Copulas and default probability
Assume a basket of 3 credits, each with some unconditional default probability $ {q_i}(t) = \Pr [{\tau _i} \le t] $ .
Consider the joint CDF $ H $ of the default times is given by $ H(t,t,t) = \Pr [{\tau _1} \le t,{\tau _2} \le t,{\tau _3} \le t] = C({q_1}(t),{q_2}(t),{q_3}(t)) $ , where $ C $ is a known copula function (e.g. Archimedan).
My question is: is there some (possibly Copula-based) representation of a function $ G $ defined as $ G(t,t,t) = \Pr [{\tau _1} > t,{\tau _2} \le t,{\tau _3} \le t] $ ?
I know a survival copula $ {\bar C} $ can be constructed from $ C $ but this is not entirely what I want as I want a joint probability of the last two names to default and the first name to survive.
Thanks
$$ \text{Pr}[\tau_1>t,\tau_2\leq t,\tau_3\leq t]=\text{Pr}[\tau_2\leq t,\tau_3\leq t] - \text{Pr}[\tau_1\leq t,\tau_2\leq t,\tau_3\leq t] $$ $$ \text{Pr}[\tau_2\leq t,\tau_3\leq t]=C(1,q_2(t),q_3(t)) $$