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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-34/5/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 34 / 5 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

Before

τ

, put

a_{s} = σ (Y_{s}) - σ (X_{s})

and

d_{s} = b_{2} (Y_{s}) - b_{1} (X_{s})

, so

d Z_{s} = a_{s} d B_{s} + d_{s} d s

. For

Z_{s} \geq 0

, the Lipschitz and drift-order hypotheses give

∣ a_{s} ∣ \leq K Z_{s}

and

d_{s} \geq b_{1} (Y_{s}) - b_{1} (X_{s}) \geq - L Z_{s}

. The derivatives of

f_{ϵ} (z) = (z + ϵ)^{- 1}

are

f_{ϵ}^{'} = - (z + ϵ)^{- 2}

and

f_{ϵ}^{''} = 2 (z + ϵ)^{- 3}

. Itô's formula gives

f_{ϵ} (Z_{t \land τ}) = f_{ϵ} (z_{0}) - \int_{0}^{t \land τ} \frac{a _{s}}{( Z _{s} + ϵ ) ^{2}} d B_{s} + \int_{0}^{t \land τ} (- \frac{d _{s}}{( Z _{s} + ϵ ) ^{2}} + \frac{a _{s}^{2}}{( Z _{s} + ϵ ) ^{3}}) d s .

The drift integrand is bounded above by

\frac{L Z _{s}}{( Z _{s} + ϵ ) ^{2}} + \frac{K ^{2} Z _{s}^{2}}{( Z _{s} + ϵ ) ^{3}} \leq \frac{L + K ^{2}}{Z _{s} + ϵ} .

The stochastic integrand has magnitude at most

K Z_{s} / (Z_{s} + ϵ)^{2} \leq K / (4 ϵ)

, so its integral on a finite horizon is a square-integrable martingale with mean zero. Global Lipschitz coefficients give finite second moments of the SDE solutions on finite horizons, which also justify the drift expectation; alternatively localize first and use boundedness of

f_{ϵ}

and Fatou. Since

Z_{t \land τ} \geq 0

, taking expectations and replacing the stopped integral by the larger positive full-time bound yields the reciprocal barrier proof of scalar diffusion comparison:

E f_{ϵ} (Z_{t \land τ}) \leq f_{ϵ} (z_{0}) + (L + K^{2}) \int_{0}^{t} E f_{ϵ} (Z_{s \land τ}) d s .

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