The electrostatic charging of the LISA test masses due to exposure of the spacecraft to energetic particles in the space environment has implications in the design and operation of the gravitational inertial sensors and can affect the quality of the science data. Robust predictions of charging rates and associated stochastic fluctuations are therefore required for the exposure scenarios expected throughout the mission. We report on detailed charging simulations with the Geant4 toolkit, using comprehensive geometry and physics models, for Galactic cosmic-ray protons and helium nuclei. These predict positive charging rates of 50+e/s (elementary charges per second) for solar minimum conditions, decreasing by half at solar maximum, and current fluctuations of up to 30+e/s/Hz1/2. Charging from sporadic solar events involving energetic protons was also investigated. Using an event-size distribution model, we conclude that their impact on the LISA science data is manageable. Several physical processes hitherto unexplored as potential charging mechanisms have also been assessed. Significantly, the kinetic emission of very low-energy secondary electrons due to bombardment of the inertial sensors by primary cosmic rays and their secondaries can produce charging currents comparable with the Monte Carlo rates.