Bankroll Management in Limit Poker and Fractional Brownian Motion

[jason1990]

Consider a typical "winning" small stakes online limit holdem player. Let X(t) denote his bankroll in BBs after t hands. If we assume that X is a Brownian motion with drift, and compute (under that assumption) risk of ruin, confidence intervals for winrate, and many other statistical quantities of interest, then we are led to all of the classical formulas that we know and love. This makes sense, of course. Brownian motion is a process whose increments are stationary, normally distributed, and independent, which are all common assumption made about the results of poker sessions.

Looking at my own graphs, though, I seem to have more swings than I ought to have if that assumption were reasonable. Granted, I am only making an "eyeball" assessment on the number and size of swings. But it seems reasonable to believe that the increments in my bankroll graph are not independent. Things such as my own personal reaction to swings as well as table image can cause correlation. If my opponents use PokerTracker or player notes, then the concept of table image can extend well beyond a single session.

A process with Gaussian increments which are positively correlated, and which become less correlated the farther apart they are, is fractional Brownian with Hurst parameter H>1/2. (See http://en.wikipedia.org/wiki/Fractional_Brownian_motion.)

My questions are these: Suppose I assume X is a fractional Brownian motion with drift. How do the classical formulas change? What is the risk of ruin formula? How about confidence intervals? In particular, if B is fractional Brownian motion, then E|B(t)|^2=t^{2H}. In other words, the standard deviation increases like t^H, which is a faster increase than Brownian motion. This would seem to indicate that confidence intervals are less reliable under this model. Any thoughts?