.
See also Proposition (5.14) of [51] for a discrete time example,
and [1] for the formula obtained this way for
the mean exit time of a diffusion on an interval conditioned to
exit at a specified boundary point.
The effect of conditioning on is simplest
for the special class of Markov killing times
introduced in the following definition:
More formally, assuming T has been set up as a stopping time
relative to a suitable enlargement 
of the filtration
,
the assumption is that, under 
for every 
, the
process 
is
an -martingale. Equivalently,
for every positive -predictable process Z and every
positive
-measurable function f.
The above definition makes sense, and the obvious analog of (29)
is valid,
if a general PCAF is substituted for 
. For example, the last exit
time from
a subset B of E will be ``killing with rate dA(t)'' (for a suitable
PCAF A)
provided
X is a strong Markov process with quasi-left-continuous sample paths (a
``Hunt
process''). In this context the PCAF A is naturally associated with the
so-called
``equilibrium distribution'' on B; see
[10, 26, 25, 28].
Note that a first passage time into a set B will be of this form
only if
the first passage occurs at the time of a jump of X.
In particular, a predictable Markov killing time T, such as the hitting
time of a
set for a process with continuous paths, will not be of this form.
Proof.
For n = 0 the result is the special case 
in (29):
For general n we proceed by induction, as in the proof of (4). Thus, define
and notice that
. Using
as
an abbreviation of 
, we have
the final equality following from (13) and (29). Thus,
, and (30) follows by induction
on n.
Remark.
Formula (30)
holds also with either 
or or both replaced by general
PCAFs. A formula could also be obtained with
instead of 
,
assuming the existence of a Lévy system for the jumps of
X and that T is a jump time. See e.g. [4, 52].
Example.
The special case of the above proposition for
an independent exponential time, when 
for
all x, is already evident in Kac [33].
Then 
is the
resolvent operator of the semi-group of X.
After cancelling the common factor of 
on both sides,
the result is as follows:
for arbitrary 
,
While the existence of left limits was assumed in the previous proposition, it is easily shown that no such hypothesis is required for (32).
