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.
Deep learning solves complex volatility equations.
problem Solving path-dependent PDEs in rough volatility.
method Interpreting PDE as BSDE, using neural network reservoir approach.
result Proved theoretical convergence for least-square regression.
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.
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.
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…
The paper develops methods to price options under rough volatility models using BSPDEs.
problem Pricing options in models with non-Markovian dynamics.
method Backward stochastic partial differential equations (BSPDEs) and deep learning for numerical approximations.
result Existence and uniqueness of weak solutions for general nonlinear BSPDEs.
A new deep learning method for option pricing in rough volatility models.
problem Efficient pricing of European options in high-dimensional rough volatility models.
method Time-stepping deep gradient flow method reformulating the option pricing PDE as an energy minimization problem.
result The method respects asymptotic behavior and known bounds for option prices.
TURB-Rot provides a large database of turbulent rotating flow snapshots for research.
problem Lack of large-scale, high-resolution datasets for turbulent rotating flows.
method Direct Numerical Simulations of Navier-Stokes equations with rotation.
result Provides a diverse set of 300K complex images and fields for testing.
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.
Model for high-frequency trading with rough volatility.
problem High-frequency trading dynamics and rough volatility modeling.
method Stochastic partial differential equation (SPDE) with rough volatility driven by a Hawkes process.
result The volatility path of the SPDE is rougher than that driven by a standard Brownian motion.
We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the con…
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…
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. 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.
We use GANs and signatures to approximate conditional laws in filtering and prediction of diffusion processes.
problem Approximating conditional laws for diffusion processes with noisy observations.
method Conditional GANs combined with signatures for approximation.
result Efficient approximation of conditional laws for diffusion processes.
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 non-Gaussian measures' concentration properties in metric spaces.
problem Concentration properties for non-linear Gaussian functionals with non-Gaussian tails.
method Prove generalised Transportation-Cost Inequalities (TCIs) for specific functionals.
result Extended TCIs for rough volatility and Parabolic Anderson Model.
This paper optimizes portfolio selection for multivariate affine and quadratic Volterra models with rough volatilities.
problem Optimizing portfolio selection for multivariate models with rough volatilities and stochastic correlations.
method Investigates continuous-time Markowitz mean-variance problem for multivariate affine and quadratic Volterra models using Riccati backward stochastic differential equations (BSDEs).
result Derives explicit solutions for BSDEs in affine Volterra models and new analytic formulae for quadratic models.
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.
In this paper we consider rough differential equations on a smooth manifold (M). The main result of this paper gives sufficient conditions on the driving vector-fields so that the rough ODE's have global (in time) solutions. The sufficient conditions involve the existence of a complete Riemannian metric …
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.
In this paper, we prove that there exists a dimensional constant δ>0 such that given any background Kähler metric ω, the Calabi flow with initial data u0 satisfying \begin{equation*} \partial \bar \partial u_0 \in L^\infty (M) \text{ and } (1- δ)ω< ω_{u_0} < (1+δ)ω, \end{equation*} admits a unique short time so…
Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
Develops multifactor approximations for SVEs with completely monotone kernels.
problem Approximating SVEs with kernels of completely monotone type.
method Multifactor approximation, Euler discretization, L2-estimation, convergence analysis. result New multifactor Euler scheme reduces computational cost and outperforms SVEs for option pricing.
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly …
New method solves PDEs for any initial condition without retraining.
problem Solving PDEs for different initial conditions requires retraining neural solvers.
method Formulate solution as conditional probability distribution.
result Approximates PDE solution for arbitrary initial conditions.
Classifies scalar second-order PDEs with low-dimensional symmetry groups.
problem Classifying differential equations with specific symmetry groups.
method Algebraic technique based on covariant form for constructing equations.
result Complete classification of quasi-linear scalar second-order PDEs with free symmetry groups of dimension ≤3.
We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data.
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.
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.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, …
The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a nove…
Estimates neural drift for stochastic equations, improving inference on noisy data.
problem Estimating drift in stochastic differential equations with neural networks.
method Non-parametric estimation using ReLU neural networks, enforcing theoretical bounds.
result Practical method for inference on noisy and rough functional data.
We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.
Notes on relative algebroids for geometric problems.
problem Geometric problems and their solutions.
method Explains how relative algebroids arise from geometric problems and introduces their structural theory.
result Relative algebroids unify Lie algebroids with partial differential equations.
Clarifies when solutions to stochastic PDEs stay near given subsets.
problem Understanding the proximity of solutions to stochastic PDEs to given subsets.
method Analyzes distance between closed sets and solutions to stochastic PDEs.
result Clarifies conditions for solutions to stay near given subsets.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
Study deep neural nets for solving complex insurance equations.
problem Solving linear and semilinear parabolic PIDEs in high dimensions.
method Deep neural network algorithms for integro-differential equations.
result Viability of deep learning for solving high-dimensional PIDEs.
We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with s>23. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…
A new unsupervised learning method calibrates rough volatility models efficiently.
problem Efficient calibration of rough volatility models with minimal data.
method Unsupervised learning using BSDE representation and neural networks.
result The proposed scheme minimizes loss and approximates BSDE solution.
We introduce stochastic normalizing flows, an extension of continuous normalizing flows for maximum likelihood estimation and variational inference (VI) using stochastic differential equations (SDEs). Using the theory of rough paths, the underlying Brownian motion is treated as a latent variable and approximated, enabl…
We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in C^n. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise u…
A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and sufficient algebraic and differential conditions for the existence of a variational…
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equati…
Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
Study minimal surfaces in Kropina 3D space, finding only planes as minimal translation surfaces.
problem Characterizing minimal surfaces in Kropina 3D space.
method Solving partial differential equations to characterize minimal surfaces.
result Only planes are minimal translation surfaces in Kropina 3D space.
In this paper, we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, and obtain the local well-posedness result in Hs,s>2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstei…