Observation noise-gain detection for Markov chains observed through scaled Brownian motion

In this preliminary article we consider the problem of estimating an unknown noise-gain for a Markov chain observed through a scaled Brownian motion. It is assumed that the unknown noise-gain is time invariant. Two objectives are addressed in this work, 1) compute an estimation scheme that is fast, and 2) compute an estimation scheme without recourse to stochastic integration. To address the first objective we avoid the Expectation Maximization (EM) algorithm, instead we develop an estimation scheme for a finite number of candidate model hypotheses. To address the second objective we develop a version of the Gauge-Transformation technique introduced by J. M. C. Clark.