General Explosions

This lecture covers the general case of continuous time Markov Chains, focusing on the pair (A, Q) where A is a probability vector and Q is a Q-matrix. It discusses the conditions for nonexplosion, the minimal chain, and the implications of explosions in the chain. The instructor explains the concept of jump chains, holding times, and the definition of processes in the presence of explosions. The lecture also delves into the theorem for a Q matrix and the minimal positive solution. Various scenarios of explosions and their impact on the Markov chain are explored, emphasizing the importance of defining the process at explosion times. The lecture concludes with insights on induction and jump times.