10. Suppose there are independent Poisson processes N1, N2 and N with respective intensity A1, 12 and A. Here, ; is the idiosyncratic shock intensity of firm i, i = 1, 2, and A is the intensity of a macro-economic shock that affects the two firms simultaneously. (a) Explain why the default time ; of firm i is given by Ti = inf {t > 0 : Ni(t) + N(t) > 0}, i = 1, 2. Find the survival function of firm i in terms of A; and 1. [2] (b) Given that all shocks are independent, explain why the joint survival function is given by S(t1, t2) = P(71 > t1, 72 > t2) = exp(-Anti - A2t2 -Amax(t1, t2)). [2] (c) We define the exponential survival copula function by ero.com C' (U1, u2) = S(ST](u1), Sz(u2)). (i) Show that C" (u1, u2) = min(uzu, -, u127 -2), where u; = Si(ti) = e-(ditalt, 0 = \\, i=1, 2. [2] (ii) For the two cases (i) 1 = 12 = 0, > > 0, (ii) A = 0, 1 > 0 and 12 > 0, explain why they correspond to perfect correlation and independent defaults, respec- tively. Find the corresponding exponential survival copula function C' (u1, u2) under each case. [2]