  This article aims to investigate the impact of noise on parameter fitting for
an Ornstein-Uhlenbeck process, focusing on the effects of multiplicative and
thermal noise on the accuracy of signal separation. To address these issues, we
propose algorithms and methods that can effectively distinguish between thermal
and multiplicative noise and improve the precision of parameter estimation for
optimal data analysis. Specifically, we explore the impact of both
multiplicative and thermal noise on the obfuscation of the actual signal and
propose methods to resolve them. First, we present an algorithm that can
effectively separate thermal noise with comparable performance to Hamilton
Monte Carlo (HMC) but with significantly improved speed. We then analyze
multiplicative noise and demonstrate that HMC is insufficient for isolating
thermal and multiplicative noise. However, we show that, with additional
knowledge of the ratio between thermal and multiplicative noise, we can
accurately distinguish between the two types of noise when provided with a
sufficiently large sampling rate or an amplitude of multiplicative noise
smaller than thermal noise. Thus, we demonstrate the mechanism underlying an
otherwise counterintuitive phenomenon: when multiplicative noise dominates the
noise spectrum, one can successfully estimate the parameters for such systems
after adding additional white noise to shift the noise balance.
