$(\F_t)_{t \in [0,T]}$ generated by these $n+1$ Brownian motions, and where the market parameters are constants $\mu_i > 0$, $\sigma_i \ge 0$, and $\nu_i \ge 0$, with $\sigma_i + \nu_i > 0$. The prices $S^i_t$ are assumed to be one-dimensional for simplicity, although we could easily extend the results to $k$-dimensional prices.

This setup covers the important special case of a *single stock*, corresponding to the situation where all the stocks are identical. That is, $\mu_i = \mu$, $\nu_i = 0$, and $\sigma_i = \sigma$, for all $i = 1, ..., n$ and for some $\mu, \sigma > 0$ independent of $i$, and so $S^i \equiv S^j$ for each $i,j$ (assuming the initial values agree). In the single stock case, the agents face the same market opportunities rather than specializing in different assets, though their risk preferences still differ.

Each agent $i$ chooses a self-financing strategy, $(\pi^i_t)_{t \in [0,T]}$, denoting the proportion of wealth invested in the stock $i$, and a consumption policy, $(c_t^i)_{t \in [0,T]}$. The wealth process of agent $i$ is then given by $$\label{processX}
    dX^i_t = \pi^i_t X_t^i \left(\mu_i dt + \nu_i dW^i_t + \sigma_i dB_t\right) - c^i_t X^i_t dt, \quad\quad X_0^i = x_0^i.$$ Note that $c^i_tX^i_t$ 