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Anderson's bounded-start reverse-time diffusion theorem

Statement

Assume AC. Fix $t_{0} < T$ , integers $d, r \geq 1$ , and a filtered probability space $(Ω, F, (F_{t})_{t_{0} \leq t \leq T}, P)$ satisfying the usual conditions. Let $W$ be an adapted standard $r$ -dimensional Brownian motion such that $W_{t} - W_{s}$ is independent of $F_{s}$ and has law $N_{r} (0, (t - s) I_{r})$ for every $t_{0} \leq s < t \leq T$ . Let $X_{t_{0}}$ be $F_{t_{0}}$ -measurable, almost surely bounded, and independent of the future Brownian increments $σ (W_{t} - W_{t_{0}} : t_{0} \leq t \leq T)$ . Let $d X_{t} = b (X_{t}, t) d t + σ (X_{t}, t) d W_{t}, X_{t} \in R^{d},$ for $t \in [t_{0}, T]$ . Assume the drift and diffusion path integrals satisfy $\int_{t_{0}}^{T} ∣ b (X_{s}, s) ∣ d s < \infty$ and $\int_{t_{0}}^{T} ∥ σ (X_{s}, s) ∥_{F}^{2} d s < \infty$ almost surely, and that this SDE has a unique nonexplosive strong solution. Put $a = σ σ^{T}$ . Also assume the strong restart property: for each deterministic $s \in [t_{0}, T]$ , the future path $(X_{r})_{s \leq r \leq T}$ is almost surely a measurable function of $X_{s}$ and $(W_{r} - W_{s})_{s \leq r \leq T}$ . Assume the coefficients and densities are smooth enough for the displayed derivatives and Itô identities to hold. Assume the marginal density $p_{t} (x)$ is positive wherever used and is a unique smooth solution of $\partial_{t} p_{t} = L_{t}^{*} p_{t}, L_{t}^{*} r = - \sum_{i} \partial_{i} (b_{i} r) + \frac{1}{2} \sum_{i, j} \partial_{i} \partial_{j} (a_{i j} r) .$ Define the $r$ -vector process $\tilde{W}$ by ${\tilde{W}}_{t_{0}} = 0$ and $d {\tilde{W}}_{t}^{k} = d W_{t}^{k} + c_{k} (X_{t}, t) d t, c_{k} (x, t) = p_{t} (x)^{- 1} \sum_{i} \partial_{i} ⁣ (p_{t} (x) σ_{i k} (x, t)) .$ Assume the augmented process $(X, \tilde{W})$ has a unique smooth joint density solving its augmented forward Kolmogorov equation for $t > t_{0}$ , and for every $t > s \geq t_{0}$ its conditional joint density given ${\tilde{W}}_{s}$ is the unique smooth solution of the corresponding conditional augmented equation; all displayed derivatives and density integrals are valid. Then the following hold.

For $t > s \geq t_{0}$ , $X_{t}$ is independent of ${\tilde{W}}_{t} - {\tilde{W}}_{s}$ , and this increment has law $N_{r} (0, (t - s) I_{r})$ .
Put $U = T - t_{0}$ , $Y_{u} = X_{T - u}$ , and $B_{u} = {\tilde{W}}_{T - u} - {\tilde{W}}_{T}$ for $0 \leq u \leq U$ . With $G_{u} = σ (Y_{v}, B_{v} : 0 \leq v \leq u)$ , $B$ is a standard $r$ -dimensional Brownian motion relative to $(G_{u})$ .
In the backward Itô convention, $Y$ satisfies $d Y_{u} = [- b (Y_{u}, T - u) + (\nabla ⁣ \cdot a) (Y_{u}, T - u) + a (Y_{u}, T - u) \nabla \log p_{T - u} (Y_{u})] d u + σ (Y_{u}, T - u) d B_{u},$ where $(\nabla ⁣ \cdot a)_{i} = \sum_{j} \partial_{j} a_{i j}$ . The score is asserted only where $p_{T - u} > 0$ ; no density or score regularity beyond the stated hypotheses is implied.

Statement references

Anderson, Sections 3–4, printed pp. 317–322: states the bounded-start theorem and gives the augmented-density proof route, including the reverse-noise increment calculation and backward-Itô correction.

Tang and Zhao, Section 2.2, PDF pp. 3–5: gives the score-form reverse SDE with a heuristic derivation, not the augmented-density proof supplied here.

Facts & Assumptions

Given: AC, the strong solution and density hypotheses in the statement, and the augmented process $Z_{t} = (X_{t}, {\tilde{W}}_{t})$ whose diffusion matrix is $Γ (x, t) = (\binom{σ (x, t)}{I_{r}})$ and whose drift is $(b (x, t), c (x, t))$ .

[A1]

AC is required by the multidimensional Itô and quadratic-covariation suppliers used in the backward-integral conversion. The Axiom of Choice

[F1]

The forward SDE is an adapted continuous strong solution on the stated usual filtration. Its strong restart property is an explicit statement hypothesis, and future Brownian increments are independent of the present filtration. Itô SDE data and strong solution $d$ -dimensional Brownian motion

[F2]

The forward marginal density satisfies the displayed adjoint equation, and the augmented density satisfies the forward equation for drift $(b, c)$ and covariance blocks $a$ , $σ$ , $σ^{T}$ , and $I_{r}$ . This is the smooth unique-density hypothesis of the statement; the operator follows from the Itô formula for $Z$ .

[F3]

For a smooth density $R (x, w, t)$ of $Z$ , the augmented forward operator is $- \partial_{i} (b_{i} R) - \partial_{w_{k}} (c_{k} R) + \frac{1}{2} \partial_{i} \partial_{j} (a_{i j} R) + \partial_{i} \partial_{w_{k}} (σ_{i k} R) + \frac{1}{2} Δ_{w} R .$ The cross derivative appears because $d [X_{i}, {\tilde{W}}_{k}]_{t} = σ_{i k} (X_{t}, t) d t$ . Multidimensional Ito formula for Brownian-driven processes Quadratic covariation of Brownian Ito processes

[F4]

For $h > 0$ , $φ_{h} (w) = (2 π h)^{- r / 2} \exp (- ∣ w ∣^{2} / (2 h))$ is a probability density satisfying $\partial_{h} φ_{h} = \frac{1}{2} Δ_{w} φ_{h}$ and converging weakly to a point mass at zero as $h ↓ 0$ ; these identities follow by differentiating the displayed expression and the Gaussian integral.

[F5]

If a conditional density has the factorization $p_{A, B ∣ C} (a, b ∣ c) = p_{A} (a) f (b - c)$ with probability densities, then $A$ is independent of $B - C$ and the latter has density $f$ . A translated conditional density factorizes after subtraction

[F6]

Left-endpoint and right-endpoint sums against a continuous integrand differ in the limit by its quadratic covariation with the continuous semimartingale integrator. For $H_{i k} (t) = σ_{i k} (X_{t}, t)$ , $d [H_{i k}, {\tilde{W}}^{k}]_{t} = \sum_{j} \partial_{j} σ_{i k} (X_{t}, t) σ_{j k} (X_{t}, t) d t,$ by Itô's formula and the Brownian-Itô covariation formula. Multidimensional Ito formula for Brownian-driven processes Quadratic covariation of Brownian Ito processes

Proof

Given: the hypotheses and notation in the Statement.

1.1F3F4A1F2

For $t > t_{0}$ set $R (x, w, t) = p_{t} (x) φ_{t - t_{0}} (w)$ . In [F3], the $x$ -drift and $x$ -diffusion terms combine to $φ_{t - t_{0}} (w) L_{t}^{*} p_{t} (x)$ . The cross term is $\partial_{i} \partial_{w_{k}} (σ_{i k} p_{t} φ) = \partial_{w_{k}} ⁣ (φ \partial_{i} (p_{t} σ_{i k})) = \partial_{w_{k}} (c_{k} p_{t} φ),$ which cancels the $- \partial_{w_{k}} (c_{k} R)$ term by the definition of $c_{k}$ . The remaining $w$ -diffusion term is $p_{t} \partial_{t} φ$ , and $\partial_{t} p_{t} = L_{t}^{*} p_{t}$ ; hence the augmented equation's right side is $\partial_{t} R$ . The initial limit is the law of $X_{t_{0}}$ times $δ_{0} (d w)$ by [F4]. Uniqueness therefore identifies $R$ with the actual joint density of $(X_{t}, {\tilde{W}}_{t})$ .

1.2F6

The definition of $\tilde{W}$ gives the forward-Itô equation $d X_{i} = (b_{i} - \sum_{k} σ_{i k} c_{k}) d t + \sum_{k} σ_{i k} d^{f} {\tilde{W}}_{k} .$ For $H_{i k} = σ_{i k} (X_{t}, t)$ , right-endpoint sums equal left-endpoint sums plus the cross-increment sums defining $[H_{i k}, {\tilde{W}}_{k}]$ . Their limits differ by the bracket in [F6]; consequently $\int_{s}^{t} H_{i k} d^{f} {\tilde{W}}_{k} = \int_{s}^{t} H_{i k} d^{b} {\tilde{W}}_{k} - \int_{s}^{t} \sum_{j} \partial_{j} σ_{i k} (X_{r}, r) σ_{j k} (X_{r}, r) d r,$ where the superscripts denote left-endpoint forward and right-endpoint backward Itô sums. This fixes the sign of the conversion.

2.1step 1.1F2F3F4

Fix $s \geq t_{0}$ and condition on ${\tilde{W}}_{s} = w_{s}$ . Step 1.1 shows that $X_{s}$ has density $p_{s}$ independently of ${\tilde{W}}_{s}$ , so the conditional initial density at time $s$ is $p_{s} (x) δ_{w_{s}} (d w)$ . The candidate $R_{s} (x, w, t) = p_{t} (x) φ_{t - s} (w - w_{s})$ has that initial limit. The calculation in step 1.1, with $φ_{t - s} (w - w_{s})$ in place of $φ_{t - t_{0}} (w)$ , proves that it solves the conditional augmented equation. By the assumed uniqueness, this is the conditional joint density of $(X_{t}, {\tilde{W}}_{t})$ given ${\tilde{W}}_{s} = w_{s}$ .

2.2F6step 1.2

The resulting backward drift in the original time variable is ${\hat{b}}_{i} = b_{i} - \sum_{k} σ_{i k} c_{k} - \sum_{j, k} \partial_{j} σ_{i k} σ_{j k} = b_{i} - \frac{1}{p_{t}} \sum_{j} \partial_{j} (p_{t} a_{i j}) .$ Indeed, substituting $c_{k} = p_{t}^{- 1} \sum_{j} \partial_{j} (p_{t} σ_{j k})$ and applying the product rule expands the last numerator as $\sum_{k} σ_{i k} \partial_{j} (p_{t} σ_{j k}) + p_{t} \sum_{j, k} (\partial_{j} σ_{i k}) σ_{j k}$ .

3.1step 2.1step 1.1F5

For $s > t_{0}$ , multiplying the conditional density in step 2.1 by the density $φ_{s - t_{0}} (w_{s})$ from step 1.1 gives the joint density of $(X_{t}, {\tilde{W}}_{t}, {\tilde{W}}_{s})$ . Apply [F5] with $A = X_{t}$ , $B = {\tilde{W}}_{t}$ , $C = {\tilde{W}}_{s}$ , and $f = φ_{t - s}$ . For $s = t_{0}$ , the same conclusion follows directly from step 2.1 with the deterministic value ${\tilde{W}}_{t_{0}} = 0$ . Thus $X_{t}$ and ${\tilde{W}}_{t} - {\tilde{W}}_{s}$ are independent and the increment has law $N_{r} (0, (t - s) I_{r})$ , proving claim 1.

4.1F1step 3.1given

For $0 \leq u < v \leq U$ , put $s = T - v$ and $t = T - u$ . Then $B_{v} - B_{u} = - ({\tilde{W}}_{t} - {\tilde{W}}_{s})$ , so by claim 1 this increment has law $N_{r} (0, (v - u) I_{r})$ . It is measurable with respect to the forward filtration at time $t$ . By the strong restart assumption in the Statement, the future $X$ path, and hence the future $\tilde{W}$ increments, are measurable functions of $X_{t}$ and the future Brownian increments of $W$ ; those Brownian increments are independent of $F_{t}$ by [F1]. Since ${\tilde{W}}_{t} - {\tilde{W}}_{s}$ is independent of $X_{t}$ by claim 1, it is independent of the pair consisting of $X_{t}$ and all future Brownian increments, and thus of the entire future segment $(X_{r}, {\tilde{W}}_{r} - {\tilde{W}}_{t})_{r \geq t}$ . The variables $Y_{q}$ and $B_{q}$ for $q \leq u$ are measurable with respect to that segment. Therefore $B_{v} - B_{u}$ is independent of $G_{u}$ . The process $B$ is continuous, starts at zero, and has these centered Gaussian increments independent of its past; hence it is standard Brownian motion in $(G_{u})$ .

5.1F6step 2.2∎

Reparameterize by $u = T - t$ and set $B_{u} = {\tilde{W}}_{T - u} - {\tilde{W}}_{T}$ . Right-endpoint sums in forward time satisfy $\int_{T - u}^{T} H_{s} d^{b} {\tilde{W}}_{s} = - \int_{0}^{u} H_{T - r} d B_{r}$ : each reversed increment is the negative of the corresponding right-endpoint forward increment. Therefore $Y_{u} - Y_{0} = - \int_{0}^{u} \hat{b} (Y_{r}, T - r) d r + \int_{0}^{u} σ (Y_{r}, T - r) d B_{r}$ . The drift is $- {\hat{b}}_{i} = - b_{i} + p_{t}^{- 1} \partial_{j} (p_{t} a_{i j}) = - b_{i} + \partial_{j} a_{i j} + \sum_{j} a_{i j} \partial_{j} \log p_{t}$ . With $t = T - u$ , this is exactly the displayed reverse SDE and completes claim 3.

Depends on

Used by

Bounded-start Brownian reversal under Anderson's hypotheses Example

Dependency tree · two levels

72 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

B. D. O. Anderson, Reverse-time diffusion equation models, Sections 3–4 (standard reference, not scraped)
Y. Tang and H. Zhao, Score-based diffusion models via stochastic differential equations: a technical tutorial, Section 2.2 (standard reference, not scraped)