Stocks

The $d$ stocks of the considered financial market are modelled by geometric brownian motions, i.e. price processes $S_j$ ( $j = 1, \ldots, d$) with

$$S_j(t) = S_j(0)e^{\mu_j t + \sigma_jW_{j+2}(t)} ,$$ (43)

where $\mu_j \in \mathbb{R}$ is the drift and $\sigma_j\in\mathbb{R}^+$ the diffusion coefficient of the brownian motion in the exponent, i.e. the pice process has the ``trend''

$$\mathbf{E}[S_j(t)] = S_j(0)e^{(\mu_j+\sigma_j^2/2)t} .$$ (44)

In terms of stochastic differential equations (SDE) we have

$$d \ln S_j = \mu_jdt + \sigma_jdW_{j+2} .$$ (45)

$W_i(t)$ is the $i$-th brownian motion at time $t$.