returns the covariance $$ V(\mathrm{x}{t_0 + \Delta t} | \mathrm{x}{t_0} = \mathrm{x}_0) $$ of the process after a time interval $ \Delta t $ according to the given discretization. This method can be overridden in derived classes which want to hard-code a particular discretization.
returns the expectation $$ E(\mathrm{x}{t_0 + \Delta t} | \mathrm{x}{t_0} = \mathrm{x}_0) $$ of the process after a time interval $ \Delta t $ according to the given discretization. This method can be overridden in derived classes which want to hard-code a particular discretization.
returns the standard deviation $$ S(\mathrm{x}{t_0 + \Delta t} | \mathrm{x}{t_0} = \mathrm{x}_0) $$ of the process after a time interval $ \Delta t $ according to the given discretization. This method can be overridden in derived classes which want to hard-code a particular discretization.
Square-root stochastic-volatility Bates process
This class describes the square root stochastic volatility process incl jumps governed by $$ \begin{array}{rcl} dS(t, S) &=& (r-d-\lambda m) S dt +\sqrt{v} S dW_1 + (e^J - 1) S dN \ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \ dW_1 dW_2 &=& \rho dt \ \omega(J) &=& \frac{1}{\sqrt{2\pi \delta^2}} \exp\left[-\frac{(J-\nu)^2}{2\delta^2}\right] \end{array} $$