下面是我的代码，但它应该允许用户指定T，n和M的输入我该怎么做？
### Simulation of a discrete process on [0, T]
# Write a program to simulate the process
process_simulation <- function(X0, t, n, mu, sigma, M) {
#initialize simulation results
X_simulated <- matrix(0, ncol=n+1, nrow=M)
dt <- t/n
#run simulations in a loop
for (m in 1:M) {
#initialize simulated process
X <- rep(0, n+1)
X[1] <- X0
#calculate sum in a loop
for (i in 0:(n-1)) {
#next Wiener process value
dz <- (dt)^0.5*rnorm(1)
#calculate next X value
dX <- mu(X[i+1], i*dt)*dt+sigma(X[i+1], i*dt)*dz
X[i+2] <- X[i+1]+dX
}
#save simulated process
X_simulated[m, ] <- X
}
#return all simulated processes
return(X_simulated)
}
#(a)
X0 <- 1
t <- 1
n <- 100
mu <- function(x, t) { return(0) }
sigma <- function(x, t) { return(1) }
M <- 1000
Xa_1 <- process_simulation(X0, t, n, mu, sigma, M)
#(i) compare graphically distribution of x(t) at t=0.5 and t=1
plot(density(Xa_1[, n/2+1]), col="blue", xlab="x", xlim=c(-5, 5), main=paste("n=", n, ", M=", M, sep=""), xaxs="i", yaxs="i")
lines(density(Xa_1[, n+1]), col="green")
#plot the true distributions (in grey color)
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 1), col="grey")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 0.5), col="grey")
legend("topright", legend=c("simulated t=0.5", "simulated t=1", "theoretical"), lty=1, col=c("blue", "green", "grey"))
#(ii) experiment with n and M
n <- 50
Xa_2 <- process_simulation(X0, t, n, mu, sigma, M)
plot(density(Xa_2[, n/2+1]), col="blue", xlab="x", xlim=c(-5, 5), main=paste("n=", n, ", M=", M, sep=""), xaxs="i", yaxs="i")
lines(density(Xa_2[, n+1]), col="green")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 1), col="grey")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 0.5), col="grey")
legend("topright", legend=c("simulated t=0.5", "simulated t=1", "theoretical"), lty=1, col=c("blue", "green", "grey"))
n <- 200
Xa_3 <- process_simulation(X0, t, n, mu, sigma, M)
plot(density(Xa_3[, n/2+1]), col="blue", xlab="x", xlim=c(-5, 5), main=paste("n=", n, ", M=", M, sep=""), xaxs="i", yaxs="i")
lines(density(Xa_3[, n+1]), col="green")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 1), col="grey")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 0.5), col="grey")
legend("topright", legend=c("simulated t=0.5", "simulated t=1", "theoretical"), lty=1, col=c("blue", "green", "grey"))
#The higher the value of n, the better fitting to the true distribution.
n <- 100
M <- 500
Xa_4 <- process_simulation(X0, t, n, mu, sigma, M)
plot(density(Xa_4[, n/2+1]), col="blue", xlab="x", xlim=c(-5, 5), main=paste("n=", n, ", M=", M, sep=""), xaxs="i", yaxs="i")
lines(density(Xa_4[, n+1]), col="green")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 1), col="grey")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 0.5), col="grey")
legend("topright", legend=c("simulated t=0.5", "simulated t=1", "theoretical"), lty=1, col=c("blue", "green", "grey"))
M <- 2000
Xa_5 <- process_simulation(X0, t, n, mu, sigma, M)
plot(density(Xa_5[, n/2+1]), col="blue", xlab="x", xlim=c(-5, 5), main=paste("n=", n, ", M=", M, sep=""), xaxs="i", yaxs="i")
lines(density(Xa_5[, n+1]), col="green")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 1), col="grey")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 0.5), col="grey")
legend("topright", legend=c("simulated t=0.5", "simulated t=1", "theoretical"), lty=1, col=c("blue", "green", "grey"))
#The higher the value of M, the better fitting to the true distribution.
n <- 200
Xa_6 <- process_simulation(X0, t, n, mu, sigma, M)
plot(density(Xa_6[, n/2+1]), col="blue", xlab="x", xlim=c(-5, 5), main=paste("n=", n, ", M=", M, sep=""), xaxs="i", yaxs="i")
lines(density(Xa_6[, n+1]), col="green")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 1), col="grey")
lines(seq(-5, 5, length.out=1000), dnorm(seq(-5, 5, length.out=1000), 1, 0.5), col="grey")
legend("topright", legend=c("simulated t=0.5", "simulated t=1", "theoretical"), lty=1, col=c("blue", "green", "grey"))
如何使用此代码，让用户决定T，n和M，然后计算代码中的内容？