The Annals of Probability

Canonical RDEs and general semimartingales as rough paths

Ilya Chevyrev and Peter K. Friz

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lépingle’s BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased a uniformly-controlled-variations (UCV) condition (Kurtz–Protter, Jakubowski–Mémin–Pagès). A number of examples illustrate the scope of our results.

Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 420-463.

Dates
Received: April 2017
Revised: December 2017
First available in Project Euclid: 13 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1544691625

Digital Object Identifier
doi:10.1214/18-AOP1264

Mathematical Reviews number (MathSciNet)
MR3909973

Zentralblatt MATH identifier
07036341

Subjects
Primary: 60H99: None of the above, but in this section
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Càdlàg rough paths stochastic and rough differential equations with jumps Marcus canonical equations general semimartingales limit theorems

Citation

Chevyrev, Ilya; Friz, Peter K. Canonical RDEs and general semimartingales as rough paths. Ann. Probab. 47 (2019), no. 1, 420--463. doi:10.1214/18-AOP1264. https://projecteuclid.org/euclid.aop/1544691625


Export citation

References

  • [1] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge Univ. Press, Cambridge.
  • [2] Applebaum, D. and Kunita, H. (1993). Lévy flows on manifolds and Lévy processes on Lie groups. J. Math. Kyoto Univ. 33 1103–1123.
  • [3] Applebaum, D. and Tang, F. (2001). Stochastic flows of diffeomorphisms on manifolds driven by infinite-dimensional semimartingales with jumps. Stochastic Process. Appl. 92 219–236.
  • [4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [5] Breuillard, E., Friz, P. and Huesmann, M. (2009). From random walks to rough paths. Proc. Amer. Math. Soc. 137 3487–3496.
  • [6] Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19–42.
  • [7] Cass, T. and Ogrodnik, M. (2017). Tail estimates for Markovian rough paths. Ann. Probab. 45 2477–2504.
  • [8] Chechkin, A. and Pavlyukevich, I. (2014). Marcus versus Stratonovich for systems with jump noise. J. Phys. A 47 342001, 15.
  • [9] Chevyrev, I. (2018). Random walks and Lévy processes as rough paths. Probab. Theory Related Fields 170 891–932.
  • [10] Chevyrev, I. and Lyons, T. (2016). Characteristic functions of measures on geometric rough paths. Ann. Probab. 44 4049–4082.
  • [11] Coutin, L., Friz, P. and Victoir, N. (2007). Good rough path sequences and applications to anticipating stochastic calculus. Ann. Probab. 35 1172–1193.
  • [12] Coutin, L. and Lejay, A. (2005). Semi-martingales and rough paths theory. Electron. J. Probab. 10 761–785.
  • [13] Davis, G. J. and Hu, T. Y. (1995). On the structure of the intersection of two middle third Cantor sets. Publ. Mat. 39 43–60.
  • [14] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [15] Friz, P., Gassiat, P. and Lyons, T. (2015). Physical Brownian motion in a magnetic field as a rough path. Trans. Amer. Math. Soc. 367 7939–7955.
  • [16] Friz, P. and Oberhauser, H. (2009). Rough path limits of the Wong–Zakai type with a modified drift term. J. Funct. Anal. 256 3236–3256.
  • [17] Friz, P. and Victoir, N. (2008). On uniformly subelliptic operators and stochastic area. Probab. Theory Related Fields 142 475–523.
  • [18] Friz, P. K. and Hairer, M. (2014). A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, Cham.
  • [19] Friz, P. K. and Shekhar, A. (2017). General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula. Ann. Probab. 45 2707–2765.
  • [20] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
  • [21] Friz, P. K. and Zhang, H. (2018). Differential equations driven by rough paths with jumps. J. Differential Equations 264 6226–6301.
  • [22] Fujiwara, T. (1991). Stochastic differential equations of jump type on manifolds and Lévy flows. J. Math. Kyoto Univ. 31 99–119.
  • [23] Fujiwara, T. and Kunita, H. (1985). Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25 71–106.
  • [24] Fujiwara, T. and Kunita, H. (1999). Canonical SDE’s based on semimartingales with spatial parameters. II. Inverse flows and backward SDE’s. Kyushu J. Math. 53 301–331.
  • [25] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin.
  • [26] Jakubowski, A., Mémin, J. and Pagès, G. (1989). Convergence en loi des suites d’intégrales stochastiques sur l’espace ${\mathbf{D}}^{1}$ de Skorokhod. Probab. Theory Related Fields 81 111–137.
  • [27] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
  • [28] Kelly, D. and Melbourne, I. (2016). Smooth approximation of stochastic differential equations. Ann. Probab. 44 479–520.
  • [29] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [30] Kunita, H. (1996). Stochastic differential equations with jumps and stochastic flows of diffeomorphisms. In Itô’s Stochastic Calculus and Probability Theory 197–211. Springer, Tokyo.
  • [31] Kurtz, T. G., Pardoux, É. and Protter, P. (1995). Stratonovich stochastic differential equations driven by general semimartingales. Ann. Inst. Henri Poincaré Probab. Stat. 31 351–377.
  • [32] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
  • [33] Kurtz, T. G. and Protter, P. E. (1996). Weak convergence of stochastic integrals and differential equations. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Math. 1627 1–41. Springer, Berlin.
  • [34] Lejay, A. and Lyons, T. (2005). On the importance of the Lévy area for studying the limits of functions of converging stochastic processes. Application to homogenization. In Current Trends in Potential Theory. Theta Ser. Adv. Math. 4 63–84. Theta, Bucharest.
  • [35] Lenglart, E., Lépingle, D. and Pratelli, M. (1980). Présentation unifiée de certaines inégalités de la théorie des martingales. In Seminar on Probability, XIV (Paris, 1978/1979) (French). Lecture Notes in Math. 784 26–52. Springer, Berlin.
  • [36] Lépingle, D. (1976). La variation d’ordre $p$ des semi-martingales. Z. Wahrsch. Verw. Gebiete 36 295–316.
  • [37] Lyons, T. and Ni, H. (2015). Expected signature of Brownian motion up to the first exit time from a bounded domain. Ann. Probab. 43 2729–2762.
  • [38] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [39] Marcus, S. I. (1978). Modeling and analysis of stochastic differential equations driven by point processes. IEEE Trans. Inform. Theory 24 164–172.
  • [40] Marcus, S. I. (1980/81). Modeling and approximation of stochastic differential equations driven by semimartingales. Stochastics 4 223–245.
  • [41] McShane, E. J. (1972). Stochastic differential equations and models of random processes. 263–294.
  • [42] Meyer, P.-A. (1972). Martingales and Stochastic Integrals. I. Lecture Notes in Mathematics. Vol. 284. Springer, Berlin.
  • [43] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge Univ. Press, Cambridge.
  • [44] Simon, T. (2003). Small deviations in $p$-variation for multidimensional Lévy processes. J. Math. Kyoto Univ. 43 523–565.
  • [45] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
  • [46] Williams, D. R. E. (1998). Differential equations driven by discontinuous paths. Ph.D. thesis, Imperial College London.
  • [47] Williams, D. R. E. (2001). Path-wise solutions of stochastic differential equations driven by Lévy processes. Rev. Mat. Iberoam. 17 295–329.