The principal subject of this thesis is hypothesis testing and related problems of estimation for stochastic processes. The thesis is concerned in particular with two areas: sequential hypothesis testing in a Bayesian setting and estimation of the parameters governing a continuous-time stochastic differential equation that drives data sampled at high-frequency. The former area is concerned with hypothesis testing for a newly developed healthcare technology and makes use of optimal stopping theory. The latter area sees the application of limit theorems for stochastic processes that allow to recover the true volatility process that can be estimated using the methods of moments estimator.