We study portfolio choice with small nonlinear price impact on general market dynamics. Using probabilistic techniques and convex duality, we show that the asymptotic optimum can be described explicitly up to the solution of a nonlinear ODE, which identifies the optimal trading speed and the performance loss due to the trading friction. Previous asymptotic results for proportional and quadratic trading costs are obtained as limiting cases. As an illustration, we discuss how nonlinear trading costs affect the pricing and hedging of derivative securities and active portfolio management.

We derive scaling limits for integral functionals of Itô processes with fast nonlinear meanreversion speeds. In these limits, the fast mean-reverting process is "averaged out" by integrating against its invariant measure. The convergence is uniformly in probability and, under mild integrability conditions, also in S p . These results are a crucial building block for the analysis of portfolio choice models with small superlinear transaction costs, carried out in the companion paper of the present study [11].Mathematics Subject Classification (2010): 60F25, 60H10.Keywords: processes with fast nonlinear mean reversion; scaling limits * We are grateful to two anonymous referees for extremely detailed and helpful comments that have greatly improved the final version of this manuscript.

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We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small nonlinear 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.