JGQD.density: Generate the Transition Density of a Scalar Jump Generalized...
In DiffusionRjgqd: Inference and Analysis for Jump Generalized Quadratic Diffusions

Description Usage Arguments Details Value Interface Warning Author(s) References See Also Examples

JGQD.density() approximates the transition density of a scalar generalized quadratic diffusion model (GQD). Given an initial value for the diffusion, Xs, the approximation is evaluated for all Xt at equispaced time-nodes given by splitting [s, t] into subintervals of length delt. JGQD.density() approximates transitional densities of jump diffusions of the form:

where

and

describes a Poisson process with jumps of the form:

arriving with intensity

subject to a jump distribition of the form:

JGQD.density(Xs = 4, Xt = seq(5, 8, 1/10), s = 0, t = 5, delt =1/100,
             Jdist = "Normal", Jtype = "Add", Dtype = "Saddlepoint",
             Trunc = c(8, 4), factorize = FALSE, factor.type = "Diffusion",
              beta, print.output = TRUE, eval.density = TRUE)

`Xs`	Initial value of the process at time s.
`Xt`	Vector of values at which the transition density is to be evaluated over the trajectory of the transition density from time s to t.
`s`	The starting time of the process.
`t`	The time horizon up to and including which the transitional density is evaluated.
`delt`	Size of the time increments at which successive evaluations are made.
`Dtype`	Character string indicating the type of density approximation (see details) to use. Types: `'Saddlepoint'` and `'Edgeworth'` are supported (default = `'Saddlepoint'`).
`Trunc`	Vector of length 2 containing the cumulant truncation order and the density truncation order respectively. May take on values 4 and 8 with the constraint that `Trunc[1] >= Trunc[2]`. Default is `c(4,4)`.
`Jdist`	Valid entries are 'Normal', 'Exponential', 'Gamma' or 'Laplace'.
`Jtype`	Valid types are 'Add' or 'Mult'.
`factorize`	Should factorization be used (default = TRUE).
`factor.type`	Can be used to envoke a special factorization in order to evaluate Hawkes processes nested within the JGQD framework.
`beta`	Variable used for a special case jump structure (for research purposes).
`print.output`	If `TRUE`, model information is printed to the console.
`eval.density`	If `TRUE`, the density is evaluated in addition to calculating the moment eqns.

vdfs

JGQD.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.

`density`	A matrix giving the density over the spatio-temporal mesh whose vertices are defined by paired permutations of the elements of `X_t` and `time`
`Xt`	A vector of points defining the state space at which the density was evaluated(recycled from input).
`time`	A vector of time points at which the density was evaluated.
`cumulants`	A matrix giving the cumulants of the diffusion. Row i gives the i-th cumulant.
`moments`	A matrix giving the moments of the diffusion. Row i gives the i-th cumulant.
`mesh`	A matrix giving the mesh used for normalization of the density.

DiffusionRjgqd uses a functional interface whereby th coefficients of a jump diffusion is defined by functions in the current workspace. By defining time-dependent functions with names that match the coefficients of the desired diffusion, DiffusionRjgqd reads the workspace and prepares the appropriate algorithm.

diffusiontable,width=100

In the case of jump diffusions, additional coefficients are required for the jump mechanism as well. Intensity coefficients and jump distributions, along with their corresponding R-names, are given in the tables below.

Intensity:

intensitytable

Jump distributions:

jumptable

Warning [1]: 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 jump 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 after which alternate density approximants may be specified through the variable Dtype.

Warning [2]: 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.

Etienne A.D. Pienaar: etiannead@gmail.com

Updates available on GitHub at https://github.com/eta21.

Daniels, H.E. 1954 Saddlepoint approximations in statistics. Ann. Math. Stat., 25:631–650.

Eddelbuettel, D. and Romain, F. 2011 Rcpp: Seamless R and C++ integration. Journal of Statistical Software, 40(8):1–18,. URL http://www.jstatsoft.org/v40/i08/.

Eddelbuettel, D. 2013 Seamless R and C++ Integration with Rcpp. New York: Springer. ISBN 978-1-4614-6867-7.

Eddelbuettel, D. and Sanderson, C. 2014 Rcpparmadillo: Accelerating R with high-performance C++ linear algebra. Computational Statistics and Data Analysis, 71:1054–1063. URL http://dx.doi.org/10.1016/j.csda.2013.02.005.

Feagin, T. 2007 A tenth-order Runge-Kutta method with error estimate. In Proceedings of the IAENG Conf. on Scientifc Computing.

Varughese, M.M. 2013 Parameter estimation for multivariate diffusion systems. Comput. Stat. Data An., 57:417–428.

See JGQD.mcmc and BiJGQD.density.

#===============================================================================

 # For detailed notes and examples on how to use the JGQD.density() function, see
 # the following vignette:

 RShowDoc('Part_2_Transition_Densities',type='html','DiffusionRjgqd')

#===============================================================================