D

Figure 7.5. To compute the probability u(x) that X(t) crosses C before 0, given X(0) = x we recognize that, in the first dt of the game, holdings will change from x to x + w, where w has a normal distribution with mean 0 and variance dt. We can thus relate u(x) at this time to the average of u(x + d W) at a slightly later time (later by dt).

(which could also be recognized as a colonization probability, using the metaphor of island biogeography) and recognize that the argument of the previous paragraph can be summarized as u(x) = EdW {u(x + dW)}

where EdW means to average over dW. Now let us Taylor expand the right hand side of Eq. (7.13) around x:

and take the average over dW, remembering that it is normally distributed with mean 0 and variance dt:

The last two equations share the same number because I want to emphasize their equivalence. To finish the derivation, we subtract w(x) from both sides, divide by dt and let dt ! 0 to obtain the especially simple differential equation uxx = 0 (7.15)

which we now solve by inspection. The second derivative is 0, so the first derivative of w(x) is a constant wx = k1 and thus w(x) is a linear function of x m(x) = k2 + k1x (7.16)

We will find these constants of integration by thinking about the boundary conditions that w(x) must satisfy.

From Eq. (7.12), we conclude that w(0) must be 0 and w(C) must be 1 since if you start with x = 0 you have hit 0 before C and if you start with C you have hit C before 0. Since w(0) = 0, from Eq. (7.16) we conclude that k2 = 0 and to make w(C) = 1 we must have k1 = 1/C so that w(x) is x w(x)=- (7.17)

What is the typical relationship between your initial holdings and those of a casino? In general C ^ x, so that w(x) ~ 0 - you are almost always guaranteed to go broke before hitting the casino limit.

But, of course, most of us gamble not to break the bank, but to have some fun (and perhaps win a little bit). So we might ask how long it will be before the game ends (i.e., your holdings are either 0 or C). To answer this question, set

T(x) = average amount of time in the game, given X(0) = x (7.18)

We derive an equation for T(x) using logic similar to that which took us to Eq. (7.15). Starting at X(0) = x, after dt the holdings will be x + d W and you will have been in the game for dt time units. Thus we conclude

and we would now proceed as before, Taylor expanding, averaging, dividing by dt and letting dt approach 0. This question is better left as an exercise.

Show that T(x) satisfies the equation — 1 = (1/2) Txx and that the general solution of this equation is T(x) = — x2 + k1x + k2. Then explain why the boundary conditions for the equation are T(0) = T(C) = 0 and use them to evaluate the two constants. Plot and interpret the final result for T(x).

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