Sequential testing of hypotheses about drift for Gaussian diffusions

In statistical inference on the drift parameter $\theta$ in the process $X_t=\theta a\left(t\right)+{\int }_0^{t}b\left(s\right)d{W}_s$, where $a\left(t\right)$ and $b\left(t\right)$ are known, deterministic functions, there is known a large number of options how to do it. We may, for example, base this inference on the differences between the observed values of the process at discrete times and their normality. Although such methods are very simple, it turns out that it is more appropriate to use sequential methods. For the hypotheses testing about the drift parameter $\theta$, it is more proper to standardize the observed process and to use sequential methods based on the first exit time of the observed process of a pre-specified interval until some given time. These methods can be generalized to the case of random part being a symmetric Itô integral or continuous symmetric martingale.