FORESTS, CUMULANTS, MARTINGALES

• 成果类型: Article
• 署名作者: Friz, Peter K.; Gatheral, Jim; Radoicic, Rados
• 署名单位: Technical University of Berlin; City University of New York (CUNY) System; Baruch College (CUNY)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1560
• 发表日期: 2022
• 页码: 1418-1445
• 关键词: differential-equations; rough

摘要

This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability space. We establish a broken exponential martingale expansion that generalizes and unifies the exponentiation result of Alos, Gatheral, and Radoicic (SSRN'17; Quant. Finance 20 (2020) 13-27) and the cumulant recursion formula of Lacoin, Rhodes, and Vargas (arXiv; (2019)). Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula, as required by many of our examples, including iterated stochastic integrals, Levy area, Bessel processes, KPZ with smooth noise, Wiener-Ito chaos, and rough stochastic (forward) variance models.