[jason1990]

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I am not very familiar with the body of literature out there, and was hoping someone could point to another situation which is similar to the one we face here.

for example that we assume that the average expected pay off on the bet is a function of time t. Now is there a similar situation, possibly somewhere in economics.

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The "Invariance Principle" or "Functional Central Limit Theorem" tells us that, after a "long time," the graph of your bankroll vs. the number of hands played will look like a Brownian motion. (Actually, you have to subtract the mean for it to look like a Brownian motion, so the actual graph looks like a Brownian motion with drift.) One way to derive the risk of ruin formula is to assume your bankroll is a Brownian motion with drift. The risk of ruin is then given in terms of the time it takes this process to hit the line -b (b is your bankroll). This is called the "hitting time."

The specific model is this: if X(t) is your bankroll after 100*t hands, then

X(t) = s*B(t) + m*t,

where B is Brownian motion, s is your standard deviation and m is your win rate (in BB/100). Written in differential notation, this is

dX = s dB + m dt.

If you want to assume that your winrate and/or standard deviation are (possibly random) functions of time, then you could write

dX = s(t) dB + m(t) dt.

This is what is called a stochastic differential equation. Depending on s and m, it may or may not have an explicit solution in terms of t and Brownian motion. You could analyze the hitting time of this solution to obtain a generalized risk of ruin. Modelling random phenomena with such processes is a common tool in mathematical finance. You can find several references if you just look up the key words Brownian motion, stochastic calculus, and mathematical finance.

However, I think it would largely be a waste of time to use this approach to analyze your poker results. It's interesting stuff and you could learn a lot by studying it, but for poker, I think it is overkill. The i.i.d. assumption about your poker results is a pretty good one, in my opinion. Where it breaks down would be when you move up or down in levels, or you switch to a different site. Your game will probably improve as you play, so that your winrate might go up within a single level at a single site. But I don't think you could observe this change without a very large sample size. And even if the changing winrate is a big factor, I think the most practical way to deal with it is to simply ignore your old results. For example, if you've played 100k hands, you may want to analyze the last 50k and the first 50k separately. Doing something like this would be much better, in my opinion, than trying something as complicated as the above.