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Subject: Stochastic differential equations ×

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    Approximation of SDEs: a stochastic sewing approach
    (Berlin ; Heidelberg ; New York, NY : Springer, 2021) Butkovsky, Oleg; Dareiotis, Konstantinos; Gerencsér, Máté
    We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. 10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler-Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is H∈(0,1) and the drift is Cα , α∈[0,1] and α>1-1/(2H) , we show the strong Lp and almost sure rates of convergence to be ((1/2+αH)∧1)-ε , for any ε>0 . Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323-2366, 2016. 10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence 1/2-ε of the Euler-Maruyama scheme for Cα drift, for any ε,α>0 .
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    Data-Driven Discovery of Stochastic Differential Equations
    (Beijing : Engineering Sciences Press, 2022) Wang, Yasen; Fang, Huazhen; Jin, Junyang; Ma, Guijun; He, Xin; Dai, Xing; Yue, Zuogong; Cheng, Cheng; Zhang, Hai-Tao; Pu, Donglin; Wu, Dongrui; Yuan, Ye; Gonçalves, Jorge; Kurths, Jürgen; Ding, Han
    Stochastic differential equations (SDEs) are mathematical models that are widely used to describe complex processes or phenomena perturbed by random noise from different sources. The identification of SDEs governing a system is often a challenge because of the inherent strong stochasticity of data and the complexity of the system's dynamics. The practical utility of existing parametric approaches for identifying SDEs is usually limited by insufficient data resources. This study presents a novel framework for identifying SDEs by leveraging the sparse Bayesian learning (SBL) technique to search for a parsimonious, yet physically necessary representation from the space of candidate basis functions. More importantly, we use the analytical tractability of SBL to develop an efficient way to formulate the linear regression problem for the discovery of SDEs that requires considerably less time-series data. The effectiveness of the proposed framework is demonstrated using real data on stock and oil prices, bearing variation, and wind speed, as well as simulated data on well-known stochastic dynamical systems, including the generalized Wiener process and Langevin equation. This framework aims to assist specialists in extracting stochastic mathematical models from random phenomena in the natural sciences, economics, and engineering fields for analysis, prediction, and decision making.