Brownian_motion_path_sampler | R Documentation |
Simulation of a path of a Brownian motion at given times
Brownian_motion_path_sampler(x, times)
x |
start value of Brownian motion |
times |
vector of real numbers to simulate Brownian motion (must be a vector of at least length two, where times[1] is the start time of the Brownian motion, and times[2:length(times)] are the times you wish to further simulate) |
Matrix of the simulated Brownian motion path at all included time points. The times are sorted. The first row are the points of the Brownian motion (named 'X') second row are corresponding times (named 'times')
# simulating path for Brownian motion starting at 0 between [0,1] path <- Brownian_motion_path_sampler(x = 0, times = seq(0, 1, 0.01)) plot(x = path['time',], y = path['X',], pch = 20, xlab = 'Time', ylab = 'X') lines(x = path['time',], y = path['X',]) # comparing the simulated distribution of simulated points to the # theoretical distribution of simulated points # set variables x <- 0 start_time <- 1.8 end_time <- 5 replicates <- 10000 paths <- list() # repeatedly simulate Brownian bridge for (i in 1:replicates) { paths[[i]] <- Brownian_motion_path_sampler(x = x, times = seq(start_time, end_time, 0.01)) } # select the points at the specified time q index <- which(seq(start_time, end_time, 0.01)==end_time) simulated_points <- sapply(1:replicates, function(i) paths[[i]]['X', index]) # calculate the theoretical mean and standard deviation of the simulated points at time q theoretical_mean <- x theoretical_sd <- sqrt(end_time-start_time) # plot distribution of the simulated points and the theoretical distribution plot(density(simulated_points)) curve(dnorm(x, theoretical_mean, theoretical_sd), add = T, col = 'red') print(paste('Theoretical variance is', end_time-start_time, 'and sample variance is', var(simulated_points)))
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