Today’s A To Z term was nominated again by @aajohannas. The other compelling nomination was from Vayuputrii, for the Mittag-Leffler function. I was tempted. But I realized I could not think of a clear way to describe why the function was interesting. Or even where it comes from that avoided being a heap of technical terms. There’s no avoiding technical terms in writing about mathematics, but there’s only so much I want to put in at once either. It also makes me realize I don’t understand the Mittag-Leffler function, but it is after all something I haven’t worked much with.
The Mittag-Leffler function looks like it’s one of those things named for several contributors, like Runge-Kutta Integration or Cauchy-Kovalevskaya Theorem or something. Not so here; this was one person, Gösta Mittag-Leffler. His name’s all over the theory of functions. And he was one of the people helping Sofia Kovalevskaya, whom you know from every list of pioneering women in mathematics, secure her professorship.
A martingale is how mathematicians prove you can’t get rich gambling.
Well, that exaggerates. Some people will be lucky, of course. But there’s no strategy that works. The only strategy that works is to rig the game. You can do this openly, by setting rules that give you a slight edge. You usually have to be the house to do this. Or you can do it covertly, using tricks like card-counting (in blackjack) or weighted dice or other tricks. But a fair game? Meaning one not biased towards or against any player? There’s no strategy to guarantee winning that.
We can make this more technical. Martingales arise form the world of stochastic processes. This is an indexed set of random variables. A random variable is some variable with a value that depends on the result of some phenomenon. A tossed coin. Rolled dice. Number of people crossing a particular walkway over a day. Engine temperature. Value of a stock being traded. Whatever. We can’t forecast what the next value will be. But we now the distribution, which values are more likely and which ones are unlikely and which ones impossible.
The field grew out of studying real-world phenomena. Things we could sample and do statistics on. So it’s hard to think of an index that isn’t time, or some proxy for time like “rolls of the dice”. Stochastic processes turn up all over the place. A lot of what we want to know is impossible, or at least impractical, to exactly forecast. Think of the work needed to forecast how many people will cross this particular walk four days from now. But it’s practical to describe what are more and less likely outcomes. What the average number of walk-crossers will be. What the most likely number will be. Whether to expect tomorrow to be a busier or a slower day.
And this is what the martingale is for. Start with a sequence of your random variables. How many people have crossed that street each day since you started studying. What is the expectation value, the best guess, for the next result? Your best guess for how many will cross tomorrow? Keeping in mind your knowledge of how all these past values. That’s an important piece. It’s not a martingale if the history of results isn’t a factor.
Every probability question has to deal with knowledge. Sometimes it’s easy. The probability of a coin coming up tails next toss? That’s one-half. The probability of a coin coming up tails next toss, given that it came up tails last time? That’s still one-half. The probability of a coin coming up tails next toss, given that it came up tails the last 40 tosses? That’s … starting to make you wonder if this is a fair coin. I’d bet tails, but I’d also ask to examine both sides, for a start.
So a martingale is a stochastic process where we can make forecasts about the future. Particularly, the expectation value. The expectation value is the sum of the products of every possible value and how probable they are. In a martingale, the expected value for all time to come is just the current value. So if whatever it was you’re measuring was, say, 40 this time? That’s your expectation for the whole future. Specific values might be above 40, or below 40, but on average, 40 is it.
Put it that way and you’d think, well, how often does that ever happen? Maybe some freak process will give you that, but most stuff?
Well, here’s one. The random walk. Set a value. At each step, it can increase or decrease by some fixed value. It’s as likely to increase as to decrease. This is a martingale. And it turns out a lot of stuff is random walks. Or can be processed into random walks. Even if the original walk is unbalanced — say it’s more likely to increase than decrease. Then we can do a transformation, and find a new random variable based on the original. Then that one is as likely to increase as decrease. That one is a martingale.
It’s not just random walks. Poisson processes are things where the chance of something happening is tiny, but it has lots of chances to happen. So this measures things like how many car accidents happen on this stretch of road each week. Or where a couple plants will grow together into a forest, as opposed to lone trees. How often a store will have too many customers for the cashiers on hand. These processes by themselves aren’t often martingales. But we can use them to make a new stochastic process, and that one is a martingale.
Where this all comes to gambling is in stopping times. This is a random variable that’s based on the stochastic process you started with. Its value at each index represents the probability that the random variable in that has reached some particular value by this index. The language evokes a gambler’s decision: when do you stop? There are two obvious stopping times for any game. One is to stop when you’ve won enough money. The other is to stop when you’ve lost your whole stake.
So there is something interesting about a martingale that has bounds. It will almost certainly hit at least one of those bounds, in a finite time. (“Almost certainly” has a technical meaning. It’s the same thing I mean when I say if you flip a fair coin infinitely many times then “almost certainly” it’ll come up tails at least once. Like, it’s not impossible that it doesn’t. It just won’t happen.) And for the gambler? The boundary of “runs out of money” is a lot closer than “makes the house run out of money”.
Oh, if you just want a little payoff, that’s fine. If you’re happy to walk away from the table with a one percent profit? You can probably do that. You’re closer to that boundary than to the runs-out-of-money one. A ten percent profit? Maybe so. Making an unlimited amount of money, like you’d want to live on your gambling winnings? No, that just doesn’t happen.
This gets controversial when we turn from gambling to the stock market. Or a lot of financial mathematics. Look at the value of a stock over time. I write “stock” for my convenience. It can be anything with a price that’s constantly open for renegotiation. Stocks, bonds, exchange funds, used cars, fish at the market, anything. The price over time looks like it’s random, at least hour-by-hour. So how can you reliably make money if the fluctuations of the price of a stock are random?
Well, if I knew, I’d have smaller student loans outstanding. But martingales seem like they should offer some guidance. Much of modern finance builds on not dealing with a stock price varying. Instead, buy the right to buy the stock at a set price. Or buy the right to sell the stock at a set price. This lets you pay to secure a certain profit, or a worst-possible loss, in case the price reaches some level. And now you see the martingale. Is it likely that the stock will reach a certain price within this set time? How likely? This can, in principle, guide you to a fair price for this right-to-buy.
The mathematical reasoning behind that is fine, so far as I understand it. Trouble arises because pricing correctly means having a good understanding of how likely it is prices will reach different levels. Fortunately, there are few things humans are better at than estimating probabilities. Especially the probabilities of complicated situations, with abstract and remote dangers.
So martingales are an interesting corner of mathematics. They apply to purely abstract problems like random walks. Or to good mathematical physics problems like Brownian motion and the diffusion of particles. And they’re lurking behind the scenes of the finance news. Exciting stuff.
Thanks for reading. This and all the other Fall 2019 A To Z posts should be at this link. Yes, I too am amazed to be halfway done; it feels like I’m barely one-fifth of the way done. For Thursday I hope to publish ‘N’. And I am taking nominations for subjects for the letters O through T, at this link.