I'm learning the Black-Scholes model as beginner. In our lecture notes, we have following setting:
Let $W=(W_t^1,\dots,W_t^d)_{t\in[0,T]}$ be a d-dim standard Brownian motion. The financial market consist of d+1 assets. The asset $S^0$ is a non-risky asset defined by $$S_0^t = e^{rt}$$ where r models the instantaneous interest rate.
The risky asset are modeled by $$S_t^i = S_0^i\text{ }exp\big(\Big(\mu^i-\frac{1}{2}\sum_{j=1}^d|\sigma^{ij}|^2\Big)t\text{ }+\text{ } \sum_{j=1}^d\sigma^{ij}W_t^j\big)$$ where $S_0^i$ are the initial prices, and $\mu^i$ and $\sigma^{ij}$ are constants, describing "drift" and correlations between the assets, respectively.
Problem
-I am clear if we are in the 1-dim BS model.
For $d\ge2$, I am confused by the index notation for risky asset, like how do $\mu^i,\sigma,\sigma^{ij}$ look like. Could anyone please explain it in detail, explanation with example will be great.
-In addition, if I want to define the price process of risky asset using BS model as a function with specific equi-distant discretization points recursively, does the following recursion seem to be correct? $$S_{t_{i+1}}=S_{t_i}\text{ }exp\big(\Big(\mu^i-\frac{1}{2}\sum_{j=1}^d|\sigma^{ij}|^2\Big)\Delta t_{i+1}\text{ }+\text{ } \sum_{j=1}^d\sigma^{ij}\underbrace{\Delta W_{t_i}^j}_{\sim\mathcal{N}(0,\Delta t_i)\stackrel{d}=\sqrt{\Delta t_i}\mathcal{N}(0,1)}\big)$$