Partial mixing and Edgeworth expansion
Given a probability space \((\Omega , {\mathcal F}, P)\), consider \({\mathcal C}\) a sub \(\sigma\)-field of \({\mathcal F}\), \(\{ {\mathcal B}_I, I\subset {\mathbb R}_+\}\) an increasing family of sub \(\sigma\)-fields of \({\mathcal F}\), and \((Z_t)_{t\in {\mathbb R}_+}\) a stochastic process such that \(Z_0\) is \({\mathcal B}_{[0]}\vee {\mathcal C}\)-measurable and \(Z_t-Z_s\) is \({\mathcal B}_{[s,t]}\vee {\mathcal C}\)-measurable, \(0\leq s < t\). The author derives asymptotic expansions for the expectation of \(f(Z_T/\sqrt{T})\), allowing the conditional mixing coefficient given \({\mathcal C}\) to have a slower than exponential decay. His general results apply to \(\sigma\)-fields of the form \[ {\mathcal B}_I = \sigma (X_t-X_s : s,t\in I)\vee \sigma (Y_t : t\in I ), \] where \((Y_t)_{t\in {\mathbb R}_+}\) is a \({\mathcal C}\)-conditional \(\epsilon\)-Markov process driven by a process \((X_t)_{t\in {\mathbb R}_+}\) with independent increments. They also apply when \((Z_t)_{t\in {\mathbb R}_+}\) is a solution to a stochastic differential equation with jumps and random coefficients that are functions of \((Y_t)_{t\in {\mathbb R}_+}\). Malliavin calculus and the support theorem for stochastic differential equations with jumps are both used to satisfy the needed conditional Cramér condition on the characteristic function of the increments of \((Z_t)_{t\in {\mathbb R}_+}\).
scientific article; zbMATH DE number 2111580
scientific article