Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting

TL;DR

This paper establishes the existence of minimal supersolutions for BSDEs with potentially infinite terminal conditions and applies these results to complex portfolio liquidation control problems, broadening previous theoretical frameworks.

Contribution

It introduces a generalized approach to minimal supersolutions for BSDEs without filtration assumptions, relaxing terminal constraints and allowing random time horizons.

Findings

Proves existence of minimal supersolutions under broad conditions.

Extends BSDE theory to include infinite terminal data.

Applies results to optimal portfolio liquidation problems.

Abstract

We study the existence of a minimal supersolution for backward stochastic differential equations when the terminal data can take the value +$\infty$ with positive probability. We deal with equations on a general filtered probability space and with generators satisfying a general monotonicity assumption. With this minimal supersolution we then solve an optimal stochastic control problem related to portfolio liquidation problems. We generalize the existing results in three directions: firstly there is no assumption on the underlying filtration (except completeness and quasi-left continuity), secondly we relax the terminal liquidation constraint and finally the time horizon can be random.