Stochastic Volatility Analysis using the Generalised Kolmogorov-Feller Equation.

We consider an approach to analysing the Stochastic Volatility of a financial time series using the Generalised Kolmogorov-Feller Equation (GKFE). After reviewing the computation of the Stochastic Volatility using a phase only condition, a Green’s function solution to the GKFE equation is derived which depends upon the ‘memory function’ used to construct the GKFE. Using the Mittag-Leffler memory function, we derive an expression for the Impulse Response Function associated with a short time window of data which is then used to derive an algorithm for computing a new index using a standard moving window process. It is shown that application of this index to both a financial time series and its corresponding Stochastic Volatility provides a correlation between the start, direction and end of a trend depending on the sampling rate of the time series and the look-back window that is used.