R: Generate the Transition Density of a Bivariate Generalized...
Generate the Transition Density of a Bivariate Generalized Quadratic Diffusion Model (2D GQD).
BiGQD.density generates approximate transitional densities for bivariate generalized quadratic diffusions (GQDs). Given a starting coordinate, (Xs, Ys), the approximation is evaluated over a lattice Xt x Yt for an equispaced discretization (intervals of width delt) of the transition time horizon [s, t] .
BiGQD.density generates approximate transitional densities for a class of bivariate diffusion processes with SDE:
x-Coordinates of the lattice at which to evaluate the transition density.
y-Coordinates of the lattice at which to evaluate the transition density.
Starting time of the diffusion.
Final time at which to evaluate the transition density.
Step size for numerical solution of the cumulant system. Also used for the discretization of the transition time-horizon. See warnings  and .
The density approximant to use. Valid types are "Saddlepoint" (default) or "Edgeworth".
If TRUE information about the model and algorithm is printed to the console.
If TRUE, the density is evaluated in addition to calculating the moment eqns.
BiGQD.density constructs an approximate transition density for a class of quadratic diffusion models. This is done by first evaluating the trajectory of the cumulants/moments of the diffusion numerically as the solution of a system of ordinary differential equations over a time horizon [s,t] split into equi-distant points delt units apart. Subsequently, the resulting cumulants/moments are carried into a density approximant (by default, a saddlepoint approximation) in order to evaluate the transtion surface.
3D Array containing approximate density values. Note that the 3rd dimension represents time.
2D Array containing approximate Xt-marginal density values (calculated using the univariate saddlepoint approximation).
2D Array containing approximate Yt-marginal density values (calculated using the univariate saddlepoint approximation).
Copy of x-coordinates.
Copy of y-coordinates.
A vector containing the time mesh at which the density was evaluated.
A matrix giving the cumulants of the diffusion. Cumulants are indicated by row-names.
The system of ODEs that dictate the evolution of the cumulants do so approximately. Thus, although it is unlikely such cases will be encountered in inferential contexts, it is worth checking (by simulation) whether cumulants accurately replicate those of the target GQD. Furthermore, it may in some cases occur that the cumulants are indeed accurate whilst the density approximation fails. This can again be verified by simulation.
The parameter delt is also used as the stepsize for solving a system of ordinary differential equations (ODEs) that govern the evolution of the cumulants of the diffusion. As such delt is required to be small for highly non-linear models in order to ensure sufficient accuracy.