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

Table of contents
- Solution i

Solution

Fix

t

and apply Itô formula to

M_{s} = U (t - s, X_{s})

for

0 \leq s \leq t

. Its drift is

- \partial_{t} U (t - s, X_{s}) + b (X_{s}) \partial_{x} U (t - s, X_{s}) + \frac{1}{2} σ (X_{s})^{2} \partial_{xx} U (t - s, X_{s}) = 0

by the Kolmogorov backward equation. Hence

M_{s} = M_{0} + \int_{0}^{s} σ (X_{u}) \partial_{x} U (t - u, X_{u}) d W_{u} .

Localization makes this a martingale, and boundedness of

U

permits passage to the limit. Conditioning the identity

M_{t} = U (0, X_{t})

on

X_{0}

gives

U (t, X_{0}) = E [U (0, X_{t}) ∣ X_{0}] .

This is the required special case of the Feynman-Kac formula, proved directly.

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