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In the present paper we address stochastic optimal control problems for astep process $(X,\mathbb{F})$ under a progressive enlargement of thefiltration. The global information is obtained adding to the referencefiltration $\mathbb{F}$ the point process $H=1_{[\tau,+\infty)}$. Here $\tau$is a random time that can be regarded as the occurrence time of an externalshock event. We study two classes of control problems, over $[0,T]$ and overthe random horizon $[0,T \wedge \tau]$. We solve these control problemsfollowing a dynamical approach based on a class of BSDEs driven by the jumpmeasure $\mu^ Z$ of the semimartingale $Z=(X,H)$, which is a step process withrespect to the enlarged filtration $\mathbb G$. The BSDEs that we consider canbe solved in $\mathbb{G}$ thanks to a martingale representation theorem whichwe also establish here. To solve the BSDEs and the control problems we need toensure that $Z$ is quasi-left continuous in the enlarged filtration$\mathbb{G}$. Therefore, in addition to the $\mathbb{F}$-quasi left continuityof $X$, we assume some further conditions on $\tau$: the {\it avoidance} of$\mathbb{F}$-stopping times and the {\it immersion} property, or alternatively{\it Jacod's absolutely continuity} hypothesis.

Progressively Enlargement of Filtrations and Control Problems for Step Processes