Sequential Testing Problems for Bessel Processes

Peter Johnson (University of Manchester, School of Mathematics)

Williamson Building G.33,

Here we will explore a way to solve the problem of sequentially testing the dimension of the Bessel process,  for dimensions $\delta\geq 2$,
through a method using a stochastic time-change. Throughout the talk we shall consider the Bayesian formulation of this
hypothesis testing problem in which we observe a Bessel process of dimension $\delta_0$ or $\delta_1$
with probability $(1-\pi)$ and $\pi$ respectively, where $\pi$ is the Bayesian prior.
The problem is then to determine as quickly and accurately as possible the dimension of the observed Bessel process. This
is achieved by minimizing the time taken to come to a decision while also minimizing costs which are assigned to wrong
terminal decisions. When the dimensions are $\delta_0=2$ and $\delta_1=3$ this problem has a
nice visual representation of detecting whether a Brownian particle is trapped in a 2-dimensional plane
or is free to move in 3-dimensional space through the observation of its distance from the origin.
Recent research has showed the importance of the signal-to-noise ratio (SNR) process in problems of this type
which is formed by considering the difference between the possible drift coefficients divided by the diffusion coefficient.
This problem is made more challenging by the fact that a Bessel process has a non-constant SNR process
which makes the natural formulation inherently two-dimensional. The optimal solution described represents the first time a sequential
testing problem of a well-known diffusion with non-constant SNR has been solved (only manufactured examples have been tackled previously).
Joint work with Goran Peskir (Manchester)

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