Simulation of Brownian motion and white noise.

Question

Let $\{W(t)\}$, $t≥0$ be a Wiener process with $ σ^2 = \operatorname{Var}\{W(1)\} = 1$. For a real constant $ε > 0$, consider the differential ratio process $∆ε = \{∆ε(t)\}$, $t>0$ given by $∆ε(t) = (W(t+ε)−W(t))/ε $ for $t > 0$. For $s > 0$ and $t ∈ R$ (the latter of which has an absoulte value small enough to make $s+t ≥ 0$), show that the autocorrelation function $R\,∆ε(t) = R∆ε(s, s+t) = E\{∆ε(s)∆ε(s+t)\}$ of $∆ε$ is a triangle like function that depends on the difference $t$ between $s > 0$ and $s+t ≥ 0$ only. Further, show that $R\,∆ε(t) → δ(t)$ (Dirac’s $δ$-function) as $ε ↓ 0$. Simulate a sample path of $\{∆ε(t)\}$, $t∈(0,1]$ for a really small $ε > 0$ and plot a graph of that sample path. Discuss the claim that the (nonexisting in the usual sense) derivative process $\{W′(t)\}$, $t≥0$ of $W$ is white noise.

Someone who can help me with this one? To show the outlined tasks and help me with some kind of pseudocode. I really do not know how to start?

Answer 1

Here is most of the code you need for the simulation, in R:

step <- 1e-5
time.s <- seq(step, 1, by=step)

delX <- rnorm(length(time.s)+10, mean=0, sd=sqrt(step))

X = cumsum(delX)

eps <- 1e-4

Deps <- (X[(eps/step+1):(length(time.s)+10)]-X[1:(length(time.s)-eps/step+10)])/eps

auto.corr <- acf(x=Deps, lag.max=ceiling(eps/step+10), type='covariance', plot=F)

plot(time.s, Deps)

plot(auto.corr)

But im not 100% sure on how to do this. Hopefully someone better at coding can finish it.