# 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

$$\text{Pr}[\tau_1>t,\tau_2\leq t,\tau_3\leq t]=\text{Pr}[\tau_2\leq t,\tau_3\leq t] - \text{Pr}[\tau_1\leq t,\tau_2\leq t,\tau_3\leq t]$$
$$\text{Pr}[\tau_2\leq t,\tau_3\leq t]=C(1,q_2(t),q_3(t))$$