Tracking volatility (stock markets)

Concerns nonlinear filtering of the volatility coefficient in a Black-Scholes type model that allows stochastic volatility. The asset price process S=(S/sub t/)/sub t/spl ges/0/ is given by dS/sub t/=rS/sub t/dt+/spl radic/v/sub t/S/sub t/dB/sub t/, where B=(B/sub t/)/sub t/spl ges/0/ is a Brownian motion and v/sub t/ is the (stochastic) volatility process. Moreover, assumed that v/sub t/=v(/spl theta//sub t/) where v is a nonnegative function and /spl theta/=(/spl theta//sub t/)/sub t/spl ges/0/ is a homogeneous Markov jump process, taking values in the finite alphabet (a/sub 1/,...,a/sub M/), with the intensity matrix /spl Lambda/=/spl par//spl lambda//sub ij//spl par/ and the initial distribution p/sub q/=P_(/spl theta//sub 0/=a/sub q/), q=1,...,M. The random process /spl theta/ is unobservable. Following Frey and Runggaldier (1999), we assume also that S/sub t/ is measured only at random times 0