Transnational Journal of Mathematical Analysis and Applications Vol. 1, Issue 1, 2013, Pages 45-55 ISSN 2347-9086 Published Online on November 30, 2013  2013 Jyoti Academic Press http://jyotiacademicpress.net 2010 Mathematics Subject Classification: 35K90, 35K50, 35K45, 47D06. Keywords and phrases: maximal regularity, non-autonomous Cauchy problem, Brownian motion. Received October 26, 2013 SEPARABILISATION PROCEDURE AND MAXIMAL REGULARITY APPLIED TO NON AUTONOMOUS CAUCHY PROBLEM AHMED SANI Department of Mathematics Ibn Zohr University Agadir Morocco e-mail: ahmedsani82@gmail.com Abstract We introduce a different method to deal with some partial stochastic and ordinary differential equations otherwise, mainly in financial mathematics, including the notions of “maximal regularity” and “random process separabilisation” using composition with appropriate random variables or Brownian motion. We have chosen as model, presented at the Introduction, the problem of Asian options, we then connect it to the Cauchy problem in autonomous and non autonomous cases. 1. Introduction A lot of stochastic differential equations (SDE) that treat replication problems by a filing (self or not) financing portfolio strategy leads to, in deterministic approach, to partial differential equation (PDE), but in interesting cases, results are restricted to the autonomous case (i.e., time independence of operators).