Suppose the forward rate F(t,T ) under the risk neutral measure is governed by Suppose the forward rate F(t,T ) under the risk neutral measure is governed by 

and consider the coupon bond with n coupon payments whose weights w_i(r, t) are defined by (8.2.12), i = 1, 2, ··· ,n, show that the stochastic duration D(t) of the coupon bond is the solution to the following equation (Munk, 1999)

Here, T_i is the payment date of the ith coupon.

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Τ dF(t, T) aFa,r) = (Σσία,τ) 'σέα,Tau dt + Σσία,r)az;), de - J j=1