Some norm estimates for semimartingales

Triet Pham (Rutgers University)
Jianfeng Zhang (University of Southern California)

Abstract


In this paper we introduce a new type of norms for semimartingales, under both linear and nonlinear expectations. Our norm is defined in the spirit of  quasimartingales, and it characterizes square integrable semimartingales. This work is motivated by our study of zero-sum stochastic differential games, whose value process is conjectured to be a semimartingale under a class of probability measures.  As a by product, we establish some a priori estimates for doubly reflected BSDEs without imposing  the  Mokobodski's condition directly.

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Pages: 1-25

Publication Date: December 30, 2013

DOI: 10.1214/EJP.v18-2406

References

  • Chassagneux, Jean-François. A discrete-time approximation for doubly reflected BSDEs. Adv. in Appl. Probab. 41 (2009), no. 1, 101--130. MR2514947
  • Crépey, Stéphane; Matoussi, Anis. Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison. Ann. Appl. Probab. 18 (2008), no. 5, 2041--2069. MR2462558
  • Cheridito, Patrick; Soner, H. Mete; Touzi, Nizar; Victoir, Nicolas. Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 (2007), no. 7, 1081--1110. MR2319056
  • Cvitanić, Jakša; Karatzas, Ioannis. Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 (1996), no. 4, 2024--2056. MR1415239
  • Cvitanic, J. and Zhang, J. : Contract Theory in Continuous Time Models, (2012), Springer Finance. MR0745449
  • Ekren, I., Keller, C., Touzi, N., and Zhang, J. : On Viscosity Solutions of Path Dependent PDEs, sl Annals of Probability, to appear, arXiv:1109.5971.
  • Ekren, I., Touzi, N., and Zhang, J. : Optimal Stopping under Nonlinear Expectation, preprint, arXiv:1209.6601.
  • Ekren, I., Touzi, N., and Zhang, J. : Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I, preprint, arXiv:1210.0006.
  • Ekren, I., Touzi, N., and Zhang, J. : Viscosity Solutions of Fully Nonlinear Path Parabolic Dependent PDEs: Part II, preprint, arXiv:1210.0007.
  • El Karoui, N.; Kapoudjian, C.; Pardoux, E.; Peng, S.; Quenez, M. C. Reflected solutions of backward SDE's, and related obstacle problems for PDE's. Ann. Probab. 25 (1997), no. 2, 702--737. MR1434123
  • Hamadène, S.; Hassani, M. BSDEs with two reflecting barriers: the general result. Probab. Theory Related Fields 132 (2005), no. 2, 237--264. MR2199292
  • Hamadène, S.; Hassani, M.; Ouknine, Y. Backward SDEs with two $rcll$ reflecting barriers without Mokobodski's hypothesis. Bull. Sci. Math. 134 (2010), no. 8, 874--899. MR2737357
  • Hu, M., Ji, S., Peng, S., and Song, Y. : Backward Stochastic Differential Equations Driven by G-Brownian Motion, preprint, arXiv:1206.5889.
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1988. xxiv+470 pp. ISBN: 0-387-96535-1 MR0917065
  • Meyer, P.-A.; Zheng, W. A. Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 20 (1984), no. 4, 353--372. MR0771895
  • Matoussi, A., Piozin, L. and Possamaï, D. : Second-order BSDEs with general reflection and Dynkin games under uncertainty, preprint, arXiv:1212.0476.
  • Nutz, Marcel. Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17 (2012), no. 24, 7 pp. MR2950190
  • Nutz, Marcel; van Handel, Ramon. Constructing sublinear expectations on path space. Stochastic Process. Appl. 123 (2013), no. 8, 3100--3121. MR3062438
  • Peng, S. Backward SDE and related $g$-expectation. Backward stochastic differential equations (Paris, 1995–1996), 141--159, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997. MR1752680
  • Peng, S. : Nonlinear Expectations and Stochastic Calculus under Uncertainty, preprint, arXiv:1002.4546.
  • Peng, Shige; Xu, Mingyu. The smallest $g$-supermartingale and reflected BSDE with single and double $L^ 2$ obstacles. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3, 605--630. MR2139035
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Pham, T. and Zhang, J. : Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation, preprint, arXiv:1209.6605.
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4 MR2020294
  • Rao, K. Murali. Quasi-martingales. Math. Scand. 24 1969 79--92. MR0275511
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 (2011), no. 2, 265--287. MR2746175
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 (2011), no. 67, 1844--1879. MR2842089
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Dual formulation of second order target problems. Ann. Appl. Probab. 23 (2013), no. 1, 308--347. MR3059237
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 (2012), no. 1-2, 149--190. MR2925572
  • Xu, Jing; Zhang, Bo. Martingale characterization of $G$-Brownian motion. Stochastic Process. Appl. 119 (2009), no. 1, 232--248. MR2485026
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