  In this paper, we provide a general approach to reformulating any
continuous-time stochastic Stackelberg differential game under closed-loop
strategies as a single-level optimisation problem with target constraints. More
precisely, we consider a Stackelberg game in which the leader and the follower
can both control the drift and the volatility of a stochastic output process,
in order to maximise their respective expected utility. The aim is to
characterise the Stackelberg equilibrium when the players adopt `closed-loop
strategies', i.e. their decisions are based solely on the historical
information of the output process, excluding especially any direct dependence
on the underlying driving noise, often unobservable in real-world applications.
We first show that, by considering the--second-order--backward stochastic
differential equation associated with the continuation utility of the follower
as a controlled state variable for the leader, the latter's unconventional
optimisation problem can be reformulated as a more standard stochastic control
problem with stochastic target constraints. Thereafter, adapting the
methodology developed by Soner and Touzi [67] or Bouchard, \'Elie, and Imbert
[14], the optimal strategies, as well as the corresponding value of the
Stackelberg equilibrium, can be characterised through the solution of a
well-specified system of Hamilton--Jacobi--Bellman equations. For a more
comprehensive insight, we illustrate our approach through a simple example,
facilitating both theoretical and numerical detailed comparisons with the
solutions under different information structures studied in the literature.
