Let Z_t denote the standard Brownian process. Show that the covariance matrix of the bivariate Gaussian random variable (Z_t ∫₀¹Z_u du) is given by

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t(1 - {/) E[(Z₁. [' Zu du)' (Zi. f' Zu du)] = (₁«₁'-5) ¹(¹79)). Zt, 1 (1 {) 3 Also, show that the conditional distribution of Z, given f Zu du = z is normal with mean 31 (1)z and variance t - 3t² (1-)². Using (4.3.29) with the choice of Y to be f Zu du and T = 1, show that (Thompson, 1999) √3n(√3z) Cfix (S, 1, 0) ≥ er ·∞ S'' max (Se³01(1-1/2)2+(r-q)1+[1-31²(1-2)²] — X, 0) dtdz, 0 where n(z)=√e-2²/2.