Reciprocal Specific Relative Entropy

• Reciprocal specific relative entropy is a divergence for continuous martingales that quantifies volatility deviations from standard Brownian motion by focusing solely on quadratic variation.
• It reverses the traditional role of the martingale law and Wiener measure compared to classical specific relative entropy, thereby isolating volatility effects from drift adjustments.
• Its variational formulation via a specific p‑Wasserstein divergence leads to the unique selection of the scaled neutral Wright–Fisher diffusion as the optimal win‑martingale.

Searching arXiv for the cited papers to ground the article in current records. arXiv Search Query: title:"Reciprocal Specific Relative Entropy between Continuous Martingales" arXiv Search Query: title:"Minimum Relative Entropy State Transitions in Linear Stochastic Systems: the Continuous Time Case" Reciprocal specific relative entropy is a divergence between continuous martingale laws that quantifies deviation from Wiener measure through volatility rather than drift. For a real continuous martingale law $\mathbb Q$ on $C([0,1];\mathbb R)$ with absolutely continuous quadratic variation $d\langle X\rangle_t=\Sigma_t\,dt$, it is defined by

$$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$

where $\mathbb W$ is Wiener measure. In the formulation of Backhoff–Zhang, this quantity is motivated from several directions: as a reciprocal variant of classical specific relative entropy, as the derivative at $p=2$ of a specific $p$-Wasserstein divergence, and as the continuous-time limit of discrete relative entropies in trinomial trees. Their central result is that minimizing this divergence over win-martingales uniquely selects the scaled neutral Wright–Fisher diffusion (Backhoff et al., 16 Feb 2026).

1. Definition and basic structure

Let $(X_t)_{t\in[0,1]}$ be a real continuous martingale under a probability law $\mathbb Q$ on $C([0,1];\mathbb R)$, and let $C([0,1];\mathbb R)$0 be Wiener measure, so that under $C([0,1];\mathbb R)$1, $C([0,1];\mathbb R)$2 is a standard Brownian motion. The setup assumes absolutely continuous quadratic variation,

$C([0,1];\mathbb R)$3

together with $C([0,1];\mathbb R)$4. Under these assumptions, reciprocal specific relative entropy is

$C([0,1];\mathbb R)$5

This definition isolates the quadratic-variation density $C([0,1];\mathbb R)$6 as the sole state variable entering the divergence (Backhoff et al., 16 Feb 2026).

A key structural feature is that $C([0,1];\mathbb R)$7 depends only on volatility. The Backhoff–Zhang formulation explicitly contrasts it with Girsanov-type entropies: reciprocal specific relative entropy measures only the volatility-distance from Brownian motion, and drift differences do not appear. This sharply distinguishes it from path-space KL functionals built from drift perturbations of Wiener dynamics (Backhoff et al., 16 Feb 2026).

The terminology “reciprocal” is tied to the role reversal between the martingale law and Wiener measure in comparison with standard specific relative entropy. This role reversal is not merely formal: on the event $C([0,1];\mathbb R)$8 almost surely, a time-change argument yields equality between the classical and reciprocal quantities. That identity is the explicit justification for the name (Backhoff et al., 16 Feb 2026).

2. Relation to classical specific relative entropy

Backhoff–Zhang compare reciprocal specific relative entropy with the classical specific relative entropy

$C([0,1];\mathbb R)$9

The two functionals involve the same basic ingredients, but with the sign of $d\langle X\rangle_t=\Sigma_t\,dt$0 reversed. This sign reversal is the defining algebraic distinction between the reciprocal and classical versions (Backhoff et al., 16 Feb 2026).

Under the additional condition $d\langle X\rangle_t=\Sigma_t\,dt$1 almost surely, one has

$d\langle X\rangle_t=\Sigma_t\,dt$2

In this sense, reciprocal specific relative entropy is not an unrelated divergence but a mirrored form of the older specific-relative-entropy paradigm. This suggests that the reciprocal construction is particularly natural when the total quadratic variation is normalized to match Brownian motion (Backhoff et al., 16 Feb 2026).

The broader literature in the data block provides a related but distinct notion of “specific relative entropy” in continuous-time linear Itô systems. In the linear stochastic control setting of Vladimirov–Petersen, the instantaneous cost density

$d\langle X\rangle_t=\Sigma_t\,dt$3

is identified as the specific relative entropy, or entropy rate, by analogy with discrete-time chain rules. There, the entropy is attached to drift distortion of the driving noise rather than to volatility distortion of a martingale law (Vladimirov et al., 2012). The juxtaposition clarifies that “specific relative entropy” is not a single invariant formula across models; rather, its concrete representation depends on which path-space deformation—drift or volatility—is being measured.

3. Variational interpretation through specific $d\langle X\rangle_t=\Sigma_t\,dt$4-Wasserstein divergence

A principal motivation for reciprocal specific relative entropy comes from the specific $d\langle X\rangle_t=\Sigma_t\,dt$5-Wasserstein divergence. For $d\langle X\rangle_t=\Sigma_t\,dt$6, Backhoff–Zhang define

$d\langle X\rangle_t=\Sigma_t\,dt$7

Using convexity of $d\langle X\rangle_t=\Sigma_t\,dt$8, they show formally that

$d\langle X\rangle_t=\Sigma_t\,dt$9

Thus, up to the additive constant $$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$0, reciprocal specific relative entropy is recovered as the derivative of the $$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$1-Wasserstein divergence at the critical exponent $$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$2 (Backhoff et al., 16 Feb 2026).

This interpretation becomes especially important in martingale transport. For

$$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$3

the case $$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$4 is degenerate: all martingales tie by Itô’s isometry. Backhoff–Zhang propose breaking this tie through

$$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$5

A direct lemma based on convexity and Fatou yields

$$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$6

and under mild moment conditions equality holds, including for the Wright–Fisher candidate identified in the win-martingale problem (Backhoff et al., 16 Feb 2026).

A plausible implication is that reciprocal specific relative entropy serves as a first-order selection criterion at a degenerate variational threshold. Rather than introducing an unrelated regularizer, it emerges as the infinitesimal correction associated with perturbing the transport cost exponent away from $$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$7.

4. Win-martingales and the control problem

The main minimization problem in Backhoff–Zhang is posed over win-martingales. Fix $$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$8. A win-martingale is a continuous martingale $$\mathfrak h(\mathbb Q\|\mathbb W) \;:=\; \frac12\;\mathbb E_\mathbb Q\Bigl[\int_0^1\bigl\{\Sigma_t\ln\Sigma_t +1-\Sigma_t\bigr\}\,dt\Bigr],$$9 on $\mathbb W$0 such that

$\mathbb W$1

and $\mathbb W$2. Denoting by $\mathbb W$3 the corresponding set of laws, the objective is

$\mathbb W$4

Because

$\mathbb W$5

is constant over $\mathbb W$6, the problem is equivalent to minimizing

$\mathbb W$7

(Backhoff et al., 16 Feb 2026).

Dynamic programming leads to the value function

$\mathbb W$8

which is a viscosity solution of the HJB equation

$\mathbb W$9

The first-order condition gives

$p=2$0

and substituting back produces the scalar PDE

$p=2$1

This HJB formulation makes the reciprocal specific relative entropy problem a stochastic control problem whose control variable is the instantaneous variance $p=2$2 rather than a drift (Backhoff et al., 16 Feb 2026).

5. Explicit solution and the Wright–Fisher minimizer

Backhoff–Zhang derive the explicit candidate

$p=2$3

for which

$p=2$4

Consequently,

$p=2$5

which identifies the candidate optimal instantaneous variance (Backhoff et al., 16 Feb 2026).

The optimizer is the scaled neutral Wright–Fisher diffusion, characterized by the instantaneous variance

$p=2$6

Backhoff–Zhang show three points: under its law $p=2$7, the cost equals $p=2$8; $p=2$9 is a viscosity solution of the HJB with the correct boundary and terminal data; and a comparison principle implies uniqueness of the viscosity solution in the class of bounded-below functions. Hence $p$0 is the unique minimizer in $p$1 (Backhoff et al., 16 Feb 2026).

This result is notable because the classical variance-minimization criterion at $p$2 does not distinguish among win-martingales. Reciprocal specific relative entropy resolves that degeneracy and selects a single law. In the language of the paper, the Wright–Fisher diffusion is therefore uniquely selected when the degenerate martingale optimal transport problem of variance minimization is suitably perturbed (Backhoff et al., 16 Feb 2026).

6. Discrete approximations and relation to reciprocal processes

Backhoff–Zhang also give a discrete trinomial-tree heuristic. At each time step $p$3, one chooses increments

$p$4

so that $p$5. The corresponding discrete relative entropy satisfies

$p$6

and letting $p$7 and $p$8 recovers exactly the reciprocal-specific-relative-entropy integrand $p$9 (Backhoff et al., 16 Feb 2026). This provides an explicit approximation scheme linking the continuous functional to familiar finite-state entropy calculations.

A separate line of work on minimum relative entropy state transitions in linear stochastic systems provides a different, but conceptually adjacent, use of entropy in continuous time. For linear Itô dynamics driven by random noise with uncertain drift,

$(X_t)_{t\in[0,1]}$0

Vladimirov–Petersen define the noise relative entropy supply

$(X_t)_{t\in[0,1]}$1

which is the KL divergence of the true noise law from the nominal Wiener law on the relevant filtration (Vladimirov et al., 2012). In that framework, the associated optimization problem over Gaussian endpoint marginals is identified with the Schrödinger bridge, and $(X_t)_{t\in[0,1]}$2 is precisely the KL divergence of the bridge measure from the nominal Wiener-driven law. The instantaneous density $(X_t)_{t\in[0,1]}$3 is called the specific relative entropy, or entropy rate, and the optimizer induces a reciprocal diffusion whose specific relative entropy rate is minimal among such bridges (Vladimirov et al., 2012).

The two theories are not identical: the martingale formulation of reciprocal specific relative entropy measures volatility distortion, whereas the linear Schrödinger-bridge formulation measures drift distortion of the driving noise. Nonetheless, both place a path-space entropy functional at the center of a reciprocal-process selection problem. This suggests a broader conceptual theme in which reciprocity is enforced by minimizing a continuous-time entropy rate subject to endpoint or terminal-law constraints.

7. Conceptual scope and common points of confusion

A common misunderstanding is to treat reciprocal specific relative entropy as a generic KL divergence on path space. In Backhoff–Zhang’s definition, it is more specialized: it is a divergence between continuous martingales relative to Wiener measure that depends only on the absolutely continuous quadratic variation density $(X_t)_{t\in[0,1]}$4. It does not encode drift mismatch, and it is therefore not interchangeable with Girsanov-type entropies (Backhoff et al., 16 Feb 2026).

Another source of confusion is the relation between the reciprocal functional and the classical specific relative entropy. The two are not universally equal. Equality is obtained under the condition $(X_t)_{t\in[0,1]}$5 almost surely through a time-change argument; outside that regime, the reciprocal terminology reflects a structural reversal rather than a blanket identity (Backhoff et al., 16 Feb 2026).

It is also important not to conflate the win-martingale minimization problem with ordinary variance minimization. For $(X_t)_{t\in[0,1]}$6, every win-martingale has the same value $(X_t)_{t\in[0,1]}$7 by Itô isometry, so no unique minimizer exists. The uniqueness of the Wright–Fisher diffusion arises only after passing to the reciprocal-entropy correction

$(X_t)_{t\in[0,1]}$8

or equivalently to reciprocal specific relative entropy (Backhoff et al., 16 Feb 2026).

Within current arXiv literature, reciprocal specific relative entropy is therefore best understood as a volatility-based entropy rate for continuous martingales that acquires its sharpest meaning in selection problems. In the win-martingale setting it yields a complete solution, with an explicit HJB, an explicit candidate value function, and a unique optimizer given by the scaled neutral Wright–Fisher diffusion (Backhoff et al., 16 Feb 2026).