Alternative models for FX: pricing double barrier options in regime-switching Lévy models with memory

This paper is a supplement to our recent paper ``Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models". We introduce the class of regime-switching L\'evy models with memory, which take into account the evolution of the stochastic parameters in the past. This generalization of the class of L\'evy models modulated by Markov chains is similar in spirit to rough volatility models. It is flexible and suitable for application of the machine-learning tools. We formulate the modification of the numerical method in ``Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models", which has the same number of the main time-consuming blocks as the method for Markovian regime-switching models.