Continuous local martingales are a particularly well behaved subset of the class of all local martingales, and the results of the previous two posts become much simpler in this case. First, the continuous local martingale property is always preserved by stochastic integration.
Theorem 1 If X is a continuous local martingale and is X-integrable, then is a continuous local martingale.
Proof: As X is continuous, will also be continuous and, therefore, locally bounded. Then, by preservation of the local martingale property, Y is a local martingale. ⬜
Next, the quadratic variation of a continuous local martingale X provides us with a necessary and sufficient condition for X-integrability.
Theorem 2 Let X be a continuous local martingale. Then, a predictable process is X-integrable if and only if
for all .
Proof: If is X-integrable then the quadratic variation is finite. Conversely, suppose that V is finite at all times. As X and, therefore, [X] are continuous, V will be continuous. So, it is locally bounded and as previously shown, is X-integrable. ⬜
In particular, for a Brownian motion B, a predictable process is B-integrable if and only if, almost surely,
for all . Then, is a continuous local martingale.
Quadratic variations also provide us with information about the sample paths of continuous local martingales.
Theorem 3 Let X be a continuous local martingale. Then,
- X is constant on the same intervals for which [X] is constant.
- X has infinite variation over all intervals on which [X] is non-constant.
Proof: Consider a bounded interval (s,t) for any , and set for k=0,1,…,n. By the definition of quadratic variation, using convergence in probability,
where V is the variation of X over the interval (s,t). By continuity, tends uniformly to zero as n goes to infinity, so and [X] is constant over (s,t) whenever the variation V is finite. This proves the second statement of the theorem, which also implies that [X] is constant on all intervals for which X is constant.
It only remains to show that whenever . Applying this also to the countable set of rational times u in (s,t) will then show that X is constant on this interval whenever [X] is.
The process is a local martingale constant up until s, with quadratic variation for . Then is a stopping time with respect to the right-continuous filtration and, by stopping, is a local martingale with zero quadratic variation . Then, as previously shown, is a martingale and, therefore, . This shows that almost surely. Finally, on the set , we have and, hence, . ⬜
Theorem 3 has the following immediate consequence.
Corollary 4 Any continuous FV local martingale is constant.
Proof: By the second statement of Theorem 3, the quadratic variation [X] is constant. Then, by the first statement, X is constant. ⬜
The quadratic covariation also tells us exactly when X converges at infinity.
Theorem 5 Let X be a continuous local martingale. Then, with probability one, the following both hold.
- exists and is finite whenever .
- and whenever .
Proof: By martingale convergence, with probability one either exists and is finite or and are both infinite. It just remains to be shown that, with probability one, exists if and only if is finite..
Let . Then, is a local martingale with quadratic variation bounded by n. So, and is an -bounded martingale which, therefore, almost surely converges at infinity. In particular, on the set
we have outside of a set of zero probability. Therefore, almost surely exists on
For the converse statement, set . Then, is a local martingale bounded by n and . Hence, is almost surely finite and is finite on the set
outside of a set of zero probability. Therefore, is almost surely finite on the set
The topology of uniform convergence on compacts in probability (ucp convergence) was introduced in a previous post, along with the stronger semimartingale topology. On the space of continuous local martingales, these two topologies are actually equivalent, and can be expressed in terms of the quadratic variation. Recalling that semimartingale convergence implies ucp convergence and that quadratic variation is a continuous map under the semimartingale topology, it is immediate that the first and third statements below follow from the second. However, the other implications are specific to continuous local martingales.
Lemma 6 Let and M be continuous local martingales. Then, as n goes to infinity, the following are equivalent.
- converges ucp to M.
- converges to M in the semimartingale topology.
- in probability, for each .
Proof: As semimartingale convergence implies ucp convergence, the first statement follows immediately from the second. So, suppose that . Write and let be the first time at which . Ucp convergence implies that tends to infinity in probability, so to prove the third statement it is enough to show that tends to zero in probability. By continuity, the stopped process is uniformly bounded by 1, so is a square integrable martingale, and Ito’s isometry gives
as n goes to infinity. The limit here follows from the fact that is bounded by 1 and tends to zero in probability. So, we have shown that tends to zero in the norm and, hence, in probability.
Now suppose that the third statement holds. This immediately gives in probability. Letting be the first time at which and be elementary predictable processes, Ito’s isometry gives
So, in particular, in probability. Finally, as whenever , which has probability one in the limit , this shows that tends to zero in probability and tends to zero in the semimartingale topology. ⬜
Applying the previous result to stochastic integrals with respect to a continuous local martingale gives a particularly strong extension of the dominated convergence theorem in this case. Note that this reduces convergence of the stochastic integral to convergence in probability of Lebesgue-Stieltjes integrals with respect to .
Theorem 7 Let X be a continuous local martingale and , be X-integrable processes. Then, the following are equivalent.
- converges ucp to .
- converges to in the semimartingale topology.
- in probability, for each .
Proof: This follows from applying Lemma 6 to the continuous local martingales and . ⬜
Theorem 7 also provides an alternative route to constructing the stochastic integral with respect to continuous local martingales. Although, in these notes, we first proved that continuous local martingales are semimartingales and used this to imply the existence of the quadratic variation, it is possible to construct the quadratic variation more directly. Once this is done, the space of X-integrable processes can be defined to be the predictable processes such that is almost surely finite for all times t. Define the topology on so that if and only if in probability as for each t, and use ucp convergence for the topology on the integrals . Then, Theorem 7 says that is the unique continuous extension from the elementary integrands to all of .