# Asymptotics for Small Nonlinear Price Impact: a PDE Approach to the Multidimensional Case

arXiv: https://arxiv.org/abs/1811.06650 · HTML: https://mlsanity.com/paper/1811.06650/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Nov 2018.

## Built on by (most cited first, 1 of 1)

- [Generative Market Equilibrium Models with Stable Adversarial Learning via Reinforcement](https://mlsanity.com/paper/2504.04300/) (2025)

## Abstract (arXiv)

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.