  We study the solution's existence for a generalized Dynkin game of switching
type which is shown to be the natural representation for general defaultable
OTC contract with contingent CSA. This is a theoretical counterparty risk
mitigation mechanism that allows the counterparty of a general OTC contract to
switch from zero to full/perfect collateralization and switch back whenever she
wants until contract maturity paying some switching costs and taking into
account the running costs that emerge over time. In this paper we allow for the
strategic interaction between the counterparties of the underlying contract,
which makes the problem solution much more tough. We are motivated in this
research by the importance to show the economic sense - in terms of optimal
contract design - of a contingent counterparty risk mitigation mechanism like
our one. In particular, we show that the existence of the solution and the game
Nash equilibrium is connected with the solution of a system of non-linear
reflected BSDE which remains an open problem. We then provide the basic ideas
to numerically search the game equilibrium via an iterative optimal stopping
approach and we show the existence of the solution for our problem under strong
condition, in the so called symmetric case.
