Second order stochastic target problems with generalized market impact

TL;DR

This paper extends stochastic target problems to include general market impacts, addressing non-concave equations and relaxing gamma constraints, and provides a dual formulation and asymptotic expansion.

Contribution

It introduces a novel approach to stochastic target problems with nonlinear, non-concave equations and general market impacts, expanding existing models and solution techniques.

Findings

Super-hedging price solves a fully nonlinear parabolic PDE

Constructs perfect hedging strategies under smooth coefficients

Provides a dual formulation and asymptotic expansion around impact-free models

Abstract

We extend the study of [7, 18] to stochastic target problems with general market impacts. Namely, we consider a general abstract model which can be associated to a fully nonlinear parabolic equation. Unlike [7, 18], the equation is not concave and the regularization/verification approach of [7] can not be applied. We also relax the gamma constraint of [7]. In place, we need to generalize the a priori estimates of [18] and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the super-hedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefficients are smooth enough. This representation leads to a general dual formulation. We finally provide an asymptotic expansion around a model without impact.