Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 4 / e

Solution

For

T_{n} = T \land n

d (e^{- λ s} V (X_{s})) = e^{- λ s} σ (X_{s}) V^{'} (X_{s}) d W_{s}

T

. After localization this is a martingale, and boundedness of

V

permits optional stopping. Thus, conditionally on

X_{0}

V (X_{0}) = E [e^{- λ T_{n}} V (X_{T_{n}}) ∣ X_{0}] .

{T < \infty}

, path continuity gives

X_{T} \in {ℓ, r}

and hence

V (X_{T}) = 1

. On

{T = \infty}

, boundedness of

V

makes

e^{- λn} V (X_{n}) \to 0

V (X_{0}) = E [e^{- λ T} ∣ X_{0}]

{ℓ < X_{0} < r}

, with the stated convention when

T = \infty

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