dt <- 0.001
W0 <- 0
t <- seq (0, 1, by=dt)

N <- length (t)
W <- rep (0, N)
W[1] <- W0
Wakt <- W0
for (k in 2:N) {
	dW <- sqrt (dt) * rnorm (1)
	Wakt <- Wakt + dW
	W[k] <- Wakt
}

plot (t, W, type="l", col="red", xlab="t", ylab="W")
abline (h = W0, lty = 2)



# ==============================================================================



dt <- 0.001
W0 <- 0
t <- seq (0, 1, by=dt)

N <- length (t)
dW <- rnorm (N - 1) * sqrt (dt)
W <- cumsum (c (W0, dW))

plot (t, W, type="l", col="red", xlab="t", ylab="W")
abline (h = W0, lty = 2)



# ==============================================================================



generuj.Wp <- function (t, dt, W0) {
	dW <- rnorm (length (t) - 1) * sqrt (dt)
	W <- cumsum (c (W0, dW))
}

W <- generuj.Wp (t, dt, W0)

plot (t, W, type = "l", col = "red", xlab = "t", ylab = "W")
abline (h = W0, lty = 2)



# ==============================================================================




M <- sapply (1:10 , function (k) {
	generuj.Wp (t, dt, W0)
})

matplot (t, M, type = "l", lty = 1, xlab = "t", ylab = "W", main = "trajektorie W")
abline (h = W0 , lty = 2)



# =============================================================================



index <- which (t == 0.6)
vyber <- M[index,]

hist (vyber, breaks = 50, freq = FALSE)

qqnorm (vyber, pch = "+")
qqline (vyber, col = "blue")

plot (ecdf (vyber))

x <- seq (-2 ,2 , by = 0.1)
y <- pnorm (x , mean = 0 , sd = sqrt (0.6))
lines (x, y, col = "red")

ks.test (vyber, pnorm, mean = 0 , sd = sqrt (0.6))
ks.test (vyber, pnorm, mean = 1 , sd = sqrt (0.6))
ks.test (vyber, pnorm, mean = 0 , sd = 1)