Reverse-time SDE for an unbounded-start Markov diffusion

For $u<v$, the joint density of $(X_{T-v},X_{T-u})$ is $p_{T-v}(x)q(T-v,x;T-u,y)$. Dividing by the strictly positive later marginal $p_{T-u}(y)$ gives the conditional law of $X_{T-v}$ given $X_{T-u}=y$, exactly $K_{u,v}(y,dx)$. Its normalization and reverse Chapman–Kolmogorov identity follow from [F1], so $Y$ is Markov with those transitions.

Choose an auxiliary nonexplosive continuous solution $Z$ of the assumed reverse martingale problem, with initial density $p_T$. First work in the usual augmentation and normalize continuous paths on its initial-measurable null exceptional set. The stopped event-test argument in [F5] preserves the given continuous local martingales in this augmentation. Compact coordinate cutoffs give $Z=Z_0+\int \tilde{b}du+N$ locally in state, with $N$ a continuous local martingale. Coordinate-product cutoffs and [F4] identify $[N{]}_u={\int }_0^u{a}_{T-v}(Z_v)dv$: subtract the two product compensators and use that a continuous finite-variation local martingale is constant. The compact-test integrals make the local drift and trace integrals finite; nonexplosion and continuity exhaust the finite interval, without a bound on $Z_0$. Left-step approximations as in [F6] give a predictable version of ${\sigma }_{T-u}(Z_u)$, and [F5] constructs the auxiliary Brownian Itô representation with the same drift and bracket. If $C=\sup \mathrm{\parallel }{\sigma }^{-1}\mathrm{\parallel }$, then $a\ge C^{-2}I$. Stop when $\mathrm{\mid }Z\mathrm{\mid }$ or the accumulated drift-plus-trace energy reaches $n$; the stops exhaust $T$ and the expected stopped variation and trace are finite. Thus [F7] supplies their occupation absolute continuity. Apply [F2], using only the original classical forward equations for $q,p$ and the actual forward path continuity: it identifies the continuous path law, all marginals and Bayes kernels of $Z$ with those of $Y$. For the SDE conclusion, only this path-law identification is used from [F2]; the compact-test identities for $Y$ are transferred on the usual augmented canonical filtration below.

Expanding the adjoint by [F3] yields diffusion coefficient $a_{T-u}$ and drift ${\tilde{b}}_u=-b_{T-u}+p_{T-u}^{-1}\mathrm{\nabla }\text{\hspace{0.17em}⁣}\cdot (p_{T-u}{a}_{T-u})=-b_{T-u}+\mathrm{\nabla }\text{\hspace{0.17em}⁣}\cdot a_{T-u}+a_{T-u}\mathrm{\nabla }\log p_{T-u}$.

Carry the continuous path law of $Y$ to the canonical space $C([0,T],{\mathbb{R}}^d)$ and use its usual augmented coordinate filtration. The law identification in [F2] transfers the compact-test stopped event identities there, where paths are everywhere continuous; normalize the continuous compensated processes before using [F4]. Pulling these versions back to the usual augmentation of the given reversed filtration preserves the identities. All stopping in this step is in that augmentation. Fix $R$ and stop when $\mathrm{\mid }Y\mathrm{\mid }$ first reaches $R$. Compactly supported cutoffs equal to $x_i$ and $x_i{x}_j$ on the ball of radius $R$ turn the compact-test martingale identities into coordinate and coordinate-product identities. Subtracting the finite-variation drift in each coordinate identity gives continuous local martingales $M^i$. The product formula in [F4] shows that $Y_i{Y}_j-Y_i(0)Y_j(0)-{\int }_0^{\cdot }({\tilde{b}}_i{Y}_j+{\tilde{b}}_j{Y}_i)dr-[M_i,M_j]$ is a local martingale. The coordinate-product test shows that the same expression with ${\int }_0^{\cdot }{a}_{ij}(T-r,Y_r)dr$ in place of $[M_i,M_j]$ is a local martingale. Their difference is a continuous finite-variation local martingale, hence constant by [F4]; it starts at zero. Thus $[M_i,M_j{]}_u={\int }_0^u{a}_{ij}(T-r,Y_r)dr$. Letting $R\uparrow \mathrm{\infty }$ is valid because the assumed solution is nonexplosive.

To handle the Borel coefficients allowed by the SDE data, choose the predictable representative $\widehat{Y}$ obtained as follows: take left-step approximations with coefficients $Y_{kT\mathrm{/}{2}^n}$, set their time-zero value to $Y_0$, and take their pointwise limit where finite and $0$ otherwise. These maps are predictable by [F6] and agree with $Y$ at every time on its common continuity event. The pair $(r,{\widehat{Y}}_r)$ is predictable, so the joint Borel coefficient gives a predictable matrix ${\mathrm{\Sigma }}_r={\sigma }_{T-r}({\widehat{Y}}_r)$; inversion is Borel on invertible matrices, and $\mathrm{\parallel }{\mathrm{\Sigma }}_r^{-1}\mathrm{\parallel }$ is uniformly bounded. Step 3.2 gives ${\int }_0^T\mathrm{\parallel }{\mathrm{\Sigma }}_r{\mathrm{\parallel }}_F^{2}dr=\mathrm{tr}[M{]}_T<\mathrm{\infty }$ almost surely, as well as $[M{]}_u={\int }_0^u{\mathrm{\Sigma }}_r{\mathrm{\Sigma }}_r^{\mathsf{T}}dr$. Apply [F5] first in the augmented filtration to construct $B=\int {\mathrm{\Sigma }}^{-1}dM$ and prove $M=\int \mathrm{\Sigma }dB$. To recover an original-adapted Brownian version, use the original predictable sigma-algebra for its elementary approximants, as in [F5], proof 1.2 and 3.1. The original-adapted compensated coordinates are defined using the progressive left-step representative of $Y$ and its drift integral (with zero assigned when that integral is not finite); on the common full event they equal the normalized coordinates. Choose increasing augmented localizers with failure probabilities at most $2^{-n}$ and original elementary inverse-coefficient approximants with stopped sup-error probabilities at most $2^{-n}$ at tolerance $2^{-n}$. Their unstopped sums against those original coordinates are original-adapted and converge uniformly almost surely to the constructed $B$. Defining the limit at each time when finite and zero otherwise gives an original-adapted version. Its increments are Brownian and independent of the earlier original sigma-algebra, a sub-sigma-algebra of the augmented one. Since $\widehat{Y}=Y$ on a common full event, this is the displayed reverse SDE with its original Borel coefficient. The assumed well-posedness and nonexplosion of the reverse martingale problem identify its law uniquely and complete the theorem.