The Annals of Probability

Strong supermartingales and limits of nonnegative martingales

Christoph Czichowsky and Walter Schachermayer

Full-text: Open access

Abstract

Given a sequence $(M^{n})^{\infty}_{n=1}$ of nonnegative martingales starting at $M^{n}_{0}=1$, we find a sequence of convex combinations $(\tilde{M}^{n})^{\infty}_{n=1}$ and a limiting process $X$ such that $(\tilde{M}^{n}_{\tau})^{\infty}_{n=1}$ converges in probability to $X_{\tau}$, for all finite stopping times $\tau$. The limiting process $X$ then is an optional strong supermartingale. A counterexample reveals that the convergence in probability cannot be replaced by almost sure convergence in this statement. We also give similar convergence results for sequences of optional strong supermartingales $(X^{n})^{\infty}_{n=1}$, their left limits $(X^{n}_{-})^{\infty}_{n=1}$ and their stochastic integrals $(\int\varphi \,dX^{n})^{\infty}_{n=1}$ and explain the relation to the notion of the Fatou limit.

Article information

Source
Ann. Probab., Volume 44, Number 1 (2016), 171-205.

Dates
Received: December 2013
Revised: July 2014
First available in Project Euclid: 2 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1454423038

Digital Object Identifier
doi:10.1214/14-AOP970

Mathematical Reviews number (MathSciNet)
MR3456335

Zentralblatt MATH identifier
1339.60045

Subjects
Primary: 60G48: Generalizations of martingales 60H05: Stochastic integrals

Keywords
Komlós’s lemma limits of nonnegative martingales Fatou limit optional strong supermartingales predictable strong supermartingales limits of stochastic integrals convergence in probability at all finite stopping times substitute for compactness

Citation

Czichowsky, Christoph; Schachermayer, Walter. Strong supermartingales and limits of nonnegative martingales. Ann. Probab. 44 (2016), no. 1, 171--205. doi:10.1214/14-AOP970. https://projecteuclid.org/euclid.aop/1454423038


Export citation

References

  • [1] Campi, L. and Schachermayer, W. (2006). A super-replication theorem in Kabanov’s model of transaction costs. Finance Stoch. 10 579–596.
  • [2] Chung, K. L. and Glover, J. (1979). Left continuous moderate Markov processes. Z. Wahrsch. Verw. Gebiete 49 237–248.
  • [3] Czichowsky, C. and Schachermayer, W. (2014). Duality theory for portfolio optimisation under transaction costs. Ann. Appl. Probab. To appear.
  • [4] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
  • [5] Delbaen, F. and Schachermayer, W. (1999). A compactness principle for bounded sequences of martingales with applications. In Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996). Progress in Probability 45 137–173. Birkhäuser, Basel.
  • [6] Dellacherie, C. (1972). Capacités et processus stochastiques. Ergebnisse der Mathematik und ihrer Grenzgebiete 67. Springer, Berlin.
  • [7] Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies 29. North-Holland, Amsterdam.
  • [8] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential B. Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam.
  • [9] Föllmer, H. and Kramkov, D. (1997). Optional decompositions under constraints. Probab. Theory Related Fields 109 1–25.
  • [10] Jacod, J. (1979). Calcul stochastique et problèmes de martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • [11] Karatzas, I. and Žitković, G. (2003). Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31 1821–1858.
  • [12] Komlós, J. (1967). A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 217–229.
  • [13] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904–950.
  • [14] Mertens, J.-F. (1972). Théorie des processus stochastiques généraux applications aux surmartingales. Z. Wahrsch. Verw. Gebiete 22 45–68.
  • [15] Perkowski, N. and Ruf, J. (2015). Supermartingales as Radon–Nikodym densities and related measure extensions. Ann. Probab. 43 3133–3176.
  • [16] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Springer, Berlin. Version 2.1, Corrected third printing.
  • [17] Schachermayer, W. (2004). Portfolio Optimization in Incomplete Financial Markets. Scuola Normale Superiore, Classe di Scienze, Pisa.
  • [18] Schwartz, M. (1986). New proofs of a theorem of Komlós. Acta Math. Hungar. 47 181–185.
  • [19] Žitković, G. (2010). Convex compactness and its applications. Math. Financ. Econ. 3 1–12.