Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case
Corresponding Author
Erhan Bayraktar
Department of Mathematics, University of Michigan, Ann Arbor, Michigan
Correspondence
Erhan Bayraktar, Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109.
Email: erhan@umich.edu
Search for more papers by this authorThomas Cayé
School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin, Ireland
Search for more papers by this authorIbrahim Ekren
Department of Mathematics, Florida State University, Tallahassee, Florida
Search for more papers by this authorCorresponding Author
Erhan Bayraktar
Department of Mathematics, University of Michigan, Ann Arbor, Michigan
Correspondence
Erhan Bayraktar, Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109.
Email: erhan@umich.edu
Search for more papers by this authorThomas Cayé
School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin, Ireland
Search for more papers by this authorIbrahim Ekren
Department of Mathematics, Florida State University, Tallahassee, Florida
Search for more papers by this authorFunding information:
Erhan Bayraktar was supported in part by the National Science Foundation (NSF) and the Susan M. Smith Professorship. Thomas Cayé was supported in part by the Swiss National Science Foundation (SNF) under grant 175133.
Abstract
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.
Open Research
DATA AVAILABILITY STATEMENT
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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