The proposed market model

Until now, the presented theory has not been fixed to a particular financial market model and was intended to give a general introduction to the portfolio optimization problem. In the following sections we apply the above ideas to a concrete model and data setup.

We model stocks and non-defaultable bonds. All stochastics evolves from a $(d+2)$-dimensional brownian motion (Wiener process) $(W_i)_{i=1,\ldots,d+2}$, where the first two components drive the dynamics of the two-factor interest rate model for the bonds and the last $d$ drive the dynamics of $d$ stocks. The brownian motions $W_i$ are correlated by a covariance matrix $\Sigma$ (see also Section 5). We assume

$$\Sigma_{i,i} = 1$$ (34)

for $i=1,\ldots,d+2$ and

$$\Sigma_{1,0} = \Sigma_{0,1}= 0,$$ (35)

i.e. each $W_i$ is a one-dimensional standard brownian motion and $W_1$ and $W_2$ are uncorrelated.