BiJGQD.density
generates approximate transitional densities for bivariate generalized quadratic jump diffusions (JGQDs). 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
].
That is, BiJGQD.density generates approximate transitional densities for a class of bivariate jump diffusion processes with SDE:
BivJumpDiff1.png
where
BivJumpDiff4.png
BivJumpDiff3.png
and
BivJumpDiff2.png
describes a bivariate Poisson process with jump matrix:
BivJumpDiff5.png
and intensity vector
BivJumpDiff6.png
BiJGQD.density(Xs, Ys, Xt, Yt, s, t, delt, Dtype, Jdist, Jtype, print.output, eval.density)
"Saddlepoint"
(default) or "Edgeworth"
.TRUE
information about the model and algorithm is printed to the console. TRUE
, the density is evaluated in addition to calculating the moment eqns.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.TransDens2.pngGQD
BiJGQD.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.
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.
BiJGQD.mcmc
and JGQD.density
.
#===============================================================================
# For detailed notes and examples on how to use the BiJGQD.density() function, see
# the following vignette:
RShowDoc('Part_3_Bivariate_Diffusions',type='html','DiffusionRjgqd')
#===============================================================================
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