  Motivated by the trade-off between exploitation and exploration in
reinforcement learning, we study a continuous-time entropy-regularized mean
variance portfolio selection problem in the presence of jumps. We propose an
exploratory SDE for the wealth process associated with multiple risky assets
which exhibit L\'evy jumps. In contrast to the existing literature, we study
the limiting behavior of the natural discrete-time formulation of the wealth
process associated to a randomized control in order to derive the
continuous-time dynamics. We then show that an optimal distributional control
of the continuous-time entropy-regularized exploratory mean-variance problem is
Gaussian. The respective optimal wealth process solves a linear SDE whose
representation is explicitly obtained.
