Hypotheses testing about the drift parameter in linear stochastic differential equation driven by stable processes

Abstract

In this paper, we consider the problem of hypotheses testing about the drift parameter $\theta$ in the process $\text {d}Y^{\delta }_{t} = \theta \dot{f}(t)Y^{\delta }_{t}\text {d}t + b(t)\text {d}L^{\delta }_{t}$ driven by symmetric $\delta$-stable Lévy process $L^{\delta }_{t}$ with $\dot{f}(t)$ being the derivative of a known increasing function f(t) and b(t) being known as well. We consider the hypotheses testing $H_{0}: \theta \le 0$ and $K_{0}: \theta =0$ against the alternatives $H_{1}: \theta >0$ and $K_{1}: \theta \ne 0$, respectively. For these hypotheses, we propose inverse methods, which are motivated by sequential approach, based on the first hitting time of the observed process (or its absolute value) to a pre-specified boundary or two boundaries until some given time. The applicability of these methods is illustrated. For the case $Y^{\delta }_{0}=0$, we are able to calculate the values of boundaries and finite observed times more directly. We are able to show the consistencies of proposed tests for $Y^{\delta }_{0}\ge 0$ with $\delta \in (1,2]$ and for $Y^{\delta }_{0}=0$ with $\delta \in (0,2]$ under quite mild conditions.