  Can a principal still offer optimal dynamic contracts that are linear in
end-of-period outcomes when the agent controls a process that exhibits memory?
We provide a positive answer by considering a general Gaussian setting where
the output dynamics are not necessarily semi-martingales or Markov processes.
We introduce a rich class of principal-agent models that encompasses dynamic
agency models with memory. From the mathematical point of view, we develop a
methodology to deal with the possible non-Markovianity and non-semimartingality
of the control problem, which can no longer be directly solved by means of the
usual Hamilton-Jacobi-Bellman equation. Our main contribution is to show that,
for one-dimensional models, this setting always allows for optimal linear
contracts in end-of-period observable outcomes with a deterministic optimal
level of effort. In higher dimension, we show that linear contracts are still
optimal when the effort cost function is radial and we quantify the gap between
linear contracts and optimal contracts for more general quadratic costs of
efforts.
