The gamblers ruin in a fair game

Many - perhaps all - books on stochastic processes or probability include a section on gambling because, let's face it, what is the point of studying probability and stochastic processes if you can't become a better gambler (see also Dubins and Savage (1976))? The gambling problem also allows us to introduce some ideas that will flow through the rest of this chapter and the next chapter.

Imagine that you are playing a fair game in a casino (we will discuss real casinos, which always have the edge, in the next section) and that your current holdings are X(t) dollars. You are out of the game when X(t) falls to 0 and you break the bank when your holdings X(t) reach the casino holdings C. If you think that this is a purely mathematical problem and are impatient for biology, make the following analogy: X(t) is the size at time t of the population descended from a propagule of size x that reached an island at time t = 0; X(t) = 0 corresponds to extinction of the population and X(t) = C corresponds to successful colonization of the island by the descendants of the propagule. With this interpretation, we have one of the models for island biogeography of MacArthur and Wilson (1967), which will be discussed in the next chapter.

Since the game is fair, we may assume that the change in your holdings are determined by a standard Brownian motion; that is, your holdings at time t and time t + dt are related by

There are many questions that we could ask about your game, but I want to focus here on a single question: given your initial stake X(0) = x, what is the chance that you break the casino before you go broke? One way to answer this question would be through simulation of trajectories satisfying Eq. (7.11). We would then follow the trajectories until X(t) crosses 0 or crosses C and the probability of breaking the casino would be the fraction of trajectories that cross C before they cross 0. The trajectories that we simulate would look like those in Figure 7.4 with a starting value of x rather than 0. This method, while effective, would be hard pressed to give us general intuition and might require considerable computer time in order for us to obtain accurate answers. So, we will seek another method by thinking along sample paths.

In Figure 7.5, I show the t — x plane and the initial value of your holdings X(0) = x. At at time dt later, your holdings will change to x + dW, where dW is normally distributed with mean 0 and variance dt. Suppose that, as in the figure, they have changed to x + w, where we can calculate the probability of dW falling around w from the normal distribution. What happens when you start at this new value of holdings? Either you break the bank or you go broke; that is, things start over exactly as before except with a new level of holdings. But what happens between 0 and dt and after dt are independent of each other because of the properties of Brownian motion. Thus, whatever happens after dt is determined solely by your holdings at dt. And those holdings are normally distributed.

To be more formal about this, let us set

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