Insider trading with penalties, entropy and quadratic BSDEs

Kyle model in continuous time where the insider may be subject to legal penalties is considered. In equilibrium the insider internalises this legal risk by trading less aggressively. The equilibrium is characterised via the solution of a backward stochastic differential equation (BSDE) whose terminal condition is determined as the fixed point of a non-linear operator in equilibrium. The insider’s expected penalties in equilibrium is non-monotone in the fee structure and is given by the relative entropy of the law of a particular h-transformation of Brownian motion.