This talk is on a particular type of self-exciting process. In focus of this talk is study of stability under coefficients previously not touched in literature while local tails are proved as lemma. The tree structure and domination structure are observed and explicitly used in proofs. The main stability result lifts condition of 1-Lipschitz continuity that was previously imposed in Brémaud-Massoulié. First result replaces 1-Lipschitz condition with continuous modulus of continuity and second result allows jumps under some additional but natural assumptions. Generalizations and ramifications are provided. At the end we discuss applications to finance.