Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems

We show the existence of a continuous-time Nash equilibrium in a financial market with risk averse market makers and an informed trader with a private information. The unwillingness of market makers to bear risk causes the informed trader to absorb large shocks in their inventories. The informed trader's optimal strategy is to drive the market price to its fundamental value while participating in the risk sharing with the market makers. The optimal strategies of the agents turn out to be solutions of a forward-backward system of partial and stochastic differential equations. In particular, the price set by the market makers is the solution to a non-standard `quadratic' backward stochastic differential equation.