Bridge sampling Monte-carlo method in R studio for variance gamma

问题

I am using trying to use bridge sampling in R studio to simulate paths for the variance gamma process. My code is:

sigma = 0.5054
theta = 0.2464 
nu = 0.1184 
mu=1
N=2^(k)
k=5
V_<-rep(NA,252)
V_[0]<-0
G_[N]<-rgamma(1, shape=N*1/nu, scale=nu)
G_<-0
V<-rnorm(theta*G[N],sigma^2*G[N])
for(l in 1:k){
n<-2^(k-l)
for(j in 1:2^i-1){
i<-(2*j-1)*n
d1<-(n)*mu^2/nu
d2<-(n)*mu^2/nu
Y<-rbeta(1,d1,d2)
G_[i]<-G_[i-1]+(G[i+n]-G[i-n])*Y
G[i]
print(G_[i])
Z<-rnorm(0,(G_[i+n]-G_[i])*sigma^2*Y)
V_[i]<-Y*V_[i+n]+(1-Y)*V_[i-n]+Z
print(V_[i])
}
}
ts.plot(V[i])

I'm not sure what I've done wrong. The algorithm I am trying to follow is as below in the picture:

回答1:

Based on your code, a numerical sequence was simulated. And it can be roughly validated by using VarianceGamma::vgFit to estimate the parameters.

Note that the time index starts from 1 due to R syntax. The sqrt of variance was used for the standard deviation in rnorm. And I probably shouldn't add the change due to interest rate vgC in the end, since it is not included in your algorithm. Please set it as 0 if it doesn't make sense.

Simulation by Brownian bridge:

# Brownian-Gamma Bridge Sampling (BGBS) of a VG process
set.seed(1) 
M <- 10
nt <- 2^M + 1 #number of observations
T <- nt - 1 #total time
T_ <- seq(0, T, length.out=nt) #fixed time increments

#random time increments
#T_ = c(0, runif(nt-2), 1)
#T_ = sort(T_) * T

r <- 1 + 0.2 #interest rate
vgC <- (r-1)
sigma <- 0.5054
theta <- 0.2464 
nu <- 0.1184

V_ <- G_ <- rep(NA,nt)
V_[1] <- 0
G_[1] <- 0
G_[nt] <- rgamma(1, shape=T/nu, scale=nu)
V_[nt] <- rnorm(1, theta*G_[nt], sqrt(sigma^2*G_[nt]))

for (k in 1:M)
  {
  n <- 2^(M-k)
  for (j in 1:2^(k-1))
    {
    i <- (2*j-1) * n
    Y <- rbeta(1, (T_[i+1]-T_[i-n+1])/nu, (T_[i+n+1]-T_[i+1])/nu)
    G_[i+1] <- G_[i-n+1] + (G_[i+n+1] - G_[i-n+1]) * Y
    Z <- rnorm(1, sd=sqrt((G_[i+n+1] - G_[i+1]) * sigma^2 * Y))
    V_[i+1] <- Y * V_[i+n+1] + (1-Y) * V_[i-n+1] + Z
    }
  }
V_ <- V_ + vgC*T_ # changes due to interest rate

plot(T_, V_)

The results roughly match with the estimation:

#Estimated parameters:
library(VarianceGamma)
dV <- V_[2:nt] - V_[1:(nt-1)]
vgFit(dV)
>    vgC   sigma   theta      nu  
> 0.2996  0.5241  0.1663  0.1184

#Real parameters:
c(vgC, sigma, theta, nu)
>    vgC   sigma   theta      nu  
> 0.2000  0.5054  0.2464  0.1184

回答2:

When I run it, I get a list of NA's

来源：https://stackoverflow.com/questions/62835198/bridge-sampling-monte-carlo-method-in-r-studio-for-variance-gamma

标签

montecarlo