Stochastic Differential Equations (SDEs)

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. ...

An Accurate Discretized Approach to Parameter Estimation in the CKLS Model via the CIR Framework ArXiv ID: 2507.10041 “View on arXiv” Authors: Sourojyoti Barick Abstract This paper provides insight into the estimation and asymptotic behavior of parameters in interest rate models, focusing primarily on the Cox-Ingersoll-Ross (CIR) process and its extension – the more general Chan-Karolyi-Longstaff-Sanders (CKLS) framework ($α\in[“0.5,1”]$). The CIR process is widely used in modeling interest rates which possess the mean reverting feature. An Extension of CIR model, CKLS model serves as a foundational case for analyzing more complex dynamics. We employ Euler-Maruyama discretization to transform the continuous-time stochastic differential equations (SDEs) of these models into a discretized form that facilitates efficient simulation and estimation of parameters using linear regression techniques. We established the strong consistency and asymptotic normality of the estimators for the drift and volatility parameters, providing a theoretical underpinning for the parameter estimation process. Additionally, we explore the boundary behavior of these models, particularly in the context of unattainability at zero and infinity, by examining the scale and speed density functions associated with generalized SDEs involving polynomial drift and diffusion terms. Furthermore, we derive sufficient conditions for the existence of a stationary distribution within the CKLS framework and the corresponding stationary density function; and discuss its dependence on model parameters for $α\in[“0.5,1”]$. ...

Unbiased simulation of Asian options ArXiv ID: 2504.16349 “View on arXiv” Authors: Bruno Bouchard, Xiaolu Tan Abstract We provide an extension of the unbiased simulation method for SDEs developed in Henry-Labordere et al. [“Ann Appl Probab. 27:6 (2017) 1-37”] to a class of path-dependent dynamics, pertaining for Asian options. In our setting, both the payoff and the SDE’s coefficients depend on the (weighted) average of the process or, more precisely, on the integral of the solution to the SDE against a continuous function with bounded variations. In particular, this applies to the numerical resolution of the class of path-dependent PDEs whose regularity, in the sens of Dupire, is studied in Bouchard and Tan [“Ann. I.H.P., to appear”]. ...