Quadratic Mean Differentiability

• Quadratic mean differentiability is defined as the property by which functions or operators admit a local quadratic expansion in the L² norm with a remainder term of lower order.
• It underpins rigorous analysis in Malliavin calculus, adaptive statistical inference, and convex optimization by connecting difference quotient techniques and generalized quadratic forms.
• Applications range from sensitivity analysis in BSDEs and asymptotic normality in MLE to differentiable solution maps in quadratic cone programs, facilitating both theoretical and computational advances.

Quadratic mean differentiability is a foundational concept in stochastic analysis, variational analysis, and statistical theory, capturing the property that a function, operator, or solution map admits a local quadratic expansion in an appropriate mean-space (typically in $L^2$). It manifests in several mathematical domains—Malliavin calculus (for stochastic processes), adaptive statistical inference (local asymptotic normality), variational second-order analysis, and differentiability of solution maps in convex optimization. This article presents a rigorous examination of quadratic mean differentiability, emphasizing its definitive criteria, connections to Malliavin calculus, statistical inference, and recent extensions in optimization and variational analysis.

1. Mathematical Foundations and Definitions

Quadratic mean differentiability (QMD) is generally characterized by the ability of a map or likelihood to be locally approximated by a linear operator in $L^2$ norm, with the remainder term of smaller order than the quadratic perturbation. In stochastic calculus, for a functional $F$ on a Wiener space and a direction $h$ in the Cameron–Martin space $\mathcal{H}$, QMD is formulated by the limit: $$\lim_{\varepsilon \to 0} \varepsilon^{-1}[F(T_{\varepsilon h}) - F] = (DF, h)_\mathcal{H}$$ in probability or in $L^p$ norm, where $DF$ is the Malliavin derivative and $T_{\varepsilon h}$ is the shift operator $T_{\varepsilon h}(\omega) = \omega + \varepsilon h$ (Mastrolia et al., 2014).

In statistical theory, QMD (also called differentiability in quadratic mean, DQM) requires that the square-root of the likelihood function, as a function of model parameters, is differentiable in $L^2$: $$\sqrt{f_{\theta+h}}(y) = \sqrt{f_\theta}(y) + \tfrac{1}{2} h^\top u_\theta(y) \sqrt{f_\theta}(y) + o(\|h\|),$$ where $u_\theta(y)$ is the score function (Christensen et al., 2023).

For optimization problems, especially those involving quadratic cone programs, QMD denotes the differentiability of the solution map that assigns problem data to the argmin or optimal solution, justified via the implicit function theorem under regularity assumptions (Healey et al., 24 Aug 2025).

2. Malliavin Calculus and BSDEs: Difference Quotient Characterization

In the context of backward stochastic differential equations (BSDEs), QMD is intertwined with the Malliavin differentiability of the solution processes $(Y_t, Z_t)$. The paper (Mastrolia et al., 2014) establishes new sufficient conditions for Malliavin differentiability in both Lipschitz and quadratic BSDE frameworks:

• (D): The terminal condition $\xi$ and the driver $f(t, \omega)$ are Malliavin differentiable uniformly in their arguments.
• (H1): For any direction $h \in \mathcal{H}$, the limit

$$\varepsilon^{-1}[f(t, \omega+\varepsilon h, Y_t, Z_t) - f(t, \omega, Y_t, Z_t)] \to (Df(t, \omega, Y_t, Z_t), h)_\mathcal{H}$$

holds in $L^p$.

• (H2): Spatial derivatives of $f$ in $(y, z)$ are jointly continuous in $L^2([0, T])$.

This framework connects QMD to the convergence of difference quotients—a direct identification of the Malliavin derivative as the Gâteaux derivative along Cameron–Martin directions. The solution $Y_t$ then belongs to the Malliavin–Sobolev space $\mathbb{D}^{1,2}$.

Typical mathematical formulations include the BSDE: $$Y_t = \xi + \int_t^T f(s, Y_s, Z_s)\, ds - \int_t^T Z_s\ dW_s$$ and the linearized equation for Malliavin derivatives: $$D_r Y_t = D_r\xi + \int_t^T [D_r f(s, Y_s, Z_s) + f_y D_r Y_s + f_z D_r Z_s]\, ds - \int_t^T D_r Z_s\ dW_s$$ for $r < t$.

In quadratic cases, additional constraints such as boundedness and continuity of $Df$ allow control over the quadratic growth in $z$. These findings extend the classical differentiability results (e.g., by Pardoux–Peng, El Karoui–Peng–Quenez) under more relaxed conditions.

3. Statistical Inference: DQM and Adaptive Designs

In parametric and nonparametric statistics, QMD is critical for establishing local asymptotic normality (LAN) and for proving the asymptotic normality of maximum likelihood estimators (MLE). The extension to adaptive designs—where covariates or design points $X_i$ are updated based on past observations—requires weaker regularity compared to classical analysis.

The central DQM (or S-DQM in adaptive designs) condition requires via (Christensen et al., 2023): $$\sum_{i=1}^n D_{(\theta, h/\sqrt{n})}(X_i) = o_p(1)$$ with $D_{(\theta, h/\sqrt{n})}(x)$ defined by

$$\int \left[\sqrt{f_{\theta+h/\sqrt{n}}}(y|x) - \sqrt{f_\theta}(y|x) - \tfrac{1}{2} h^\top u_\theta(y,x) \sqrt{f_\theta}(y|x)\right]^2 d\mu(y)$$

This summable differentiability ensures that the cumulative error in the quadratic approximation vanishes, even for dependent or design-adapted observations.

Through S-DQM, LAN results are established for complex adaptive designs—such as the Bruceton "up-and-down" design, Robbins–Monro, and Markovian Langlie—without requiring third-order differentiability or domination conditions. Log-likelihood expansions are achieved: $A_n(h) = h^\top U_n - h^\top J_n h + o_p(1)$ where $U_n$ is the normalized score and $J_n$ the sample information matrix.

4. Second-Order Variational Analysis: Generalized Differentiability and Quadratic Bundles

The notion of QMD has been extended to nonsmooth variational analysis through generalized twice differentiability and quadratic bundles (Khanh et al., 3 Jan 2025). For a function $f$ at $(\bar{x}, \bar{v})$, generalized twice differentiability is posited if:

• $f$ is twice epi-differentiable at $\bar{x}$ for $\bar{v}$;
• The second-order subderivative $d^2 f(\bar{x}|\bar{v})$ is a generalized quadratic form:

$q(x) = \langle x, A x \rangle + \delta_L(x)$

for symmetric $A$ and indicator $\delta_L$ of a linear subspace $L$.

For prox-regular and subdifferentially continuous functions, the quadratic bundle $\operatorname{quad} f(\bar{x}, \bar{v})$ is nonempty and dense within certain localizations. Moreau envelopes serve as an analytical bridge: the generalized twice differentiability of $f$ at $(\bar{x}, \bar{v})$ is equivalent to the classical twice differentiability of $e_\lambda f$ at $\bar{x}+\lambda \bar{v}$.

Generalized QMD ensures that robust second-order information is available even in the absence of classical Hessians, with major implications for sensitivity analysis, tilt stability, and primal-dual optimality conditions.

5. Differentiability of Solution Maps in Quadratic Cone Programs

In convex optimization, QMD manifests as the differentiability of the solution map in quadratic cone programs—problems structured via quadratic objectives and cone constraints (Healey et al., 24 Aug 2025). The solution map is justified using a homogeneous primal-dual embedding: $Q(u) = v$ with variables $u = (x, y, \tau)$, $v = (0, s, \kappa)$, and normalization function $\mathcal{N}(z, \theta) = 0$. Under regularity (invertible Jacobian $D_z \mathcal{N}$ and differentiability of cone projections), the implicit function theorem yields: $$D s(\theta) = -[D_z \mathcal{N}(z,\theta)]^{-1} D_\theta \mathcal{N}(z,\theta)$$ The differentiation framework enables efficient Jacobian–vector and vector–Jacobian product computations through iterative linear solvers (e.g., LSMR), especially suitable for large-scale problems and GPU-accelerated environments (diffqcp implementation).

These methods generalize earlier differentiability results for linear cone programs, provide tools for sensitivity analysis in optimization and machine learning applications, and enable seamless integration of optimization layers within deep architectures.

6. Applications and Implications Across Domains

Quadratic mean differentiability underpins analysis and numerical methods in several advanced domains:

• Stochastic calculus and financial mathematics: QMD assures that BSDE solutions are Malliavin differentiable, facilitating density estimates and computation of derivatives (e.g., Greeks via explicit representations of $DY$ and $DZ$) (Mastrolia et al., 2014, Imkeller et al., 2022).
• Statistical inference and experiment design: S-DQM paves the way for LAN and asymptotic normality of estimators even under adaptive, dependent sampling procedures, supporting efficient and statistically sound experimental designs (Christensen et al., 2023).
• Variational analysis and optimization: Generalized QMD yields second-order conditions critical for stability, numerics, and tilt analysis in nonsmooth optimization, often via quadratic bundles and Moreau envelopes (Khanh et al., 3 Jan 2025).
• Machine learning and optimization layers: Differentiable solution maps in quadratic cone programs, implemented efficiently (e.g., in diffqcp), allow embedding convex optimization within neural network architectures for advanced learning tasks (Healey et al., 24 Aug 2025).

A plausible implication is that further relaxations of regularity conditions (e.g., in S-DQM or via generalized quadratic forms) will continue broadening the applicability of QMD to increasingly complex, adaptive, or nonsmooth models.

7. Future Directions and Interconnections

Recent developments suggest several directions:

• Further weakening of regularity conditions—such as in quadratic BSDEs with rough drift or in adaptive designs with noncompact or time-varying covariate spaces.
• Exploration of QMD in high-dimensional and nonconvex optimization, leveraging variational analysis tools (quadratic bundles, second-order epi-differentiability).
• Integration of QMD-regime solution maps with scalable differentiable programming frameworks, expanding the scope of machine learning architectures and sensitivity analysis in control and finance.
• Investigating higher-order differentiability and second-order QMD properties in both stochastic and variational contexts, potentially yielding refined criteria for robustness and convergence.

Quadratic mean differentiability thus serves as a central pillar for rigorous analysis, sensitivity, and computation in stochastic processes, statistical inference, and modern nonlinear optimization, with domain-specific generalizations underpinning much of today's advanced mathematical and engineering frameworks.