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. (more…)