[1] , Global solutions to rough differential equations with unbounded vector fields (2010), hal:inria-00451193.
☞ This article revisits the Euler scheme proposed by A.M. Davie in [20] in a general setting and gives existence, uniqueness to solutions of RDEs with unbounded coefficients.
[2] , Controlled differential equations as Young integrals: a simple approach (2009), hal:inria-00402397.
☞ This article covers many results (existence, uniqueness, continuity, ...) on rough differential equations driven by paths of finite p-variations with p < 2.
[3] , Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310.
[4] T. Lyons and Z. Qian, System Control and Rough Paths, Oxford Mathematical Monographs, Oxford University Press, 2002.
[5] A. Lejay, An introduction to rough paths, Séminaire de probabilités XXXVII, Lecture Notes in Mathematics, vol. 1832, Springer-Verlag, 2003, pp. 1–59.
[6] T. Lyons, M. Caruana, and T. Lévy, Differential Equations Driven by Rough Paths, École d’été des probabilités de Saint-Flour XXXIV — 2004 (J. Picard, ed.), Lecture Notes in Math., vol. 1908, Springer, Berlin, 2007.
[7] , Yet another introduction to rough paths (2009). To appear in Séminaire de probabilités, Lecture Notes in Mathematics, Springer-Verlag.
[8] P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press, 2009.
The following references allows one to better understand the algebraic and
geometric structure used the in theory of rough paths.
[9] F. Baudoin, An introduction to the geometry of stochastic flows,
Imperial College Press, London, 2004.
☞ This in not a book about rough paths, but it gives some nice insight
about the algebraic and geometric structure used the the theory of
rough paths.
[10] , The Magnus expansion and
some of its applications, Phys. reports 470 (2009), 151–238.
[11] , Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86–140.
☞ The point of view of M. Gubinelli focuses on an algebraic version of the lemma that gives a rough path from an almost rough paths.
[12] , Curvilinear integrals along enriched paths, Electron. J. Probab. 11 (2006), no. 35, 860–892.
☞ The point of view of D. Feyel and A. de La Pradelle is mainly based on the Green-Riemann formula.
[13] , Rough path analysis via fractional calculus, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2689–2718.
☞ In this article, the authors use fractional calculus to develop a notion of stochastic integral in the spirit of rough path theory.
[14] , An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), 251–282.
☞ The original article where the so-called Young integral is constructed.
[15] , Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré Sect. B (N.S.) 13 (1977), no. 2, 99–125.
[16] , On the gap between deterministic and stochastic ordinary differential equations, Ann. Probability 6 (1978), no. 1, 19–41.
☞ These two articles [15, 16] presents a way to solve a controlled differential equations in dimension one or when the vector fields commute.
[17] P. Friz, Continuity of the Itô-Map for Hölder rough paths with applications to the support theorem in Hölder norm, Probability and Partial Differential Equations in Modern Applied Mathematics, IMA Volumes in Mathematics and its Applications, vol. 140, Springer, 2005, pp. 117–135.
☞ This article show that one may use a more precise norm than the p-variation norm and show a support theorem.
[18] , On (p,q)-rough paths, J. Differential Equations 225 (2006), no. 1, 103–133.
☞ This article defines the notion of (p,q)-rough path which allows one to define a rough differential equation with a drift. Also, it brings a proof of the existence of a solution to some rough differential equation when the vector field is not regular enough using a Brower fixed point theorem and bring some precisions on the difference between geometric and non geometric rough paths.
[19] , A Note on the Notion of Geometric Rough Paths, Probab. Theory Related Fields 136 (2006), no. 3, 395–416.
☞ This article deals with the notion of geometric rough paths and correct some results in [3].
[20] , Differential Equations Driven by Rough Signals: an Approach via Discrete Approximation, Appl. Math. Res. Express. AMRX 2 (2007), Art. ID abm009, 40.
☞ This article presents an alternative notion of solution to rough differential equation and prove the convergence of the Euler scheme. It also gives counterexamples to the uniqueness when the coefficients are not smooth enough.
[21] , An Extension Theorem to Rough Paths, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 5, 835–847.
☞ This article is about the way to lift a Hölder continuous path into a rough path. See also [25].
[22] , Euler Estimates of Rough Differential Equations, J. Differential Equations 244 (2008), no. 2, 388–412.
☞ The authors develop another view on Rough Differential Equation using sub-Riemanniann geodesics, which borrows some ideas from [20].
[23] , A non-commutative sewing lemma, Electron. Commun. Probab. 13 (2008), 24–34.
☞ This article presents a nice and elegant proof of the transformation of an almost rough path into a rough path.
[24] , On rough differential equations, Electron. J. Probab. 14 (2009), no. 12, 341–364.
☞ This article deal with the case of unbounded vector field. Yet it does not cover the case of vector field with linear growth.
[25] , Hölder-continuous rough paths with arbitrary by Fourier normal ordering (2009), arxiv:0903.2716.
☞ This article is about the way to lift a Hölder continuous path into a rough path. Its content is completely different from [21].
A lot of work has been done for developing a theory of Stochastic Differential
Equations driven by a fractional Brownian motion, and several approaches are
possible (see [26,27]). Rough path is only one of these construction.
[26] L. Coutin, An introduction to (stochastic) calculus with respect fo
fractional Brownian motion, Séminaire de Probabilités XL, 2007,
pp. 3–65.
[27] Yu. Mishura, Stochastic calculus for fractional Brownian motion
and related processes, Lecture Notes in Mathematics, vol. 1929,
Springer-Verlag, Berlin, 2008.
[28] , Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields 122 (2002), no. 1, 108–140.
☞ This article shows existence of iterated integrals for the fBM with H > 1∕4 and the non existence of such integrals when H < 1∕4 as a limit of piecewise linear approximations of the path. To overcome this difficulty, other constructions have been provided to cover this case [41,42,45].
[29] , Large deviations for rough paths of the fractional Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 2, 245–271.
☞ This article shows a large deviation principle for the fBM.
[30] , Operators associated with stochastic differential equations driven by fractional Brownian motions, Stochastic Proces. Appl. 117 (2007), no. 5, 550–574.
[31] , Self-similarity and fractional Brownian motions on Lie groups, Electron. J. Probab. 13 (2008), no. 38, 1120–1139.
☞ These articles [30, 31] study the properties in small time of the semi-group associated to a SDE driven by a fBM using some algebraic properties on controlled differential equations and differential equations on Lie group (see for example [9]).
[32] , A version of Hörmander’s theorem for the fractional Brownian motion, Probab. Theory Related Fields 139 (2007), no. 3-4, 373–395.
[33] A. Millet and M. Sanz-Solé, Approximation of rough paths of fractional Brownian motion, Seminar on Stochastic Analysis, Random Fields and Applications V, Progr. Probab., vol. 59, Birkhäuser, Basel, 2008, pp. 275–303.
[34] , Good Rough Path Sequences and Applications to Anticipating & Fractional Stochastic Calculus, Ann. Probab. 35 (2007), no. 3, 1172–1193.
☞ This article shows that the fBM solves some anticipative stochastic differential equation.
[35] , Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion, J. Theoret. Probab. 20 (2007), no. 4, 871–899.
☞ The Doss-Sussmann theory is used to study the rate of convergence of several schemes.
[36] , The rough path associated to the multidimensional analytic fbm with any Hurst parameter (2008), hal:hal-00327355.
☞ This article shows the existence of a rough path can be constructed for any value of H by using an analytical rough path proposed by J. Unterberger [42].
[37] , Delay equations driven by rough paths, Electron. J. Probab. 13 (2008), no. 67, 2031–2068.
[38] , Some differential systems driven by a fBm with Hurst parameter greater than 1/4 (2009), arxiv:0901.2010.
☞ These articles [38,49] study some delay equations driven by fBM.
[39] , Rough Volterra equations 1: the algebraic integration setting (2008), arxiv:0809.2000.
☞ This paper and its companion [40] develop a rough path theory for Volterra equations and gives examples of applications to the fBM.
[40] , Rough Volterra equations 2: convolutional generalized integrals (2008), arxiv:0810.1824.
[41] , Stochastic calculus for fractional Brownian motion with Hurst exponent H > 1∕4: a rough path method by analytic extension, Ann. Probab. 37 (2009), no. 2, 565–614.
☞ An alternative construction of the rough path above the fBM using an analytic regularization scheme
[42] , A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index (2009), arxiv:0906.1406.
☞ An expository article on the lifting method proposed in [43]
[43] , A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering (2009), arxiv:0906.2716.
☞ An application of the general Fourier normal ordering method of [25] to the case of fractional Brownian motion. See also [36].
[44] , Moment estimates for solutions of linear stochastic differential equations driven by analytic fractional Brownian motion (2009), arxiv:0905.0782.
[45] , A construction of the rough path above fractional Brownian motion using Volterra’s representation (2009), arxiv:0909.1307.
☞ This article shows how to construct a rough path above a fBM for any Hurst index using a construction alternative to the one based on the analytic fBM [41,42].
[46] , Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations, ESAIM Probab. Stat. 9 (2005), 165–184.
☞ This article is an exemple of the use of the continuity theorem in the study of the asymptotic behavior of some random process.
[47] , Convergence of multi-dimensional quantized SDE’s (2008), arxiv:0801.0726.
☞ This article develops a way of approximating a solution of some SDE using quantization (i.e., the replacement of a random variable by a discrete one) of the coefficients in the Karhunen-Loève decomposition. It may then be used for any Gaussian process.
[48] , Weak approximation of a fractional SDE (2007), arxiv:0790.0805.
☞ This article studies an approximation of a fractional SDE when the fBM is approximated by a Kac-Stroock approximation.
[49] , Discretizing the fractional Levy area (2009), arxiv:0902.0497.
[50] , Trees and asymptotic developments for fractional stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 157–174.
We present here some results on SPDE related where fractional noise is used as
an example. Recently, other constructions have been proposed.
[51] , Young integrals and SPDEs,
Potential Anal. 25 (2006), no. 4, 307–326.
☞ This article was a first attempt to deal with Stochastic Partial
Differential Equations driven by rough paths. Here, a notion of mild
solution is developped which can be used for fractional noise with
enough regularity.
[52] , The 1-d stochastic wave equation
driven by a fractional Brownian motion, Stochastic Process. Appl. 117
(2007), no. 10, 1448–1472.
[53] , Rough evolution equation (2008),
arxiv:0803.0552.
☞ This article develops a notion of SPDE using the theory of rough
paths and proposed as an example a SPDE driven by a space-time
fractional Brownian motion.
[54] , Differential Equations Driven by Gaussian Signals I (2007), arxiv:0707.0313.
☞ This paper and its companion [55] relate the regularity of a Gaussian process to the covariance function and construct a rough path using the Karhunen-Loève decomposition.
[55] , Differential Equations Driven by Gaussian Signals II (2007), arxiv:0711.0668.
[56] , Enhanced Gaussian Processes and Applications, ESAIM Probab. Stat. 13 (2009), 247–269, DOI 10.1051/ps:2008007.
☞ This article also uses the Karhunen-Loève decomposition to construct an approximation of some Gaussian process and shows results of Wong-Zakai type.
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