Exercise 3.11 (Approximation of an Itô Integral). In this example, the stochastic integral $\int^t_0tW(t)dW(t)$ is considered. The expected value of the integral and the expected value of the square of the integral are estimated using $M=1,000,000$ sample paths. The number of intervals used on $[0,1]$ is $N$ where $N=4,8,16,\dots,256$. The preceding approximate method is used, which for this problem, has the explicit form \begin{align*} E((I(f))^\alpha)&=E\left(\int^1_0tW(t)dW(t)\right)^\alpha\\ &\approx\frac{1}{M}\sum^M_{j=1}\left(\sum^{N-1}_{i=0}t_i^{(N)}W^{(j)}(t_i^{(N)})(W^{(j)}(t^{(N)}_{t+1})-W^{(j)}(t^{(N)}_i))\right)^\alpha\quad(2)\\ \end{align*} for $\alpha=1$ or $2$ and $t_i^{(N)}=i/N$ for $i=0,1,2,\dots,N$. Notice that $(W^{(j)}(t^{(N)}_{i+1})-W^{(j)}(t_i^{(N)}))=\eta_i^{(j)}/\sqrt{N}$ where $\eta_i^{(j)}\sim N(0,1)$ and also that $(W^{(j)}(t^{(N)}_{i})-W^{(j)}(t_{i-1}^{(N)}))=\eta_{i-1}^{(j)}/\sqrt{N}$. A computer program that performs this calculation is listed at the end of this chapter. The calculational results are given in Table 3.2 for $M=1,000,000$ sample paths. Notice that the results improve as $N$ increases. The exact values are $E(I(f))=0$ and $E((I(f))^2)=0.25.$ \begin{array}{ccc} \hline \text{Value of N}&|E(I(f))|&|E((I(f))^2)|\\ \hline \color{red}{2^2}&\color{red}{0.00107}&\color{red}{0.14071}\\ 2^3&0.00012&0.19151\\ 2^4&0.00069&0.21906\\ 2^5&0.00038&0.23508\\ 2^6&0.00007&0.24209\\ 2^7&0.00001&0.24486\\ 2^8&0.00002&0.24821\\ \hline \end{array} Source: Modeling with Itô Stochastic Differential Equations by Edward Allen, pg. 74
My Question
There is a code provided on pg. 87, but I would like to calculate the value $N$, specifically $2^2$ for $|E(I(f))|$and$|E((I(f))^2)$ above in $\color{red}{\text{red}}$, manually. My assumption is to use $\text{(2)}$ to get these values, but I'm unsure where to begin.
Would anybody be able to provide a hint?
Firstly, you must use a gaussian random number generator, since $W^{(j)}(t_i^{(N)}) \sim \mathcal{N}(0,t_i^{(N)})$, with $t_i^{(N)} = \frac{i}{N}$, and then you must simply apply Eq. (2) for $N = 2$ straighforwardly.
Nevertheless, I don't know what you expect. If you mean 'manually' as by hand, it won't be feasible, because the sums include millions of terms and draws. If you intend to reproduce the results by yourself (but with a computer), your own computations can still lead to (close but) different values, since the outcomes of a random variable are by nature... random.
Notwithstanding, the moments of the considered stochastic process, namely $X_t = \int_0^t sW_s \,\mathrm{d}W_s$, can be determined analytically. Indeed, one has $\mathbb{E}[X_t] = 0$, since $W_s$ and $\mathrm{d}W_s$ are independent and have zero mean, while Itô's isometry gives $$ \mathrm{Var}[X_t] = \mathbb{E}[X_t^2] = \mathbb{E}\left[\int_0^t s^2W_s^2 \,\mathrm{d}s\right] = \int_0^t s^2\,\mathbb{E}[W_s^2] \,\mathrm{d}s = \int_0^t s^3 \,\mathrm{d}s = \frac{1}{4}t^4, $$ which is equal to $1/4$ when $t=1$.