  We consider a quasi-metric topological structure for the construction of a
new reinforcement learning model in the framework of financial markets. It is
based on a Lipschitz type extension of reward functions defined in metric
spaces. Specifically, the McShane and Whitney extensions are considered for a
reward function which is defined by the total evaluation of the benefits
produced by the investment decision at a given time. We define the metric as a
linear combination of a Euclidean distance and an angular metric component. All
information about the evolution of the system from the beginning of the time
interval is used to support the extension of the reward function, but in
addition this data set is enriched by adding some artificially produced states.
Thus, the main novelty of our method is the way we produce more states -- which
we call "dreams" -- to enrich learning. Using some known states of the
dynamical system that represents the evolution of the financial market, we use
our technique to simulate new states by interpolating real states and
introducing some random variables. These new states are used to feed a learning
algorithm designed to improve the investment strategy by following a typical
reinforcement learning scheme.
