Study finds eigenvalue patterns on rough manifolds with measurable metrics.
problem Eigenvalue patterns on rough Riemannian manifolds with measurable metrics.
method Demonstrated a Weyl law for eigenvalues of Laplacian and weighted Laplace equations.
result Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds.
Extends Whitney's theorem for functions on rough boundaries.
problem Global extension of manifold-valued functions on domains with rough boundaries.
method Using locally convex spaces of compactly-supported sections of vector bundles, proving the existence of an extension operator.
result The restriction map from everywhere-defined functions is a submersion, allowing local linear splittings.
Volterra square-root process boundary behavior and martingale measures
problem Boundary behavior of the Volterra square-root process
method Comparison principles for Volterra integral equations and generalized Riemann-Liouville fractional equations
result Finiteness of negative p-moments and atom at the boundary for rough kernels Deep neural network approximates flow averages for rough walls in multiscale simulations.
problem Approximating flow averages in rough-wall Stokes flow simulations.
method Fourier neural operator for local averages, parameterized by local wall geometry.
result Stable and accurate HMM solution with reduced micro problem solving cost.
We establish both local and global well-posedness for the heat flow of polyharmonic maps from Rn to a compact Riemannian manifold without boundary for initial data with small BMO norms.
This paper establish the local (or global, resp.) well-posedness of the heat flow of biharmonic maps from Rn to a compact Riemannian manifold without boundary with small local BMO (or BMO, resp.) norms.
Maps preserve distances in non-positively curved spaces.
problem Understanding continuous maps between boundaries of Hadamard manifolds.
method Circumcenter extension maps, cross ratio preservation, visibility conditions.
result Circumcenter extension maps are rough isometries under certain conditions.
Comparison results for rough and non-rough Heston models, tighter bounds on moment explosion times.
problem Comparing Heston models with and without roughness.
method Comparison principle for non-linear Volterra integral equations.
result Tighter bounds on moment explosion times for rough Heston models.
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.
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.
We investigate the well-posedness of (i) the heat flow of harmonic maps from Rn to a compact Riemannian manifold without boundary for initial data in BMO; and (ii) the hydrodynamic flow (u,d) of nematic liquid crystals on Rn for initial data in BMO−1×BMO.
Derives a rough SABR formula for short maturities.
problem Modeling volatility smiles under rough volatility.
method Derives an ODE and solves it numerically.
result Develops a very accurate approximation called the rough SABR formula.
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.
Develops a new method for quantizing rough volatility for volatility derivatives pricing.
problem Pricing volatility derivatives in rough volatility models.
method Functional quantization of rough volatility using offline computable quantizers.
result Pricing VIX Futures in the rough Bergomi model shows competitive results.
Researchers compute Greeks for rough Volterra SV models using Malliavin calculus.
problem Computing Greeks under rough Volterra stochastic volatility models.
method Malliavin calculus techniques, extending integration by parts to non-square integrable functionals.
result Formulas for computing Greeks (Delta, Gamma, Rho, Vega) under various rough Volterra SV models.
Measures of implied volatility roughness corrected for bias.
problem Bias in measuring implied volatility roughness.
method Examined implied volatility of short-term options and VIX index, corrected for bias.
result Corrected measures indicate appropriate proxies for underlying volatility.
This study simplifies rough Heston model's conditional density equation.
problem Analyzing rough volatility in financial models.
method Pathwise transformation and Fokker-Planck formulation of conditional density equation.
result Transformed equation yields deterministic PDE with path-dependent coefficients.
Explains affine rough models and their transform formulas.
problem Rough volatility models in financial markets.
method Explains affine structure and transforms formulas.
result Shows exponential-affine transform formulas for rough models.
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. New methods price American options in rough volatility models.
problem Pricing American options under rough volatility.
method Integrating deep-signature and signature-kernel learning into optimal stopping problem solutions.
result Performance comparison in rough Heston and rough Bergomi models.
Study approximates weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
problem Approximating weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
method Used Euler type scheme with integrated kernels to study weak convergence rate.
result Obtained weak convergence rate of min(3α−1,1) for discretised rough Ornstein-Uhlenbeck process and stochastic rough volatility model. Study approximates rough stochastic volatility models using diffusion processes.
problem High computational cost in simulating rough stochastic volatility models.
method Approximates stochastic Volterra equations with an N-dimensional diffusion process.
result Approximations converge strongly with superpolynomial rate in N.
Integrates rough geometric forms on manifolds.
problem Integrating rough forms on complex manifolds.
method Combines Whitney's geometric integration and sewing approaches.
result Introduced distributional k-forms for integration.
Volatility models must be rough to match market skew.
problem Inconsistent non-rough volatility models with power law volatility skew.
method Asymptotic expansion and continuous price dynamics analysis.
result Volatility must be rough to align with market skew.
Let p be a real number greater than one and let G be a connected graph of bounded degree. In this paper we introduce the p-harmonic boundary of G. We use this boundary to characterize the graphs G for which the constant functions are the only p-harmonic functions on G. It is shown that any continuous func…
The paper designs multi-factor models for rough volatility, making them easier to simulate.
problem Efficient simulation of rough volatility models due to their non-Markovian and non-semimartingale nature.
method Designs tractable multi-factor stochastic volatility models with Markovian structure.
result Derives a numerical method for solving fractional Riccati equations in rough Heston models.
Estimates roughness of volatility from discrete variance data.
problem Estimating roughness exponent of stochastic volatility from discrete observations of integrated variance.
method Pathwise estimator based on fractional Brownian motion with drift.
result Strong consistency theorems for rough volatility models.
Study finds Bitcoin volatility exhibits roughness, not constant over time.
problem Investigating roughness in Bitcoin volatility.
method Multifractal detrended fluctuation analysis and shuffled time series analysis.
result Bitcoin volatility exhibits roughness (generalized Hurst exponent < 1/2).
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…
The paper proves global existence of solutions for rough differential equations on smooth manifolds.
problem Global existence of solutions for rough differential equations on smooth manifolds.
method Sufficient conditions involving a complete Riemannian metric and bounded covariant derivatives of driving fields and their commutators.
result Global solutions exist for rough differential equations under certain conditions.
A hybrid framework for American option pricing under time-varying rough volatility.
problem Pricing American options under time-varying rough volatility.
method Signature method combined with gradient-boosted ensemble for Hurst parameter estimation, regime switch, and Random Fourier Features for acceleration.
result The proposed hybrid framework improves performance over fixed-roughness baselines and reduces duality gaps in some regimes.
New method analyzes volatility models for option prices, especially in rough volatility.
problem Analyzing option prices in rough volatility models.
method Introducing a new methodology to analyze stochastic volatility models, focusing on asymptotics and numerics.
result Detailed expansion and numerical evidence for implied volatility in rough volatility models.
Volatility roughness studied using fractional noise-driven models.
problem Volatility roughness interpretation.
method Data-reconstructed fractional volatility model with fractional noise.
result Option pricing equation and solution derived using Malliavin calculus.
New mathematical tools help analyze rough volatility in financial markets.
problem Mathematical models of rough volatility lost Markovianity and semi-martingality.
method Use of Hairer's regularity structures, an extension of rough path theory.
result Shows regularity structures can analyze rough volatility models.
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.
Efficient simulation scheme for rough Heston model reduces computational cost.
problem Accurate and efficient simulation of the rough Heston model for option pricing.
method Weak simulation scheme based on Markovian approximations of the rough Heston process.
result The new scheme exhibits second order weak convergence with linear computational cost.
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.
The paper develops a new model for rough volatility in commodity markets.
problem Calibration of rough volatility models for commodity futures prices.
method Developed a general rough volatility model with automatic calibration and treatment of the Samuelson effect.
result Calibrated rBergomi and rHeston models to WTI Crude Oil futures options data.
Sharp bounds on weak convergence rate for rough volatility models.
problem Understanding the convergence rate in discretizing rough volatility models.
method Analyzing general and linear models to derive bounds.
result Sharper bound of \(H + 1/2\) for linear models.
Bitcoin volatility shows multifractal structure, contradicting rough volatility models.
problem Applying rough volatility models to Bitcoin volatility data.
method Normalised p-variation framework, multifractal Detrended Fluctuation Analysis, log-log moment scaling, wavelet leaders.
result Bitcoin volatility exhibits multifractal structure, violating rough volatility model assumptions.
Develops efficient Monte Carlo methods for rough volatility models.
problem Challenges in simulating rough volatility processes.
method Extension of Donsker's approximation to fractional Brownian motion.
result Justifies the validity of simple Monte Carlo methods for rough volatility models.
Develops rough set classifiers using confusion matrices.
problem Evaluating classifier quality in machine learning.
method Combines rough set theory with confusion matrices.
result Defines indices and classifiers based on rough confusion matrices.
Perfect hedging of options with a dynamic portfolio in rough volatility models.
problem Hedging options in rough volatility models.
method Presented a simple but general result showing perfect hedging with a dynamic portfolio of underlying and variance swap.
result Rough volatility models significantly reduce hedging error compared to diffusion-based models.
The paper explores how score-driven models can approximate rough volatility.
problem Modeling rough volatility with long memory structures.
method Extending score-driven models to include infinite-lag structures and heavy-tailed decay.
result Score-driven models converge to fractional Ornstein-Uhlenbeck processes under appropriate scaling.
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.
Analyzes how rough volatility affects stock pricing and risk premium.
problem Impact of non-deterministic volatility risk on stock pricing.
method Rough volatility model under historical measure, analysis of stochastic volatility risk.
result Impact of non-deterministic volatility risk on pricing is significant.
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.
No-arbitrage leads to power-law market impact and rough volatility.
problem Understanding market impact and volatility dynamics.
method Mathematical proof and analysis of stochastic Volterra equations.
result Market impact function is power-law, implying rough volatility.