Summary: We propose a deep learning-based methodology to investigate the complex dynamics of electricity prices observed in power markets. The aims are: (a) to process missing data in power price time series with irregular observation times; (b) to detect a Gaussian component in the log-return empirical distributions if there is one; (c) to define suitable stochastic models of the dynamics of power prices. We apply this methodology to US wholesale electricity price time series which are characterized by missing data, high volatility, jumps and spikes. To this end, a multi-layer neural network is built and trained based on a dataset containing information on market prices, traded volumes, numbers of trades and counterparties. The forecasts of the trained neural network are used to fill the gaps in the electricity price time series. Starting with the no-gap reconstructed electricity price time series, clustering techniques are then used to identify the largest Gaussian cluster in the log-return empirical distribution. In each market under investigation, we found that log-returns show considerably large Gaussian clusters. This fact allows us to decouple normal stable periods in which log-returns present Gaussian behavior from turbulent periods in which jumps and spikes occur. The decoupling between the stable motion and the turbulent motion enabled us to define suitable mean-reverting jump-diffusion models of power prices and provide an estimation procedure that makes use of the full information contained in both the Gaussian component and the jumpy component of the log-return distribution. The results obtained demonstrate an interesting agreement with empirical data.