  We generalize the results of Bielecki and Rutkowski (2015) on funding and
collateralization to a multi-currency framework and link their results with
those of Piterbarg (2012), Moreni and Pallavicini (2017), and Fujii et al.
(2010b).
  In doing this, we provide a complete study of absence of arbitrage in a
multi-currency market where, in each single monetary area, multiple interest
rates coexist. We first characterize absence of arbitrage in the case without
collateral.
  After that we study collateralization schemes in a very general situation:
the cash flows of the contingent claim and those associated to the collateral
agreement can be specified in any currency. We study both segregation and
rehypothecation and allow for cash and risky collateral in arbitrary currency
specifications. Absence of arbitrage and pricing in the presence of collateral
are discussed under all possible combinations of conventions.
  Our work provides a reference for the analysis of wealth dynamics, we also
provide valuation formulas that are a useful foundation for cross-currency
curve construction techniques. Our framework provides also a solid foundation
for the construction of multi-currency simulation models for the generation of
exposure profiles in the context of xVA calculations.
