Most textbooks define CVA as being independent of whether the bank survives. Does it not makes sense to simultaneously consider counterparty and bank default?I am trying to ascertain whether or not "real traders" are doing this and/or should be doing this?Also, from a mathematical point of view, how does the evaluation of the integral change for CVA when you include the bank survival probabilities?thanks

That consideration is referred to as DVA and is extensively discussed in the post-crisis literature.

Thanks bearishI am aware of CVA's counterpart DVA, however there seems to be a school of thought which complicates the matter by incorporating both (i) counterparty default and (ii) bank survival in CVA as well as (iii) bank default and (iv) counterparty survival in DVA. As a result CVA becomes a function of two sets of probabilities instead of the usual one. The same holds for DVA.I.e, this defines CVA as the (risk-neutral) expected loss for the bank due to counterparty default subject to survival of the bank. Conversely, this notion leads to DVA as the (risk-neutral) expected loss for the counterparty due to bank default subject to survival of the counterparty.To oversimplify it, this means that CVA = Loss * PD changes to CVA = Loss * PD * SP. Here I have PD as the counterparty default probability and SP as the bank survival probability.hope this is more clearthanks

You can use a first to default probability for the counterparty default instead of the standard one that is independent on the bank default. Then you will have to take into account correlation of course with a copula for example.