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

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})

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.

ii

Solution

The stochastic-integral identity obtained in part (i), evaluated at

s = t

, is

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

Since

d X_{s} - b (X_{s}) d s = σ (X_{s}) d W_{s}

, this has the requested form with the previsible process

θ_{s} = \partial_{x} U (t - s, X_{s}) .

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