The fundamental theorem of asset pricing under default and collateral in finite discrete time

We consider a financial market where time and uncertainty are modeled by a finite event-tree. The event-tree has a length of N, a unique initial node at the initial date, and a continuum of branches at each node of the tree. Prices and returns of J assets are modeled, respectively, by a R2J×R2J-valued stochastic process . In this framework we prove a version of the Fundamental Theorem of Asset Pricing which applies to defaultable securities backed by exogenous collateral suffering a contingent linear depreciation.