Electronic Journal of Probability

A Generalized Ito's Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals

Chunrong Feng and Huaizhong Zhao

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In this paper, a generalized It${\hat {\rm o}}$'s formula for continuous functions of two-dimensional continuous semimartingales is proved. The formula uses the local time of each coordinate process of the semimartingale, the left space first derivatives and the second derivative $\nabla _1^- \nabla _2^-f$, and the stochastic Lebesgue-Stieltjes integrals of two parameters. The second derivative $\nabla _1^- \nabla _2^-f$ is only assumed to be of locally bounded variation in certain variables. Integration by parts formulae are asserted for the integrals of local times. The two-parameter integral is defined as a natural generalization of both the Ito integral and the Lebesgue-Stieltjes integral through a type of It${\hat {\rm o }}$ isometry formula.

Article information

Electron. J. Probab., Volume 12 (2007), paper no. 57, 1568-1599.

Accepted: 23 December 2007
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 60J55: Local time and additive functionals

local time continuous semimartingale generalized It$hat {rm o}$'s formula stochastic Lebesgue-Stieltjes integral

This work is licensed under aCreative Commons Attribution 3.0 License.


Feng, Chunrong; Zhao, Huaizhong. A Generalized Ito's Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals. Electron. J. Probab. 12 (2007), paper no. 57, 1568--1599. doi:10.1214/EJP.v12-468. https://projecteuclid.org/euclid.ejp/1464818528

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  • Ash, Robert B. Probability and measure theory. Second edition. With contributions by Catherine Doléans-Dade. Harcourt/Academic Press, Burlington, MA, 2000. xii+516 pp. ISBN: 0-12-065202-1 MR1810041 (2001j:28001)
  • Azéma, J.; Jeulin, T.; Knight, F.; Yor, M. Quelques calculs de compensateurs impliquant l'injectivité de certains processus croissants. (French) [Some compensator calculations implying the injectivity of certain increasing processes] Séminaire de Probabilités, XXXII, 316–327, Lecture Notes in Math., 1686, Springer, Berlin, 1998. MR1655302 (2000b:60090)
  • Bouleau, Nicolas; Yor, Marc. Sur la variation quadratique des temps locaux de certaines semimartingales. (French) C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 9, 491–494. MR0612544 (82d:60143)
  • Chung, K. L.; Williams, R. J. Introduction to stochastic integration. Second edition. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1990. xvi+276 pp. ISBN: 0-8176-3386-3 MR1102676 (92d:60057)
  • Eisenbaum, Nathalie. Integration with respect to local time. Potential Anal. 13 (2000), no. 4, 303–328. MR1804175 (2002e:60085)
  • N. Eisenbaum, Local time-space calculus for revisible semimartingales, Seminaire de Probabilites, Vol XL, Lecture Notes in Mathematics 1899, Springer-Verlag, (2007),137-146.
  • K. D. Elworthy, A. Truman and H. Z. Zhao, Generalized Ito Formulae and space-time Lebesgue-Stieltjes integrals of local times, Seminaire de Probabilites, Vol XL, Lecture Notes in Mathematics 1899, Springer-Verlag, (2007),117-136.
  • K. D. Elworthy, A. Truman and H. Z. Zhao, Asymptotics of Heat Equations with Caustics in One-Dimension, Preprint (2006).
  • Feng, Chunrong; Zhao, Huaizhong. Two-parameter $p,q$-variation paths and integrations of local times. Potential Anal. 25 (2006), no. 2, 165–204. MR2238942
  • C. R. Feng and H. Z. Zhao, Rough Path Integral of Local Time, C.R.Acad, Sci. Paris, Ser.I Math (to appear).
  • Flandoli, Franco; Russo, Francesco; Wolf, Jochen. Some SDEs with distributional drift. II. Lyons-Zheng structure, Itô's formula and semimartingale characterization. Random Oper. Stochastic Equations 12 (2004), no. 2, 145–184. MR2065168 (2006a:60105)
  • Föllmer, Hans; Protter, Philip; Shiryayev, Albert N. Quadratic covariation and an extension of Itô's formula. Bernoulli 1 (1995), no. 1-2, 149–169. MR1354459 (96k:60121)
  • Föllmer, Hans; Protter, Philip. On Itô's formula for multidimensional Brownian motion. Probab. Theory Related Fields 116 (2000), no. 1, 1–20. MR1736587 (2001b:60097)
  • Ghomrasni, R.; Peskir, G. Local time-space calculus and extensions of Itô's formula. High dimensional probability, III (Sandjberg, 2002), 177–192, Progr. Probab., 55, Birkhäuser, Basel, 2003. MR2033888 (2005j:60106)
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. xiv+464 pp. ISBN: 0-444-86172-6 MR0637061 (84b:60080)
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 MR1121940 (92h:60127)
  • Kunita, Hiroshi. Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1990. xiv+346 pp. ISBN: 0-521-35050-6 MR1070361 (91m:60107)
  • Lyons, Terry; Qian, Zhongmin. System control and rough paths. Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2002. x+216 pp. ISBN: 0-19-850648-1 MR2036784 (2005f:93001)
  • Lyons, Terence J.; Zheng, Wei An. A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). Astérisque No. 157-158 (1988), 249–271. MR0976222 (90a:60103)
  • McShane, Edward James. Integration. Princeton University Press, Princeton, N. J., 1944, 1957. viii+394 pp. (Fourth printing 1957). MR0082536 (18,567a)
  • Meyer, P. A. Un cours sur les intégrales stochastiques. (French) Séminaire de Probabilités, X (Seconde partie: Théorie des intégrales stochastiques, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), pp. 245–400. Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976. MR0501332 (58 #18721)
  • Moret, S.; Nualart, D. Generalization of Itô's formula for smooth nondegenerate martingales. Stochastic Process. Appl. 91 (2001), no. 1, 115–149. MR1807366 (2002b:60097)
  • Peskir, Goran. A change-of-variable formula with local time on curves. J. Theoret. Probab. 18 (2005), no. 3, 499–535. MR2167640 (2006k:60096)
  • G. Peskir, A change-of-variable formula with local time on surfaces, Seminaire de Probabilites, Vol XL, Lecture Notes in Mathematics 1899, Springer-Verlag, (2007), 69-96.
  • Peskir, Goran. On the American option problem. Math. Finance 15 (2005), no. 1, 169–181. MR2116800 (2005i:91066)
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1994. xii+560 pp. ISBN: 3-540-57622-3 MR1303781 (95h:60072)
  • Rogers, L. C. G.; Walsh, J. B. Local time and stochastic area integrals. Ann. Probab. 19 (1991), no. 2, 457–482. MR1106270 (92g:60107)
  • Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 2. Itô calculus. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xiv+480 pp. ISBN: 0-521-77593-0 MR1780932 (2001g:60189)
  • Russo, F.; Vallois, P. Itô formula for $C\sp 1$-functions of semimartingales. Probab. Theory Related Fields 104 (1996), no. 1, 27–41. MR1367665 (96m:60125)
  • Tanaka, Hiroshi. Note on continuous additive functionals of the $1$-dimensional Brownian path. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 1962/1963 251–257. MR0169307 (29 #6559)
  • Walsh, John B. An introduction to stochastic partial differential equations. École d'été de probabilités de Saint-Flour, XIV–-1984, 265–439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. MR0876085 (88a:60114)
  • Wang, Albert T. Generalized Ito's formula and additive functionals of Brownian motion. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), no. 2, 153–159. MR0488327 (58 #7876)