Recall from the previous post that a cadlag adapted process is a local martingale if there is a sequence of stopping times increasing to infinity such that the stopped processes are martingales. Local submartingales and local supermartingales are defined similarly.
An example of a local martingale which is not a martingale is given by the `double-loss’ gambling strategy. Interestingly, in 18th century France, such strategies were known as martingales and is the origin of the mathematical term. Suppose that a gambler is betting sums of money, with even odds, on a simple win/lose game. For example, betting that a coin toss comes up heads. He could bet one dollar on the first toss and, if he loses, double his stake to two dollars for the second toss. If he loses again, then he is down three dollars and doubles the stake again to four dollars. If he keeps on doubling the stake after each loss in this way, then he is always gambling one more dollar than the total losses so far. He only needs to continue in this way until the coin eventually does come up heads, and he walks away with net winnings of one dollar. This therefore describes a fair game where, eventually, the gambler is guaranteed to win.
Of course, this is not an effective strategy in practise. The losses grow exponentially and, if he doesn’t win quickly, the gambler must hit his credit limit in which case he loses everything. All that the strategy achieves is to trade a large probability of winning a dollar against a small chance of losing everything. It does, however, give a simple example of a local martingale which is not a martingale.
The gamblers winnings can be defined by a stochastic process representing his net gain (or loss) just before the n’th toss. Let be a sequence of independent random variables with . Here, represents the outcome of the n’th toss, with 1 referring to a head and -1 referring to a tail. Set and
This is a martingale with respect to its natural filtration, starting at zero and, eventually, ending up equal to one. It can be converted into a local martingale by speeding up the time scale to fit infinitely many tosses into a unit time interval
This is a martingale with respect to its natural filtration on the time interval . Letting then the optional stopping theorem shows that is a uniformly bounded martingale on , continuous at , and constant on . This is therefore a martingale, showing that is a local martingale. However, , so it is not a martingale. Alternatively, an example of a continuous local martingale which is not a martingale can be constructed as follows. Let be a standard Brownian motion and be the first time at which it hits one, which is almost surely finite. Then, by optional stopping, is a martingale starting at 0 and ending up at 1. Rescaling the time index of the Brownian motion,
defines a local martingale with respect to its natural filtration, in a similar way as above. Again, however, , so is not a martingale.
Martingale and submartingale criteria
The first question we might ask is, when is a local martingale actually a martingale?
Theorem 1 A local martingale is a martingale if and only if it is of class (DL).
This result is a simple application of uniform integrability to the limits of for some localizing sequence . However, it also follows from Theorem 4 below applied to both and .
Even though the martingale property can fail, nonnegative local martingales are, at least, supermartingales. In particular, if is one of the examples of local martingales given above, then is nonnegative and, hence, a supermartingale. Consequently, the local martingale examples above are, in fact, submartingales. Furthermore, as they are not proper martingales, Theorem 1 shows that they are examples of submartingales which are not of class (DL).
Lemma 2 A nonnegative local supermartingale such that is integrable is a supermartingale.
Proof: Let be a localizing sequence for (w.r.t. the supermartingale property). Then, for any times and the supermartingale property gives
for all . Letting go to infinity, bounded convergence on the left hand side and Fatou’s lemma on the right gives,
Then, increasing to infinity, monotone convergence gives . In particular, putting gives , so is a supermartingale. ⬜
The class (DL) property also gives a criterion for a nonnegative local submartingale to be a proper submartingale. Considering the examples of submartingales which are not of class (DL) above, we see that the nonnegativity condition is required here.
Lemma 3 A nonnegative local submartingale is a submartingale if and only if it is of class (DL).
Proof: Nonnegative cadlag submartingales are of class (DL), so only the converse statement is required. Suppose that is a class (DL) local submartingale. Then there is a localizing sequence such that for times . As is of class (DL), uniform integrability can be used to take the limit on both sides of this inequality, showing that is integrable and . ⬜
Finally, the following gives a criterion for a general local submartingale to be a proper martingale.
Theorem 4 A local submartingale is a submartingale if and only if is integrable and is of class (DL).
Similarly, a local supermartingale is a supermartingale if and only if is integrable and is of class (DL).
Proof: By applying the result to , only the supermartingale case needs to be proven. If is a cadlag supermartingale then is integrable by definition and, is a nonnegative submartingale and hence of class (DL).
Conversely, suppose that is integrable and is of class (DL). Then, is a nonnegative supermartingale and, by Lemma 2 above, is a supermartingale. Similarly, is a class (DL) local submartingale and, by Lemma 3, is a submartingale. Therefore, is a supermartingale. ⬜
Limits of martingales
One way in which local martingales arise is as limits of local martingales. In general, limits of martingales are not martingales. Consider, for example, any local martingale which is not a proper martingale, and let be a localizing sequence. Then, the martingales converge uniformly on compacts to the non-martingale . So, the local conditions cannot be dropped from the following.
Theorem 5 Let be a sequence of continuous local martingales converging ucp to a limit . Then, is a continuous local martingale.
In general, ucp limits of cadlag martingales need not even be local martingales. However, such limits will indeed be local martingales if a local integrability condition is applied to the jumps of the martingales. In particular, recalling that ucp limits of continuous processes are themselves continuous, Theorem 5 above is an immediate consequence of the following.
Theorem 6 Let be a sequence of local martingales (resp. local submartingales, local supermartingales) converging ucp to a limit . If
is locally integrable then is a local martingale.
Proof: It is enough to prove the submartingale case, as the martingale and supermartingale cases follow fro applying this to .
First, as it is a ucp limit of cadlag adapted processes, will be cadlag and adapted. Passing to a subsequence if necessary, we may suppose that converges to uniformly on compacts. Then,
is cadlag, adapted, and increasing. It has jumps which, by the condition of the theorem, is locally integrable. Therefore, is locally integrable. Let be a localizing sequence, so that is integrable. Then, are local submartingales bounded by and, in particular, are of class (DL). So, they are proper submartingales converging to and, applying bounded convergence to this limit, is a submartingale. Therefore, is a localizing sequence for , showing that it is a local martingale. ⬜