We consider the problem of learning Stochastic Differential Equations of the form dXt=f(Xt)dt+σ(Xt)dWt from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions f, σ, and stochastic process dWt representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion and data adapted kernels learned via a new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the time-increment map Xt→Xt+dt as a Computational Graph in which f, σ and dWt appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.

### One shot learning of stochastic differential equations with data adapted kernels.

#### Abstract

We consider the problem of learning Stochastic Differential Equations of the form dXt=f(Xt)dt+σ(Xt)dWt from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions f, σ, and stochastic process dWt representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion and data adapted kernels learned via a new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the time-increment map Xt→Xt+dt as a Computational Graph in which f, σ and dWt appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.
##### Scheda breve Scheda completa Scheda completa (DC)
2023
Settore SECS-S/06 - Metodi mat. dell'economia e Scienze Attuariali e Finanziarie
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2209.12086.pdf

embargo fino al 28/02/2025

Descrizione: Accepted version available on ArXiv: https://arxiv.org/abs/2209.12086
Tipologia: Accepted version (post-print)
Licenza: Solo Lettura
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Tipologia: Published version
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11384/125423`
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