Under the risk neutral measure Q, the dynamics of the price process of an asset S(t) and the discount bond price process are governed by

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Under the risk neutral measure Q, the dynamics of the price process of an asset S(t) and the discount bond price process are governed by 

where Z₁(t) and Z₂(t) are uncorrelated standard Q-Brownian processes, σ_B(t, t) = 0 and the volatilities are time dependent functions.

(a) Find cov(S(t), B(t, T′)).

(b) Suppose the bond price B(t,T) is used as the numeraire in the T-forward measure Q_T. Show that the solutions of the above stochastic differential equations are given by (Nielsen and Sandmann, 1996)

where Z^T₁ (t) and Z^T₂ (t) are uncorrelated standard Wiener processes under the T-forward measure Q^T.

dS(t) = S(t) [r(t) dt + 0 (t) dZ (t) + 02 (t) dZ2 (t)] dB(t, T')= B(t, T')[r(t) dt+oB (t, T') dZ (t)],