  We propose a deep learning approach to study the minimal variance pricing and
hedging problem in an incomplete jump diffusion market. It is based upon a
rigorous stochastic calculus derivation of the optimal hedging portfolio,
optimal option price, and the corresponding equivalent martingale measure
through the means of the Stackelberg game approach. A deep learning algorithm
based on the combination of the feedforward and LSTM neural networks is tested
on three different market models, two of which are incomplete. In contrast, the
complete market Black-Scholes model serves as a benchmark for the algorithm's
performance. The results that indicate the algorithm's good performance are
presented and discussed.
  In particular, we apply our results to the special incomplete market model
studied by Merton and give a detailed comparison between our results based on
the minimal variance principle and the results obtained by Merton based on a
different pricing principle. Using deep learning, we find that the minimal
variance principle leads to typically higher option prices than those deduced
from the Merton principle. On the other hand, the minimal variance principle
leads to lower losses than the Merton principle.
