We derive an asymptotic expansion for the quadratic variation of a stochastic process satisfying a stochastic differential equation driven by a fractional Brownian motion, based on the theory of asymptotic expansion of Skorohod integrals converging to a mixed normal limit. In order to apply the general theory, it is necessary to estimate functionals that are a randomly weighted sum of products of multiple integrals of the fractional Brownian motion, in expanding the quadratic variation and identifying the limit random symbols. To overcome the difficulty, we introduce two types of exponents by means of the "weighted graphs" capturing the structure of the sum in the functional, and investigate how the exponents change by the action of the Malliavin derivative and its projection.
翻译:我们得出了一个无症状的扩展,用于满足由分数布朗运动驱动的随机差异方程式的二次变异,该方程式以Skorohod整体体的无症状扩展理论为根据,融合到混合的正常极限。为了应用一般理论,我们有必要估计成分数布朗运动的多重组合体产品随机加权总和的功能,以扩大二次变异和确定限随机符号。为了克服这一困难,我们引入了两类“加权图 ”, 以捕捉功能中总和的结构, 并调查马利亚温衍生物及其投影的动作如何改变指数。