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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. 

maximal regularity, non-autonomous Cauchy problem, Brownian motion.