Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 1 / b

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- Solution b

Solution

The mean first-passage time obeys the backward equation

D τ^{''} + α τ^{'} = - 1, τ (0) = 0, τ^{'} (L) = 0.

For

α \neq = 0

, direct integration gives

τ (y) = \frac{D}{α ^{2}} e^{αL / D} (1 - e^{- α y / D}) - \frac{y}{α} .

In particular,

τ (L) = \frac{D}{α ^{2}} (e^{αL / D} - 1) - \frac{L}{α} .

This increases monotonically with drift away from the target, so the constrained optimum is

α_{*} = - \overset{α}{ˉ} .

Expanding the exponential at zero drift yields

τ_{0} = α \to 0 lim τ (L) = \frac{L ^{2}}{2 D} .

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