Convergence of an Euler discretisation scheme for the Heston stochastic-local volatility model with CIR interest rates

We consider the Heston-CIR stochastic-local volatility model in the context of foreign exchange markets, which contains both a stochastic and a local volatility component for the exchange rate combined with the Cox-Ingersoll-Ross dynamics for the domestic and foreign interest rates. We study a full truncation scheme for simulating the stochastic volatility component and the two interest rates and derive the exponential integrability of full truncation Euler approximations for the square root process. Under a full correlation structure and a realistic set of assumptions on the so-called leverage function, we prove strong convergence of the exchange rate approximations and then deduce the convergence of Monte Carlo estimators for a number of vanilla and path-dependent options.