Convoluted Brownian motion: a semimartingale approach

Sylvie Rœlly^a, Pierre Vallois<sup>b

a</sup> Universität Potsdam, Institut für Mathematik, Karl-Liebknecht-Str. 24-25, 14476 Potsdam OT Golm, Germany
^b Universitacuté de Lorraine, Institut de Mathématiques Elie Cartan, INRIA-BIGS, CNRS UMR 7502, BP 239, 54506 Vanduvre-lès-Nancy Cedex, France

Abstract: In this paper we analyse semimartingale properties of a class of Gaussian periodic processes, called convoluted Brownian motions, obtained by convolution between a deterministic function and a Brownian motion. A classical example in this class is the periodic Ornstein-Uhlenbeck process. We compute their characteristics and show that in general, they are never Markovian nor satisfy a time-Markov field property. Nevertheless, by enlargement of filtration and/or addition of a one-dimensional component, one can in some case recover the Markovianity. We treat exhaustively the case of the bidimensional trigonometric convoluted Brownian motion and the multidimensional monomial convoluted Brownian motion.

Keywords: Periodic Gaussian process, periodic Ornstein-Uhlenbeck process, Markov-field property, enlargement of filtration.

MSC: 60G10, 60G15, 60G17, 60H10, 60H20

Language: English

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