  This paper investigates the moral hazard problem in finite horizon with both
continuous and lump-sum payments, involving a time-inconsistent sophisticated
agent and a standard utility maximiser principal. Building upon the so-called
dynamic programming approach in Cvitani\'c, Possama\"i, and Touzi [18] and the
recently available results in Hern\'andez and Possama\"i [43], we present a
methodology that covers the previous contracting problem. Our main contribution
consists in a characterisation of the moral hazard problem faced by the
principal. In particular, it shows that under relatively mild technical
conditions on the data of the problem, the supremum of the principal's expected
utility over a smaller restricted family of contracts is equal to the supremum
over all feasible contracts. Nevertheless, this characterisation yields, as far
as we know, a novel class of control problems that involve the control of a
forward Volterra equation via Volterra-type controls, and infinite-dimensional
stochastic target constraints. Despite the inherent challenges associated to
such a problem, we study the solution under three different specifications of
utility functions for both the agent and the principal, and draw qualitative
implications from the form of the optimal contract. The general case remains
the subject of future research.
