Bernoulli

  • Bernoulli
  • Volume 19, Number 5B (2013), 2414-2436.

Stochastic integration with respect to additive functionals of zero quadratic variation

Alexander Walsh

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Abstract

We consider a Markov process $X$ associated to a nonnecessarily symmetric Dirichlet form $\mathcal{E}$. We define a stochastic integral with respect to a class of additive functionals of zero quadratic variation and then we obtain an Itô formula for the process $u(X)$, when $u$ is locally in the domain of $\mathcal{E}$.

Article information

Source
Bernoulli, Volume 19, Number 5B (2013), 2414-2436.

Dates
First available in Project Euclid: 3 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1386078608

Digital Object Identifier
doi:10.3150/12-BEJ457

Mathematical Reviews number (MathSciNet)
MR3160559

Zentralblatt MATH identifier
1286.60049

Keywords
additive functional Dirichlet form Fukushima decomposition Itô formula Markov process stochastic calculus quadratic variation zero energy process

Citation

Walsh, Alexander. Stochastic integration with respect to additive functionals of zero quadratic variation. Bernoulli 19 (2013), no. 5B, 2414--2436. doi:10.3150/12-BEJ457. https://projecteuclid.org/euclid.bj/1386078608


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References

  • [1] Albeverio, S., Ru-Zong, F., Röckner, M. and Stannat, W. (1995). A remark on coercive forms and associated semigroups. In Partial Differential Operators and Mathematical Physics (Holzhau, 1994). Oper. Theory Adv. Appl. 78 1–8. Basel: Birkhäuser.
  • [2] Bardina, X. and Jolis, M. (1997). An extension of Ito’s formula for elliptic diffusion processes. Stochastic Process. Appl. 69 83–109.
  • [3] Bardina, X. and Rovira, C. (2007). On Itô’s formula for elliptic diffusion processes. Bernoulli 13 820–830.
  • [4] Bouleau, N. and Yor, M. (1981). Sur la variation quadratique des temps locaux de certaines semimartingales. C. R. Acad. Sci. Paris Sér. I Math. 292 491–494.
  • [5] Chen, Z.Q., Fitzsimmons, P.J., Kuwae, K. and Zhang, T.S. (2008). Perturbation of symmetric Markov processes. Probab. Theory Related Fields 140 239–275.
  • [6] Chen, Z.Q., Fitzsimmons, P.J., Kuwae, K. and Zhang, T.S. (2008). Stochastic calculus for symmetric Markov processes. Ann. Probab. 36 931–970.
  • [7] Eisenbaum, N. (2000). Integration with respect to local time. Potential Anal. 13 303–328.
  • [8] Eisenbaum, N. (2006). Local time–space stochastic calculus for Lévy processes. Stochastic Process. Appl. 116 757–778.
  • [9] Eisenbaum, N. and Kyprianou, A.E. (2008). On the parabolic generator of a general one-dimensional Lévy process. Electron. Commun. Probab. 13 198–209.
  • [10] Eisenbaum, N. and Walsh, A. (2009). An optimal Itô formula for Lévy processes. Electron. Commun. Probab. 14 202–209.
  • [11] Fitzsimmons, P.J. (1997). Absolute continuity of symmetric diffusions. Ann. Probab. 25 230–258.
  • [12] Fitzsimmons, P.J. and Kuwae, K. (2004). Non-symmetric perturbations of symmetric Dirichlet forms. J. Funct. Anal. 208 140–162.
  • [13] Föllmer, H. (1981). Calcul d’Itô sans probabilités. In Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French). Lecture Notes in Math. 850 143–150. Berlin: Springer.
  • [14] Föllmer, H. (1981). Dirichlet processes. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Math. 851 476–478. Berlin: Springer.
  • [15] Föllmer, H. and Protter, P. (2000). On Itô’s formula for multidimensional Brownian motion. Probab. Theory Related Fields 116 1–20.
  • [16] Föllmer, H., Protter, P. and Shiryayev, A.N. (1995). Quadratic covariation and an extension of Itô’s formula. Bernoulli 1 149–169.
  • [17] Fukushima, M. (1979). A decomposition of additive functionals of finite energy. Nagoya Math. J. 74 137–168.
  • [18] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19. Berlin: de Gruyter.
  • [19] Ghomrasni, R. and Peskir, G. (2003). Local time–space calculus and extensions of Itô’s formula. In High Dimensional Probability, III (Sandjberg, 2002). Progress in Probability 55 177–192. Basel: Birkhäuser.
  • [20] Graversen, S.E. and Rao, M. (1985). Quadratic variation and energy. Nagoya Math. J. 100 163–180.
  • [21] Hu, Z.C., Ma, Z.M. and Sun, W. (2006). Extensions of Lévy–Khintchine formula and Beurling–Deny formula in semi-Dirichlet forms setting. J. Funct. Anal. 239 179–213.
  • [22] Hu, Z.C., Ma, Z.M. and Sun, W. (2010). On representations of non-symmetric Dirichlet forms. Potential Anal. 32 101–131.
  • [23] Kim, J.H. (1987). Stochastic calculus related to nonsymmetric Dirichlet forms. Osaka J. Math. 24 331–371.
  • [24] Kuwae, K. (1998). Functional calculus for Dirichlet forms. Osaka J. Math. 35 683–715.
  • [25] Kuwae, K. (2008). Maximum principles for subharmonic functions via local semi-Dirichlet forms. Canad. J. Math. 60 822–874.
  • [26] Kuwae, K. (2010). Stochastic calculus over symmetric Markov processes without time reversal. Ann. Probab. 38 1532–1569.
  • [27] Lyons, T.J. and Zheng, W.A. (1988). A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Astérisque 157–158 249–271. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987).
  • [28] Ma, Z.-M. and Röckner, M. (1992). Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Berlin: Springer.
  • [29] Nakao, S. (1985). Stochastic calculus for continuous additive functionals of zero energy. Z. Wahrsch. Verw. Gebiete 68 557–578.
  • [30] Oshima, Y. (1988). Lectures on Dirichlet spaces. Lecture notes, Univ. Erlangen Nürnberg.
  • [31] Protter, P.E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Berlin: Springer. Version 2.1, corrected third printing.
  • [32] Russo, F. and Vallois, P. (1995). The generalized covariation process and Itô formula. Stochastic Process. Appl. 59 81–104.
  • [33] Russo, F. and Vallois, P. (2000). Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics Stochastics Rep. 70 1–40.
  • [34] Sato, K.i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press. Translated from the 1990 Japanese original, revised by the author.
  • [35] Walsh, A. (2011). Extended Itô calculus. Thèse, Univ. Pierre et Marie Curie, available on: http://tel.archives-ouvertes.fr/tel-00627558_v1/.
  • [36] Walsh, A. (2011). Local time–space calculus for symmetric Lévy processes. Stochastic Process. Appl. 121 1982–2013.
  • [37] Walsh, A. (2012). Extended Itô calculus for symmetric Markov processes. Bernoulli 18 1150–1171.