A Markov diffusion is a continuous-path <Markov process> whose local evolution is described by a drift vector and diffusion matrix. For sufficiently regular coefficients, its <infinitesimal generator> is a second-order differential operator and its conditional expectations satisfy a <Kolmogorov backward equation>.

Markov diffusion

Fix $t$ and apply <Itô formula> to $M_s=U(t-s,X_s)$ for $0\leq s\leq t$. Its drift is
$$
-\partial_tU(t-s,X_s)+b(X_s)\partial_xU(t-s,X_s)
+\frac12\sigma(X_s)^2\partial_{xx}U(t-s,X_s)=0
$$
by the <Kolmogorov backward equation>. Hence
$$
M_s=M_0+\int_0^s\sigma(X_u)\partial_xU(t-u,X_u)dW_u.
$$
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
$$
U(t,X_0)=\mathbb E[U(0,X_t)\mid X_0].
$$
This is the required special case of the <Feynman-Kac formula>, proved directly.

Solution

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

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For a diffusion with <infinitesimal generator> $L$, the Kolmogorov backward equation is $\partial_tu=Lu$. Its solutions propagate terminal observables backward through the transition semigroup.

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