Taylor Expansion

Noise estimation of SDE from a single data trajectory ArXiv ID: 2509.25484 “View on arXiv” Authors: Munawar Ali, Purba Das, Qi Feng, Liyao Gao, Guang Lin Abstract In this paper, 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 allows us to recover the underlying $\mathbb P$ Brownian motion increments. 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. ...

Higher order approximation of option prices in Barndorff-Nielsen and Shephard models ArXiv ID: 2401.14390 “View on arXiv” Authors: Unknown Abstract We present an approximation method based on the mixing formula (Hull & White 1987, Romano & Touzi 1997) for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das & Langrené (2022). We obtain asymptotic results for the error of the $N^{"\text{th"}}$ order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein-Uhlenbeck process or a gamma Ornstein-Uhlenbeck process. ...