Maximum likelihood drift estimation for a threshold diffusion: Maximum likelihood drift estimation for a threshold diffusion
2019
We study the maximum likelihood estimator of the drift parameters of a
stochastic differential equation, with both drift and diffusion coefficients
constant on the positive and negative axis, yet discontinuous at zero.
This threshold diffusion is called drifted Oscillating Brownian motion. For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term.
We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations.
In long time, the asymptotic behaviors of
the positive and negative occupation times rule the ones of the estimators.
Differently from
most known results in the literature, we do not restrict ourselves to the
ergodic framework: indeed, depending on the signs of the drift, the process may
be ergodic, transient or null recurrent. For each regime, we establish whether
or not the estimators are consistent; if they are, we prove the convergence in
long time of the properly rescaled difference of the estimators towards a normal or
mixed normal distribution. These theoretical results are backed by numerical
simulations.
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