SIMloglik function

Pedersen's approximation of the likelihood

Pedersen's approximation of the likelihood of a process solution of a stochastic differential equation. This function is useful to calculate approximated maximum likelihood estimators when the transition density of the process is not known. It is computationally intensive.

SIMloglik(X, theta, d, s,  M=10000, N=2, log=TRUE)

Arguments

• X: a ts object containing a sample path of an sde.
• theta: vector of parameters.
• d,s: drift and diffusion coefficients; see details.
• log: logical; if TRUE, the log-likelihood is returned.
• N: number of subintervals; see details.
• M: number of Monte Carlo simulations, which should be an even number; see details.

Details

The function SIMloglik returns the simulated log-likelihood obtained by Pedersen's method. The functions s and d are the drift and diffusion coefficients with arguments (t,x,theta).

Returns

• x: a number

Author(s)

Stefano Maria Iacus

References

Pedersen, A. R. (1995) A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations, Scand. J. Statist., 22, 55-71.

Examples

## Not run:

set.seed(123)
d <- expression(-1*x)
s <- expression(2) 
sde.sim(drift=d, sigma=s,N=50,delta=0.01) -> X

S <- function(t, x, theta) sqrt(theta[2])
B <- function(t, x, theta) -theta[1]*x

true.loglik <- function(theta) {
 DELTA <- deltat(X)
 lik <- 0
 for(i in 2:length(X))
  lik <- lik + dnorm(X[i], mean=X[i-1]*exp(-theta[1]*DELTA), 
   sd = sqrt((1-exp(-2*theta[1]*DELTA))*
              theta[2]/(2*theta[1])),TRUE)
 lik  
}

xx <- seq(-10,10,length=20)
sapply(xx, function(x) true.loglik(c(x,4))) -> py
sapply(xx, function(x) EULERloglik(X,c(x,4),B,S)) -> pz
sapply(xx, function(x) SIMloglik(X,c(x,4),B,S,M=10000,N=5)) -> pw

plot(xx,py,type="l",xlab=expression(beta),
   ylab="log-likelihood",ylim=c(0,15)) # true
lines(xx,pz, lty=2) # Euler
lines(xx,pw, lty=3) # Simulated
## End(Not run)