Dazed and Confused

In probability theory, a (discrete-time) martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies the identity

E(X_{n+1}\mid X_1,\dots,X_n)=X_n,

i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the last observation. As is frequent in probability theory, the term was adopted from the language of gambling.

Somewhat more generally, a sequence Y1, Y2, Y3, ... is said to be a martingale with respect to another sequence X1, X2, X3, ... if

E(Y_{n+1}\mid X_1,\dots,X_n)=Y_n, for every n.

A continuous-time martingale is a zero-drift stochastic process. That is, a random variable θ follows a continuous-time martingale with a.s. continuous sample paths iff

d\theta = \sigma \, dz

where z is a Wiener process and the variable σ is a constant or a stochastic process that may depend on θ or other stochastic variables.

Here is an example:

Let's say you are [playing poker. You determine that your probability of winning a hand is 60%, your probability of losing a hand is 40%. You decide that you will risk 10 cents on each hand.

You have plotted a histogram of strings of winners and losers, and you determine that 3 losers in a row is the most probable string. So you DOUBLE your bet on the 4th hand after three losers. I have shown in my trading strategy that it improves my returns by 20% annually.