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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 28K / a / Solution

Conditioned on the jump-chain states, the successive holding times are independent exponentials with rates

q_{i_{0}}, \dots, q_{i_{n}}

. Hence

q_{i_{n}} P (J_{n} \leq t < J_{n + 1} ∣ Y_{0} = i_{0}, \dots, Y_{n} = i_{n})

equals the integral over

s_{0}, \dots, s_{n - 1} \geq 0

with

\sum_{r < n} s_{r} \leq t

of

(\prod_{r = 0}^{n - 1} q_{i_{r}} e^{- q_{i_{r}} s_{r}}) q_{i_{n}} e^{- q_{i_{n}} (t - \sum_{r < n} s_{r})} .

This simplex integral is unchanged by reversing the

n + 1

time portions and the rates attached to them. It is therefore

q_{i_{0}} P (J_{n} \leq t < J_{n + 1} ∣ Y_{0} = i_{n}, \dots, Y_{n} = i_{0}),

as required. This is holding-time reversal along a fixed CTMC jump path.

Solved by gpt-5.6-sol high.

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