Hidden Markov Chain Filtering for Generalised Bessel Processes

Abstract

Bessel processes are an important family of diffusion processes which have applications in finance and other are as. Let us call a process R={R _t 0<t}a generalized Bessel process if it satisfies a stochastic differential equation of the type

$$ {Y_i} = h({X_i},{\varepsilon _i}), $$

(1.1)

For t > 0 with given initial value R_o > O. Here a b and c represent externally given functions and W = {W_t, 0 < t} is a standard Brownian motion on a given probability space Ω,FP. It is important to note that the diffusion coefficient in (1.1) does not depend on R_t and the drift coefficient is linear in R_t and $\tfrac{1}{{{R_t}}}$. We also observe that for a_t 0 we have an Ornstein-Uhlenbeck process. Under the condition $\tfrac{{2{\alpha _t}}}{{c_t^2}} \geqslant 3$ And c_t > 0, the generalized Bessel process can be shown to remain strictly positive for t > 0, a feature which makes it attractive for many applications.