Stability in quadratic variation (2405.20839v1)
Abstract: Consider a sequence of cadlag processes ${Xn}_n$, and some fixed function $f$. If $f$ is continuous then under several modes of convergence $Xn\to X$ implies corresponding convergence of $f(Xn)\to f(X)$, due to continuous mapping. We study conditions (on $f$, ${Xn}_n$ and $X$) under which convergence of $Xn\to X$ implies $\left[f(Xn)-f(X)\right]\to 0$. While interesting in its own right, this also directly relates (through integration by parts and the Kunita-Watanabe inequality) to convergence of integrators in the sense $\int_0t Y_{s-}df(Xn_s)\to\int_0t Y_{s-}df(X_s)$. We use two different types of quadratic variations, weak sense and strong sense which our two main results deal with. For weak sense quadratic variations we show stability when $f\in C1$, ${Xn}_n,X$ are Dirichlet processes defined as in \cite{NonCont} $Xn\xrightarrow{a.s.}X$, $[Xn-X]\xrightarrow{a.s.}0$ and ${(Xn)*_t}_n$ is bounded in probability. For strong sense quadratic variations we are able to relax the conditions on $f$ to being the primitive function of a cadlag function but with the additional assumption on $X$, that the continuous and discontinuous parts of $X$ are independent stochastic processes (this assumption is not imposed on ${Xn}_n$ however), and ${Xn}_n,X$ are Dirichlet processes with quadratic variations along any stopping time refining sequence. To prove the result regarding strong sense quadratic variation we prove a new It^o decomposition for this setting.