Asymptotics for Small Nonlinear Price Impact : A PDE Homogenization Approach to the Multidimensional Case

Using ideas from homogenization theory and stability of viscosity solutions, we provide an asymptotic expansion of the value function of a multidimensional utility maximization problem with small non-linear price impact. In our model cross-impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second order PDE related to an ergodic control problem. We illustrate our result on a multivariate geometric Brownian motion price model