Frontiers of Mathematics in China >
Finite dimensional characteristic functions of Brownian rough path
Received date: 23 Sep 2015
Accepted date: 17 Apr 2017
Published date: 06 Jul 2017
Copyright
The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2. Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in and obtain an explicit formula for the case when d = 2.
Xi GENG , Zhongmin QIAN . Finite dimensional characteristic functions of Brownian rough path[J]. Frontiers of Mathematics in China, 2017 , 12(4) : 859 -877 . DOI: 10.1007/s11464-017-0648-x
| 1 |
FrizP, VictoirN. Multidimensional Stochastic Processes as Rough Paths. Cambridge: Cambridge Univ Press, 2010
|
| 2 |
GaveauB. Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math, 1977, 139(1): 95–153
|
| 3 |
HaraK, IkedaN. Quadratic Wiener functionals and dynamics on Grassmannians. Bull Sci Math, 2001, 125(6): 481–528
|
| 4 |
HelmesK, SchwaneA. Lévy’s stochastic area formula in higher dimensions.J Funct Anal, 1983, 54(2): 177–192
|
| 5 |
HidaT. Quadratic functionals of Brownian motion. J Multivariate Anal, 1971, 1(1): 58–69
|
| 6 |
IkedaN, KusuokaS, ManabeS. Lévy’s stochastic area formula for Gaussian processes. Comm Pure Appl Math, 1994, 47(3): 329–360
|
| 7 |
IkedaN, ManabeS. Asymptotic formulae for stochastic oscillatory integrals. In: Elworthy K D, Ikeda N, eds. Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics. Pitman Res Notes Math Ser, 284. Boston: Pitman, 1993, 136–155
|
| 8 |
LevinD, WildonM. A combinatorial method for calculating the moments of Lévy area. Trans Amer Math Soc, 2008, 360(12): 6695–6709
|
| 9 |
LevinJ. On the matrix Riccati equation. Proc Amer Math Soc, 1959, 10(4): 519–524
|
| 10 |
LévyP. Le mouvement brownien plan. Amer J Math, 1940, 62: 487–550
|
| 11 |
LyonsT. Differential equations driven by rough signals. Rev Mat Iberoam, 1998, 14(2): 215–310
|
| 12 |
LyonsT, QianZ. System Control and Rough Paths. Oxford: Oxford Univ Press, 2002
|
| 13 |
ReidW T. A matrix differential equation of Riccati type. Amer J Math, 1946, 68: 237–246
|
| 14 |
ReidW T. Riccati Differential Equations. New York: Academic Press, 1972
|
| 15 |
SipiläinenE M. A Pathwise View of Solutions of Stochastic Differential Equations. Ph D Thesis. University of Edinburgh, 1993
|
/
| 〈 |
|
〉 |