[jason1990]

This is the original question from the original thread:

[ QUOTE ]
Can someone show me some sort of equation that will tell me how often (every x number of hands) I should expect to encounter a downswing of 100 BB, 200 BB, etc... Standard deviation of 12 BB/100 hands, win rate 1.3 BB/100 hands.

[/ QUOTE ]
Below is the relevant material from the paper I referenced earlier which provides the answer in the case that we use the Brownian motion with drift model.

Suppose you have a win rate of 1.3 BB/100 and a standard deviation of 12 BB/100. Then you can expect to play 13965 hands before you see a 100 BB downswing. Here's a general formula.

Let N be the number of hands you must play before you see a downswing of 100 BB. To analyze N, we model your bankroll by a Brownian motion with drift, i.e. let X(n) be your net profit in BB after n*100 hands. Then

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

where s=12 and w=1.3. Let M(t) be the maximum value of X on the interval [0,t] and define T to be the first time that M(t)-X(t)=a, where a=100. Then N=100*T.

If we let g=w/s^2, then

E[T] = (e^{2ga} - 1)/(2gw) - a/w
= (e^{2aw/s^2} - 1)*s^2/(2w^2) - a/w

In our example,

E[T] = (e^{260/144} - 1)*144/(2*1.69) - 100/1.3
= 139.65,

so E[N]=13965.

If you want to answer more interesting questions such as this:

[ QUOTE ]
given a probability like 99%, say P, how many hands would have to be played to have a probability P that a 100BB slide will occur somewhere during the play of those hands?

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you need to know the distribution of T, i.e. you need to have a formula for the function F(t)=P(T<t). Unfortunately, I could not find such a formula. But the article does give this: for all b>0,

E[e^{-bT}]
= f(b)e^{-ga}/(f(b)cosh(f(b)a) - g sinh(f(b)a)),

where

f(b) = sqrt{g^2 + 2b/s^2}

and cosh and sinh are, respectively, the hyperbolic cosine and sine. This is the Laplace transform of T and someone with enough computing experience may be able to (at least numerically) invert this to provide us with a formula or a graph of the function F(t). However, even without an inversion, you could use the Laplace transform to compute the moments of T. For example, you could answer this question:

We know the mean number of hands needed to see a 100 BB downswing (in this example) is 13965; that is, E[N]=13965. What is the standard deviation of N?

Anyone want to take a crack at this? Anyone want to derive a general formula for E[T^2]?