Today's decisions on climate change mitigation affect the damage that future generations will bear. Discounting future benefits and costs of climate change mitigation is one of the most important components of assessing efficient climate mitigation pathways. We extend the DICE model with stochastic discount rates to reflect the dynamic nature of discount rates. Stochastic rates give rise to a mitigation strategy, resulting in all model quantities becoming stochastic. We show that the classical calibration of the DICE model induces intergenerational inequality: future generations have to bear higher costs relative to GDP and that this effect worsens under stochastic discount rates. Accounting for additional financing risks, we investigate two modifications of DICE. We find that allowing financing of abatement costs and considering non-linear financing effects for large damages improves intergenerational effort sharing. We propose a modified optimisation to keep costs below 3% of GDP, resulting in more equal efforts between generations.