DiffusionRjgqd (version 0.1.1)

JGQD.density: Generate the Transition Density of a Scalar Jump Generalized Quadratic Diffusion (GQD).

Description

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:

ScalarEqn1.png

where

ScalarEqn2.png

ScalarEqn3.png

and

ScalarEqn4.png

describes a Poisson process with jumps of the form:

ScalarEqn6.png

arriving with intensity

ScalarEqn5.png

subject to a jump distribition of the form:

ScalarEqn7.png

Usage

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)

Arguments

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.

Value

Interface

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. difftable.pngdiffusiontable,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.pngintensitytable Jump distributions: jumptable.pngjumptable

Warning

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.

Details

TransDens1.pngvdfs

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.

References

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 Also

See JGQD.mcmc and BiJGQD.density.

Examples

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

 # 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')

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