Conditioning Degenerate Diffusion Models

Conditioning Degenerate Diffusion Models

arXiv:2609.04090v1 [cs.LG] 3 Sep 2026

Uğur Aydın & Tamer Başar Department of Electrical and Computer Engineering and Coordinated Science Laboratory University of Illinois Urbana Champaign Illinois, IL 61801, USA {uaydin2, basar1}@illinois.edu

Abstract Current conditioned generative models heavily rely on score functions for guidance during training. When the generative model is a diffusion process with a singular diffusion coefficient and the underlying (conditional) densities either do not exist or are not smooth, we use causal optimal transport to define approximate loss functions that identify a minimum-entropy control for guidance under minimal assumptions. Our approach relies on causal optimal transport and its characterization through the predictable representation property of (conditioned) diffusion processes whose associated martingale problem is well posed, à la Üstünel.

1

Introduction

Conditioned diffusion processes appear in physical chemistry Bolhuis et al. (2002), genetics Wang et al. (2011), economics Elerian et al. (2001), and guided generative models Zhang et al. (2023); Domingo i Enrich et al. (2025). In practice, a common feature of these problems is that the conditioned process must be constructed from data generated by an unconditioned process X = (Xt )0≤t≤T together with a terminal condition Baudoin (2002); Tzen & Raginsky (2019); Pidstrigach et al. (2025). Hence, at time t, the available data are the state history and the condition. In particular, consider an n-dimensional stochastic differential equation (SDE) of the form dXt = b(t, Xt ) dt + σ(t, Xt ) dBt ,

X0 = η,

(1)

where B is a d-dimensional Brownian motion, and a terminal condition Y = G(XT ), where G : Rn → Rk and η is potentially a random variable. Degeneracy refers to the case in which σ(t, x) ∈ Rn×d does not have full rank (which can be also extended to the case where the corresponding Malliavin covariance operator is also singular). When the conditional likelihood is sufficiently smooth, the Doob h-transform identifies a controlled equation
dXtu = b(t, Xtu ) + σ(t, Xtu )ut (Xtu , y) dt + σ(t, Xtu ) dBt , X0u ∼ L(η | Y = y), (2) whose state-path law agrees with L(X[0,T ] | Y = y) Pidstrigach et al. (2025); Tzen & Raginsky (2019). If Zty = ht (Xt ; y) is a smooth conditional likelihood, then the Brownian-space shift is ut = σ(t, Xt )⊤ ∇x log ht (Xt ; y), and the added state drift is σu = σσ ⊤ ∇x log h. This shift is the Föllmer drift of the corresponding Brownian realization, and it is characterized by a minimumentropy property Lassalle (2018), which also has connections with causal optimal transport. When σ is not invertible, this problem is considerably more difficult than in the invertible case. First, the filtration F (X) need not coincide with that of the driving (ambient) Brownian motion B. 1

As suggested by the notation, we require the control (ut (Xt ))t to be F(X)-adapted, i.e. a closed-loop control. This is important because adaptiveness of u to the path of X u is crucial for the controlled SDE equation 2 to remain closed-loop; otherwise, its implementation would require information external to F(X) ∨ σ(Y ) generated from equation 1. Furthermore, when G is the identity map, as is the case, for instance, when equation 1 is used as a bridge model, the corresponding conditional law at the terminal time is a Dirac measure and is therefore singular rather than smooth.

2

Related Work and Contributions

2.1

Related Work

In this work, we introduce loss functions that yield a control u such that X and X u , introduced in equation 1 and equation 2, respectively, have the same law. This result relies on the adapted representation property of SDEs satisfying weak uniqueness Üstünel (2019), which was first established in Stroock & Yor (1980); see also Elworthy et al. (2010) for a special case on Riemannian manifolds. Using this property, Üstünel (2019) also characterized the associated causal optimal transport problem. We use these results to derive a closed-loop variant of the so-called Föllmer drift (ut )t under minimal conditions, which has minimum energy among the admissible shifts (that satisfy Girsanov’s theorem). Moreover, through a localization argument, we relax the strict positivity condition imposed in Üstünel (2019) in their treatment of the causal optimal transport problem. With the advancements of generative models, there has been recent interest on the behavior of degenerate diffusion models. The works Dockhorn et al. (2021); Blessing et al. (2025) study bridge models under degeneracy when the densities are sufficiently smooth. A theory for manifoldbased generative models has been developed in De Bortoli (2022); De Bortoli et al. (2022), which are inherently degenerate and satisfy less restrictive smoothness conditions on the corresponding densities compared to their Euclidean variants. In the presence of degeneracy or non-smooth densities, recently Malliavin calculus based approaches have been used Pidstrigach et al. (2025); Mirafzali et al. (2025). However, Malliavin calculus based approaches still require sufficient smoothness and regularity (as in non-degeneracy of the corresponding Malliavin covariance operator) of b and σ. Several different approaches have also appeared Conforti et al. (2025); Blessing et al. (2025). However, all these works either assume that L(Xt |Y = y) is continuously differentiable, or suppose that b and σ are sufficiently smooth/regular.

2.2

Contributions

Our contributions in this paper are summarized as follows. 1. We extend the predictable-representation result for degenerate diffusions in Üstünel (2019) to random initializations. 2. In Section 4, we introduce local regression losses that do not require an ambient density or score. Under local square integrability of the conditional Radon–Nikodym derivative and a state-feedback factorization, the corresponding minimizer is the unique minimum-energy Brownian perturbation. 3. We present this loss function as a solution of a control problem (Subsection 5.1). Our loss functions depend on choosing sufficient increments of the data, say ε. To demonstrate the 2

performance of our loss function under different ε, in Subsection 5.2 we provide a numerical experiment. We also provide an application to an image generation problem in Subsection 5.3.

3

Notation

For a process X, we let Ft0 (X) := σ{Xs : 0 ≤ s ≤ t} and let Ft (X) denote its usual P-augmentation. For another probability measure Q, FtQ (X) denotes the corresponding Q-augmentation. The ambient filtration is denoted by (Gt )t≤T . For a measure Q and a σ-field H, Q|H denotes restriction to H. We write LQ (U ) for the law of U under Q and DKL (Q∥P) for relative entropy. The space RT L2a (F(X), Q; Rd ) consists of F (X)-predictable processes v such that EQ 0 |vt |2 dt < ∞. The notation L2a ([0, r], F(X), Q; Rd ) uses the same definition with the integral restricted to [0, r], and L2a,loc ([0, T ), F(X), Q; Rd ) means that this condition holds on every interval [0, r] with r < T . The symbol A† denotes the Moore–Penrose pseudoinverse and ∥A∥F the Frobenius norm. Given a matrix valued curve a : [0, T ] → Rn×d , we use
e a) := Π(a(t)) := a(t)⊤ a(t)a⊤ (t) † a(t), Π(t,

e σ(·, X· )) = Π(σ(t, Xt )), Pt (X) := Π(t,

i.e., Π(a(t)) is the orthogonal projection onto Ran(a(t)⊤ ) = (ker a(t))⊥ ; hence Pt (X) is an orthogonal projection onto Ran(σ(t, Xt )⊤ ) = (ker σ(t, Xt ))⊥ . All process equalities are understood up to the product measure explicitly specified in the statement.

4

Loss Functions for Degenerate Conditioned Diffusions

In this section, we introduce our main regression formulae and explain how they can be used as population losses. The predictable-representation and minimum-energy arguments are given in Section 6, and the detailed proofs are deferred to the appendix. Assumption 1 (Regularity and well-posedness). On a complete filtered probability space (Ω, F, (Gt )t≤T , P) satisfying the usual conditions, B is a d-dimensional (Gt )-Brownian motion and X0 is G0 -measurable. Equation 1 has a pathwise-continuous strong solution that is unique up to indistinguishability. The maps b : [0, T ] × Rn → Rn and σ : [0, T ] × Rn → Rn×d are bounded and Borel measurable. If µ0 = LP (X0 ), then for µ0 -a.e. x the equation started from x has a weak solution that is unique in law. A regular conditional family (Px )x∈Rn of P given X0 exists. Global Lipschitz continuity in the state variable together with linear growth is a standard sufficient condition for strong well-posedness for SDEs. Boundedness is used below only for simple integrability estimates and can be replaced by corresponding localized moment assumptions. We only use pathwise uniqueness for the minimum entropy property of the adapted shift. Assumption 2 (Regularity of conditional density). Let Py := P( · | Y = y) be a regular conditional probability. For LP (Y )-a.e. y and every t < T , Py |Ft (X) ≪ P|Ft (X) ,

Zty (X) :=

d(Py |Ft (X) ) ∈ L2 (Ft (X), P). d(P|Ft (X) )

When y is fixed, we suppress the superscript and write Zt (X) for Zty (X). The only restrictve condition that is used in this work is the square integrability of the process Z y , which is needed to apply the martingale representation theorem. 3

Remark 1 (Square integrability and entropy). The L2 (P) condition is used to invoke the predictablerepresentation theorem, which is the primary additional restriction in our work. Since for x > 0 it holds that x log x ≤ C(1+x2 ), Assumption 2 also implies that we have finite entropy: EP [Zty log Zty ] < ∞ for every t < T . Assumption 3. There exists a Borel measurable function G such that Y = G(XT ) and solutions of equation 1 are Markov processes. Remark 2. We need Assumpton 3 to have loss functions that rely on (Xt , Y ) at time t rather than (X[0,t] , Y ), where X[0,t] denotes the whole trajectory of the process X up to time t. For the rest of this section, Assumptions 1, 2, and 3 are in force. Theorem 1 (Degenerate Tweedie formula-1). For LP (Y )-a.e. y and every r < T , there is a unique realized process u(·, X· , y) ∈ L2a,loc ([0, T ), F(X), Py ; Rd ) satisfying ut (Xt , y) = lim EP ε↓0

y

 Z t+ε   Z t+ε  1 1 Ps (X) dBs Xt = lim EP Ps (X) dBs Xt , Y = y ε↓0 ε t ε t

(3)

where both limits are in L2 ([0, r] × Ω, dt ⊗ dPy ). Moreover, Pt (X)ut (Xt , y) = ut (Xt , y) dt ⊗ dPy -a.e. There is a canonical weak realization of equation 2 whose state-path law on [0, r] is LP (X[0,r] | Y = Rr y). Among the admissible Brownian perturbations, u uniquely minimizes EQ 0 |vt |2 dt. Remark 3 (Independent simulation). The proof relies on Girsanov theorem, which proves existence of a canonical weak controlled solution with the desired law. If the controlled equation 2 is weakly unique, then every weak solution of that equation has the same law. Weak uniqueness of the original equation alone does not imply weak uniqueness after adding a merely measurable control. Corollary 1 (Degenerate Tweedie formula-2). Equation 3 can also be written as     Bt+ε − Bt Bt+ε − Bt y ut (Xt , y) = lim EP Xt = lim EP Xt , Y = y . ε↓0 ε↓0 ε ε

(4)

A Girsanov shift changes the state drift only through σu. A component of u in ker σ would therefore leave the state law unchanged while increasing its energy. The minimum-energy control must consequently satisfy Pt (X)ut = ut . Corollary 1 is useful in simulation because the raw Brownian increments are already available and no numerical projection is required. If only the state trajectory of X and the condition Y are observed, one can instead learn the induced state-space drift. Theorem 2 (Degenerate Tweedie formula-3). Let ut (Xt , y) be the control obtained from equation 3. Then, for every r < T ,   Py Xt+ε − Xt σ(t, Xt )ut (Xt , y) = lim E − b(t, Xt ) Xt (5) ε↓0 ε
exists in L2 [0, r] × Ω, dt ⊗ dPy . Under the conditioned measure Py , enlarging the information by Y changes the semimartingale decomposition of X. The predictable-representation result below (Theorem 4) identifies the new finite-variation term as σu and rules out an additional orthogonal martingale contribution.

4

Although σu does not identify an arbitrary Brownian control modulo ker σ, it does identify the canonical control. Indeed, σ(t, Xt )† σ(t, Xt )ut = Pt (X)ut = ut , where σ † = σ ⊤ (σσ ⊤ )† . Hence, the state-space loss recovers u through the Moore–Penrose inverse even when σ is noninjective. The next corollary recovers the classical Tweedie formula when it is available. Notice that the preceding recovery of the control does not recover R · the entire Brownian path. In a degenerate model, B cannot generally be reconstructed from 0 σ(s, Xs ) dBs . The missing kernel component is irrelevant to the minimum-energy control and to the controlled state equation 2. Corollary 2. Suppose that Zty = h(t, Xt ; y) with h > 0, and that h admits the Itô martingale differential D E dh(t, Xt ; y) = σ(t, Xt )⊤ ∇x h(t, Xt ; y), dBt ; for example, this holds for a C 1,2 solution of the corresponding backward equation. Then, ut (Xt , y) = σ(t, Xt )⊤ ∇x log h(t, Xt ; y)

dt ⊗ dPy -a.e.

The desired control u can be approximated by the following local regression problem. Proposition 1. Fix r < T and 0 < ε < T − r. The problem inf

v∈L2a (F (X),dt⊗dPy ;Rd )

Py

Z r

E

0

1 vt − ε

Z t+ε

2

Ps (X)dBs dt.

(6)

t

has the unique minimizer given by the product-space conditional expectation that satisfies  Z t+ε  Py 1 uε (t, Xt , y) = E Ps (X) dBs Xt dt ⊗ dPy -a.e. ε t Moreover, u∗ε → u(·, X· , y) in L2 ([0, r] × Ω, dt ⊗ dPy ). Proof. Conditional expectation is the orthogonal projection onto the closed product-space subspace L2a (F(X), dt ⊗ dPy ; Rd ). The convergence is from Theorem 1. Remark 4. Analogous projection statements hold for Corollary 1 and Theorem 2. Replacing the targets Stij in Algorithm 1 leads to the corresponding algorithms. In Algorithm 1 we adapt (Pidstrigach et al., 2025, Algorithm 1) to the degenerate setting under our loss functions. For example, for an overparameterized ReLU parametrization with finite-dimensional separated inputs, vanishing finite-sample optimization error can be justified under the random-initialization, width, data-separation, and learning-rate conditions of Allen-Zhu et al. (2019), which are sufficient to establish convergence of the algorithm considered in Tzen & Raginsky (2019). In particular, Allen-Zhu et al. (2019) establish linear convergence of gradient descent and stochastic gradient descent for ℓ2 regression. When these specific conditions are not imposed, we retain vanishing optimization error as an explicit assumption. We discuss the corresponding implementation details in Appendix G. We emphasize, however, that the regression step equation 8 must in practice be approximated using a neural network.

5

Algorithm 1 Training with equation 6 Require: SDE equation 1, batch size N , current control approximation uθ , uniform grid tk = kh, window ε = qh, and r < T − ε. 1: for i = 1 to N do 2: Sample a path X i and its corresponding Brownian path B i from equation 1. 3: Compute the corresponding condition Y i = G(XTi ). 4: For every tj ≤ r, compute the discrete projected-noise target Stij =

j+q−1
1 X Ptk (X i ) Btik+1 − Btik . qh

(7)

k=j

5:

Calculate the single-path loss X

ℓi (θ) = h

uθtj (Xtij , Y i ) − Stij

2

.

(8)

{j: tj ≤r}

6: end for 7: Calculate the Monte Carlo full-batch loss N

LN (θ) =

1 X ℓi (θ). N i=1

8: Take a gradient step on LN (θ).

5

Applications

5.1

A Stochastic Control Problem

The variational formula of Boué & Dupuis (1998) is interpreted in Tzen & Raginsky (2019) as a stochastic control problem with a quadratic running cost. This control problem approach has been used in the generative model for conditioning in recent works, for instance, see Domingo i Enrich et al. (2025). We next extend this interpretation to the degenerate setting. The statement is informal; Theorem 10 in Appendix D gives the precise version for an arbitrary target endpoint law. Theorem 3 (Informal). Suppose that X0 = x0 is deterministic, and let zry :=

dLP (Xr | Y = y) . dLP (Xr )

Define the endpoint density process and its control by Z t y,r P y Zt := E [zr (Xr ) | Ft (X)] = 1 + ⟨ξsy,r , dBs ⟩, 0

Consider

uy,r t :=

ξty,r Zty,r

 Z r  1 2 y |vt | dt − log zr (Xr ) , V (r, y) := inf E 2 0 (Q,v) Q

where the infimum is taken over admissible Brownian perturbations.

6

on {Zty,r > 0}.

(9)

Then, V (r, y) = 0. The endpoint control uy,r is the unique minimizer in the admissible one-sided Girsanov class, and its canonical measure Qy,r satisfies LQy,r (Xr ) = LP (Xr | Y = y). Among all admissible Brownian perturbations realizing this conditional terminal law, uy,r is the unique minimum-energy perturbation: Z Z 1 Qy,r r y,r 2 1 Q r |vt |2 dt. E |ut | dt = inf E 2 2 (Q,v): 0 0 LQ (Xr )=LP (Xr |Y =y)

5.2

Degenerate Nonlinear Diffusion

We consider the nonlinear diffusion dXt = b(Xt ) dt + σ dBt ,

X0 = (1, 0, 0.4)⊤ ,

(1)

Y = XT ,

(10)

where Bt is a standard three-dimensional Brownian motion and     20x1 (1 − x21 ) 1 1 1 b(x) = −x2 + 0.35x1  , σ = 0 1 2 . (11) −0.8x3 0 2 4   3 3 6 Since rank(σ) = 2 < 3, the instantaneous covariance σσ ⊤ = 3 5 10 is singular. The 6 10 20   5/6 1/3 −1/6 corresponding projection to Ran(σ ⊤ ) is P = σ † σ =  1/3 1/3 1/3  . −1/6 1/3 5/6 For each ε ∈ {0.005, 0.01, 0.1}, we train a separate control using the fixed-ε loss " # Bt+ε − Bt 2 Lε (θ) = E P uθ (t, Xt , Y ) − P . (12) ε The training pool contains 80,000 independent reference trajectories. Each model is trained for 20,000 Adam updates with batch size 2048. At evaluation time, we impose the terminal condition Y = −1 and simulate h i bt = b(X bt ) + σP uθ (t, X bt , −1) dt + σ dBt , dX (13) where P uθ is obtained from equation 12 through a neural network Tzen & Raginsky (2019). Be(1) cause Xt is autonomous, the ground-truth can be computed from the one-dimensional backward Kolmogorov equation. Writing   (1) (1) h(t, x1 ) = p XT = −1 Xt = x1 , (14) the corresponding backward Kolmogorov equation has diffusion term 23 ∂x1 x1 h. The exact Brownianspace bridge control is   ∂x1 log h(t, x1 ) u⋆ (t, x) = σ ⊤ ∇x log h(t, x1 ) = ∂x1 log h(t, x1 ) , (15) ∂x1 log h(t, x1 ) 7

Figure 1: Sample paths from the numerical ground-truth Doob bridge and from the learned controls with ε = 0.005, ε = 0.01, and ε = 0.1. Thin curves represent individual trajectories, the bold black curve is the empirical mean, and the horizontal dashed line denotes the terminal target (1) XT = −1. and the corresponding state-space drift correction is σu⋆ (t, x) = (3, 3, 6)⊤ ∂x1 log h(t, x1 ),

(16)

which we use to compare our empirical results against the ground-truth. We evaluate each method using 15,000 independently simulated trajectories. Sample trajectories obtained from this experiment can be seen on Figure 1.

5.3

Conditioned Image Generation

We next give a proof-of-concept application to image inpainting. In contrast to Algorithm 1, the terminal samples in this experiment are drawn from an empirical data law rather than from the terminal law of the reference SDE. Our implementation of this experiment closely follows the one in Baker et al. (2024). Degenerate models are natural when the variability relevant to the conditioning task is intrinsically lower-dimensional Li & He (2026). For example, in an inpainting problem in which only a localized feature is missing, one can use the accessible directions of the diffusion to represent the variability of the missing region while conditioning on the observed context. In our numerical illustration, we train on the Shirt class of Fashion-MNIST Xiao et al. (2017), condition on the left half of a 28 × 28 image, and generate its missing right half. The forward model is dXt = AS dBt , X0 = x0 , where the deterministic x0 is the empirical training mean, A embeds the accessible latent directions into the image space, and S rescales these directions according to their empirical standard deviations. The latent coordinates comprise the 392 whitened visible pixels and 256 whitened principal components of the hidden half. Before training, each hidden half is projected onto these retained principal components, so the reconstructed terminal target lies in the affine support of the model. Hence, the accessible dimension is 648 < 784, and thus every generated path remains on a fixed affine subspace and no ambient image-space density or score is required.

8

Figure 2: Conditioned image-generation results for eight held-out examples. First column is the ground truth, the second column is the given condition, and the last column is the generated image. The SDE we used for conditioned model is degenerate, cannot be “time-reverted” via the methods used in the existing literature and ∇x log p(Y = y|Xt = x), and σ −1 do not exist. Let νT be the empirical law of the projected terminal images. We first sample a target E ∼ νT and then sample an additive Gaussian bridge from x0 to E. This defines a bridge-mixture law RνT ; it is not the reference law of dXt = AS dBt . Although νT is atomic and singular at T , for every observed condition y and every r < T , the conditional bridge-mixture marginal νry = LRνT (Xr | Y = y) is absolutely continuous with respect to the reference marginal µr . The regression expectation in this experiment is taken under RνT , not under the reference path law P. We use the finite target Bt+εε−Bt , . This choice does not send ε to zero. Since the diffusion is supported on a 648-dimensional affine subspace of the 784-dimensional image space, Xt is singular with respect to the ambient Lebesgue measure and there is no canonical ambient Lebesgue score ∇x log pt (x). Our construction requires neither such a score nor a smooth ambient extension of the conditional likelihood. Since the left half of the terminal image is observed, its corresponding control is available analytically. A small U-Net therefore learns only the unknown hidden component. We train its parameters using AdamW. Its inputs are the current decoded image, the observed left half, the inpainting mask, and time, and its image-shaped output is mapped onto the retained PCA coordinates of the hidden half. At generation time, an exponential moving average of the trained network repeatedly predicts the hidden control. Each Euler–Maruyama step combines the predicted control with a new Gaussian increment in the accessible coordinates and decodes the updated coordinates back into image space. The time grid is refined near the terminal time, producing multiple right-half completions consistent with the supplied left half. Results are shown in Figure 2.

6

Predictable Representation Property and Causal Optimal Transport

The loss functions we derived rely on the predictable representation property of diffusion processes when the corresponding martingale problem is well-posed. To derive those loss functions, we extend the martingale representation result of Üstünel (2019) to a random initialization in the conditioned setting. Related constant-rank results for manifold-valued diffusions appear in Elworthy et al. (2010). However, when the rank is not constant, the associated geometric distributions need not retain the same smoothness; see (Lee, 2013, Theorem 10.34) and (Elworthy et al., 2010, Subsection 9.2). Theorem 4. Let (X, B) be a weak solution of equation 1 on a complete filtered probability space and b, and σ have linear growth, where B is a (Gt )-Brownian motion and X0 is G0 -measurable. Let 9

µ0 = LP (X0 ), and suppose that for µ0 -a.e. x equation 1 with X0 = x has a weak solution that is unique in law. Then, every F ∈ L2 (FT (X), P; R) has a representation Z T F = EP [F | X0 ] +

⟨Hs , dBs ⟩,

(17)

0

for a unique H ∈ L2a (F(X), P; Rd ) satisfying Pt (X)Ht = Ht dt ⊗ dP-a.e. Conversely, if g(X0 ) ∈ RT L2 (σ(X0 ), P) and H has these properties, then g(X0 ) + 0 ⟨Hs , dBs ⟩ belongs to L2 (FT (X), P; R). Proof. See Appendix C for a complete proof. In particular, predictability and square integrability of the integrand is not sufficient to ensure that F is FT (X) measurable. In our next result, we essentially show that the corresponding minimum entropy Brownian shifts therefore needs to be of the norm σu as in equation 2, which generalizes (Üstünel, 2019, Theorem 3) when the diffusion has random initialization setting. Theorem 5. Under the hypotheses of Theorem 4, let Ht := σ(X0 ) ∨ Ft (B), with the usual augmentation, and suppose F ∈ L2 (HT , P; R). Write Z T P

F = E [F | X0 ] +

⟨ξs , dBs ⟩,

(18)

0

where ξ ∈ L2a ((Ht )t≤T , P; Rd ). Then, Z T ⟨Ps (X)EP [ξs |Fs (X)], dBs ⟩.

EP [F | FT (X)] = EP [F | X0 ] + 0

Proof. See Appendix C for a complete proof. The representation in equation 18 is satisfied for any such F . Using this representation, we derive a nonsmooth analogue of the Doob h-transform and connect it to causal optimal transport Lassalle (2012, 2018). In particular, using Theorem 5, we present a method to approximate the corresponding minimum entropy shift. As shown in Corollary 2, this naturally generalizes the Doob h-transform while preserving the desired properties. Theorem 6. Suppose that Assumptions 1–2 hold. Let the locally square-integrable integrand ξ be defined by Z t

⟨ξs , dBs ⟩,

Zt (X) = Z0 (X) +

Ps (X)ξs = ξs ,

0

which exists and is unique by Assumption 2 and Theorem 4. Set ut = ξt /Zt (X) on {Zt (X) > 0} and define it arbitrarily on {Zt (X) = 0}. Then, for every r < T ,  Z t+ε  Py 1 lim E Ps (X) dBs Xt = ut (Xt , y) (19) ε↓0 ε t in L2 ([0, r] × Ω, dt ⊗ dPy ). Proof. See Appendix C for a complete proof. Our next result verifies the minimum-energy property without requiring Z to be strictly positive under dt ⊗ dP, which extends (Üstünel, 2019, Theorem 7) to our setting. In particular, we verify that the limit equation 19 has the minimum entropy property, as desired. 10

Theorem 7. Suppose that Assumptions 1–2 hold, fix r < T , and let ξ be the local integrand from Lemma 2. Set
ut = ξt /Zt (X) on {Zt (X) > 0} and ut = 0 otherwise. Define H0y := DKL Py |F0 (X) ∥P|F0 (X) . Let Ayr be the class of pairs (Q, v) such that Q ≪ P on Gr , d(Q|G0 ) = Z0 (X), d(P|G0 ) Rt Rr v is (Gt )-predictable, BtQ := Bt − 0 vs ds is a (Gt , Q)-Brownian motion, EQ 0 |vt |2 dt < ∞, and Z 1 Q r y |vt |2 dt. DKL (Q∥P) = H0 + E 2 0 For every (Q, v) ∈ Ayr whose corresponding controlled equation
dXt = b(t, Xt ) + σ(t, Xt )vt dt + σ(t, Xt )dBtQ

(20)

has state-path law LP (X[0,r] | Y = y), Z r Q

Py

2

Z r

|vt | dt ≥ E

E

0

|ut |2 dt.

(21)

0

The canonical pair (Qy,r , u), defined by d(Qy,r |Gr )/d(P|Gr ) = Zr (X), belongs to Ayr and attains equality. Equality holds for equation 21 if, and only if, Q = Qy,r and vt = ut dt ⊗ dQy,r -a.e. Proof. See Appendix C for a complete proof. Remark 5. The preceding theorem compares perturbations on the fixed ambient space and therefore does not require uniqueness of the controlled equation. Weak uniqueness of equation 2 is needed only to transfer the canonical law to every independently constructed controlled solution; see Remark 3. Assumption 3 is not needed for this energy comparison; it is imposed in the regression results only to express the path-adapted shift as a function of (t, Xt , y).

7

Conclusion

In this work, we have introduced regression losses for computing minimum-entropy adapted shifts when the diffusion is degenerate and the conditional law may be nonsmooth or singular. The construction uses the predictable representation property of the diffusion and does not require an ambient score or an invertible diffusion coefficient. Under mild conditions, we have demonstrated that the obtained controls satisfy the minimum-entropy in a class of “admissible controls”. Extensions to manifold-valued and infinite-dimensional diffusions are natural directions for future work.

8

Acknowledgments

Research of the authors was supported in part by the AFOSR Grant FA9550-24-1-0152.

11

References Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via overparameterization. In Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 242–252, 2019. Jürgen Amendinger. Martingale representation theorems for initially enlarged filtrations. Stochastic Processes and Their Applications, 89(1):101–116, 2000. Elizabeth L Baker, Gefan Yang, Michael L Severinsen, Christy A Hipsley, and Stefan Sommer. Conditioning non-linear and infinite-dimensional diffusion processes. In Advances in Neural Information Processing Systems, volume 37, pp. 10801–10826, 2024. Fabrice Baudoin. Conditioned stochastic differential equations: theory, examples and application to finance. Stochastic Process. Appl., 100:109–145, 2002. Denis Blessing, Julius Berner, Lorenz Richter, and Gerhard Neumann. Underdamped diffusion bridges with applications to sampling. In Y. Yue, A. Garg, N. Peng, F. Sha, and R. Yu (eds.), International Conference on Learning Representations, volume 2025, pp. 2970–3002, 2025. Peter G Bolhuis, David Chandler, Christoph Dellago, and Phillip L Geissler. Transition path sampling: Throwing ropes over rough mountain passes, in the dark. Annual Review of Physical Chemistry, 53(1):291–318, 2002. Michelle Boué and Paul Dupuis. A variational representation for certain functionals of Brownian motion. The Annals of Probability, 26(4):1641–1659, 1998. Giovanni Conforti, Alain Durmus, and Marta Gentiloni Silveri. KL convergence guarantees for score diffusion models under minimal data assumptions. SIAM Journal on Mathematics of Data Science, 7(1):86–109, 2025. Valentin De Bortoli. Convergence of denoising diffusion models under the manifold hypothesis. Transactions on Machine Learning Research, 2022. ISSN 2835-8856. Valentin De Bortoli, Emile Mathieu, Michael Hutchinson, James Thornton, Yee Whye Teh, and Arnaud Doucet. Riemannian score-based generative modelling. In Advances in Neural Information Processing Systems, volume 35, pp. 2406–2422, 2022. Tim Dockhorn, Arash Vahdat, and Karsten Kreis. Score-based generative modeling with criticallydamped Langevin diffusion. arXiv preprint arXiv:2112.07068, 2021. Carles Domingo i Enrich, Michal Drozdzal, Brian Karrer, and Ricky TQ Chen. Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control. In International Conference on Learning Representations, volume 2025, pp. 53791–53846, 2025. Ola Elerian, Siddhartha Chib, and Neil Shephard. Likelihood inference for discretely observed nonlinear diffusions. Econometrica, 69(4):959–993, 2001. K David Elworthy, Yves Le Jan, and Xue-Mei Li. The Geometry of Filtering. Springer Science & Business Media, 2010. Martin Hairer. Advanced stochastic analysis. Lecture Notes, https://www. hairer. org, 2026.

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Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, second edition, 2013. Nobuyuki Ikeda and Shinzo Watanabe. Stochastic Differential Equations and Diffusion Processes, volume 24. Elsevier, 2014. Peter E. Kloeden and Eckhard Platen. Numerical Solution of Stochastic Differential Equations, volume 23 of Applications of Mathematics (New York). Springer-Verlag, Berlin, 1992. Hui-Hsiung Kuo. Gaussian measures in Banach spaces. In Gaussian Measures in Banach Spaces, pp. 1–109. Springer, 2006. Thomas Kurtz. The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities. Electron. J. Probab., 12, 2007. Rémi Lassalle. Invertibility of adapted perturbations of the identity on abstract Wiener space. Journal of Functional Analysis, 262(6):2734–2776, 2012. Rémi Lassalle. Causal transport plans and their Monge–Kantorovich problems. Stochastic Analysis and Applications, 36(3):452–484, 2018. John M Lee. Smooth Manifolds. Springer, 2013. Tianhong Li and Kaiming He. Back to basics: Let denoising generative models denoise. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 36115– 36125, 2026. Ehsan Mirafzali, Utkarsh Gupta, Patrick Wyrod, Frank Proske, Daniele Venturi, and Razvan Marinescu. Malliavin calculus for score-based diffusion models. arXiv preprint arXiv:2503.16917, 2025. Ashkan Nikeghbali. An essay on the general theory of stochastic processes. Probab. Surv., 3: 345–412, 2006. Jakiw Pidstrigach, Elizabeth Louise Baker, Carles Domingo-Enrich, George Deligiannidis, and Nikolas Nüsken. Conditioning diffusions using Malliavin calculus. In Forty-second International Conference on Machine Learning, 2025. Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer Science & Business Media, 2013. Daniel W Stroock and Marc Yor. On extremal solutions of martingale problems. Annales Scientifiques de l’École Normale Supérieure, 13(1):95–164, 1980. Belinda Tzen and Maxim Raginsky. Theoretical guarantees for sampling and inference in generative models with latent diffusions. In Conference on Learning Theory, pp. 3084–3114. PMLR, 2019. A. S. Üstünel. Martingale representation for degenerate diffusions. J. Funct. Anal., 276(11):3468– 3483, 2019. Jin Wang, Kun Zhang, Li Xu, and Erkang Wang. Quantifying the Waddington landscape and biological paths for development and differentiation. Proceedings of the National Academy of Sciences, 108(20):8257–8262, 2011. 13

Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-MNIST: A novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017. Lvmin Zhang, Anyi Rao, and Maneesh Agrawala. Adding conditional control to text-to-image diffusion models. In 2023 IEEE/CVF International Conference on Computer Vision (ICCV), pp. 3813–3824. IEEE, 2023.

A

A Primer on Stochastic Analysis

We recall the notions of weak and strong well-posedness used in the paper. Definition 1 (Brownian motion). Let (Ω, F, (Gt )t , P) be a complete filtered probability space. A d-dimensional Brownian motion is a (Gt )-adapted process B with continuous paths such that 1. B0 = 0 P-a.s.; 2. for every 0 ≤ s < t, the increment Bt − Bs is independent of Gs ; 3. for every 0 ≤ s < t, Bt − Bs ∼ N (0, (t − s)Id ). Definition 2 (Weak solution). A weak solution of equation 1 consists of a filtered probability space (Ω, F, (Gt )t , P), a (Gt )-Brownian motion B, and a continuous adapted process X with the prescribed initial law such that Z t Z t Xt = X0 + b(s, Xs ) ds + σ(s, Xs ) dBs P-a.s. 0

0

Remark 6. In a weak solution, X need not be adapted to the filtration generated by B alone; the ambient filtration may contain additional randomness. Definition 3 (Weak uniqueness). Equation 1 has weak uniqueness, or uniqueness in law, if any two weak solutions with the same initial law induce the same law for the state process. When needed, joint uniqueness refers to the law of (X, B). Definition 4 (Strong solution). A solution is strong if X is adapted to the usual augmentation of σ(X0 ) ∨ Ft (B); equivalently, it is constructed from the given initial variable and driving Brownian motion without additional randomness. Definition 5 (Pathwise uniqueness). Equation 1 has pathwise uniqueness if any two solutions on the same filtered probability space, driven by the same Brownian motion and with the same initial variable, are indistinguishable. Theorem 8 (Yamada-Watanabe). A weak solution exists and pathwise uniqueness holds if, and only if, a strong solution exists and uniqueness in law holds. The classical statement is for deterministic initial conditions; extensions to more general stochastic equations are discussed in Kurtz (2007) and the references therein. In general, weak uniqueness need not imply pathwise uniqueness or strong existence. A standard example is Tanaka’s equation dXt = sign(Xt )dBt .

14

Another example is Tsirelson’s equation, which instead has a nonanticipative path-dependent drift, and has the following form dXt = b(t, X[0,t] ) dt + dBt . Our martingale representation theorem requires weak existence and uniqueness in law for almost every deterministic initial state x under the conditioned measure P(·|X0 = x); thus, it is applicable for the SDEs above although they do not admit a strong solution. Definition 6. Let (Ft0 )t≥0 be a filtration and let N P denote the collection of all subsets of P-null 0 . The P-usual augmentation of (F 0 ) P sets in F∞ t t≥0 is the filtration (Ft )t≥0 defined by  \ FtP := Fs0 ∨ N P , t ≥ 0. s>t

It is the smallest right-continuous and P-complete filtration containing (Ft0 )t≥0 .

B

A Discussion of the Tweedie Formulae

We first recall the Kunita–Watanabe decomposition, which explains the potential orthogonal martingale component when the diffusion is degenerate. Theorem 9 (Kunita–Watanabe decomposition). Let (Ω, F, (Ft )t≥0 , P) satisfy the usual conditions. Let M = (M 1 , . . . , M d ) be a continuous square-integrable martingale and let N be a square-integrable martingale. Then, there exist a predictable M -integrable process H and a squareintegrable martingale L such that Z t Nt = N0 + ⟨Hs , dMs ⟩ + Lt , 0

with ⟨L, M i ⟩t = 0,

i = 1, . . . , d,

t ≥ 0.

The decomposition is unique up to the usual d⟨M ⟩⊗dP equivalence for H and indistinguishability for L, where ⟨A, B⟩t denotes the quadratic variation between A and B. When (Ft )t is a Brownian filtration, the Brownian martingale-representation theorem of Itô forces L ≡ 0 when M = B. For a degenerate equation, however, Ft (X) can be strictly smaller than the filtration generated by the driving Brownian motion. It is therefore not obvious that the orthogonal term vanishes after conditioning. The martingale representation result for degenerate diffusions shows that the relevant Brownian integrand is unique; its image σ(t, Xt )ut is the added state drift in equation 2, where (ut )t is (Ft (X)) R · t -adapted. Thus, adaptiveness of (ut )t is not sufficient for a stochastic integral of the form 0 ⟨us , dBs ⟩ to be (Ft (X))t -adapted.

C

Proofs of the Theorems in Section 6

We first show that we can choose the projection (Pt (X))t as a Borel function of the coefficient. This choice is independent of any completion of the state filtration and can therefore be used under both P and the conditional measures Px = P( · | X0 = x). This distinction is important, because in the original results of Üstünel (2019), the process (Pt (X))t is chosen via a measurable selection theorem; thus, it is not implementable in practice. 15

C.1

Auxiliary Results

Lemma 1. Let K = (Kt ) be a (Gt )t -predictable process. Let z(·, ·) : [0, T ] × Rn → Rn×d be Borel measurable. Define qt (·) := z(t, Kt (·))⊤ (z(t, Kt (·))z(t, Kt (·))⊤ )† z(t, Kt (·)). Then, q = (qt )t is (Gt )-predictable. Moreover, qt (·) is the orthogonal projection onto Ran(z(t, Kt (·))⊤ ) = (ker z(t, Kt (·)))⊥ . Proof. For a ∈ Rn×d , recall that Π(a) := a⊤ (aa⊤ )† a, where (aa⊤ )† is the Moore–Penrose inverse of aa⊤ (Horn & Johnson, 2013, 7.3P7). By (Horn & Johnson, 2013, 7.3P15),
−1 Π(a) = lim a⊤ aa⊤ + k −1 In a, k→∞

and for each k, the map on the right-hand side is continuous in a. Consequently, Π is Borel measurable. Thus, q = (Π(z(t, Kt )))t is predictable (Revuz & Yor, 2013, Chapter IX, Proposition 1.1). The projection property follows directly from the Moore–Penrose pseudoinverse identities (Horn & Johnson, 2013, 7.3P7). Remark 7. Since (t, ω) 7→ (t, Xt (ω)) is predictable and σ is Borel, (σ(t, Xt ))t is predictable. The projection P· (X) need not be continuous when the rank of σ is not constant, but predictability is sufficient to have well-defined stochastic integrals. We also remark that, we did not assume that (Gt )t is augmented. Second, for completeness, we show that (Zt (X))t is a martingale. Lemma 2. Suppose that Assumptions 1–2 hold. For LP (Y )-a.e. y, the process Zt (X) :=

d(Py |Ft (X) ) , d(P|Ft (X) )

0 ≤ t < T,

is a nonnegative (Ft (X), P)-martingale. Moreover, for every R < T , it admits a continuous version of the form Z t

⟨ξs , dBs ⟩,

Zt (X) = Z0 (X) +

0 ≤ t ≤ R,

0

where Ps (X)ξs = ξs dt ⊗ dP-a.e., and Z0 (X) =

d(Py |F0 (X) ) . d(P|F0 (X) )

Proof. For 0 ≤ s ≤ t < T and A ∈ Fs (X), using the tower property and Assumption 2, we obtain EP [Zt (X)1A ] = Py (A) = EP [Zs (X)1A ] . Hence EP [Zt (X) | Fs (X)] = Zs (X)

P-a.s.,

and thus Z is a nonnegative (Ft (X), P)-martingale. Fix R < T . By Assumption 2, ZR (X) ∈ L2 (FR (X)). Therefore, Theorem 4 gives an F (X)predictable process ξ R ∈ L2a (F(X), P; Rd ) such that Z R ZR (X) = EP [ZR (X) | X0 ] + ⟨ξsR , dBs ⟩. 0

16

By the martingale property, EP [ZR (X) | X0 ] = Z0 (X). The stochastic integral is an F (X)-martingale. Taking conditional expectation with respect to Ft (X) therefore yields Z t Zt (X) = Z0 (X) +

⟨ξsR , dBs ⟩,

0 ≤ t ≤ R.

0

The right-hand side is continuous. Uniqueness of the integrand shows that these representations agree on overlapping intervals, and thus they define a single locally square-integrable process ξ on [0, T ).

C.2

Proof of Theorem 4

Conditional versions of (Üstünel, 2019, Theorems 1 and 2), even under sufficiently regular assumptions, require significant technical care because the projection process (Pt (Xt ))t need not be continuous. In particular, switching between probability measures changes the corresponding completed filtrations and therefore requires care when passing to enlargements of the filtration generated by the diffusion process. In the proof of Theorem 4, for this reason we use predictable σ-field associated with the raw filtration generated by X to obtain the desired predictable representation property of X. The proof is organized into several parts. We start from an ambient probability space (Ω, F, P) such that a solution (X, B) to equation 1 lies on. Then, we show that (X, B) is also a solution to equation 1 under the conditioned measure Px ( · ) = P( · | X0 = x). This requires B to be a Brownian motion under Px too. Proofs of these statements are relatively standard; however, since we were not able to find a source for these statements, we provide complete proofs. Lemma 3. Let (X, B) be defined on (Ω, F, (Gt )t≤T , P), where B is a d-dimensional (Gt , P)-Brownian motion, X is continuous and (Gt )-adapted, and X0 is G0 -measurable. Let µ0 := LP (X0 ), and let (Px )x∈Rn be a regular conditional probability of P given X0 . Define the raw σ-algebra Gt0 := σ(Xs , Bs : 0 ≤ s ≤ t),

(22)

and let (Gtx )t≤T be the Px -usual augmentation of (Gt0 )t≤T . Then, for µ0 -a.e. x, B is a d-dimensional (Gtx , Px )-Brownian motion. Proof. Since X and B are (Gt )-adapted, Gt0 ⊆ Gt for every t ≤ T . Hence, under P, every increment Bt − Bs is independent of Gs0 . For each rational s ∈ [0, T ], choose a countable π-system Cs generating Gs0 . Such a class may be constructed from finite-dimensional cylinder events of (Xq , Bq ), q ∈ Q ∩ [0, s], since X and B have continuous paths Kuo (2006). Fix rational 0 ≤ s < t ≤ T , A ∈ Cs , and θ ∈ Qd . For every bounded Borel function f : Rn → R, the random variable f (X0 )1A is Gs -measurable. Therefore, h i 1 2 EP f (X0 )1A ei⟨θ,Bt −Bs ⟩ = e− 2 |θ| (t−s) EP [f (X0 )1A ] . (23) Disintegrating with respect to X0 yields Z Z h i Px i⟨θ,Bt −Bs ⟩ f (x)E 1A e µ0 (dx) = Rn

Rn

17

1

2

f (x)e− 2 |θ| (t−s) Px (A) µ0 (dx).

(24)

Since this holds for every bounded Borel f , h i 1 2 EPx 1A ei⟨θ,Bt −Bs ⟩ = e− 2 |θ| (t−s) Px (A)

(25)

for µ0 -a.e. x. There are only countably many choices of (s, t, A, θ),

s, t ∈ Q ∩ [0, T ],

A ∈ Cs ,

θ ∈ Qd .

(26)

Consequently, there exists a single µ0 -null set N such that, for every x ∈ / N , all of the preceding identities hold simultaneously. Enlarging N if necessary, disintegration of the P-a.s. events {B0 = 0} and {B has continuous paths} also gives, for every x ∈ / N , Px (B0 = 0) = 1 and B has continuous paths Px -a.s. Fix x ∈ / N . For rational s < t, the monotone class theorem extends the preceding identity from A ∈ Cs to every A ∈ Gs0 . Continuity in θ then extends it from θ ∈ Qd to every θ ∈ Rd . Thus, h i 1 2 EPx 1A ei⟨θ,Bt −Bs ⟩ = Px (A)e− 2 |θ| (t−s) , A ∈ Gs0 . (27) By uniqueness of characteristic functions, Bt − Bs ∼ N (0, (t − s)Id ) under Px , and Bt − Bs is independent of Gs0 . Now let 0 ≤ s < t ≤ T be arbitrary. Choose rational sequences sk ↓ s and tk → t such that sk < tk . If A ∈ Gs0 , then A ∈ Gs0k , and hence i h 1 2 EPx 1A ei⟨θ,Btk −Bsk ⟩ = Px (A)e− 2 |θ| (tk −sk ) . (28) By continuity of B and dominated convergence, letting k → ∞ gives i h 1 2 EPx 1A ei⟨θ,Bt −Bs ⟩ = Px (A)e− 2 |θ| (t−s) .

(29)

Thus Bt − Bs is independent of Gs0 and has law N (0, (t − s)Id ) for every 0 ≤ s < t ≤ T . Finally, these properties are preserved under the Px -usual augmentation of the raw filtration. Therefore, B is a d-dimensional (Gtx , Px )-Brownian motion. Next, we show that a solution of (X, B) of equation 1 under P is also a solution under Px . This essentially amounts to picking predictable approximations of (X, B) under the raw filtration, and showing that these approximations also work under Px Lemma 4. Suppose, in addition to the hypotheses of Lemma 3, that b : [0, T ] × Rn → Rn and σ : [0, T ] × Rn → Rn×d are bounded Borel functions and that Z t Z t Xt = X0 + b(s, Xs ) ds + σ(s, Xs ) dBs , 0 ≤ t ≤ T, (30) 0

0

holds P-a.s. Then, for µ0 -a.e. x, Px (X0 = x) = 1, and, with respect to the filtration (Gtx )t≤T , Z t Z t Xt = x + b(s, Xs ) ds + σ(s, Xs ) dBs , 0

0 ≤ t ≤ T,

Px -a.s.

(31)

0

Consequently, for µ0 -a.e. x, (X, B) is a weak solution of equation 1 with deterministic initial condition X0 = x under Px . 18

Proof. We first identify the initial condition. Since (Px )x is a regular conditional probability of P given X0 , for every bounded Borel function f : Rn → R, EPx [f (X0 )] = f (x) for µ0 -a.e. x. Taking f from a countable determining class for B(Rn ) and removing a single µ0 -null set gives LPx (X0 ) = δx , and hence Px (X0 = x) = 1 for µ0 -a.e. x. It remains to verify that the stochastic integral appearing in the original P-SDE agrees, on almost every fiber, with the Itô integral defined under Px . Since X is continuous and adapted to the raw filtration (Gt0 )t , the process σ = (σ(t, Xt ))t is G 0 -predictable (Revuz & Yor, 2013, Chapter IX, Proposition 1.1). Since σ is bounded, there exists a sequence of bounded elementary G 0 -predictable matrix-valued processes (σ m )m≥1 such that, after passing to a subsequence, Z T ∞ X P ∥σ m (s, Xs ) − σ(s, Xs )∥2F ds < ∞. (32) E 0

m=1

Rt For each m, define Itm := 0 σ m (s, Xs ) dBs . Since σ m is an elementary predictable process, Itm is given by a finite sum of the form X
Itm = Hjm Bt∧tj+1 − Bt∧tj , j

and therefore represents the same random variable whether it is viewed under P or under Px . Let Z t

It :=

σ(s, Xs ) dBs 0

denote the Itô integral under P. By Itô’s isometry and equation 32, Z T ∞ ∞ X X P m 2 P ∥σsm − σs ∥2F ds < ∞ E |It − It | ≤ E m=1

(33)

0

m=1

for every t ≤ T . Disintegrating equation 32 with respect to X0 and using Tonelli’s theorem gives Z T Z X Z T ∞ ∞ X ∥σ m (s, Xs ) − σ(s, Xs )∥2F ds < ∞. ∥σ m (s, Xs ) − σ(s, Xs )∥2F ds µ0 (dx) = EP EPx Rn m=1

0

m=1

0

(34) Hence, outside a µ0 -null set, Z T Px

E

∥σ m (s, Xs ) − σ(s, Xs )∥2F ds −→ 0.

(35)

0

By Lemma 3, after enlarging the exceptional set if necessary, B is a (Gtx , Px )-Brownian motion. Therefore, σ is square-integrable and predictable under Px . Itô’s isometry under Px and equation 35 imply Z t

Itm −→ Itx :=

σs dBs

in L2 (Px )

(36)

0

for every t ≤ T . We now identify Itx with the stochastic integral It appearing in the original P-SDE. Fix t ∈ Q ∩ [0, T ]. Disintegrating equation 33 yields Z X ∞ ∞ X EP |Itm − It |2 < ∞. (37) EPx |Itm − It |2 µ0 (dx) = Rn m=1

m=1

19

Thus, for µ0 -a.e. x, Itm −→ It

in L2 (Px ).

Comparing this with equation 36 and using uniqueness of the L2 (Px ) limit gives Z t σ(s, Xs ) dBs Px -a.s. It =

(38)

0

for µ0 -a.e. x. Since Q ∩ [0, T ] is countable, a single µ0 -null set can be chosen so that equation 38 holds simultaneously for all rational t ∈ [0, T ]. For each rational t ∈ [0, T ], the original SDE identity Z t b(s, Xs ) ds + It Xt = X0 + 0

holds P-a.s. Disintegrating this probability-one event with respect to X0 and using countability of the rational times gives, outside one µ0 -null set, Z t Z t Xt = x + b(s, Xs ) ds + σ(s, Xs ) dBs 0

0

for every rational t ∈ [0, T ], Px -a.s., which follows from the definition of regular conditional probability directly. Rt Rt Finally, X has continuous paths, t 7−→ 0 b(s, Xs ) ds is continuous, and t 7−→ 0 σ(s, Xs ) dBs has a continuous version under Px . Hence the preceding identity extends from rational times to every t ∈ [0, T ]. Since X is adapted to (Gtx )t and, by Lemma 3, B is a (Gtx , Px )-Brownian motion, (X, B) is a weak solution of equation 1 under Px with initial condition x. Proof of Theorem 4. Let (X, B) be the given weak solution and let (Px )x∈Rn be a regular conditional probability of P given X0 : Px ( · ) = P( · | X0 = x), whose existence follows from the running assumptions, see (Ikeda & Watanabe, 2014, Theorem I.3.3). For each x ∈ Rn , let Gt0 := σ(Xs , Bs : 0 ≤ s ≤ t), and let (Gtx )t≤T be its Px -usual augmentation. By Lemma 3, for µ0 -a.e. x, B is a d-dimensional (Gtx , Px )-Brownian motion. Moreover, by Lemma 4, after removing a further µ0 -null set, Px (X0 = x) = 1 and Z t Z t Xt = x + b(s, Xs ) ds + σ(s, Xs ) dBs , 0 ≤ t ≤ T, Px -a.s. 0

0

Therefore, for µ0 -a.e. x, (X, B) is a weak solution of equation 1 with initial condition x on (Ω, F, (Gtx )t≤T , Px ). For a probability measure Q, write F Q (X) for the usual augmentation of the raw state filtration under Q. We argue by orthogonality as in (Üstünel, 2019, Theorem 1), now disintegrating with respect to Px . Let K be the closed subspace of L2 (FTP (X), P; R) generated by random variables of the form Z T g(X0 ) + ⟨Ht , dBt ⟩, 0

where g(X0 ) ∈ L2 (F0P (X)), H is F P (X)-predictable, square-integrable, and Ht = Pt (X)Ht dt ⊗ dPa.e. It suffices to show K⊥ = {0}, where K⊥ is the orthogonal complement of K under the L2 norm. Fix Z ∈ K⊥ . Choose an FT0 (X)-measurable representative equal to Z P-a.s. and continue 20

to denote it by Z. Disintegration transfers this equality to Px -a.s. for µ0 -a.e. x; in particular, Z is FTPx (X)-measurable on those fibers. Since F0P (X) = σ(X0 ) ∨ NP , orthogonality to every g(X0 ) gives EP [Z | X0 ] = 0. Let P 0 (X) denote the predictable σ-field associated with the raw filtration (Ft0 (X))t∈[0,T ] (Revuz & Yor, 2013, Chapter I, Exercise 4.20). Since the continuous path space C([0, T ]; Rn ) is Polish, the raw state filtration (Ft (X))t and its predictable σ-field are countably generated (Nikeghbali, 2006, Proposition 2.19). Hence, we may choose a countable family of bounded, Rd -valued, P 0 (X)measurable simple processes generated by a countable algebra (because the underlying L2 space is separable). We define (J m )m≥1 to be an enumeration of all rational finite linear combinations of this family. Then, (J m )m≥1 itself is dense in
L2 P 0 (X), dt ⊗ Q; Rd for every probability measure Q. For m ≥ 1, set Htm := Pt (X)Jtm . Since X is continuous and adapted to its raw filtration (Ft0 (X))t∈[0,T ] , it is F 0 (X)-predictable. Since (t, x) 7→ σ(t, x) is Borel measurable, (σ(t, Xt ))t is P 0 (X)-measurable. Moreover, as argued in the proof of Lemma 1, the map a 7→ a⊤ (aa⊤ )† a is Borel measurable. Hence (Pt (X))t is P 0 (X)-measurable (and predictable). Therefore, H m is P 0 (X)-measurable, and hence predictable with respect to F P (X). Since J m is bounded and Pt (X) is an orthogonal projection, H m is square-integrable. Furthermore, Pt (X)Htm = Pt (X)2 Jtm = Pt (X)Jtm = Htm ,

dt ⊗ dP-a.e.

For each m, since H m is P 0 (X)-measurable and square-integrable, there exists a sequence of raw elementary predictable processes (H m,k )k≥1 such that Z T |Htm,k − Htm |2 dt −→ 0, EP 0

see (Hairer, 2026, Proposition 4.6). Passing to a subsequence, we may assume that Z T ∞ X P E |Htm,k − Htm |2 dt < ∞. k=1

0

Disintegration and Tonelli’s theorem then show, for all m, that H m,k → H m in R simultaneously m 2 H dB as the L (P) limit of the corresponding raw elementary integrals. Applying the same summable subsequence argument to the integral errors and using Itô’s isometry under both P and Px shows that this fixed version is also the Px -Itô integral for µ0 -a.e. x. Intersecting the resulting countably many full-measure sets makes this choice valid for every m below. Hence, for every Borel set A ⊂ Rn , L2 -orthogonality of Z to K gives    Z T  Z T 0 = EP Z 1{X0 ∈A} ⟨Htm , dBt ⟩ = EP Z ⟨1{X0 ∈A} Htm , dBt ⟩ . L2 (dt⊗dPx ) for µ0 -a.e. x. Choose the P-version of

0

0

Therefore,  Z T   Z T  m Px m E Z ⟨Ht , dBt ⟩ X0 = 0 =⇒ E Z ⟨Ht , dBt ⟩ = 0. P

0

0

21

(39)

Since the family (H m )m≥1 is countable, there exists a single set N ⊂ Rn , with µ0 (N ) = 0, such that the preceding equality holds simultaneously for every m ≥ 1 and every x ∈ / N. Now fix x ∈ / N such that EPx
[Z] = 0, Z ∈ L2 (Px ), and the fixed-initial-condition hypotheses hold. Let H ∈ L2a F Px (X), Px ; Rd satisfy Pt (X)Ht = Ht dt ⊗ dPx -a.e., where (Pt (X))t is F Px (X)adapted by Lemma 1. Since the predictable σ-field of the augmented filtration is the dt ⊗ Px completion of P 0 (X), the process H admits a P 0 (X)-measurable representative. In particular, by density of (J m )m≥1 , there exists a sequence (J mk )k≥1 such that Z T Px |Jtmk − Ht |2 dt −→ 0. E 0

Since Pt (X) is an orthogonal projection and Pt (X)Ht = Ht dt ⊗ dPx -a.e., Z T Z T |Pt (X)Jtmk − Pt (X)Ht |2 dt |Htmk − Ht |2 dt = EPx EPx 0 0 Z T ≤ EPx |Jtmk − Ht |2 dt −→ 0. 0

By Itô’s isometry under Px , Z T 0

⟨Htmk , dBt ⟩ −→

Z T ⟨Ht , dBt ⟩

in L2 (Px ).

0

Since Z ∈ L2 (Px ), by Cauchy-Schwarz and Itô isometry, it follows that Px

E

 Z T   Z T  mk Px Z ⟨Ht , dBt ⟩ = 0. Z ⟨Ht , dBt ⟩ = lim E k→∞

0

(40)

0

RT Thus Z is orthogonal under Px to every stochastic integral 0 ⟨Hs , dBs ⟩ with H ∈ L2a (F Px (X), Px ; Rd ) and Pt (X)Ht = Ht dt ⊗ dPx -a.e. Under Px , X0 = x. The fixed-initial representation theorem (Üstünel, 2019, Theorem 1), together with equation 39–equation 40, therefore gives Z = 0 Px -a.s. for µ0 -a.e. x. Integrating with respect to µ0 , Z P(Z ̸= 0) =

Px (Z ̸= 0) µ0 (dx) = 0.

Thus K⊥ = {0}. It remains to show that closure of K in L2 consist of terms of the form Z T g(X0 ) + ⟨Hs , dBs ⟩, 0

where Pt (X)Ht = Ht dt ⊗ dP-a.e. To show this, we closely follow the  proof of (Üstünel, 2019, Theorem 2). Let Fe := F − EP [F | X0 ], i.e., EP Fe | X0 = 0. By the preceding density argument, there exists a sequence (F m )m≥1 ⊂ L2 (FTP (X), P) such that F m → Fe in L2 (P), and, for every m, Z T Fm = ⟨Htm , dBt ⟩, Htm = Pt (X)Htm dt ⊗ dP-a.e. 0

Since F m → Fe in L2 (P), Itô’s isometry gives Z T lim EP |Htm − Htk |2 dt = m,k→∞

0

22

lim EP |F m − F k |2 = 0.

m,k→∞

Hence (H m )m≥1 converges in L2 (dt ⊗ dP; Rd ) to some predictable process H. Since Pt (X) is an orthogonal projection and Pt (X)Htm = Htm , we also have Pt (X)Ht = Ht dt ⊗ dP-a.e. Indeed, Z T P

Z T

2

P

|Pt (X)Ht − Ht | dt ≤ 4 lim E

E

m→∞

0

|Ht − Htm |2 dt = 0.

0

Applying Itô’s isometry once more, Z T

⟨Htm , dBt ⟩ −→

Z T

in L2 (P).

⟨Ht , dBt ⟩ 0

0

Therefore, Z T ⟨Ht , dBt ⟩

Fe =

P-a.s.,

0

and consequently Z T P

⟨Ht , dBt ⟩,

F = E [F | X0 ] +

Ht = Pt (X)Ht

dt ⊗ dP-a.e.

0

This proves the assertion.

C.3

Proof of Theorem 5

Proof of Theorem 5. Since X0 is G0 -measurable and B is a (Gt )-Brownian motion, X0 ⊥⊥ σ(Bs : 0 ≤ s ≤ T ). Thus, the enlargement of the Brownian filtration by X0 is independent, and the equivalence condition in (Amendinger, 2000, Theorem 4.2) is satisfied. Consequently, equation 18 follows in (Ht )t≤T ; equivalently, one may apply the Brownian martingale representation theorem conditionally on X0 . For the rest of the proof, we follow the projection argument of (Üstünel, 2019, Theorem 3). Let ξb be the L2 (dt⊗dP) predictable projection of ξ onto the closed subspace of F (X)-predictable processes (Revuz & Yor, 2013, Exercise 5.14). Then, Z T P

E

|ξbt |2 dt ≤ EP

Z T

0

|ξt |2 dt < ∞,

0

and, for every square-integrable F(X)-predictable process H, P

Z TD

E

E ξt − ξbt , Ht dt = 0.

(41)

0

Define Kt := Pt (X)ξbt . By Lemma 1, Pt (X) is F (X)-predictable. Hence K is an F(X)predictable process. Moreover, since Pt (X) is an orthogonal projection with respect to the Euclidean metric, we have |Kt | ≤ |ξbt | dt ⊗ P-a.e., and therefore K is square-integrable. Also, Pt (X)Kt = Pt (X)2 ξbt = Kt Set

dt ⊗ dP-a.e.

Z T Fe := EP [F | X0 ] +

⟨Kt , dBt ⟩. 0

23

(42)

By Theorem 4, since K is F(X)-predictable, square-integrable, and equation 42 holds, we have Fe ∈ L2 (FT (X), P). We claim that Fe = EP [F | FT (X)]. It is therefore enough to prove that for every Z ∈ L2 (FT (X), P; R).

EP [(F − Fe)Z] = 0

Fix such a Z. By Theorem 4, there exists a square-integrable F(X)-predictable process H satisfying Pt (X)Ht = Ht dt ⊗ dP-a.e. such that Z T ⟨Ht , dBt ⟩. (43) Z = EP [Z | X0 ] + 0

On the other hand, by the initially enlarged-filtration Brownian martingale representation established above, Z T Z T ⟨ξt − Kt , dBt ⟩. ⟨ξt , dBt ⟩ =⇒ F − Fe = F = EP [F | X0 ] + 0

0

In particular, EP [F − Fe | X0 ] = 0 P-a.e. Using equation 43, we therefore obtain h i EP [(F − Fe)Z] = EP (F − Fe)EP [Z | X0 ]   Z T Z T P ⟨Ht , dBt ⟩ . ⟨ξt − Kt , dBt ⟩ +E 0

0

The first term is zero by conditioning on X0 . By Itô’s isometry, we have Z T  Z T  Z T ⟨ξt − Kt , Ht ⟩dt. EP ⟨ξt − Kt , dBt ⟩ ⟨Ht , dBt ⟩ = EP 0

0

0

Since Kt = Pt (X)ξbt , the symmetry of the orthogonal projection Pt (X) and the invariance of Ht under Pt (X) yield D E ⟨Kt , Ht ⟩ = Pt (X)ξbt , Ht D E = ξbt , Pt (X)Ht = ⟨ξbt , Ht ⟩. Consequently, Z T P

E [(F − Fe)Z] = EP

⟨ξt − ξbt , Ht ⟩dt. 0

Since H is F (X)-predictable, the defining orthogonality property of equation 41 gives EP [(F − Fe)Z] = 0. Since this holds for every Z ∈ L2 (FT (X)) and Fe ∈ L2 (FT (X)), the Hilbert-space characterization of conditional expectation implies Fe = EP [F | FT (X)] P-a.s. Therefore, Z TD E EP [F | FT (X)] = EP [F | X0 ] + Pt (X)ξbt , dBt . 0

This proves the result. 24

C.4

Proof of Theorem 6

Proof of Theorem 6. Fix r < T and choose R such that r < R < T . By Lemma 2, on [0, R], Z t ⟨ξs , dBs ⟩,

Zt (X) = Z0 (X) +

Ps (X)ξs = ξs .

0

Set

ξt Zt (X)

ut :=

on {Zt (X) > 0},

and set ut = 0 on {Zt (X) = 0}. This set is dt⊗dPy -null because Py (Zt (X) = 0) = EP [Zt (X)1{Zt (X)=0} ] = 0. For n ≥ 1, define the stopping time    1 R τn := R ∧ inf t ≥ 0 : Zt (X) ∈ / ,n , n with inf ∅ = ∞. On [0, τnR ], 1/n ≤ Zt (X) ≤ n, and dZt (X) = Zt (X)⟨ut , dBt ⟩. The change of measure separates into an initial change by the G0 -measurable density Z0 and a dynamic change generated by u. The initial change leaves Brownian increments Brownian, since they are independent of G0 ; on {Z0 > 0}, the normalized density Zt /Z0 generates the dynamic Girsanov shift. Thus, Girsanov’s theorem over (Ft (X))t shows that Z t∧τnR

Z t∧τnR Ps (X) dBs −

Mt∧τnR :=

0

us ds 0

is an (Ft (X), Py )-local martingale. It is in fact square-integrable, since Py

Z τnR

Z τnR

2

P

|us | ds = E

E

0

0

|ξs |2 ds ≤ n EP Zs (X)

Z R

|ξs |2 ds < ∞.

0

We next remove the localization on the fixed interval [0, R]. Since Z is a nonnegative continuous martingale, it remains zero after its first hitting time of zero. Therefore, for   AR := inf Zs (X) = 0 , 0≤s≤R

we have ZR (X) = 0 on AR , and thus Py (AR ) = EP [ZR (X)1AR ] = 0. On AcR , continuity gives 0 < inf s≤R Zs (X) ≤ sups≤R Zs (X) < ∞. On AR , the stopped values converge to zero, which equals ZR because a nonnegative continuous martingale remains zero after its first hitting time of zero. Consequently, pointwise, ZτnR (X) −→ ZR (X) and τnR ↗ R Py -a.s. 25

P-a.s.,

Moreover, Itô’s formula for log Z on [0, τnR ] gives 1 Py E 2

Z τnR

2

Py



|us | ds = E 0

 i h ZτnR (X) log = EP ZτnR (X) log ZτnR (X) − EP [Z0 (X) log Z0 (X)] . Z0 (X)

By Doob’s Optional Sampling Theorem, we obtain ZτnR = EP [ZR | FτnR (X)]. In particular, conditional Jensen’s inequality gives 2 Zτ2R ≤ EP [ZR | FτnR (X)]. n

The conditional expectations on the right form a uniformly integrable family. Hence (Zτ2R )n is n uniformly integrable. Since (z log z)+ ≤ z 2 and (z log z)− ≤ e−1 , the family (ZτnR log ZτnR )n is uniformly integrable as well. Together with the almost-sure convergence above, by Vitali’s convergence theorem, EP [ZτnR log ZτnR ] −→ EP [ZR log ZR ]. R τR R∞ R τR Using monotone convergence on 0 n |us |2 ds = 0 |us |2 1{s≤τnR } ds, we obtain limn→∞ 0 n |us |2 ds = RR 2 0 |us | ds. Thus, we obtain both square integrability and the exact identity 1 Py E 2

Z R

|us |2 ds = EP [ZR log ZR ] − EP [Z0 log Z0 ].

(44)

0 R

Since τnR ↗ R Py -a.s. and M τn is a martingale for every n, the process Z t

Z t Ps (X) dBs −

Mt := 0

us ds,

0 ≤ t ≤ R,

0

is a continuous (Ft (X), Py )-local martingale. Its matrix quadratic variation is Z t ⟨M ⟩t =

⊤

Z R

Z t Ps (X) ds =⇒ tr⟨M ⟩R =

Ps (X)Ps (X) ds =

0

0

0

tr Ps (X) ds ≤ d R.

Hence, by (Revuz & Yor, 2013, Chapter IV, Proposition 1.23) (applied componentwise), M is a square-integrable (Ft (X), Py )-martingale on [0, R]. Define the canonical ambient measure Qy,R on GR by d(Qy,R |GR ) = ZR (X). d(P|GR ) The representation of Z is a square-integrable (Gt , P)-martingale, and thus Zt (X) is the density process of Qy,R . Moreover, Qy,R |FR (X) = Py |FR (X) . The preceding localization and Girsanov’s theorem show that Z t y,R Bt := Bt − us ds, 0 ≤ t ≤ R, 0

is a (Gt , Qy,R )-Brownian motion. Since Qy,R ≪ P on GR , the original SDE formulation (equation 1) is preserved. Substitution gives
dXt = b(t, Xt ) + σ(t, Xt )ut dt + σ(t, Xt ) dBty,R , Consequently, LQy,R (X[0,R] ) = LP (X[0,R] | Y = y). 26

0 ≤ t ≤ R.

(45)

For 0 < ε < R − r and 0 ≤ t ≤ r, the martingale property of M together with Theorem 5 give  Z t+ε    Z t+ε 1 1 y y Ps (X) dBs Xt = EP us ds Xt . EP ε t ε t Assumption 3 gives the σ(Xt )-measurable version ut = ut (Xt , y). As u ∈ L2 ([0, R] × Ω, dt ⊗ dPy ), by the Lebesgue differentiation theorem, Z 1 t+ε us ds −→ ut in L2 ([0, r] × Ω, dt ⊗ dPy ). ε t Since ut is σ(Xt )-measurable and conditional expectation is an L2 -contraction, the same convergence holds after conditioning. Therefore  Z t+ε  ξt Py 1 lim E Ps (X) dBs Xt = ut = ε↓0 ε t Zt (X) in L2 ([0, r]×Ω, dt⊗dPy ). Since r < T was arbitrary, the result follows on every compact subinterval of [0, T ), as desired.

C.5

Proof of Theorem 7

Proof of Theorem 7. Fix r < T . Throughout the proof, let (Ω, F, (Gt )0≤t≤r , P) be the filtered probability space carrying a weak solution (X, B) of equation 1, where B is a (Gt )-Brownian motion and Ft (X) ⊆ Gt , 0 ≤ t ≤ r. Thus, under P, dXt = b(t, Xt ) dt + σ(t, Xt ) dBt . Let Zt (X) =

d(Py |Ft (X) ) , d(P|Ft (X) )

ut :=

ξt Zt (X)

(46)

on {Zt (X) > 0}.

By Lemma 2 and Theorem 4, there exists a square-integrable integrand ξ = (ξs )s on [0, r] such that Z t d(Py |F0 (X) ) Zt (X) = Z0 (X) + ⟨ξs , dBs ⟩, Z0 (X) = , Ps (X)ξs = ξs . d(P|F0 (X) ) 0 In particular, since the integrand is (Gt )-adapted, (Zt (X))0≤t≤r is also a (Gt , P)- martingale. We now construct an ambient probability measure which realizes the conditional state law. Define Qy,r on (Ω, Gr ) by d(Qy,r |Gr ) := Zr (X). (47) d(P|Gr ) Since Zr (X) is Fr (X)-measurable and EP [Zr (X)] = 1, this defines a probability measure. Moreover, for every A ∈ Fr (X), Qy,r (A) = EP [Zr (X)1A ] = Py (A). Hence Qy,r |Fr (X) = Py |Fr (X) . Furthermore, because Z is a (Gt , P)-martingale with terminal value Zr (X), Zt (X) = EP [Zr (X) | Gt ] , 27

0 ≤ t ≤ r.

(48)

Thus, Z is the Radon-Nikodym derivative of Qy,r with respect to P on (Gt )t . As proved by the localization argument in Theorem 6, under Qy,r , the process Z t y,r us ds, 0 ≤ t ≤ r, Bt := Bt − 0

is a (Gt )-Brownian motion. Since equation 46 holds P-a.s. and Qy,r ≪ P, the same stochastic identity holds Qy,r -a.s. Substituting dBt = dBty,r + ut dt gives
dXt = b(t, Xt ) + σ(t, Xt )ut dt + σ(t, Xt )dBty,r . (49) Consequently, Ω, F, (Gt )t≤r , Qy,r , X, B y,r

is a weak solution of equation 2 with control u. By equation 48, LQy,r (X[0,r] ) = LPy (X[0,r] ) = LP (X[0,r] | Y = y). Thus, the canonical controlled realization has the conditioned state-path law on [0, r]. We next prove its minimum-energy property. Applying the exact limit identity equation 44 with R = r yields Z

1 Py r y y DKL P |Fr (X) P|Fr (X) = DKL P |F0 (X) P|F0 (X) + E |ut |2 dt. (50) 2 0 Rr Since 0 |ut |2 dt is Fr (X)-measurable and Qy,r and Py agree on Fr (X), equivalently, Z

1 Qy,r r E |ut |2 dt = DKL Py |Fr (X) P|Fr (X) − DKL Py |F0 (X) P|F0 (X) . 2 0 Let (Q, v) ∈ Ayr have the conditioned state-path law. Then Q|Fr (X) = Py |Fr (X) , and data processing inequality gives
DKL Py |Fr (X) P|Fr (X) ≤ DKL (Q∥P) Z r 1 |vt |2 dt. = H0y + EQ 2 0 Combining this inequality with equation 50 yields Z r Z r 2 Py Q E |vt | dt ≥ E |ut |2 dt. 0

0 y,r

density Z0 (X), and the construction above shows that B = B − R · The canonical pair has initial y,r P 0 us ds is Brownian under Q ; indeed, E [Zr (X) | G0 ] = Z0 (X). Identity present in equation 50 gives the required entropy identity, and thus (Qy,r , u) ∈ Ayr and it attains equality. If another admissible adapted shift attains the same value, then the data-processing inequality used above must hold with equality. By the equality case of conditional Jensen’s inequality,   dQ P dQ =E Fr (X) P-a.s. dP dP Thus, dQ/dP admits an Fr (X)-measurable version. Since its restriction to Fr (X) has density Zr (X), it follows that dQ = Zr (X) P-a.s., dP 28

and hence Q = Qy,r

on Gr .

Theorem 4 was used above to identify the canonical shift u through the martingale R · representation of the density process, and the corresponding Girsanov argument shows that B− 0 us ds is a Brownian R· motion under Qy,r . By admissibility, B− 0 vs ds is a Brownian motion under Q. Under the common measure Q = Qy,r , uniqueness of the continuous semimartingale decomposition therefore gives vt = ut

dt ⊗ dQy,r -a.e.

This proves the minimum-energy property and uniqueness.

D

Statement and Proof of Theorem 3

First, we provide a formal version of Theorem 3. For simplicity, we assume that X0 is deterministic. Theorem 10. Suppose that X0 = x0 is deterministic and Assumption 1 holds. Fix r < T , set µr := LP (Xr ), and let νr be a probability measure such that νr ≪ µr ,

zr :=

dνr ∈ L2 (µr ). dµr

Let Ar be the class of pairs (Q, v) such that Q ≪ P on Gr , v is (Gt )-predictable, Z t Q vs ds Bt := Bt − 0

Rr is a (Gt , Q)-Brownian motion, EQ 0 |vt |2 dt < ∞. Consider the stochastic optimal control problem  Z r  Q 1 2 V (r, νr ) := inf E |vt | dt − log zr (Xr ) , (51) 2 0 (Q,v)∈Ar where the objective is interpreted in the extended sense. Define Ztν,r := EP [zr (Xr ) | Ft (X)] ,

0 ≤ t ≤ r.

Then, Z0ν,r = 1, Zrν,r = zr (Xr ), and there exists a unique ξ ν,r ∈ L2a ([0, r], F(X), P; Rd ) such that Z t ν,r Zt = 1 + ⟨ξsν,r , dBs ⟩, Pt (X)ξtν,r = ξtν,r dt ⊗ dP-a.e. 0

Set uν,r t := and define

ξtν,r Ztν,r

on {Ztν,r > 0},

uν,r t := 0

d(Qν,r |Gr ) = zr (Xr ). d(P|Gr )

ν,r Then (Qν,r , uν,r ) ∈ Ar , Pt (X)uν,r dt ⊗ dQν,r -a.e., and t = ut Z t Btν,r := Bt − uν,r s ds 0

29

otherwise,

is a (Gt , Qν,r )-Brownian motion. Under this measure, the coordinate process satisfies ν,r dXt = (b(t, Xt ) + σ(t, Xt )uν,r t ) dt + σ(t, Xt )dBt .

(52)

Moreover, LQν,r (Xr ) = νr ,

V (r, νr ) = 0,

and uν,r is the unique minimum-energy Brownian perturbation realizing νr : Z Z 1 Qν,r r ν,r 2 1 Q r |vt |2 dt. E |ut | dt = inf E 2 2 (Q,v)∈A : r 0 0 LQ (Xr )=νr

In particular, under Assumption 2, one may take νr = LP (Xr | Y = y). We then write zry , Z y,r , uy,r , and Qy,r for the corresponding objects. Remark 8. Pairs induced by strictly positive true stochastic exponentials are special cases of Ar . Allowing Q to be merely absolutely continuous, rather than equivalent, includes canonical densities that may vanish under P. Proof of Theorem 10. Since zr (Xr ) ∈ L2 (Fr (X), P) and X0 is deterministic, Z0ν,r = 1,

Zrν,r = zr (Xr ).

Theorem 4 implies that there exists F(X)-adapted square integrable ξ ν,r such that Z t ν,r ⟨ξsν,r , dBs ⟩, Ps (X)ξsν,r = ξsν,r . Zt = 1 + 0

Define uν,r s :=

ξsν,r 1 ν,r . Zsν,r {Zs >0}

Then dZtν,r = Ztν,r ⟨uν,r t , dBt ⟩,

ν,r Pt (X)uν,r t = ut .

Moreover, {Ztν,r = 0} is dt ⊗ dQν,r -null. The localization and uniform-integrability argument in the proof of Theorem 6 applies with terminal density zr (Xr ). It shows that B ν,r is Brownian under Qν,r and that Z 1 Qν,r r ν,r 2 E |ut | dt = DKL (Qν,r ∥P). 2 0 Thus (Qν,r , uν,r ) ∈ Ar . Since dQν,r /dP = zr (Xr ), LQν,r (Xr ) = νr . The original SDE identity holds Qν,r -a.s.; substituting dBt = dBtν,r + uν,r t dt proves equation 52. For any (Q, v) ∈ Ar , let λr := LQ (Xr ). Data processing inequality for relative entropy gives DKL (Q∥P) ≥ DKL (λr ∥µr ). 30

Therefore, with the standard extended-value convention,  Z r  Z 1 |vt |2 dt − log zr (Xr ) = DKL (Q∥P) − log zr dλr EQ 2 0 ≥ DKL (λr ∥νr ) ≥ 0. The canonical pair attains zero, and hence V (r, νr ) = 0. Now restrict to competitors satisfying λr = νr . Then Z Z 1 Q r 1 Qν,r r ν,r 2 2 |vt | dt ≥ DKL (νr ∥µr ) = E |ut | dt. E 2 2 0 0 If equality holds, equality also holds in data processing. Its equality case shows that dQ/dP is σ(Xr )-measurable. Since its pushforward density is zr , necessarily d(Q|Gr ) d(Qν,r |Gr ) = zr (Xr ) = . d(P|Gr ) d(P|Gr ) R· R· Thus Q = Qν,r . Under this common measure, both B − 0 vs ds and B − 0 uν,r s ds are Brownian ν,r motions. Uniqueness of the continuous semimartingale decomposition gives vt = uν,r t dt ⊗ dQ -a.e. Note that, for any bounded Borel function φ, h i EP φ(Xr )EP [Zry (X) | Xr ] = EP [φ(Xr )Zry (X)] y

= EP [φ(Xr )] Z = φ(x)zry (x) LP (Xr )(dx). Thus, LP (Xr | Y = y) ≪ µr , and

zry (Xr ) = EP [Zry | Xr ].

The L2 -contraction and Assumption 2 give zry (Xr ) ∈ L2 (P), and thus all preceding conclusions apply.

E

Proofs of the degenerate Tweedie formulae

E.1

Proof of Degenerate Tweedie formula-1

Proof of Theorem 1. Apply Theorem 6. It gives equation 3 in the stated L2 space with ut (Xt , y) = ξt /Zt (X). Since Pt (X)ξt = ξt dt ⊗ dP-a.e., we also have Pt (X)ut (Xt , y) = ut (Xt , y) dt ⊗ dPy -a.e., and uniqueness follows from the uniqueness of the integrand in the martingale representation by Theorem 4. The conditional-law and minimum-entropy claims on every [0, r], r < T , follow directly from Theorem 7.

E.2

Proof of Degenerate Tweedie formula-2

Proof of Corollary 1. Fix r < T , 0 ≤ t ≤ r, and 0 < ε < T −r. We apply Theorem 5 componentwise to Bt+ε − Bt . For each j ∈ {1, . . . , d}, j th -component of Bt+ε − Bt satisfies Z T (Bt+ε − Bt )j = 1(t,t+ε] (s)ej , dBs , 0

31

  and EP (Bt+ε − Bt )j | X0 = 0, since Bt+ε −Bt is independent of Gt , and hence of σ(X0 ). Therefore, Theorem 5 gives Z t+ε   P j ⟨Ps (X)ej , dBs ⟩ . E (Bt+ε − Bt ) | FT (X) = t

Applying this componentwise yields Z t+ε P

E [Bt+ε − Bt | FT (X)] =

Ps (X) dBs .

(53)

t

Since Y = G(XT ), σ{Xt , Y } ⊆ FT (X). Hence, by the tower property and equation 53, h i EP [Bt+ε − Bt | Xt , Y ] = EP EP [Bt+ε − Bt | FT (X)] Xt , Y  Z t+ε  P Ps (X) dBs Xt , Y . =E t

Thus, the result follows from Theorem 1.

E.3

Proof of Degenerate Tweedie formula-3

Proof of Theorem 2. Fix R with r < R < T . Under the canonical measure Qy,R of Theorem 7, for 0 < ε < R − r and 0 ≤ t ≤ r, Z t+ε Z t+ε
Xt+ε − Xt = b(s, Xs ) + σ(s, Xs )us (Xs , y) ds + σ(s, Xs ) dBsy,R . t

t

The stochastic integral is a square-integrable (Gt , Qy,R )-martingale increment, so its conditional expectation given Xt is zero. Since Qy,R and Py agree on FR (X), this gives    Z t+ε 
Py Xt+ε − Xt Py 1 E − b(t, Xt ) Xt = E b(s, Xs ) − b(t, Xt ) ds Xt ε ε t   Z t+ε 1 y σ(s, Xs )us (Xs , y) ds Xt . + EP ε t By Assumption 1 and Theorem 1, Z r Z r y y |ut (Xt , y)|2 dt < ∞. |σ(t, Xt )ut (Xt , y)|2 dt ≤ ∥σ∥2∞ EP EP 0

0

Thus, by Assumption 1, the bound above, and the L2 -Lebesgue differentiation theorem, Z
1 t+ε b(s, Xs ) − b(t, Xt ) ds −→ 0, ε t Z 1 t+ε σ(s, Xs )us (Xs , y) ds −→ σ(t, Xt )ut (Xt , y) ε t in L2 ([0, r] × Ω, dt ⊗ dPy ). For any such pair Htε → Ht , Z r Z r 2 y Py Py ε E E [Ht | Xt ] − Ht dt ≤ EP |Htε − Ht |2 dt −→ 0. 0

0

Thus both convergences remain valid after conditioning on Xt . Passing to the limit in the preceding identity proves equation 5. 32

F

Terminal-Increment Regression for Additive Diffusions

For additive diffusions with a fixed noise matrix, the local Brownian target can be replaced exactly by a terminal-increment target. This removes the auxiliary limit ε ↓ 0 required in the the definition of the loss function. Theorem 11. Let B be a d-dimensional (Gt )-Brownian motion, let X0 be G0 -measurable, and suppose that Z t

Xt = X0 +

b(s) ds + σBt , 0

where b : [0, T ] → Rn is deterministic and integrable and σ ∈ Rn×d is fixed, possibly rank deficient. Let Y = G(XT ) for a Borel map G. Fix r < T , let Hr be the σ-field on [0, r] × Ω generated by (t, ω) 7→ (t, Xt (ω), Y (ω)). For Qt := (BT − Bt )/(T − t), define the product-space conditional expectation ū := Edt⊗dP [Q | Hr ]. Choose a Borel representative ū : [0, r] × Rn × Y → Rd for this conditional expectation. Then, ū(t, Xt , Y ) = EP [Qt | Xt , Y ] dt ⊗ dP-a.e. For every 0 < ε < T − r,     P P (BT − Bt ) P Bt+ε − Bt ū(t, Xt , Y ) = E Xt , Y = E Xt , Y (54) T −t ε for dt ⊗ dP-a.e. (t, ω) ∈ [0, r] × Ω. Moreover, Z r P

E

0

BT − B t 2 T < ∞, dt = d log T −t T −r

and the regression problem Z r inf

P

v∈L2 (Hr ,dt⊗dP;Rd )

vt −

E

0

BT − Bt 2 dt T −t

(55)

has the unique minimizer ū. If Assumptions 1–2 also hold, let ξ y be the integrand from Lemma 2. For LP (Y )-a.e. y, let upath,y := ξty /Zty (X) on {Zty (X) > 0} and set it to zero otherwise. Then ū(t, Xt , y) = upath,y in t t 2 L ([0, r] × Ω, dt ⊗ dPy ). Consequently, the path-adapted control admits this state-feedback version, and the terminal-increment loss recovers its canonical controlled state law on [0, r] and its minimumenergy property. Proof. Fix 0 ≤ t ≤ r and 0 < ε < T − r, and write ∆Bt,T := BT − Bt ,

∆Bt,ε := Bt+ε − Bt .

The Brownian bridge decomposition is ∆Bt,ε =

ε ∆Bt,T + Rt,ε , T −t

where Rt,ε is centered Gaussian and independent of Gt ∨ σ(∆Bt,T ). Since Z T XT = Xt +

b(s) ds + σ∆Bt,T , t

33

(56)

the pair (Xt , Y ) is measurable with respect to Gt ∨ σ(∆Bt,T ). Hence EP [Rt,ε | Xt , Y ] = 0, and conditioning equation 56 gives     P BT − B t P Bt+ε − Bt Xt , Y = E Xt , Y . E ε T −t Since P is the orthogonal projection onto Ran(σ ⊤ ), we have σ(I −P ) = 0. The Gaussian vectors P ∆Bt,T and (I −P )∆Bt,T are independent, and the latter is jointly independent of Gt ∨σ(P ∆Bt,T ). Since Y depends on the future increment only through P ∆Bt,T , EP [(I − P )∆Bt,T | Xt , Y ] = 0. This proves all identities in equation 54. The stated integrability follows from EP |BT − Bt |2 = d(T − t). The regression claim is the orthogonal-projection characterization of conditional expectation in the Hilbert space Vr . Finally, choose a deterministic sequence εk ↓ 0 and disintegrate the exact identities in equation 54 simultaneously outside one LP (Y )-null set. The same independence argument with conditioning on Ft (X) ∨ σ(Y ) gives     P Bt+εk − Bt P BT − Bt E Ft (X) ∨ σ(Y ) = E Xt , Y . εk T −t The path-feedback form of Corollary 1 therefore yields ū(t, Xt , y) = upath,y in the stated fiberwise t L2 space. In particular, the additive model supplies its own state-feedback version without Assumption 3. The controlled-law and minimum-energy conclusions follow from the path-feedback form of Theorem 7.

G

Implementation of the Training Procedure

This section studies the procedure for the convergence to the control u appearing in equation 3. We consider both the projected target used in Proposition 1 and the unprojected Brownian-increment target from Corollary 1.

G.1

Population regression

Assume throughout this subsection that Assumptions 1, 2, and 3 hold. Fix r < T and 0 < ε < T −r. Define Z 1 t+ε Bt+ε − Bt Mtε,proj := Ps (X) dBs and Mtε,raw := . ε t ε Let Hr be the product σ-field on [0, r] × Ω generated by the map (t, ω) 7→ (t, Xt (ω), Y (ω)), and let Vr := L2 (Hr , dt ⊗ dP; Rd ). This is the closed regression space corresponding to state feedback controls that appears in equation 2. For either choice Mtε ∈ {Mtε,proj , Mtε,raw }, consider the population loss Z r Lε (v) = EP |vt − Mtε |2 dt. 0

34

Its unique minimizer over Vr is the product-space conditional expectation. Choose a Borel representative u∗ε : [0, r] × Rn × Y → Rd ; then, dt ⊗ dP-a.e., u∗ε (t, Xt , Y ) = EP [Mtε | Xt , Y ] . When no arguments are displayed, u∗ε denotes the realized process u∗ε (t, Xt , Y ). Indeed, the orthogonality property of conditional expectation gives Z r ∗ P |vt − u∗ε (t, Xt , Y )|2 dt. Lε (v) − Lε (uε ) = E

(57)

0

Consequently, minimizing the squared regression loss is equivalent to approximating the appropriate conditional expectation. After disintegration with respect to Y , Proposition 1 and Corollary 1 give, for LP (Y )-a.e. y, u∗,proj −→ u ε

u∗,raw −→ u ε

in L2 ([0, r] × Ω, dt ⊗ dPy ),

in L2 ([0, r] × Ω, dt ⊗ dPy ).

Let gε (y) be such that Z r EP

|u∗ε (t, Xt , Y ) − u(t, Xt , Y )|2 dt =

0

Z gε (y) LP (Y )(dy).

We additionally assume convergence in L2 ([0, r]×Ω, dt⊗dP); it follows, for example, if gε (y) → 0 for LP (Y )-a.e. y and the family (gε )ε is uniformly integrable under LP (Y ). Under this joint regularity conditions, both losses recover the same minimum-energy control.

G.2

Discretization of the projected target

that appears in equation 7, In this subsection, we provide an approximation scheme for Mtε,proj j whose convergence is not obvious since the projection (Pt (X))t is not continuous. Let tk = kh be a uniform grid, assume T = Kh for an integer K, and suppose that ε = qh for an integer q ≥ 1. Let X h be the Euler–Maruyama approximation of X, and let ηh (0) = 0 and ηh (s) = tk for s ∈ (tk , tk+1 ]. Define the predictable piecewise-constant approximation h

P s := Pηh (s) (X h ),

0 ≤ s ≤ T,

h

where P 0 = P0 (X h ). Equivalently, h

P s = Ptk (X h ),

s ∈ (tk , tk+1 ].

For 0 ≤ t ≤ r, set 1 Mtε,proj,h := ε

Z t+ε t

h

P s dBs =⇒ Mtε,proj,h = j

j+q−1
1 X Ptk (X h ) Btk+1 − Btk . qh k=j

By Itô’s isometry and Fubini’s theorem, Z Z r 2 2 1 P r+ε h ε,proj,h ε,proj P E Mt − Mt dt ≤ E P s − Ps (X) ds, ε F 0 0

35

(58)

where ∥ · ∥F is the Frobenius norm. Therefore, for fixed ε, the discrete target converges whenever Z r+ε 2 h P E P s − Ps (X) ds −→ 0. F

0

For example, suppose that the state-dependent projection P (t, x) satisfies
∥P (t, x) − P (s, z)∥F ≤ LP |x − z| + |t − s|1/2 ,

(59)

and that the numerical approximation of X has mean-square error of order h. Then, Z r+ε Z r 2 2 h h EP Mtε,proj,h − Mtε,proj dt ≤ C . P s − Ps (X) ds ≤ Ch =⇒ EP ε F 0 0 The displayed bound is driven to zero by the sufficient conditions ε ↓ 0 and h/ε → 0. Below, we provide a sufficient condition for equation 59. Proposition 2. Let X h be the Euler–Maruyama approximation of X. Suppose that b and σ are globally Lipschitz in the state variable, 1/2-Hölder continuous in time, and of linear growth, with EP |X0 |2 < ∞. Assume further that, for some ρ ≥ 1 and κ > 0,
rank σ(t, x) = ρ, sρ σ(t, x) ≥ κ for every (t, x), where sρ (σ) denotes the smallest nonzero singular value of σ. Then, there is a constant C, independent of h and ε, such that Z r+ε 2 h P E P s − Ps (X) ds ≤ Ch. F

0

Consequently,

Z r P

E

0

2 h Mtε,proj,h − Mtε,proj dt ≤ C . ε

Proof. The constant-rank assumption and the uniform lower bound on the smallest nonzero singular value imply that the orthogonal projection Π(a) = a† a onto Ran(a⊤ ) is Lipschitz continuous as a function of a on this class of matrices. For a, e a ∈ Rn×d satisfying rank(a) = rank(e a) = ρ,

sρ (a), sρ (e a) ≥ κ,

let Π(a) := a† a,

Π(e a) := e a† e a

denote the orthogonal projections onto Ran(a⊤ ) and Ran(e a⊤ ), respectively. Since
e a I − Π(e a) = 0, we have

Π(a) I − Π(e a) = a† (a − e a) I − Π(e a) . Moreover, ∥a† ∥op =

1 1 ≤ , sρ (a) κ

and therefore ∥Π(a)(I − Π(e a))∥F ≤

1 1 ∥a − e a∥F =⇒ ∥(I − Π(a))Π(e a)∥F ≤ ∥a − e a∥F . κ κ 36

Since Π(a) − Π(e a) = Π(a)(I − Π(e a)) − (I − Π(a))Π(e a), we obtain ∥Π(a) − Π(e a)∥F ≤

2 e σ(·, x)) − Π(s, e σ(·, z))∥F ≤ 2 ∥σ(t, x) − σ(s, z)∥F . ∥a − e a∥F =⇒ ∥Π(t, κ κ

Using the Lipschitz continuity of σ in the state variable and its 1/2-Hölder continuity in time, we conclude that
e σ(·, x)) − Π(s, e σ(·, z))∥F ≤ C |x − z| + |t − s|1/2 , ∥Π(t, where C depends only on κ and the corresponding regularity constants of σ. The standard strong Euler–Maruyama estimate and the increment estimate for X yield EP Xs − Xηhh (s)

2

≤ Ch,

see (Kloeden & Platen, 1992, Theorem 10.2.2.). Hence,   2 2 h P P h E P s − Ps (X) ≤ C E Xηh (s) − Xs + |s − ηh (s)| F

≤ Ch. Integrating over [0, r + ε] proves the first assertion. Finally, Itô’s isometry and Fubini’s theorem give Z r Z Z 2 2 1 r P t+ε h ε,proj,h ε,proj P Mt − Mt E dt = 2 E P s − Ps (X) ds dt ε 0 F 0 t Z r+ε 2 1 h P s − Ps (X) ds ≤ EP ε F 0 h ≤C . ε

G.3

The unprojected loss function

When Corollary 1 is used, the discrete target is Bt − Btj Mtε,raw = j+q = j qh

j+q−1
1 X Btk+1 − Btk . qh

(60)

k=j

This equality is exact. Hence, unlike the projected case, there is no stochastic-integral discretization error and no regularity assumption on Pt (X) is needed. A time discretization is still required for the integral in the loss and, when X is simulated numerically, for the state process itself.

G.4

Monte Carlo and optimization consistency

To approximate the loss functions we defined for the Tweedie’s formula, a standard approach is to use neural networks. We refer to (Tzen & Raginsky, 2019, Section 3) for a theoretical study of the feasibility of this approach. Their analysis is directly applicable in our setting. 37

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