stochastic differential games. After showing that this equation admits a unique and separable classical solution, the fixed point is resolved via a system of non-linear ordinary differential equations. Despite many similarities with [@lacker2017mean], the consumption renders the arguments substantially more involved.

The paper is organized as follows. In Section \[finite population\], we formulate and solve the $n$-agent model described above. Then, in Section $3$, we study the infinite population counterpart of this problem, arguing that the $n\to\infty$ limit results in a simpler form of the equilibrium. Finally, Section $4$ discusses and interprets the form of the equilibrium and its dependence on the model parameters.

The n-agent game {#finite population}
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In this section, we consider the $n$-player game, where each agent trades in a common investment horizon $[0,T]$. Agents may invest in their own specific stocks or in a common riskless bond which offers zero interest rate. The price of stock $i$, in which only agent $i$ trades, is given by the dynamics $$\label{prices}
    \frac{dS^i_t}{S^i_t} = \mu_i dt + \nu_i dW^i_t + \sigma_i dB_t,$$ where the Brownian motions $W^1, ..., W^n, B$ are independent and defined on a probability space $(\Omega, \F, \PP)$, which we endow with the natural filtration 