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

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- Solution a

Solution

The backward generator of the drift--diffusion is

L = α \partial_{y} + D \partial_{y}^{2}

. The survival probability satisfies the Kolmogorov backward equation

\partial_{t} Q = D \partial_{y}^{2} Q + α \partial_{y} Q, 0 < y < L,

with

Q (0, t) = 0, \partial_{y} Q (L, t) = 0, Q (y, 0) = 1.

The target is absorbing, while reflection gives the Neumann boundary condition at

L

.

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