time changes for continuous time vs. Euler approximation of stochastic jump processes

stochasticEuler.R

# continuous time rate
R <- 1/5

# mean time to next event in continuous time
mean(rexp(n = 1e6,rate = R))

# get a discrete time rate to plug into Euler approximation of continuous time process
get_r <- function(R,dt){
  log(1 + R*dt) / dt
}

dt <- 0.5
r <- get_r(R = R,dt = dt)

# check the Euler process gives the same expected waiting time as continuous process
mean(rgeom(n = 1e6,prob = 1-exp(-r*dt)) * dt)

# alternatively we can start with r and want R (discrete to continuous)
get_R <- function(r,dt){
  (exp(r*dt)-1) / dt
}

R <- get_R(r,dt)

mean(rexp(n = 1e6,rate = R))
mean(rgeom(n = 1e6,prob = 1-exp(-r*dt)) * dt)