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Weak Collocation Regression for Inferring Stochastic Dynamics with Lévy Noise (2403.08292v1)

Published 13 Mar 2024 in math.NA, cs.AI, cs.NA, and math.DS

Abstract: With the rapid increase of observational, experimental and simulated data for stochastic systems, tremendous efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the broad applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extracting stochastic dynamics with L\'{e}vy noise are relatively few. In this work, we propose a Weak Collocation Regression (WCR) to explicitly reveal unknown stochastic dynamical systems, i.e., the Stochastic Differential Equation (SDE) with both $\alpha$-stable L\'{e}vy noise and Gaussian noise, from discrete aggregate data. This method utilizes the evolution equation of the probability distribution function, i.e., the Fokker-Planck (FP) equation. With the weak form of the FP equation, the WCR constructs a linear system of unknown parameters where all integrals are evaluated by Monte Carlo method with the observations. Then, the unknown parameters are obtained by a sparse linear regression. For a SDE with L\'{e}vy noise, the corresponding FP equation is a partial integro-differential equation (PIDE), which contains nonlocal terms, and is difficult to deal with. The weak form can avoid complicated multiple integrals. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems. Numerical experiments demonstrate that our method is accurate and computationally efficient.

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References (49)
  1. Non-gaussian analytic option pricing: A closed formula for the lévy-stable model. Available at SSRN 2828673, 2017.
  2. David Applebaum. Lévy processes and stochastic calculus. Cambridge university press, 2009.
  3. Harish S Bhat. Drift identification for lévy alpha-stable stochastic systems. arXiv e-prints, pages arXiv–2212, 2022.
  4. The pricing of options and corporate liabilities. Journal of political economy, 81(3):637–654, 1973.
  5. Sparse learning of stochastic dynamical equations. The Journal of chemical physics, 148(24):241723, 2018.
  6. Weak solutions to a fractional fokker–planck equation via splitting and wasserstein gradient flow. Applied Mathematics Letters, 42:30–35, 2015.
  7. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 113(15):3932–3937, 2016.
  8. Efficient solution of the first passage problem by path integration for normal and poissonian white noise. Probabilistic Engineering Mechanics, 41:121–128, 2015.
  9. A unified meshfree pseudospectral method for solving both classical and fractional pdes. SIAM Journal on Scientific Computing, 43(2):A1389–A1411, 2021.
  10. Solving inverse stochastic problems from discrete particle observations using the fokker–planck equation and physics-informed neural networks. SIAM Journal on Scientific Computing, 43(3):B811–B830, 2021.
  11. Nonparametric inference of stochastic differential equations based on the relative entropy rate. Mathematical Methods in the Applied Sciences, 46(2):2986–2996, 2023.
  12. Detecting the maximum likelihood transition path from data of stochastic dynamical systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(11):113124, 2020.
  13. The discovery of dynamics via linear multistep methods and deep learning: Error estimation. SIAM Journal on Numerical Analysis, 60(4):2014–2045, 2022.
  14. Jinqiao Duan. An introduction to stochastic dynamics, volume 51. Cambridge University Press, 2015.
  15. Optimal portfolios when stock prices follow an exponential lévy process. Finance and Stochastics, 8(1):17–44, 2004.
  16. An end-to-end deep learning approach for extracting stochastic dynamical systems with α𝛼\alphaitalic_α-stable lévy noise. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(6), 2022.
  17. Michael I Ganzburg. Polynomial interpolation, an l-function, and pointwise approximation of continuous functions. Journal of approximation theory, 153(1):1–18, 2008.
  18. Dynamical inference for transitions in stochastic systems with α𝛼\alphaitalic_α-stable lévy noise. Journal of Physics A: Mathematical and Theoretical, 49(29):294002, 2016.
  19. Fokker–planck equations for stochastic dynamical systems with symmetric lévy motions. Applied Mathematics and Computation, 278:1–20, 2016.
  20. Nonparametric estimation of stochastic differential equations with sparse gaussian processes. Physical Review E, 96(2):022104, 2017.
  21. Monte carlo fpinns: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations. Computer Methods in Applied Mechanics and Engineering, 400:115523, 2022.
  22. Revealing hidden dynamics from time-series data by odenet. Journal of Computational Physics, 461:111203, 2022.
  23. Ronald L Iman. Latin hypercube sampling. Encyclopedia of quantitative risk analysis and assessment, 3, 2008.
  24. Sde-net: Equipping deep neural networks with uncertainty estimates. In 37th International Conference on Machine Learning, ICML 2020, pages 5361–5371. International Machine Learning Society (IMLS), 2020.
  25. Mateusz Kwaśnicki. Ten equivalent definitions of the fractional laplace operator. Fractional Calculus and Applied Analysis, 20(1):7–51, 2017.
  26. Christiane Lemieux. Monte carlo and quasi-monte carlo sampling. 2009.
  27. A data-driven approach for discovering stochastic dynamical systems with non-gaussian lévy noise. Physica D: Nonlinear Phenomena, 417:132830, 2021.
  28. Extracting stochastic dynamical systems with α𝛼\alphaitalic_α-stable lévy noise from data. Journal of Statistical Mechanics: Theory and Experiment, 2022(2):023405, 2022.
  29. Pde-net: Learning pdes from data. In International conference on machine learning, pages 3208–3216. PMLR, 2018.
  30. Weak collocation regression method: fast reveal hidden stochastic dynamics from high-dimensional aggregate data. Journal of Computational Physics, 502:112799, 2024.
  31. Learning stochastic behaviour from aggregate data. In International Conference on Machine Learning, pages 7258–7267. PMLR, 2021.
  32. Turbulence and financial markets. Nature, 383(6601):587–588, 1996.
  33. Andrew Matacz. Financial modeling and option theory with the truncated lévy process. International Journal of Theoretical and Applied Finance, 3(01):143–160, 2000.
  34. Michael D. McKay. Latin hypercube sampling as a tool in uncertainty analysis of computer models. In Proceedings of the 24th Conference on Winter Simulation, page 557–564, New York, NY, USA, 1992. Association for Computing Machinery.
  35. Learning mean-field equations from particle data using wsindy. Physica D: Nonlinear Phenomena, 439:133406, 2022.
  36. Manfred Opper. Variational inference for stochastic differential equations. Annalen der Physik, 531(3):1800233, 2019.
  37. fpinns: Fractional physics-informed neural networks. SIAM Journal on Scientific Computing, 41(4):A2603–A2626, 2019.
  38. Umberto Picchini. Inference for sde models via approximate bayesian computation. Journal of Computational and Graphical Statistics, 23(4):1080–1100, 2014.
  39. A robust nonparametric framework for reconstruction of stochastic differential equation models. Physica A: Statistical Mechanics and its Applications, 450:294–304, 2016.
  40. Fractional fokker–planck equation for nonlinear stochastic differential equations driven by non-gaussian lévy stable noises. Journal of Mathematical Physics, 42(1):200–212, 2001.
  41. Governing equations for probability densities of marcus stochastic differential equations with lévy noise. Stochastics and Dynamics, 17(05):1750033, 2017.
  42. Well-posedness for the fractional fokker-planck equations. Journal of Mathematical Physics, 56(3):031502, 2015.
  43. Generative ensemble-regression: learning stochastic dynamics from discrete particle ensemble observations. 2020.
  44. Lévy anomalous diffusion and fractional fokker–planck equation. Physica A: Statistical Mechanics and its Applications, 282(1-2):13–34, 2000.
  45. Onsagernet: Learning stable and interpretable dynamics using a generalized onsager principle. Physical Review Fluids, 6(11):114402, 2021.
  46. Stochastic dynamics driven by combined lévy–gaussian noise: fractional fokker–planck–kolmogorov equation and solution. Journal of Physics A: Mathematical and Theoretical, 53(38):385001, 2020.
  47. First-passage problem for stochastic differential equations with combined parametric gaussian and lévy white noises via path integral method. Journal of Computational Physics, 435:110264, 2021.
  48. Adaptive sampling methods for learning dynamical systems. In Mathematical and Scientific Machine Learning, pages 335–350. PMLR, 2022.
  49. The maximum likelihood climate change for global warming under the influence of greenhouse effect and lévy noise. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(1):013132, 2020.

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