Composition of processes and related partial differential equations

TL;DR

This paper investigates compositions of fractional Brownian motions, derives associated PDEs, examines iterated processes, and explores compositions with Cauchy processes, providing new insights into their distributions and equations.

Contribution

It introduces new PDEs for composed fractional Brownian motions, analyzes iterated processes, and establishes factorizations and equations involving Cauchy processes.

Findings

Derived PDEs for distributions of composed fractional Brownian motions.

Calculated moments of iterated Brownian motion processes.

Proved factorization of complex compositions into products of fractional Brownian motions.

Abstract

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.