  We study finite-time horizon continuous-time linear-convex reinforcement
learning problems in an episodic setting. In this problem, the unknown linear
jump-diffusion process is controlled subject to nonsmooth convex costs. We show
that the associated linear-convex control problems admit Lipchitz continuous
optimal feedback controls and further prove the Lipschitz stability of the
feedback controls, i.e., the performance gap between applying feedback controls
for an incorrect model and for the true model depends Lipschitz-continuously on
the magnitude of perturbations in the model coefficients; the proof relies on a
stability analysis of the associated forward-backward stochastic differential
equation. We then propose a novel least-squares algorithm which achieves a
regret of the order $O(\sqrt{N\ln N})$ on linear-convex learning problems with
jumps, where $N$ is the number of learning episodes; the analysis leverages the
Lipschitz stability of feedback controls and concentration properties of
sub-Weibull random variables. Numerical experiment confirms the convergence and
the robustness of the proposed algorithm.
