This work uses multiple three-dimensional Direct Numerical Simulations (DNSs) to i) investigate the ignition process of a cold lean premixed mixture at atmospheric conditions by a jet of hot burnt gases that may be cooled before injection ii) evaluate models able to predict the outcome of such a scenario in terms of ignition. Understanding and being able to model ignition of cold premixed mixtures by hot burnt gases is essential to design systems like engines (to ensure ignition) and flameproof enclosures (to prevent ignition). Limited work has focused on the combined effects of the jet injection speed and temperature on ignition. This is difficult to do by using experiments only and DNS is a natural approach to gain knowledge on that point. By varying the hot jet injection speed and temperature, the three-dimensional, kinetically detailed, DNSs allow a parametric study of the impact of these parameters on the ignition process and provide data to build and test models. Simulations prove that jet injection speed and temperature (usually less than the adiabatic flame temperature because of cooling effects through the injection hole) directly govern ignition. Chemical Explosive Mode Analysis (CEMA) is used to characterize the reacting flow structure which is strongly impacted by the jet injection speed. Based on the DNSs conclusions, a zero-dimensional Lagrangian model where a small element of the jet burnt gases mixes at a certain rate with the fresh gases while it potentially ignites is found to be a good candidate to predict the outcome of an ignition sequence (success or failure).