  Pricing and hedging exotic options using local stochastic volatility models
drew a serious attention within the last decade, and nowadays became almost a
standard approach to this problem. In this paper we show how this framework
could be extended by adding to the model stochastic interest rates and
correlated jumps in all three components. We also propose a new fully implicit
modification of the popular Hundsdorfer and Verwer and Modified Craig-Sneyd
finite-difference schemes which provides second order approximation in space
and time, is unconditionally stable and preserves positivity of the solution,
while still has a linear complexity in the number of grid nodes.
