Suppose we define the modified forward LIBOR L m i (t) and futures LIBOR L f i

Question:

Suppose we define the modified forward LIBOR L^m_i (t) and futures LIBOR L^f _i (t) by

1 1 + i+1 L" (t) = EQT; QTi B(Ti, Ti+1) [B +D] Ti+D)]: 1 1 + +1 L (1) = E'g [B(T, T+1). { di+14 respectively. Here, Q_T_i and Q are the T_i-forward measure and risk neutral measure, respectively. Assuming that the discount bond price B(t,T) follows the Gaussian HJM process, show that L_m(t) and L^f(t) satisfy the following stochastic differential equations:

dLm (t) = dL (1): 1 + i+1Lm (t) i+1 1++1L i+1 m [og(t, Tj) og(t, Tj+1)]dzi (t), j=1 (1) - [of(t, Tj) og(t,

where Z^Ti(t) = (Z^Ti₁ (t)···Z^Ti_m (t))^T is an m-dimensional Q_Tj-Brownian process and Z(t) = (Z1(t)···Z_m(t))^T is an m-dimensional Q-Brownian process.