In michaelgordy/basicAffineProcess: Basic Affine Processes

This vignette shows how to verify the analytic solution to the Laplace transform in laplaceBAP() via simulation.

First we set our parameters:

library(basicAffineProcess)

tt <- seq(0.05,15,by=0.05)
mu <- 0.25
kappa <- 0.30  
sigma <- 0.20  
lambda <- 0.5  # jump rate
zeta <- 0.25   # mean jump size
w <- -0.5
X0 <- 0.1
simtrials <- 5000  # number of trials in simulation

Next we calculate the Laplace transform analytically and by simulation:

L <- laplaceBAP(tt,mu,kappa,sigma,lambda,zeta,w)
psi.analytic <- exp(L$A0 + L$B0*X0)

# Laplace transform for BAP process by simulation
laplaceBAPmontecarlo <- function(X0,tt,mu,kappa,sigma,lambda=0,zeta=1,w=-1,trials) {
  psi <- rep(0,length(tt))
  dt <- diff(c(0,tt))
  for (j in 1:trials) {
    Yt <- rep(0,length(tt))
    XY <- list(X=X0,Y=0)
    for (k in 1:length(tt)) {
      XY <- rBAPtransition2(XY,mu,kappa,sigma,lambda,zeta,dt[k])
      Yt[k] <- XY$Y
    }
    psi <- psi+exp(w*Yt)
  }
  return(psi/trials)
}

psi.sim <- laplaceBAPmontecarlo(X0,tt,mu,kappa,sigma,lambda,zeta,w,simtrials)

Now let's plot the analytical and simulated transforms with log-scaling. If everything goes well, the two curves will be roughly linear as time increases and very close.

library(ggplot2)
library(scales)
library(tidyr)

data.frame(time=tt, analytic=psi.analytic, simulated=psi.sim) %>%
         gather(method,Laplace,-time) %>%
         ggplot(aes(x=time, y=Laplace, color=method)) + geom_line() +
           scale_y_continuous(trans=log_trans()) 

michaelgordy/basicAffineProcess documentation built on May 22, 2019, 9:50 p.m.