Distress propagation in financial systems may be modeled by epidemics on a random graph in which nodes represent financial institutions and edges the exposures between them. Cascade dynamics may be reduced to the evolution of a multi‐dimensional Markov chain that corresponds to a sequential discovery of exposures and determines at any time the size of contagion. The end of contagion becomes a stopping time with respect to the history of the Markov chain. We study the optimal intervention strategy by a lender of last resort with objective to minimize the size of contagion under budget constraints. Our results show that, in the case of non‐anticipative information, the optimal strategy strongly depends on the proportion of banks that use short-term financial instruments for funding.