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.
We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption 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 and a linear parabolic PDE . We illustrate our result on a multivariate geometric Brownian motion price model.
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Using ideas from homogenization theory and stability of viscosity solutions, we
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from consumption
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and a linear parabolic PDE