  We develop a unified valuation theory that incorporates credit risk
(defaults), collateralization and funding costs, by expanding the replication
approach to a generality that has not yet been studied previously and reaching
valuation when replication is not assumed. This unifying theoretical framework
clarifies the relationship between the two valuation approaches: the adjusted
cash flows approach pioneered for example by Brigo, Pallavicini and co-authors
([12, 13, 34]) and the classic replication approach illustrated for example by
Bielecki and Rutkowski and co-authors ([3, 8]). In particular, results of this
work cover most previous papers where the authors studied specific replication
models.
