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is the conditional probability of exiting by time t without hitting 0. Thus

is the mean time to exit (0, L] given that X(0) = x without returning to 0. Our goal is to find this time.

The probability of escape by time t without returning to the origin satisfies the differential equation

with the initial condition w(x, 0) = 0 and the boundary conditions w(0, t) = 0 and w(L, t) = 1. We know that w(x) satisfies the time-independent version of Eq. (8.59) with the same boundary conditions (w(0) = 0, w(L) = 1).

Exercise 8.12 (E)

Show that

exp(s2 )ds

exp(s2 )ds

We now derive an equation for h(x) using what I like to call the Kimura Maneuver, since it was popularized by M. Kimura in his work in population genetics (Kimura and Ohta 1971).

We begin by differentiating Eq. (8.59) with respect to time, multiplying by t and integrating:

tWtyx dt X

We then exchange the order of integration and differentiation on the right hand side of Eq. (8.61), and that, for example J0°°tMtxx dt = (S2/Sx2) J0°°twt dt = hxx and which allows us to rewrite Eq. (8.61) as

and we now integrate the right hand side of Eq. (8.62) by parts, keeping in mind that both w(x, 0) = 0 and that the time derivative of w(x, t) goes to 0 as t so that w x

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