61.11 Hitting This Before That
be a real valued martingale for
= 0. In
assume the conditions of Theorem 61.10.4
are satisfied. Thus there exists
are equiintegrable. With the Doob optional sampling theorem it is
possible to estimate the probability that
before it hits b
where a <
0 < b
There is no loss of generality in assuming T
since if it is less than ∞,
you could just
t > T.
In this case, the equiintegrability of the M
t < T,
and from Theorem 61.9.5
Definition 61.11.1 Let M be a process adapted to the filtration ℱt and let τ be a
stopping time. Then Mτ, called the stopped process is defined by
With this definition, here is a simple lemma.
Lemma 61.11.2 Let M be a right continuous martingale adapted to the normal
filtration ℱt and let τ be a stopping time. Then Mτ is also a martingale adapted to
the filtration ℱt.
Proof:Let s < t. By the Doob optional sampling theorem,
Theorem 61.11.3 Let
be a continuous real valued martingale adapted to the
normal filtration ℱt and let
Then if a < 0 < b the following inequalities hold.
In words, P
is the probability that M
hits b no later than when it hits a.
(Note that if τa
= τb then you would have
Proof: For x ∈ ℝ, define
with the usual convention that inf
. Let a <
0 < b
Then the following claim will be important.
Proof of the claim: Let t > 0. Then by the Doob optional sampling theorem,
Observe the martingale Mτ
must be bounded because it is stopped when M
. There are two cases according to whether τ
has values between
In this case
On the other hand, you could have τ < ∞.
Then in this case
is eventually equal to either
depending on which it hits first. In either case,
the martingale Mτ
is bounded and by the martingale convergence theorem, Theorem
, there exists Mτ
and since the Mτ
are bounded, the dominated convergence theorem implies
This proves the claim.
Also note that
. Now from the claim,
The last term equals 0. By continuity, M
is either equal to
whether τa < τb
or τb < τa
Consider this last term. By the definition,
. Since M
this can only happen if M
has values in
Therefore, this last term satisfies
The following diagram may help in keeping track of the various substitutions.
From 61.11.44, this yields on substituting for P
and so since
Next use 61.11.44 to substitute for P
From 61.11.44, used to substitute for P
Next use 61.11.44 to substitute for the term P
Now the boxed in formulas in 61.11.45 - 61.11.48 yield the conclusion of the theorem.
This proves the theorem.
before it hits b
with other occurrences of similar
expressions being defined similarly.