Paper explores rough path theory for frictionless markets, linking NCFL to unbiased rough integrators.
problem Tackles the limits of rough path theory in frictionless markets.
method Investigates the capacity of rough path theory to support No Free Lunch markets.
result Establishes a 'Rough Kreps-Yan' theorem linking NCFL to unbiased rough integrators.
Foundation for robust finance using rough path theory.
problem Mathematical models of financial markets under Knightian uncertainty.
method Introducing Property (RIE) for càdlàg paths, proving existence of rough integrals, verifying admissibility of trading strategies.
result Existence and stability of rough path integrals for non-gradient integrands.
Develops a new model-free approach to portfolio theory using rough paths.
problem Handles more general portfolios without probabilistic assumptions.
method Rough path theory for stochastic portfolio theory (SPT).
result Asymptotic growth rates of various portfolios match.
Unified approach to stochastic control, filtering, and stopping using rough paths.
problem Addressing gaps in classical problems of stochastic control, filtering, and stopping.
method Combining rough path theory with controlled rough paths to provide a pathwise deterministic framework.
result Established rigorous connection between candidate solutions and Hamilton-Jacobi-Bellman equation.
Novel approach to financial derivatives pricing using rough path theory.
problem No-arbitrage conditions in financial markets necessitating precise integration methods.
method Developed a polynomial-based approximation class for rough path functionals, extending to non-geometric rough paths.
result Motivated a hypothesis for payoff functionals in financial markets, facilitating analysis.
The paper introduces surface signatures for irregular surfaces and rough surfaces.
problem Characterizing and integrating highly irregular paths and surfaces.
method Introducing surface signatures and proving extension theorems.
result Surface signatures are universal for surface holonomy and rough surfaces.
Framework for training stochastic spiking neural networks with rough signals.
problem Training stochastic spiking neural networks with noisy spike timing and dynamics.
method Rough path theory and signature kernels for gradient computation.
result Pathwise gradients of SSNNs' trajectories and event times exist and satisfy a recursive relation.
Model rough volatility using RDEs with correlated Brownian motion and fractional Brownian motion.
problem Modeling rough volatility with correlated stochastic processes.
method Developed a method to lift Brownian motion and rough paths, applying it to fractional Brownian motion to model rough volatility.
result Calibrated a new rough volatility model to market data.
Study of gamma-hedging using rough paths for European and exotic options.
problem Applying rough paths to gamma-hedging strategies for derivatives.
method Rough-path theory applied to discrete-time gamma-hedging strategy.
result Sure replication of European and exotic derivatives under regular pricing signals.
We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of Itô's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlle…
Develops pathwise analysis for log-optimal portfolios using rough paths theory.
problem Analyzing stability and approximation of log-optimal portfolios.
method Pathwise approach based on càdlàg rough paths theory.
result Establishes pathwise stability and error estimates for log-optimal portfolios.
Study large deviations for hypoelliptic diffusion on sub-Riemannian manifolds.
problem Large deviations for hypoelliptic diffusion measures on sub-Riemannian manifolds.
method Rough path theory and manifold-valued Malliavin calculus.
result Proved a large deviation principle for pinned hypoelliptic diffusion measures.
New method uses LSTM and signature theory to solve complex financial PDEs.
problem Solving path-dependent PDEs for financial derivatives pricing.
method Combining LSTM networks and rough paths theory.
result Efficient algorithms for pricing and hedging path-dependent derivatives.
A new model adapts Hurst parameter in real-time for volatility forecasting.
problem Capturing volatility dynamics and clustering in financial markets.
method Rough Bergomi model with EWMA-driven time-dependent Hurst parameter.
result Empirical validation shows superior performance in diverse asset classes.
Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. It draws on the analysis of LC Young and the geometric algebra of KT Chen. The concepts and the uniform estimates, have widespread application and have simplified proofs of basic questions fr…
The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.
problem Approximating functionals of non-geometric rough paths.
method Extending rough paths with time and quadratic variation terms, proving uniform approximation.
result Linear functionals of extended signatures uniformly approximate continuous functionals.
We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of [Bayer et al; A regularity structure for rough vola…
New methods for volatility modeling using rough paths and signatures.
problem Calibrating implied volatility surfaces in various stochastic models.
method Analytical approximations and signature-based models based on rough path theory.
result Signature-based models achieve comparable accuracy to analytical expansions and can capture more complex dynamics.
Estimates roughness of financial volatility paths using horizontal visibility graphs.
problem Estimating roughness in financial volatility models.
method Introduces L+(t) for first-passage horizons, treating uncensored observations as first-passage times.
result Estimates roughness through a single tail exponent θ, separating rough Bergomi volatility from classical models.
Signature Isolation Forest removes constraints from FIF by using rough path theory's signature transform.
problem Challenges in FIF's linear inner product and dictionary choices leading to unreliable results.
method Introduces Signature Isolation Forest using rough path theory's signature transform to remove linearity constraints.
result Demonstrates relevance of methods through numerical experiments and real-world applications.
Derives functional Itô formula for non-anticipative maps of rough paths.
problem Functional Itô formula for non-anticipative maps of càdlàg rough paths.
method Approximation properties of the signature and Marcus transformation.
result Functional Taylor expansion for sufficiently regular non-anticipative maps.
Framework combines random features with CDEs for efficient time-series learning.
problem Efficient training of time-series models with strong inductive bias.
method Random Fourier CDEs and Random Rough DEs using continuous-time reservoirs and log-ODE discretization.
result Unified perspective on random-feature reservoirs and path-signature theory.
Scalable machine learning with path signatures for time series and graphs.
problem Challenges in real-world time series and graph data.
method Combines rough path theory with probabilistic, deep, and kernel methods.
result Scalable models for time series and graph data.
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
New method solves optimal stopping problems using rough path signatures.
problem Optimal stopping problems in finance and other fields.
method Using rough path signatures and deep neural networks.
result Solves optimal stopping problems efficiently under minimal assumptions.
New algorithm reduces MFGs with common noise complexity.
problem Prohibitive computational cost in solving MFGs with common noise.
method Signatured deep fictitious play based on rough path theory.
result Significantly reduced computational complexity and improved efficiency.
We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough-paths allow us to generali…
A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extr…
Universal approximation for stochastic processes using Brownian motion.
problem Approximating stochastic processes with linear functionals.
method Establishing Lp-type universal approximation theorems for rough path spaces. result Linear functionals on the signature of time-extended Brownian motion can approximate any p-integrable stochastic process. Neural RDEs extend CDEs to irregular time series.
problem Modeling long irregular time series efficiently.
method Representing time series through log-signature and solving RDEs.
result Significant training speed-ups and improved model performance.
Study rough volatility models using path-dependent PDEs and fractional Brownian motions.
problem Modeling and analyzing rough volatility in financial markets.
method Showed conditional expectations are unique classical solutions to path-dependent PDEs derived from functional Itô formula. Leverage these to study weak rates of convergence for discretized stochastic integrals.
result Obtained optimal weak error rates for approximating log-stock prices in rough volatility models.
Develops a new solver for path-dependent PDEs using signature kernels.
problem Solving path-dependent PDEs (PPDEs) efficiently and accurately.
method Uses signature kernels to solve PPDEs by approximating the solution with minimal norm in a reproducing kernel Hilbert space.
result Proves the consistency of the numerical scheme, ensuring convergence to PPDE solutions as the number of collocation points increases.
The study examines insurance demand under rough volatility and path-dependent shocks.
problem Optimal insurance and investment strategies under rough volatility and path-dependent shocks.
method Rough volatility model and Hawkes process with power kernel, Functional Ito formula extension.
result Individuals demand more catastrophe insurance when path-dependent effects are considered.
Develops a test to distinguish between standard and rough volatility.
problem Determining whether asset volatility follows a standard semimartingale or a rough process.
method Uses sample autocovariance of high-frequency asset return data to detect negative autocorrelation at high frequencies.
result Evidence of rough volatility in SPY high-frequency data.
Study finds rough volatility models underperform in SPX option pricing.
problem Inconsistency of rough volatility models with SPX option prices.
method Empirical study using SPX options data, comparing rough and Markovian models.
result Rough volatility models with H∈(0,1/2) are inconsistent with SPX smiles, especially at short maturities. We provide a draft of a theory of geometric integration of rough differential forms which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of r…
We bring the theory of rough paths to the study of non-parametric statistics on streamed data. We discuss the problem of regression where the input variable is a stream of information, and the dependent response is also (potentially) a stream. A certain graded feature set of a stream, known in the rough path literature…
We introduce a notion of p-rough integrator on any Banach manifolds, for any p≥1, which plays the role of weak geometric Holder p-rough paths in the usual Banach space setting. The awaited results on rough differential equations driven by such objects are proved, and a canonical representation is given if the man…
We introduce time-inhomogeneous stochastic volatility models, in which the volatility is described by a nonnegative function of a Volterra type continuous Gaussian process that may have very rough sample paths. The main results obtained in the paper are sample path and small-noise large deviation principles for the log…
Universal approximation for rough paths and Lévy processes.
problem Approximating continuous functionals of càdlàg paths.
method Linear functionals of time-extended signatures.
result Universal approximation theorem for continuous functionals of càdlàg paths.
Study finds roughness in volatility despite diffusive instantaneous volatility.
problem Determining the roughness of volatility in financial assets.
method Non-parametric method based on normalized p-th variation for estimating roughness of sample paths. result Realized volatility exhibits rough behavior with a significantly smaller Hurst exponent than instantaneous volatility.
Paper introduces branched signature model for efficient computation and data-driven applications.
problem Efficient computation and data-driven modeling of branched rough paths.
method Develops a universal approximation theorem and constructs an extension map to realize branched signatures.
result Explicit construction of branched signatures via an extension map for efficient computation.
The paper develops a deep signature approach for option pricing under non-Markovian stochastic volatility models.
problem Pricing options under non-Markovian stochastic volatility models is challenging due to the dependence on historical paths.
method Reformulate the asset dynamics as a rough stochastic differential equation and represent rough paths via signatures. Apply standard analytical tools to solve the transformed equation.
result The deep signature approach provides a theoretically grounded and computationally efficient framework for option pricing.
2DSig-Detect detects adversarial perturbations in images.
problem Adversarial attacks degrade image model performance.
method 2D-signature embedded semi-supervised framework using rough path theory.
result 2DSig-Detect outperforms other methods in detecting adversarial perturbations.
We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in R2. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they e…
New method for pricing European options in rough LSV models.
problem Pricing European options in non-Markovian local stochastic volatility models.
method Conditional LSV dynamics, rough path theory, rough partial differential equations (RPDEs).
result Established a PDE pricing method for non-Markovian models.
Two signature-based methods solve optimal stopping in non-Markovian frameworks.
problem Optimal stopping in non-Markovian frameworks, particularly pricing American options.
method Primal and dual formulations using linear functionals of rough path signatures.
result Both primal and dual methods converge and provide numerical examples.
The study uses response theory to understand RNNs processing input signals.
problem Understanding how RNNs process sequential data.
method Deriving a Volterra series representation for SRNNs output using response theory from nonequilibrium statistical mechanics.
result SRNNs can be viewed as kernel machines operating on a reproducing kernel Hilbert space associated with the response feature.