Some linear fractional stochastic equations

Ivan Nourdin (1) , Ciprian A. Tudor (2, 3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 SAMOS - Statistique Appliquée et MOdélisation Stochastique
3 CES - Centre d'économie de la Sorbonne

Using the multiple stochastic integrals we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one parameter and two parameter cases. When the drift is zero, we show that in the one-parameter case the solution in an exponential, thus positive, function while in the two-parameter settings the solution is negative on a non-negligible set.

Mots clés: fractional Brownian motion; fractional Brownian sheet; multiple stochastic integral; Girsanov transform

Domaines: Probabilités [math.PR]

Ivan Nourdin, Ciprian A. Tudor. Some linear fractional stochastic equations. Stochastics: An International Journal of Probability and Stochastic Processes, 2006, 78 (2), pp.51-65. ⟨10.1080/17442500600688997⟩. ⟨hal-00013903v2⟩