An extension theorem to rough paths. (English) Zbl 1134.60047

Can every continuous path of finite \(p\)-variation in a Banach space \(V\) be lifted to a geometric \(q\)-rough path, where \(q>p\)? The authors give an affirmative answer through this paper. In fact, as the major result, they prove the following theorem.
Theorem. Fix \(p \in [1,+\infty)\). Let \(V\) be a Banach space and \(K\) a closed subgroup of \(G^{([p])}(V)\). If \(x\) is a \((G^{([p])}(V) / K, \| \cdot\| _{G}^{([p])(V) / K})\) continuous path of finite \(p\)-variation, with \( p \notin N \setminus \{0,1\}\), then one can lift \(x\) to a weak geometric \(p\)-rough path.


60H99 Stochastic analysis
34F05 Ordinary differential equations and systems with randomness
34G99 Differential equations in abstract spaces
46N30 Applications of functional analysis in probability theory and statistics
60G17 Sample path properties
93C99 Model systems in control theory
Full Text: DOI Numdam EuDML


[1] Bass, R.F.; Hambly, B.M.; Lyons, T.J., Extending the wong – zakai theorem to reversible Markov processes, J. eur. math. soc., 4, 237-269, (2002) · Zbl 1010.60070
[2] Capitaine, M.; Donati-Martin, C., The Lévy area process for the free Brownian motion, J. funct. anal., 179, 1, 153-169, (2001) · Zbl 0979.60044
[3] Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. theory related fields, 122, 1, 108-140, (2002) · Zbl 1047.60029
[4] Doss, H., Liens entre équations différentielles stochastiques et ordinaires, Ann. inst. H. Poincaré, 13, 99-125, (1977) · Zbl 0359.60087
[5] Folland, G.B.; Stein, E.M., Hardy spaces on homogeneous groups, Math. notes, 28, (1982) · Zbl 0508.42025
[6] P. Friz, N. Victoir, On the notion of geometric rough paths, preprint, 2004 · Zbl 1108.34052
[7] Gromov, M., Carnot – caratheodory spaces seen from within, (), 79-323 · Zbl 0864.53025
[8] Hambly, B.M.; Lyons, T.J., Stochastic area for Brownian motion on the sierpinski gasket, Ann. probab., 26, 1, 132-148, (1998) · Zbl 0936.60073
[9] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, Graduate texts in mathematics, vol. 113, (1991), Springer-Verlag New York · Zbl 0734.60060
[10] Ledoux, M.; Lyons, T.; Qian, Z., Lévy area of Wiener processes in Banach spaces, Ann. probab., 30, 2, 546-578, (2002) · Zbl 1016.60071
[11] Lejay, A., Introduction to rough paths, Séminaire de probabilités, Lecture notes in mathematics, vol. XXXVII, (2003) · Zbl 1041.60051
[12] Lyons, T., Differential equations driven by rough signals, Rev. mat. iberoamericana, 14, 2, 215-310, (1998) · Zbl 0923.34056
[13] Lyons, T.; Qian, Z., System control and rough paths, (2002), Oxford University Press · Zbl 1029.93001
[14] Reutenauer, C., Free Lie algebras, London mathematical society monographs (N.S.), vol. 7, (1993), Oxford Science Publications · Zbl 0798.17001
[15] Serre, J.P., Lie algebras and Lie groups, Lecture notes in mathematics, vol. 1500, (1992)
[16] Sussman, H.J., On the gap between deterministic and stochastic ordinary differential equations, Ann. probab., 6, 19-41, (1978) · Zbl 0391.60056
[17] Varadarajan, V.S., Lie groups, Lie algebras, and their representations, Graduate texts in mathematics, vol. 102, (1984) · Zbl 0955.22500
[18] N.B. Victoir, Levy area for the free Brownian motion: existence and non-existence, J. Funct. Anal., in press · Zbl 1062.46055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.