Switching type valuation and design problems in general OTC contracts with CVA, collateral and funding issue.

In this work, we think to a possible real situation where two defaultable counterparties want to de fine a CSA in which both of them need the flexibility and the possibility to activate the collateralization during the life of the underlying claim/contract. We refer speci cally to a contingent risk mitigation mechanism that allows the counterparty to switch from zero to full/perfect collateralization (or even partial) and switch back whenever she wants until maturity paying some switching costs and taking into account the running costs that emerge over time. The running costs that we model and consider in the analysis of this problem are - by one side - those related to CVA namely counterparty risk hedging costs and - by the other side - the collateral and funding/liquidity costs that emerge when collateralization is active. We show the solution existence and uniqueness for the problem so defined, we build also an algorithm to determine numerically the value function and the optimal switching strategy. We also generalize the analysis allowing the strategic interaction between the parties and facing in the end the pricing problem of a general contract of this kind through reflected backward SDE technique.