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We formulate and investigate a general stochastic control problem under aprogressive enlargement of filtration. The global information is enlarged froma reference filtration and the knowledge of multiple random times together withassociated marks when they occur. By working under a density hypothesis on theconditional joint distribution of the random times and marks, we prove adecomposition of the original stochastic control problem under the globalfiltration into classical stochastic control problems under the referencefiltration, which are determined in a finite backward induction. Our methodrevisits and extends in particular stochastic control of diffusion processeswith finite number of jumps. This study is motivated by optimization problemsarising in default risk management, and we provide applications of ourdecomposition result for the indifference pricing of defaultable claims, andthe optimal investment under bilateral counterparty risk. The solutions areexpressed in terms of BSDEs involving only Brownian filtration, and remarkablywithout jump terms coming from the default times and marks in the globalfiltration.

Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management