In this paper different types of compositions involving independent fractional Brownian motions B^j_{ H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{ H_1}(|B^2_{ H_2}(t)|), t>0 and J_F(t)=B^1_{ H_1}(|B^2_{ H_2}(t)|^{ 1/H_1}), t>0 are derived by different methods and compared with those existing in the literature and with those related to B^1(|B^2_{ H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0 is examined in detail and its moments are calculated. Furthermore for J^{ n-1}_F(t)=B^1_{ H}(|B^2_H(...|B^n_H(t)|^{ 1/H}...)|^{ 1/H}), t>0 the following factorization is proved J^{ n-1}_F(t)=\prod_{ j=1}^{ n} B^j_{ \frac{ H}{ n}}(t), t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.
Categories: math.PR math-ph math.MP