Composition of Processes and Related Partial Differential Equations

In this paper various types of compositions involving independent fractional Brownian motions B(Hj)(j)(t), t > 0, j=1,2, are examined. The partial differential equations governing the distributions of I(F)(t) = B(H1)(1) (broken vertical bar B(H2)(2) (t)broken vertical bar), t > 0, and J(F)(t) = B(H1)(1)(broken vertical bar B(H2)(2) (t)broken vertical bar(1/H1)), t > 0, are derived by various methods and compared with those existing in the literature and with those related to B(1) (broken vertical bar B(H2)(2) (t)broken vertical bar), t > 0. The process of iterated Brownian motion I(F)(n) (t), t > 0, is examined in detail and its moments are calculated. Furthermore, for J(F)(n-1) (t) = B(H)(1) (broken vertical bar B(H)(2) (.....vertical bar B(H)(n)(t)vertical bar(1/H) ....)vertical bar(1/H)), t > 0. the following factorization is proved: J(F)(n-1) (t) = Pi(n)(j=1) B(H/n)(j) (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.