DIFFERENTIABILITY OF QUADRATIC FORWARD-BACKWARD SDES WITH ROUGH DRIFT

• 成果类型: Article
• 署名作者: Imkeller, Peter; Pellat, Rhoss likibi; Menoukeu-pamen, Olivier
• 署名单位: Humboldt University of Berlin; University of Liverpool
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2079
• 发表日期: 2024
• 页码: 4758-4798
• 关键词: numerical-simulation; convex generators; path regularity; BSDEs; EQUATIONS; approximation

摘要

In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for which the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system possesses nonlinearity of type f (|y|)|z|2, where f is any locally integrable function. We prove both the Malliavin and classical differentiability of solutions to this type of QFBSDEs and provide representations of these derivatives processes. As a by-product, we derive a representation formula of the control variable Zt as a conditional expectation of the terminal value, the driver and the Malliavin weights, when the drift term is only bounded and H & ouml;lder continuous. We study a numerical approximation of this system in the sense of Imkeller and Dos Reis ( Stochastic Process. Appl. 120 (2010) 2286-2288) in which the authors assume that the drift is Lipschitz and the driver of the BSDE is globally Lipschitz in y and quadratic in the traditional sense in z (i.e., f is a positive constant). We show that the rate of convergence is the same as in ( Stochastic Process. Appl. 120 (2010) 2286-2288).