# On a class of path-dependent singular stochastic control problems

## TL;DR

This paper introduces a new probabilistic representation for a class of non-Markovian singular stochastic control problems, linking their solutions to Z-constrained BSDEs and analyzing their regularity and applications.

## Contribution

It provides a novel probabilistic framework connecting non-Markovian control problems with Z-constrained BSDEs, extending to degenerate diffusions and applications in utility maximization.

## Key findings

- Representation of solutions via Z-constrained BSDEs
- Quantification of solution regularity in terms of initial data
- Application to utility maximization with transaction costs

## Abstract

This paper studies a class of non$-$Markovian singular stochastic control problems, for which we provide a novel probabilistic representation. The solution of such control problem is proved to identify with the solution of a $Z-$constrained BSDE, with dynamics associated to a non singular underlying forward process. Due to the non$-$Markovian environment, our main argumentation relies on the use of comparison arguments for path dependent PDEs. Our representation allows in particular to quantify the regularity of the solution to the singular stochastic control problem in terms of the space and time initial data. Our framework also extends to the consideration of degenerate diffusions, leading to the representation of the solution as the infimum of solutions to $Z-$constrained BSDEs. As an application, we study the utility maximisation problem with transaction costs for non$-$Markovian dynamics.