ПрепринтЗовнішня публікація5 серпня 2010 р.

Equations of Mathematical Physics and Compositions of Brownian and Cauchy processes

Luisa BeghinORCIDEnzo OrsingherLyudmyla SakhnoORCID

Анотація

We consider different types of processes obtained by composing Brownian motion $B(t)$, fractional Brownian motion $B_{H}(t)$ and Cauchy processes $% C(t)$ in different manners. We study also multidimensional iterated processes in $\mathbb{R}^{d},$ like, for example, $\left( B_{1}(|C(t)|),...,B_{d}(|C(t)|)\right) $ and $\left( C_{1}(|C(t)|),...,C_{d}(|C(t)|)\right) ,$ deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly we prove that some processes like $% C(|B_{1}(|B_{2}(...|B_{n+1}(t)|...)|)|)$ are governed by fractional diffusion equations.

Класифікація

Домен

Social SciencesPhysical Sciences

Галузь

Economics, Econometrics and FinanceMathematics

Підгалузь

FinanceMathematical Physics

Тема

★Stochastic processes and financial applicationsadvanced mathematical theories

Ідентифікатори

DOI: 10.48550/arxiv.1008.0928

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