Purba Das

Abstract

We propose a data-driven framework for model discovery of stochastic differential equations (SDEs) from a single trajectory, without requiring the ergodicity or stationary assumption on the underlying continuous process. By combining (stochastic) Taylor expansions with Girsanov transformations, and using the drift function’s initial value as input, we construct drift estimators while simultaneously recovering the model noise. This drift is unique up to the fixed underlying Brownian sample path, in line with pathwise unique Rough-Smooth decomposition (Ananova-Cont 2017). This allows us to recover the underlying P-Brownian motion increments for the observer path ω. Building on these estimators, we introduce the first stochastic Sparse Identification of Stochastic Differential Equation (SSISDE) algorithm, capable of identifying the governing SDE dynamics from a single observed trajectory without requiring ergodicity or stationarity. To validate the proposed approach, we conduct numerical experiments with both linear and quadratic drift–diffusion functions. Among these, the Black–Scholes SDE is included as a representative case of a system that does not satisfy ergodicity or stationarity. If time permits, we will talk about the extension of these results in the context of pathwise Föllmer SDE. 

Our speaker

Purba Das is a Lecturer in Financial Mathematics in the Department of Mathematics, King's College London. Before that, she was a Byrne Research Assistant Professor of Mathematics at the University of Michigan (2022-2023). Purba completed her DPhil in Mathematical Institute at the University of Oxford in 2022. Purba’s research interests lie in the fields of stochastic analysis and its applications to mathematical finance. In particular exploring ‘roughness’ via pathwise methods. 

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