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Fix $t>0$ and define $w(s,y)=u(t-s,|y|)$ for $0\le s\le t$. The assumed smooth extension and the Neumann boundary condition at zero make $w$ a $C^{1,2}$ function. The heat equation givesThe Itô formula therefore makes $w(s,B_s)$ a local martingale. Stop first when $|B|$ leaves a large compact interval. The exponential growth bound and the finite exponential moments of the maximum of Brownian motion on $[0,t]$ give uniform integrability, so localization and the dominated convergence theorem yieldThis is the Feynman-Kac formula for the Neumann heat problem.

The bounded local martingale $\pi$ is a true martingale. Its terminal condition is ${\pi }_T=U(T,v_T,S_T)=g(S_T)$, soThis is also the Feynman-Kac formula for the displayed backward equation.

The pair $(v_t,Y_t)$ is a Markov diffusion with infinitesimal generatorPart d shows that the terminal condition in the equation for $\tilde{U}$ is exactlywhen evaluated at $(v_T,Y_T)$. The Feynman-Kac formula applied to the displayed backward equation therefore givesPart c identifies the right side with ${\pi }_0$.

Fix $t$ and apply Itô formula to $M_s=U(t-s,X_s)$ for $0\le s\le t$. Its drift isby the Kolmogorov backward equation. HenceLocalization makes this a martingale, and boundedness of $U$ permits passage to the limit. Conditioning the identity $M_t=U(0,X_t)$ on $X_0$ givesThis is the required special case of the Feynman-Kac formula, proved directly.