Self-organized regime switching in null-recurrent dynamics

Abstract

Based on discrete observations $X_0,X_{\Delta},\dots, X_{n\Delta}$ for $\Delta=n^{-\gamma}$ with $\gamma\in [0,1)$ of the null-recurrent dynamic $dX_t = \sigma(X_t)dW_t$ with a Brownian motion $W$ and $\sigma(x)=\alpha\mathbb{1}\{x<\rho\} + \beta\mathbb{1}\{x\geq \rho\}$, we derive rate of convergence and limiting distribution of the profile MLE for $\rho$. This includes low-frequency asymptotics ($\gamma=0$) for which the observations form a null-recurrent Markov chain. The derived non-standard limit is the argsup over a doubly stochastic drifted Poisson process explicitly involving the local time of oscillating Brownian motion. Its dependence on $\rho$ as well as the unknown volatility levels $\alpha$ and $\beta$ is shown to be continuous w.r.t. the topology of weak convergence, enabling statistical inference. Whereas this limit is independent of the sampling frequency, the profile MLE's rate of convergence equals $n^{-(1+\gamma)/2}$ and is proven to be minimax optimal. The surprising idea of the proof of the limit theorem is to relate the long-term behavior of the null-recurrent Markov chain to the infill asymptotics on a fixed time interval. Indeed, in the very special case that $(X_t)_{t\geq 0}$ is started in the true parameter $X_0=\rho_0$, the process $(X_t-\rho_0)_{t\geq 0}$ is shown to possess a desirable distributional self-similarity. On basis of the strong Markov property, the artificial constallation of starting in $\rho_0$ is finally overcome by a coupling argument.