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Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case
Author(s) -
Bayraktar Erhan,
Cayé Thomas,
Ekren Ibrahim
Publication year - 2021
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12283
Subject(s) - utility maximization problem , nonlinear system , mathematics , geometric brownian motion , limit (mathematics) , asymptotic expansion , bellman equation , ergodic theory , function (biology) , multivariate statistics , maximization , brownian motion , mathematical optimization , econometrics , mathematical economics , economics , mathematical analysis , utility maximization , diffusion process , statistics , physics , economy , quantum mechanics , evolutionary biology , biology , service (business)
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.