Changepoint detection problems on a finite interval

We consider several settings of changepoint detection problems for Brownian motion and diffusion processes. A changepoint is an unknown moment of time when parameters (e.g. the drift) of the observable process change. We do not know exactly when it happens, but can detect it through changes in the behavior of the process.
We will be interested in changepoint detection problems when the changepoint takes values in a finite interval, which turns out to be a more difficult setting than when it is unbounded. For these problems we obtain analytical solutions for particular penalty criteria. It will be also shown how this theory can be applied in finance to construct trading strategies.