1d:T7bb,# Pedersen's simulated transition density Simulated transition density $X(t) | X(t_0) = x_0$X(t) | X(t0) = x0 of a diffusion process based on Pedersen's method. ```r dcSim(x0, x, t, d, s, theta, M=10000, N=10, log=FALSE) ``` ## Arguments - `x0`: the value of the process at time `0`. - `x`: value in which to evaluate the conditional density. - `t`: lag or time. - `theta`: parameter of the process; see details. - `log`: logical; if TRUE, probabilities $p$ are given as $log(p)$. - `d`: drift coefficient as a function; see details. - `s`: diffusion coefficient as a function; see details. - `N`: number of subintervals; see details. - `M`: number of Monte Carlo simulations, which should be an even number; see details. ## Details This function returns the value of the conditional density of $X(t) | X(0) = x0$ at point `x`. The functions `d` and `s`, must be functions of `t`, `x`, and `theta`. ## Returns - **x**: a numeric vector ## 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. ## Author(s) Stefano Maria Iacus ## Examples ```r ## Not run: d1 <- function(t,x,theta) theta[1]*(theta[2]-x) s1 <- function(t,x,theta) theta[3]*sqrt(x) from <- 0.08 x <- seq(0,0.2, length=100) sle10 <- NULL sle2 <- NULL sle5 <- NULL true <- NULL set.seed(123) for(to in x){ sle2 <- c(sle2, dcSim(from, to, 0.5, d1, s1, theta=c(2,0.02,0.15), M=50000,N=2)) sle5 <- c(sle5, dcSim(from, to, 0.5, d1, s1, theta=c(2,0.02,0.15), M=50000,N=5)) sle10 <- c(sle10, dcSim(from, to, 0.5, d1, s1, theta=c(2,0.02,0.15), M=50000,N=10)) true <- c(true, dcCIR(to, 0.5, from, c(2*0.02,2,0.15))) } par(mar=c(5,5,1,1)) plot(x, true, type="l", ylab="conditional density") lines(x, sle2, lty=4) lines(x, sle5, lty=2) lines(x, sle10, lty=3) legend(0.15,20, legend=c("exact","N=2", "N=5", "N=10"), lty=c(1,2,4,3)) ## End(Not run) ```

dcSim function