On the utility problem in a market where price impact is transient
Lóránt Nagy, Miklós Rásonyi
TL;DR
This work tackles utility maximization in a discrete-time market with transient price impact, modeled via resilience $r_t$ and depth $\delta_t$, where the attainable terminal wealth set can be nonconvex. It develops a novel dynamic programming framework that uses backward induction, random subsequences, and essential supremum arguments to construct an optimal trading strategy without the monotonicity constraints previously assumed on depth and resilience. A key contribution is proving the existence of an optimizer even when the wealth-domain is nonconvex, extending prior results that required convexity. The results provide a rigorous existence theory for optimal investment with transient liquidity frictions in discrete time and outline how solvency constraints or continuous-time limits might be incorporated or differ.
Abstract
We consider a discrete-time model of a financial market where a risky asset is bought and sold with transactions having a transient price impact. It is shown that the corresponding utility maximization problem admits a solution. We manage to remove some unnatural restrictions on the market depth and resilience processes that were present in earlier work. A non-standard feature of the problem is that the set of attainable portfolio values may fail the convexity property.