Path-Dependent Score Adjustments

• Path-dependent score adjustments are analytic and probabilistic techniques that update scores based on full historical trajectories, integrating stochastic analysis and functional Itô calculus.
• They employ advanced methods such as path-dependent stochastic differential equations, rough differential equations, and signature techniques to capture history-dependent dynamics.
• These methods are applied in financial modeling, machine learning, risk management, and control theory, enabling robust calibration and dynamic score consistency.

Path-dependent score adjustments are analytic and probabilistic techniques for dynamically updating a score or value functional in systems whose evolution depends not only on the current state, but also on the entire past trajectory. Such mechanisms arise naturally in stochastic analysis, financial modeling, stochastic control, machine learning, and dynamical systems with memory. The theory blends path-dependent stochastic differential equations, functional Itô calculus, partial differential equations (PDEs) on path spaces, and advanced numerical methods, all addressing the challenge of accounting for the accumulated information along a process's path when calibrating, estimating, or controlling the evolution of system performance or risk scores.

1. Foundations of Path-Dependent Differential Equations

A central framework for path-dependent score adjustments is the theory of path-dependent stochastic and rough differential equations. Unlike Markovian equations, these allow the coefficients (drift, volatility, vector fields) to depend functionally on the whole past trajectory up to the current time, e.g., if $y_{[0,t]}$ denotes the path up to $t$: $d y_t = V(y_{[0,t]})\,dh_t + F(y_{[0,t]})\,dX_t,$ where $V, F$ are path-dependent vector fields, $h$ is an $\alpha$-Hölder path (deterministic component), and $X$ is a weak geometric $p$-rough path (possibly a rough stochastic signal). This formulation captures systems where score adjustments are determined by the history of the process, not just its instantaneous value.

The well-posedness for such equations is established via the machinery of $C^1$-approximate flows: families of smooth maps $M_{t,s}$ nearly satisfying the flow property, with controlled $C^1$-norm errors, allowing one to "sew" local approximations into a global flow via a form of the sewing lemma. This approach gives existence, uniqueness, and stability for the solution flow, and underpins Euler-type expansion estimates for numerical schemes (Bailleul, 2013).

2. Path-Dependent Additive Functionals and Martingale Problems

Extending from state-dependent martingales and additive functionals to the path-dependent case, probability laws (the so-called path-dependent canonical class) are constructed so that, for any initial path, the future evolution remains conditionally consistent with the observed past. Additive functionals (e.g., accumulated score or quadratic variation) are path functionals $A_{t,u}$ measurable in $u$, admitting càdlàg versions $A^{s,\eta}$ along each trajectory so that $A_{t,u} = A^{s,\eta}_u - A^{s,\eta}_t$ almost surely under the appropriate law, encoding cumulative score adjustments over time.

Solving path-dependent martingale problems, one requires that for every adapted process (often of the form "test function minus integral with respect to a non-local generator"), the martingale property is preserved, yielding dynamic consistency and a sound probabilistic basis for path-dependent adjustment rules (Barrasso et al., 2018).

3. Path-Dependent PDEs and Functional Itô Calculus

Path-dependent scores driven by semimartingales can be characterized analytically through path-dependent (pseudo-)PDEs (PPDEs). A canonical PPDE, associated with option pricing, is: $$D_t u(t, \gamma_t) + \langle b_t(\gamma_t), D_x u(t, \gamma_t) \rangle + \frac{1}{2} \mathrm{tr}\left[ \sigma_t(\gamma_t) D_{xx} u(t, \gamma_t) \sigma_t^\mathsf{T}(\gamma_t) \right] = 0,$$ with $u_T(\gamma_T) = g(\gamma_T)$. Here $D_t$ denotes the "horizontal" (Dupire's time) derivative and $D_x$, $D_{xx}$ the vertical and vertical–second derivatives, all in the sense of functional Itô calculus (Lee et al., 2022). This structure enables direct computation of sensitivities (Greeks) with respect to both time and path, critical for dynamic score adjustment mechanisms.

Functional Itô calculus provides the needed Taylor-type expansions, revealing how an adjusted score evolves: $$F(t, X_t) - F(0, X_0) = \int_0^t D F(s, X_s)\,dA_s + \int_0^t V F(s, X_s)\,dX_s + \frac{1}{2}\int_0^t V^2 F(s, X_s)\,d[X]^2_s + \ldots,$$ with $D$ and $V$ the horizontal and vertical derivatives, accommodating both regularity and jump (non-continuity) behavior.

4. Calibration, Model Estimation, and Signature-Based Representations

Modern methodologies use path signatures—the collection of all coordinate-iterated integrals of a path—as a universal, nonparametric feature set to encode history for both model calibration and score adjustment. In the signature SDE approach, coefficients are modeled as linear functionals of signature terms, yielding the "signature SDE": $$dY_t = A_{\theta}(S(Y)_t)\,dt + B_{\theta}(S(Y)_t)\circ dW_t,$$ where $S(Y)_t$ is the truncated signature of the process $Y$ up to $t$ (Semnani et al., 28 May 2025). Parameter estimation is executed by matching expected restricted signatures from empirical data to their polynomial expansions via the Expected Signature Matching Method (ESMM), achieving consistent recovery of path-dependent dynamics.

Algebraic expansions in the signature tensor algebra linearize even Volterra-type and distributed delay equations, facilitating both analysis and numerically efficient computation of path-dependent scores (Jaber et al., 6 Jul 2024).

5. Control Theory and Path-Dependent Hamilton-Jacobi-BeLLMan Equations

Stochastic control problems with history-dependent value functionals are governed by stochastic path-dependent Hamilton–Jacobi–BeLLMan (SPHJB) equations on path spaces: $$-\mathcal{O}_t u(t,x) - H\bigg( t, x, D_x u, D_{xx} u, dw\,u \bigg) = 0$$ with terminal condition $u(T,x) = G(x)$, where $\mathcal{O}_t$ is an adapted horizontal derivative and $H$ encodes the Hamiltonian minimized over control parameters (Qiu, 2020). Uniqueness and well-posedness are established in the viscosity solution framework, adapted to the infinite-dimensional setting of path space, allowing for robust theoretical and computational treatments of optimal, path-dependent score adjustment policies.

6. Algorithms and Score Adjustments in Practice

Implementation in applied settings, including finance, machine learning, and risk management, leverages both analytic and data-driven procedures for updating scores under path-dependence:

• Algorithms based on recurrent (LSTM) neural networks and signatures approximate solutions to PPDEs for both evaluation and dynamic adjustment (Sabate-Vidales et al., 2020).
• Optimal transport and model calibration schemes extend to path-dependent constraints, reducing infinite-dimensional PPDEs to finite-dimensional PDEs via semifiltration techniques (Guo et al., 2018).
• In probabilistic classification, general adjustment methods (UGA, BGA) guarantee loss-reduction on test data under proper scoring rules, even under approximate knowledge of the class distribution, making the strategies robust under path-dependent dataset shift (Heiser et al., 2021).
• Integration of score-based models with Metropolis-Hastings schemes via detailed balance-enforced loss functions allows path-dependent corrections in high-dimensional generative modeling, especially effective under heavy-tailed or multimodal distributions (Aloui et al., 31 Dec 2024).

7. Martingale and Semigroup Approaches: Analytic–Probabilistic Synthesis

Recent advances establish an overview between semigroup theory and path-dependent martingale characterizations. A process $V(t)$ is an $\mathbb{E}$-martingale (for a general expectation operator) if and only if its time-shifted version $U(t) = \Theta_{-t} V(t)$ solves a final value problem involving a path-dependent generator $A$: $$\partial_t U(t) \in -A U(t) + \Phi(t),\qquad U(T) = F,$$ where the compensator $\Psi$ for the original process is identified with an $\mathbb{E}$-derivative, often coinciding with Dupire’s time derivative. This framework enables analytic identification of the necessary score adjustment to enforce the martingale property in systems with path-dependent terminal conditions and links directly to the solution of path-dependent PDEs (Denk et al., 2 Jul 2025).

---

In summary, path-dependent score adjustments blend the analysis of path-dependent differential equations, functional Itô calculus, signature methods, control theory, semigroup frameworks, and scalable algorithms. These approaches allow estimation, calibration, and adjustment of dynamically evolving scores that reflect the complete historical trajectory of the underlying process, providing both a robust theoretical foundation and practical computational tools across a wide range of applications.