Stochastic calculus with respect to Gaussian processes - Lebovits, Joachim (Potential analysis, 2019)
Stochastic integration wrt Gaussian processes has raised strong interest in recent years, motivated in particular by its applications in Internet traffic modeling, biomedicine and finance. The aim of this work is to define and develop a White Noise Theory-based anticipative stochastic calculus with respect to all Gaussian processes that have an integral representation over a real (maybe infinite) interval. Very rich, this class of Gaussian processes contains, among many others, Volterra processes (and thus fractional Brownian motion) as well as processes the regularity of which varies along the time (such as multifractional Brownian motion). A systematic comparison of the stochastic calculus (including Itô formula) we provide here, to the ones given by Malliavin calculus in Aloś (Ann. Probab. 29(2), 766-801 2001), Mocioalca and Viens (J. Funct. Anal. 222(2), 385-434 2005), Nualart and Taqqu (Stoch. Anal Appl. 24(3), 599-614 2006), Kruk et al. (J. Funct. Anal. 249(1), 92-142 2007), Kruk and Russo (2010), Lei and Nualart (Commun. Stoch. Anal. 6(3), 379-402 2012) and Sottinen and Viitasaari (2014), and by Itô stochastic calculus is also made. Not only our stochastic calculus fully generalizes and extends the ones originally proposed in Mocioalca and Viens (J. Funct. Anal. 222(2), 385-434 2005) and in Nualart and Taqqu (Stoch. Anal Appl. 24(3), 599-614 2006) for Gaussian processes, but also the ones proposed in Bender (Stoch. Process. Appl. 104, 81-106 2003), Biagini et al. (2004) and Elliott and Van der Hoek (Math. Financ. 13(2), 301-330 2003) for fractional Brownian motion (resp. in Lebovits, Ann. Univ. Buchar. Math. Ser. 4(LXII)(1), 397-413 2013; Lebovits and Lévy Véhel Stoch. Int. J. Probab. Stoch. Processes 86(1), 87-124 2014; Lebovits et al. Stoch. Process. Appl. 124(1), 678-708 2014 for multifractional Brownian motion).
Published online: 20 December 2017
Gesehen am 22.07.2019