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We develop a kernel-based solver for path-dependent PDEs (PPDEs) along with a convergence theory.
Theory of reproducing kernels
N. Aronszajn · 1950
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Fractional Brownian motions, fractional noises and applications
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Climate dynamics as a nonlinear Brownian motion
D. Sonechkin · 1998
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Scattered data approximation
H. Wendland · 2004
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Ergodicity of stochastic differential equations driven by fractional Brownian motion
M. Hairer · 2005
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Consistency and robustness of kernel-based regression in convex risk minimization
A. Christmann and I. Steinwart · 2007
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Differential equations driven by rough paths
T. J. Lyons, M. Caruana, and T. Lévy · 2007
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Derivative reproducing properties for kernel methods in learning theory
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Multidimensional stochastic processes as rough paths: theory and applications
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Asymptotic theory for Brownian semi-stationary processes with application to turbulence
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Assessing relative volatility/intermittency/energy dissipation
O. E. Barndorff-Nielsen, M. S. Pakkanen, and J. Schmiegel · 2014
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C. Bayer, P. Friz, and J. Gatheral · 2016
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Probabilistic numerical methods for partial differential equations and Bayesian inverse problems
J. Cockayne, C. Oates, T. Sullivan, and M. Girolami · 2016
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Hybrid scheme for Brownian semistationary processes
M. Bennedsen, A. Lunde, and M. S. Pakkanen · 2017
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On the convergence of monotone schemes for path-dependent PDEs
Z. Ren and X. Tan · 2017
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Signature moments to characterize laws of stochastic processes
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Solving high-dimensional partial differential equations using deep learning
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DGM: A deep learning algorithm for solving partial differential equations
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Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization: From a Game Theoretic Approach to Numerical Approximation and Algorithm Design
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A martingale approach for fractional Brownian motions and related path dependent PDEs
Functional limit theorems for the fractional Ornstein–Uhlenbeck process
J. Gehringer and X.-M. Li · 2022
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Slow-fast systems with fractional environment and dynamics
X.-M. Li and J. Sieber · 2022
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Neural stochastic PDEs: Resolution-invariant learning of continuous spatiotemporal dynamics
C. Salvi, M. Lemercier, and A. Gerasimovics · 2022
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Path dependent Feynman–Kac formula for forward backward stochastic Volterra integral equations
H. Wang, J. Yong, and J. Zhang · 2022
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Error analysis of kernel/GP methods for nonlinear and parametric PDEs
P. Batlle, Y. Chen, B. Hosseini, H. Owhadi, and A. M. Stuart · 2023
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Rough volatility
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Sample paths estimates for stochastic fast-slow systems driven by fractional Brownian motion
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Rough volatility, path-dependent PDEs and weak rates of convergence
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Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes
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Neural signature kernels as infinite-width-depth-limits of controlled ResNets
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Global universal approximation of functional input maps on weighted spaces
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A neural RDE-based model for solving path-dependent PDEs
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New directions in the applications of rough path theory
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Optimal stopping via distribution regression: a higher rank signature approach
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Non-adversarial training of neural SDEs with signature kernel scores
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Deep curve-dependent PDEs for affine rough volatility
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Gaussian Volterra processes as models of electricity markets
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Path-dependent PDEs for volatility derivatives
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Weighted signature kernels
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