arXiv subjects

Paul P. Hager

Publications and source records attributed to Paul P. Hager.

10 recordsLinked to original sources

arXiv2026

Expected signatures via partial integration, coordinate change and symmetrization

We study signature transformations of heterogeneous paths $Y=(A,X)$ whose components may differ in regularity and probabilistic structure. We introduce an invertible change of coordinates $Ψ$ such that the transformed signature $Ψ\circ\mathrm{Sig}$ eliminates mixed integration against the irregular component $X$ and admits a representation in terms of signature coordinates of $X$ and iterated integration against the regular component $A$. In addition, we exploit this representation to further represent partially symmetrized signatures. Our main application concerns new expected signature formulas for processes with deterministic augmentation. On the analytical side, these formulas are leveraged to study moment problems. On the numerical side, they enable accurate computation of expected signatures, thereby overcoming typical computational bottlenecks in applications. We illustrate these advantages in a signature-based stochastic control problem driven by fractional Brownian motion.

math.PR↗

arXiv2026

Dynamic Universal Approximation via Signature Controlled Differential Equations

We study signature controlled differential equations (Sig-CDEs), that is, path-dependent controlled differential equations (CDEs) whose vector fields factor through the signature map. Working on spaces of stopped Hölder paths, we develop an existence, uniqueness, and stability theory for general path-dependent CDEs, and translate these pathwise well-posedness criteria into conditions on the corresponding signature functionals. We then prove dynamic universality: simply parametrized Sig-CDEs approximate the solution path of any well-posed path-dependent CDE arbitrarily well, uniformly over bounded sets of controls and initial histories, with global variants obtained using weighted spaces. Within this framework, entire maps of group-like elements provide a specific class of Sig-CDEs. Using a new class of limiting tensor spaces, we recast Sig-CDEs as infinite-dimensional classical CDEs and prove their well-posedness via a gauge-type scaling argument, thereby establishing a principled way to lift generic path-dependent dynamics. Lastly, we study truncated Sig-CDEs as finite-dimensional differential equations on step$-N$ Lie groups under intrinsic conditions, that is, with well-posedness formulated in terms of the underlying group metric.

math.CA↗

arXiv2026

Expanding the rough Heston model in $H$

We study the dependence of the fractional Riccati equation in the rough Heston model on the Hurst parameter $H$. For each expansion point $H_0\in(-1/2,1/2]$, we derive a Taylor expansion of the Riccati solution in $H$, whose coefficients are characterized recursively as solutions of linear Volterra equations with fractional-logarithmic kernels. We prove local uniform convergence of the resulting Taylor series and, in particular, analyticity of the fractional Riccati solution in the Hurst parameter. Through the affine transform formula, this yields approximations of the rough Heston characteristic function and Fourier prices. Numerically, once a reference solution at $H_0$ is available, the expansion coefficients can be computed recursively and evaluated for many nearby values of $H$. We implement the method around $H_0=1/2$, using the classical Heston solution, and around $H_0=0$, using a Padé approximation. Experiments for European call options indicate that low expansion orders already provide accurate implied volatilities across a wide range of Hurst parameters, including the hyper-rough regime.

q-fin.MF↗

arXiv2026

The Volterra signature

Modern approaches for learning from non-Markovian time series, such as recurrent neural networks, neural controlled differential equations or transformers, typically rely on implicit memory mechanisms that can be difficult to interpret or to train over long horizons. We propose the \emph{Volterra signature} $\mathrm{VSig}(x;K)$ as a principled, explicit feature representation for history-dependent systems. By developing the input path $x$ weighted by a temporal kernel $K$ into the tensor algebra, we leverage the associated Volterra--Chen identity to derive rigorous learning-theoretic guarantees. Specifically, we prove an \emph{injectivity} statement (identifiability under augmentation) that leads to a \emph{universal approximation} theorem on the infinite dimensional path space, which in certain cases is achieved by \emph{linear functionals} of $\mathrm{VSig}(x;K)$. Moreover, we demonstrate applicability of the \emph{kernel trick} by showing that the inner product associated with Volterra signatures admits a closed characterization via a two-parameter integral equation, enabling numerical methods from PDEs for computation. For a large class of exponential-type kernels, $\mathrm{VSig}(x;K)$ solves a linear state-space ODE in the tensor algebra. Combined with inherent invariance to time reparameterization, these results position the Volterra signature as a robust, computationally tractable feature map for data science. We demonstrate its efficacy in dynamic learning tasks on real and synthetic data, where it consistently improves classical path signature baselines.

stat.ML↗

arXiv2026

Computational aspects of the Volterra Signature

The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive Picard iterates of linear controlled Volterra equations, making their exact computation of additional mathematical interest. However, the kernel introduces substantial algorithmic challenges. We provide a resolution by first decomposing the Chen-type convolution relation established in [arXiv:2603.04525] into analytic and arithmetic parts, and then introducing several efficient algorithms: a general approximative scheme with quadratic complexity $O(J^2)$ in the number of time steps $J$, an FFT-based acceleration with complexity $O(J\log J)$ for convolution kernels on uniform grids, and an exact recursion with complexity $O(JR^2)$ for kernels admitting a state-space representation of dimension $R$; retaining standard signature complexity in the path dimension and truncation level $N$. We further show that the number of factors in matrix-valued kernels of the form $K(t,s)=\sum_p k_p(t-s)A_p$ do not increase the asymptotic complexity in $J$ and $N$. Finally, we derive a finite-difference predictor--corrector scheme for the associated Volterra signature kernel. All algorithms are implemented in the publicly available JAX-based package "tensordev".

math.NA↗

arXiv2026

Microstructural Foundation of Rough Log-Normal Volatility Models

We establish a microstructural foundation of the rough Bergomi model. Specifically, we consider a sequence of order driven financial market models where orders to buy or sell an asset arrive according to a Poisson process and have a long lasting impact on volatility. Using a recently established C-tightness result for càdlàg processes we establish the weak convergence of the price-volatility process to a log-normal rough volatility model. Our weak convergence result is accompanied by weak error rates that employ a recently established Clark-Ocone formula for Poisson processes and turn our microstructure model into viable alternative to classical simulation schemes. The weak error rates strongly hinge on Poisson arrival dynamics and are novel to the rough microstructure literature.

q-fin.MF↗

arXiv2025

Expected Signature Kernels for Lévy Rough Paths

The expected signature kernel arises in statistical learning tasks as a similarity measure of probability measures on path space. Computing this kernel for known classes of stochastic processes is an important problem that, in particular, can help reduce computational costs. Building on the representation of the expected signature of (inhomogeneous) Lévy processes with absolutely continuous characteristics as the development of an absolutely continuous path in the extended tensor algebra [F.-H.-Tapia, Forum of Mathematics: Sigma (2022), "Unified signature cumulants and generalized Magnus expansions"], we extend the arguments developed for smooth rough paths in [Lemercier-Lyons-Salvi, "Log-PDE Methods for Rough Signature Kernels"] to derive a PDE system for the expected signature of inhomogeneous Lévy processes. As a specific example, we see that the expected signature kernel of Gaussian martingales satisfies a Goursat PDE.

math.PR↗

arXiv2025

A Mean-Field Game of Market Entry: Portfolio Liquidation with Trading Constraints

We consider both $N$-player and mean-field games of optimal portfolio liquidation in which the players are not allowed to change the direction of trading. Players with an initially short position of stocks are only allowed to buy while players with an initially long position are only allowed to sell the stock. Under suitable conditions on the model parameters we show that the games are equivalent to games of timing where the players need to determine the optimal times of market entry and exit. We identify the equilibrium entry and exit times and prove that equilibrium mean-trading rates can be characterized in terms of the solutions to a highly non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium both in the mean-field and the $N$-player game.

q-fin.MF↗

arXiv2025

On expected signatures and signature cumulants in semimartingale models

The concept of signatures and expected signatures is vital in data science, especially for sequential data analysis. The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors [Unified signature cumulants and generalized Magnus expansions, FoM Sigma '22] we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity.

stat.ML↗

arXiv2023

Mean-Field Liquidation Games with Market Drop-out

We consider a novel class of portfolio liquidation games with market drop-out ("absorption"). More precisely, we consider mean-field and finite player liquidation games where a player drops out of the market when her position hits zero. In particular round-trips are not admissible. This can be viewed as a no statistical arbitrage condition. In a model with only sellers we prove that the absorption condition is equivalent to a short selling constraint. We prove that equilibria (both in the mean-field and the finite player game) are given as solutions to a non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium in the MFG and the existence of a unique equilibrium in the $N$-player game. We establish the convergence of the equilibria in the finite player games to the obtained mean-field equilibrium and illustrate the impact of the drop-out constraint on equilibrium trading rates.

q-fin.MF↗