Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A path-dependent PDE solver based on signature kernels (2403.11738v3)

Published 18 Mar 2024 in math.NA, cs.NA, and q-fin.CP

Abstract: We develop a provably convergent kernel-based solver for path-dependent PDEs (PPDEs). Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal recovery problem by approximating the solution of a PPDE with an element of minimal norm in the signature reproducing kernel Hilbert space (RKHS) constrained to satisfy the PPDE at a finite collection of collocation paths. In the linear case, we show that the optimisation has a unique closed-form solution expressed in terms of signature kernel evaluations at the collocation paths. We prove consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases. Finally, several numerical examples are presented, in particular in the context of option pricing under rough volatility. Our numerical scheme constitutes a valid alternative to the ubiquitous Monte Carlo methods.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (55)
  1. E. Abi Jaber and S. Villeneuve. Gaussian agency problems with memory and linear contracts. Available at SSRN 4226543, 2022.
  2. N. Aronszajn. Theory of reproducing kernels. Transactions of the American mathematical society, 68(3):337–404, 1950.
  3. Assessing relative volatility/intermittency/energy dissipation. Electron. J. Statist., 8(2):1996–2021, 2014.
  4. O. E. Barndorff-Nielsen and J. Schmiegel. Brownian semistationary processes and volatility/intermittency. Advanced financial modelling, 8:1–26, 2009.
  5. Error analysis of kernel/GP methods for nonlinear and parametric PDEs. arXiv preprint arXiv:2305.04962, 2023.
  6. Pricing under rough volatility. Quantitative Finance, 16(6):887–904, 2016.
  7. Rough volatility. SIAM, 2023.
  8. Hybrid scheme for Brownian semistationary processes. Finance and Stochastics, 21:931–965, 2017.
  9. Generalizing from several related classification tasks to a new unlabeled sample. Advances in neural information processing systems, 24, 2011.
  10. Rough volatility, path-dependent PDEs and weak rates of convergence. arXiv preprint arXiv:2304.03042, 2023.
  11. Weighted signature kernels. The Annals of Applied Probability, 34(1A):585–626, 2024.
  12. Solving and learning nonlinear PDEs with gaussian processes. arXiv preprint arXiv:2103.12959, 2021.
  13. Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes. arXiv preprint arXiv:2304.01294, 2023.
  14. I. Chevyrev and H. Oberhauser. Signature moments to characterize laws of stochastic processes. arXiv preprint arXiv:1810.10971, 2018.
  15. A. Christmann and I. Steinwart. Consistency and robustness of kernel-based regression in convex risk minimization. Bernoulli, 13(3):799–819, 2007.
  16. Neural signature kernels as infinite-width-depth-limits of controlled ResNets. International Conference on Machine Learning, 2023.
  17. Sk-tree: a systematic malware detection algorithm on streaming trees via the signature kernel. In 2021 IEEE international conference on cyber security and resilience (CSR), pages 35–40. IEEE, 2021.
  18. Probabilistic numerical methods for partial differential equations and Bayesian inverse problems. arXiv preprint arXiv:1605.07811, 2016.
  19. Asymptotic theory for brownian semi-stationary processes with application to turbulence. Stochastic processes and their applications, 123(7):2552–2574, 2013.
  20. Global universal approximation of functional input maps on weighted spaces. arXiv preprint arXiv:2306.03303, 2023.
  21. B. Dupire and V. Tissot-Daguette. Functional expansions. arXiv preprint arXiv:2212.13628, 2022.
  22. Sample paths estimates for stochastic fast-slow systems driven by fractional brownian motion. Journal of Statistical Physics, 179(5):1222–1266, 2020.
  23. A neural rde-based model for solving path-dependent PDEs. arXiv preprint arXiv:2306.01123, 2023.
  24. New directions in the applications of rough path theory. IEEE BITS the Information Theory Magazine, 2023.
  25. Multidimensional stochastic processes as rough paths: theory and applications, volume 120. Cambridge University Press, 2010.
  26. J. Gehringer and X.-M. Li. Functional limit theorems for the fractional ornstein–uhlenbeck process. Journal of Theoretical Probability, 35(1):426–456, 2022.
  27. M. Hairer. Ergodicity of stochastic differential equations driven by fractional brownian motion. Ann. Probab., 33(2):703–758, 2005.
  28. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018.
  29. Optimal stopping via distribution regression: a higher rank signature approach. arXiv preprint arXiv:2304.01479, 2023.
  30. Non-adversarial training of neural SDEs with signature kernel scores. Advances in Neural Information Processing Systems, 2023.
  31. A. Jacquier and M. Oumgari. Deep curve-dependent PDEs for affine rough volatility. SIAM Journal on Financial Mathematics, 14(2):353–382, 2023.
  32. Deep signature transforms. Advances in Neural Information Processing Systems, 32, 2019.
  33. F. J. Király and H. Oberhauser. Kernels for sequentially ordered data. Journal of Machine Learning Research, 20, 2019.
  34. Distribution regression for sequential data. In International Conference on Artificial Intelligence and Statistics, pages 3754–3762. PMLR, 2021.
  35. X.-M. Li and J. Sieber. Slow-fast systems with fractional environment and dynamics. The Annals of Applied Probability, 32(5):3964–4003, 2022.
  36. Differential equations driven by rough paths. Springer, 2007.
  37. Signature kernel conditional independence tests in causal discovery for stochastic processes. arXiv preprint arXiv:2402.18477, 2024.
  38. Gaussian volterra processes as models of electricity markets. arXiv preprint arXiv:2311.09384, 2023.
  39. Neural rough differential equations for long time series. In International Conference on Machine Learning, pages 7829–7838. PMLR, 2021.
  40. A. Pannier. Path-dependent PDEs for volatility derivatives. arXiv preprint arXiv:2311.08289, 2023.
  41. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686–707, 2019.
  42. Z. Ren and X. Tan. On the convergence of monotone schemes for path-dependent PDEs. Stochastic Processes and their Applications, 127(6):1738–1762, 2017.
  43. Solving path dependent PDEs with LSTM networks and path signatures. arXiv preprint arXiv:2011.10630, 2020.
  44. The signature kernel is the solution of a Goursat PDE. SIAM Journal on Mathematics of Data Science, 3(3):873–899, 2021.
  45. SigGPDE: Scaling sparse Gaussian processes on sequential data. In International Conference on Machine Learning, pages 6233–6242. PMLR, 2021.
  46. Neural stochastic PDEs: Resolution-invariant learning of continuous spatiotemporal dynamics. Advances in Neural Information Processing Systems, 35:1333–1344, 2022.
  47. Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems, 34:16635–16647, 2021.
  48. Y. F. Saporito and Z. Zhang. Path-dependent deep Galerkin method: A neural network approach to solve path-dependent partial differential equations. SIAM Journal on Financial Mathematics, 12(3):912–940, 2021.
  49. J. Sirignano and K. Spiliopoulos. DGM: A deep learning algorithm for solving partial differential equations. Journal of computational physics, 375:1339–1364, 2018.
  50. D. Sonechkin. Climate dynamics as a nonlinear brownian motion. International Journal of Bifurcation and Chaos, 8(04):799–803, 1998.
  51. Hilbert space embeddings and metrics on probability measures. The Journal of Machine Learning Research, 11:1517–1561, 2010.
  52. F. Viens and J. Zhang. A martingale approach for fractional Brownian motions and related path dependent PDEs. The Annals of Applied Probability, 29(6):3489–3540, 2019.
  53. Path dependent Feynman–Kac formula for forward backward stochastic Volterra integral equations. Annales de l’Institut Henri Poincaré: Probabilités et Statistiques, 58(2):603–638, 2022.
  54. H. Wendland. Scattered data approximation, volume 17. Cambridge university press, 2004.
  55. D.-X. Zhou. Derivative reproducing properties for kernel methods in learning theory. Journal of computational and Applied Mathematics, 220(1-2):456–463, 2008.
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now