The main contribution of this thesis is the full characterisations and solutions to the sequential testing and quickest detection problems for Bessel processes.These are the first known solutions for problems considering well-studied diffusions with a non-constant signal-to-noise ratio (SNR) process which are inherently two-dimensional. Chapter 3 is a base for a joint paper with G. Peskir which describes the solution to the sequential testing problem for Bessel processes, of dimension greater than or equal to two, through a method of stochastic time-change. This passes the problem in to the time-space domain, meaning the auxiliary problem is well suited to being considered for a finite time horizon. This is then fully characterised and solved using local time-space calculus on curves, before letting this horizon tend to the infinite horizon case. This solution is followed by a description of general diffusion processes that this method will also work for.Chapter 4 is the base for a paper in which the solution to the sequential testing of the dimension of a Bessel process is used to give the solution to a natural optimal stopping problem concerning the sequential testing of the mean of the CIR process with exponential time costs through a time-space change. The CIR process considered has a non-constant SNR but the optimal stopping problem is also time inhomogeneous meaning that the problem also requires knowledge of the running time making the problem three-dimensional. The optimal surfaces are calculated for a given example.The quickest detection problem considered in Chapter 6 is based on the joint paper with G. Peskir. This paper characterises and finds solution to the Bayesian quickest detection problem for a Bessel process of dimension greater than or equal to two. It contains many results that are applicable to all quickest detection problems of this type and particularly when the observed processes have a non-constant SNR process. This includes a description of a change of measure which allows the sufficient statistics to become uncoupled in the second component, also abandoning the need to include the innovation process. This simplifies the problems and allows for subsequent analysis. Additionally, a change-of-variable formula is shown to reduce the sufficient statistic in such problems to the observed process and a process of bounded variation. In combination with the measure change this also removesany mixed derivatives in the resulting partial differential equation (PDE) formulation. The methodologies used in providing the solution to problem concerning Bessel processes also address a number of key issues in dealing with fully two-dimensional problems of optimal stopping for diffusions and their connections with (locally) parabolic PDEs. This makes the work of general probabilistic interest.The remaining chapters include (i) a review in detection theory (Chapter 2), (ii) a connection between the observed processes of traditional disorder problems andself-exciting processes via a stochastic time change (Chapter 5), and (iii) a novel signal processing application for spectrogram data which uses optimal detection methodsto show how frequency-phase tracking can provide great gains in spread of detection with equal accuracy (Chapter 7). It is the intent of these chapters that follow up research in thedescribed directions could lead to publishable results in the near future.