The Wiener process $(W_t)$ is a continuous stochastic process that satisfies the following there conditions:
- $W_0 = 0$,
- the increments $\mathrm{d}W_t = W_{t + \mathrm{d}t} - W_t$ are normally distributed with mean $0$ and variance $\mathrm{d}t$,
- the increments are mutually independents, i.e. $\mathrm{d}W_i$ is independent from $\mathrm{d}W_j$ for every $i$ different from $j$.
I now want to discretize the Wiener process in order to simulate it as described in the beginning of “An algorithmic introduction to numerical simulatiom of stochastic differential equation” by Higham (2001).
- First, I have to discretize the time interval $[0,T]$ in $N$ sub-intervals of equal length $\delta_t = \frac{T}{N}$. In this way, each of the $N+1$ time instants are given by $t_i = i \cdot \delta_t$.
- Then, at each time instant $t_i$, the discretized version of the Wiener process is \begin{align*} W_i = W_{i-1} + \mathrm{d}W_{i-1}, \end{align*} where $\mathrm{d} W_{i-1}\sim N(0,\delta_t)$ and $W_0=0$.
I would like to undestand how proprieties 2 and 3 imply the above iteration formula.