DiffusionRgqd (version 0.1.2)

GQD.passage: Calculate the First Passage Time Density of a Time-Homogeneous GQD Process to a Fixed Barrier.

Description

GQD.passage uses the cumulant truncation procedure of Varughese (2013) in conjunction with a Volterra-type integral equation developed by Buonocore et al. (1987) in order to approximate the first passage time density of a time-homogeneous univariate GQD $$dX_t = (theta[1]+theta[2]X_t+theta[3]X_t^2)dt+\sqrt{theta[4]+theta[5]X_t+theta[6]X_t^2}dW_t,$$ to a fixed barrier.

Usage

GQD.passage(Xs, B, theta, t, delt)

Arguments

Xs

Initial value of the process.

B

Fixed barrier (>Xs).

theta

Parameter vector giving the coefficients of the time-homogeneous GQD.

t

The time horizon up to and including which the density is to be evaluated.

delt

Stepsize for the solution of the first passage time density.

Value

• density
  The approximate first passage time density.
• time
  The time points at which the approximation is evaluated.
• prob.coverage
  The approximate cumulative probability coverage.

Warning

Warning [1]: Some instability may occur when delt is large or where the saddlepoint approximation fails. As allways it is important to check both the validity of the diffusion process under the given parameters and the quality of the sadlepoint approximation. For a given set of parameters the latter can be checked using GQD.density. Warning [2]: The first passage time problem is considered from one side only i.e. XsB one may equivalently consider Yt=-X_t, apply Ito's lemma and proceed as above.

References

Updates available on GitHub at https://github.com/eta21. A. Buonocore, A. Nobile, and L. Ricciardi. 1987 A new integral equation for the evaluation of first-passage- time probability densities. Adv. Appl. Probab. 19:784--800. 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 Scientific Computing. R. G. Jaimez, P. R. Roman and F. T. Ruiz. 1995 A note on the volterra integral equation for the first-passage-time probability density. Journal of applied probability, 635--648. Varughese, M.M. 2013 Parameter estimation for multivariate diffusion systems. Comput. Stat. Data An., 57:417--428.

See Also

GQD.density for functions that generate the transitional density of GQDs. GQD.mcmc and GQD.remove for MCMC procedures to perform inference on GQDs.

Examples

#===============================================================================
# Calculate the first passage time from state X_0 = 7 to X_T =10 for
# various diffusions, with T the first passage time.
#===============================================================================
  
  res1 <- GQD.passage(7,10,c(0.1*10,-0.1,0,0.2,0,0),25,1/100)
  res2 <- GQD.passage(7,10,c(0.1*10,-0.1,0,0,0.2,0),25,1/100)
  res3 <- GQD.passage(7,10,c(0,0.1*10,-0.1,0.5^2,0,0),25,1/100)
  res4 <- GQD.passage(7,10,c(0,0.1*10,-0.1,0,0,0.1^2),25,1/100) 

  expr1 <- expression(dX[t]==(1-0.1*X[t])*dt+sqrt(0.2)*dW[t])
  expr2 <- expression(dX[t]==(1-0.1*X[t])*dt+sqrt(0.2*X[t])*dW[t])
  expr3 <- expression(dX[t]==(1*X[t]-0.1*X[t]^2)*dt+0.5*dW[t])
  expr4 <- expression(dX[t]==(1*X[t]-0.1*X[t]^2)*dt+0.1*X[t]*dW[t])


#===============================================================================
# Plot the resulting first passage time densities.
#===============================================================================

  par(mfrow=c(2,2))
  plot(res1$density~res1$time,type='l',main=expr1,xlab='Time (t)',ylab='density',col='blue')
  plot(res2$density~res2$time,type='l',main=expr2,xlab='Time (t)',ylab='density',col='blue')
  plot(res3$density~res3$time,type='l',main=expr3,xlab='Time (t)',ylab='density',col='blue')
  plot(res4$density~res4$time,type='l',main=expr4,xlab='Time (t)',ylab='density',col='blue')