Random time-dependent Brownian motion a new approach to fractals of order n

This paper goes in quest of physical models to generate real-valued fractional Brownian motion with independent increments. A detailed analysis, which exhibits some relations between Poissonian white noise and Kramers–Moyal expansion, suggests considering Brownian motions which depends upon random time. One so obtains an Itôʹs lemma of order four, but generalization to higher order is straightforward. As an application, one derives a generalization of the Black–Scholes equation in mathematical finance.