Forward density duality identifies an auxiliary reverse process

Fix $0\le s<T$ and $x$. Set $h_{s,x}(r,y)=q(s,x;r,y)\mathrm{/}{p}_r(y)$ for $r>s$. The two classical forward equations and the exact adjoint product identity give ${\mathrm{\partial }}_r{h}_{s,x}=D_r{h}_{s,x}$. This is $C^{1,2}$ in the evolving variables, because $p>0$ and both density solutions are classical; it requires no joint coefficient-derivative bound. On every closed interval $0\le u\le T-s-\delta$, [F2] applied after spatial localization gives $H_u^x=h_{s,x}(T-u,Z_u)$ a nonnegative local martingale. Its initial expectation is $\int q(s,x;T,y)dy=1$, since $Z_0$ has density $p_T$. Thus it is a supermartingale before $T-s$ by [F3]. This also applies at $s=0$, using the assumed transition density from forward time zero.

For any Borel $0\le f\le 1$, put $H_u^f=\int p_s(x)f(x)H_u^{x}dx=B_{T-u,s}f(Z_u)$. Integrate each fixed-$x$ supermartingale inequality against $p_s(x)f(x)dx$, first tested on an arbitrary nonnegative bounded earlier-filtration event variable. Tonelli in [F3] proves the mixture supermartingale inequality without a common choice of conditional versions in $x$. The mixture lies in $[0,1]$. The complementary mixture $H^{1-f}$ is also a supermartingale and $H^f+H^{1-f}=1$ by the marginal identity, so both inequalities must be equalities. Hence $H^f$ is a true martingale on $[0,T-s)$, and ${\displaystyle E{B}_{r,s}f(Z_{T-r})=\int p_{s}fdx}$ for $s<r\le T$. Linearity extends this identity to bounded real Borel $f$.

Fix a compact smooth $f$ and put $F_{\delta }(r,y)=\mathrm{\mid }{B}_{r,r-\delta }f(y)-f(y)\mathrm{\mid }$. It is bounded by $2\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$, and $\int p_r{F}_{\delta }(r,\cdot )\le E\mathrm{\mid }f(X_{r-\delta })-f(X_r)\mathrm{\mid }$. Integrating over any compact interior time interval and using continuity of $X$ and [F5] makes this tend to zero. On every compact time-state cylinder $p$ has a positive minimum, so $F_{\delta }\to 0$ in local Lebesgue $L^1$. Apply [F4] to the occupation measure ${\mu }_n$ and the time-reversed function $F_{\delta }(T-u,y)$, first on a bounded state cylinder and then exhausting state balls; boundedness controls the tails of the finite measure ${\mu }_n$. Thus ${\displaystyle E{\int }_a^b{F}_{\delta }(T-u,Z_u)1_{u\le {\tau }_n}du\to 0}$ for every compact interior $[a,b]$. Since $E(T-{\tau }_n)\to 0$ and $F_{\delta }$ is uniformly bounded, [F5] removes stopping. No unknown marginal of $Z$ has been replaced by $p$ in this argument.

In the expectation identity in step 2.1 take $r=s+\delta$, multiply by any continuous time test $\zeta$ compactly supported in $(0,T)$, and integrate over $s$. Step 3.1, with $u=T-s-\delta$, permits replacement of $B_{s+\delta ,s}f(Z_{T-s-\delta })$ by $f(Z_{T-s-\delta })$. Continuity of $Z$ and boundedness of $f$ then let $\delta \downarrow 0$, giving $\int \zeta (s)Ef(Z_{T-s})ds=\int \zeta (s)\int p_{s}fdxds$. Both integrands are continuous in $s$, the second because it equals $Ef(X_s)$, so they agree at every interior time. Compact smooth tests determine finite Borel measures by [F6]. Continuity of both processes extends the equality to $s=0,T$. Thus $\mathrm{Law}(Z_{T-s})=p_{s}dx$ for every $s$.

Fix $0\le s<r\le T$. The conditional martingale identity of step 2.1 gives ${\displaystyle E[B_{s+\delta ,s}f(Z_{T-s-\delta })\mid {\mathcal{F}}_{T-r}^Z]=B_{r,s}f(Z_{T-r})}$. Now step 4.1 supplies the law $p_{s+\delta }$ for the random argument on the left, so ${\displaystyle E\mathrm{\mid }{B}_{s+\delta ,s}f(Z_{T-s-\delta })-f(Z_{T-s-\delta })\mathrm{\mid }\le E\mathrm{\mid }f(X_s)-f(X_{s+\delta })\mathrm{\mid }\to 0}$. Continuity of $Z$ and the $L^1$ contraction in [F5] therefore prove ${\displaystyle E[f(Z_{T-s})\mid {\mathcal{F}}_{T-r}^Z]=B_{r,s}f(Z_{T-r})}$. First use compact smooth $f$, then [F6] extends to bounded Borel tests. The endpoint limit uses the marginal identity only after it has been proved in step 4.1.

Step 5.1 identifies all Bayes transitions of $Z$, and its initial law is $p_T$. The actual reversed process $Y$ has those kernels by [F1]. Iterated conditioning gives the same finite-dimensional laws; [F7] gives the same continuous path law. It remains to transfer compact-test local martingales without completing the natural filtration ${\mathcal{G}}_u=\sigma (Y_r:0\le r\le u)$. We specify raw-progressive compensated-test representatives below; their identities with the usual pathwise expressions hold simultaneously in time on a probability-one event. No assertion about arbitrary exceptional-path modifications of stopped processes is needed.

For either process $V=Y$ or $V=Z$, take its past dyadic step approximants, equal to $V_{kT\mathrm{/}{2}^m}$ on $(kT\mathrm{/}{2}^m,(k+1)T\mathrm{/}{2}^m]$ and to $V_0$ at zero. Define ${\widehat{V}}_u$ as their finite vector limit when it exists, and zero otherwise. Each step process is progressive for the raw natural filtration: on $[0,t]$ its rectangles use only coordinates at times at most $t$. The limit rule preserves this measurability and adaptedness. On the common continuity event, $\widehat{V}=V$ at every time, including the endpoints. For $\varphi \in C_c^{\mathrm{\infty }}$, define $J_u^{\varphi }(V)={\int }_0^u{D}_{T-r}\varphi ({\widehat{V}}_r)dr$ when the positive and negative integrals are both finite, and zero otherwise; parameter integration in [F3] makes this a finite progressive process, as in the drift convention of [F2]. Use the progressive compensated representative $M_u^{\varphi }(V)=\varphi ({\widehat{V}}_u)-\varphi (V_0)-J_u^{\varphi }(V)$.

The following finite-prefix construction supplies literal raw stopping times even on exceptional paths. Put $g(r,x)=\mathrm{\mid }{c}_{T-r}(x)\mathrm{\mid }+{\sum }_{i,j}\mathrm{\mid }{a}_{T-r,ij}(x)\mathrm{\mid }$ and $A_u(V)={\int }_0^{u}g(r,{\widehat{V}}_r)dr\in [0,\mathrm{\infty }]$. It is adapted by parameter integration. Define $K_u(V)$ as the supremum of $0$, all finite $A_q(V)$ with rational $0\le q\le u$, and $A_u(V)$ if finite. This is adapted, using only countably many past coordinates of $A$ and its current value. Pathwise, $\{u:A_u<\mathrm{\infty }\}$ is an initial interval containing zero. If its supremum is $\zeta >0$, then $A$ is continuous before $\zeta$, and Tonelli over disjoint time intervals tending to $\zeta$ gives $A_{\zeta }={\lim }_{u\uparrow \zeta }{A}_u=\mathrm{\ell }$, possibly infinite (the endpoint singleton has zero time measure). If $\mathrm{\ell }<\mathrm{\infty }$, $K$ follows $A$ up to $\zeta$ and then stays equal to $\mathrm{\ell }$; if $\mathrm{\ell }=\mathrm{\infty }$, it tends continuously to infinity there and stays infinite. If $\zeta =0$, every positive-time $A$ is infinite and $K\equiv 0$. The same argument includes $\zeta =T$ and the case of finite $A_T$. Consequently $K\wedge n$ is everywhere continuous and nondecreasing, with initial value zero, for every finite $n$.

Set ${\rho }_n(V)=\inf \{u\in [0,T]:K_u(V)\ge n\}\wedge T$, with empty infimum infinite. For $u<T$, continuity and monotonicity proved in step 8.1 give exactly $\{{\rho }_n(V)\le u\}=\{K_u(V)\ge n\}$; at $T$ the stopping event is the whole space. Thus these are increasing raw-natural-filtration stopping times. For $Z$, its finite drift and diffusion energy imply $A_T(Z)<\mathrm{\infty }$ almost surely: its bracket density is positive semidefinite for almost every time along almost every path, so ${\sum }_{i,j}\mathrm{\mid }{a}_{ij}\mathrm{\mid }\le d\mathrm{tr}a$ there. Equal continuous path laws give $A_T(Y)<\mathrm{\infty }$ almost surely as well. On those continuity and finite-energy events, $K=A$, $A_{u\wedge {\rho }_n}\le n$, and ${\rho }_n=T$ eventually. Bounded derivatives of $\varphi$ give $\mathrm{\mid }{D}_{T-r}\varphi (x)\mathrm{\mid }\le C_{\varphi }g(r,x)$, hence $\mathrm{\mid }{M}_{u\wedge {\rho }_n}^{\varphi }(V)\mathrm{\mid }\le 2\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}+C_{\varphi }n$ almost surely. The stopped values are raw-adapted and measurable: compose the progressive restriction of $M^{\varphi }$ on $[0,u]$ with the measurable map $\omega \mapsto (u\wedge {\rho }_n(\omega ),\omega )$.

In the auxiliary usual filtration, [F2] makes $M^{\varphi }(Z)$ a continuous local martingale up to indistinguishability. Normalize its continuity exceptional set in that initial sigma-algebra before applying [F3]. Its stop at the already constructed ${\rho }_n(Z)$ is bounded almost surely by step 9.1, hence a true martingale. The path integrals, clocks and stopping times above are Borel functions of a continuous coordinate path: parameter integration gives the integrals, countable suprema give $K$, and $\{{\rho }_n\le u\}=\{K_u\ge n\}$ for $u<T$ gives measurability of ${\rho }_n$; evaluation of a continuous path at a measurable time is Borel. The finite-integral convention also makes the stopped compensated functionals Borel. Equal path laws therefore transfer their event-integral martingale identities to every cylinder event $\{(Y_{r_1},\dots ,Y_{r_k})\in H\}$ with $r_i\le s$, $H$ Borel, and $s\le t$. For fixed $s,t,n,\varphi$, the events for which the identity holds form a lambda-system, because the stopped variables are integrable; these cylinders form a pi-system generating the original ${\mathcal{G}}_s$. Thus [F6] extends the identity to every ${\mathcal{G}}_s$ event, without replacing that filtration by the one generated by $\widehat{Y}$. Hence each $M^{\varphi }(Y{)}^{{\rho }_n(Y)}$ is a raw martingale. Exhaustion in step 9.1 proves the local martingale problem in the stated raw natural filtration, using the specified progressive compensated representatives and no additional coefficient or filtration assumptions.