The existence of dominating local martingale measures

Abstract

We prove that for locally bounded processes the absence of arbitrage opportunities of the first kind is equivalent to the existence of a dominating local martingale measure. This is related to and motivated by results from the theory of filtration enlargements.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Acciaio, B., Fontana, C., Kardaras, C.: Arbitrage of the first kind and filtration enlargements in semimartingale financial models (2014). arXiv preprint arXiv:1401.7198

  2. 2.

    Aksamit, A., Choulli, T., Deng, J., Jeanblanc, M.: Arbitrages in a progressive enlargement setting (2013). arXiv preprint arXiv:1312.2433

  3. 3.

    Amendinger, J., Imkeller, P., Schweizer, M.: Additional logarithmic utility of an insider. Stoch. Process. Appl. 75, 263–286 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Amendinger, J.: Martingale representation theorems for initially enlarged filtrations. Stoch. Process. Appl. 89, 101–116 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Ankirchner, S.: Information and semimartingales. Ph.D. thesis, Humboldt-Universität zu Berlin (2005). Available online http://www.stochastik.uni-jena.de/stochastik_multimedia/Publikationen+Ankirchner/Diss(homepage).pdf

  6. 6.

    Ankirchner, S., Dereich, S., Imkeller, P.: The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Probab. 34, 743–778 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Ankirchner, S., Dereich, S., Imkeller, P.: Enlargement of filtrations and continuous Girsanov-type embeddings. In: Donati-Martin, C., et al. (eds.) Séminaire de Probabilités XL. Lecture Notes in Math., vol. 1899, pp. 389–410. Springer, Berlin (2007)

    Google Scholar 

  8. 8.

    Becherer, D.: The numeraire portfolio for unbounded semimartingales. Finance Stoch. 5, 327–341 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Brannath, W., Schachermayer, W.: A bipolar theorem for \(L^{0}_{+}(\varOmega, \mathcal{F},\mathbb{P})\). In: Azéma, J., et al. (eds.) Séminaire de Probabilités, XXXIII. Lecture Notes in Math., vol. 1709, pp. 349–354. Springer, Berlin (1999)

    Google Scholar 

  10. 10.

    Carr, P., Fisher, T., Ruf, J.: On the hedging of options on exploding exchange rates. Finance Stoch. 18, 115–144 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Cvitanić, J., Schachermayer, W., Wang, H.: Utility maximization in incomplete markets with random endowment. Finance Stoch. 5, 259–272 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Delbaen, F., Schachermayer, W.: Arbitrage possibilities in Bessel processes and their relations to local martingales. Probab. Theory Relat. Fields 102, 357–366 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Delbaen, F., Schachermayer, W.: The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5, 926–945 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Dellacherie, C.: Ensembles aléatoires. I. In: Meyer, P.-A. (ed.) Séminaire de Probabilités, III, Univ. Strasbourg, 1967/68. Lecture Notes in Math., vol. 88, pp. 97–114. Springer, Berlin (1969)

    Google Scholar 

  16. 16.

    Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. B: Theory of Martingales. North-Holland Mathematics Studies, vol. 72. North-Holland, Amsterdam (1982)

    Google Scholar 

  17. 17.

    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)

    Google Scholar 

  18. 18.

    Föllmer, H.: The exit measure of a supermartingale. Probab. Theory Relat. Fields 21, 154–166 (1972)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Föllmer, H., Gundel, A.: Robust projections in the class of martingale measures. Ill. J. Math. 50, 439–472 (2006)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Föllmer, H., Imkeller, P.: Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space. Ann. Inst. Henri Poincaré Probab. Stat. 29, 569–586 (1993)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Föllmer, H., Kramkov, D.: Optional decompositions under constraints. Probab. Theory Relat. Fields 109, 1–25 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20, 381–408 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Harrison, J.M., Pliska, S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11, 215–260 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Imkeller, P., Pontier, M., Weisz, F.: Free lunch and arbitrage possibilities in a financial market model with an insider. Stoch. Process. Appl. 92, 103–130 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979)

    Google Scholar 

  26. 26.

    Jacod, J.: Grossissement initial, hypothèse (H ′) et théorème de Girsanov. In: Jeulin, Th., Yor, M. (eds.) Grossissements de Filtrations: Exemples et Applications. Lecture Notes in Mathematics, vol. 1118, pp. 15–35. Springer, Berlin (1985)

    Google Scholar 

  27. 27.

    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  28. 28.

    Jarrow, R.A., Protter, P., Shimbo, K.: Asset price bubbles in incomplete markets. Math. Finance 20, 145–185 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Karatzas, I., Lehoczky, J.P., Shreve, S.E., Xu, G.-L.: Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29, 702–730 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Kardaras, C.: Finitely additive probabilities and the fundamental theorem of asset pricing. In: Chiarella, C., Novikov, A. (eds.) Contemporary Quantitative Finance. Essays in Honour of Eckhard Platen, pp. 19–34. Springer, Berlin (2010)

    Google Scholar 

  33. 33.

    Kardaras, C.: Market viability via absence of arbitrage of the first kind. Finance Stoch. 16, 651–667 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Kardaras, C., Kreher, D., Nikeghbali, A.: Strict local martingales and bubbles. Ann. Appl. Probab. (2015, to appear). arXiv preprint arXiv:1108.4177. doi:10.1214/14-AAP1037

  35. 35.

    Kardaras, C., Platen, E.: On the semimartingale property of discounted asset-price processes. Stoch. Process. Appl. 121, 2678–2691 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Kunita, H.: Absolute continuity of Markov processes. In: Meyer, P.-A. (ed.) Séminaire de Probabilités X. Lecture Notes in Mathematics, vol. 511, pp. 44–77. Springer, Berlin (1976)

    Google Scholar 

  38. 38.

    Larsen, K., Žitković, G.: Stability of utility-maximization in incomplete markets. Stoch. Process. Appl. 117, 1642–1662 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Loewenstein, M., Willard, G.A.: Local martingales, arbitrage, and viability—free snacks and cheap thrills. Econom. Theory 161, 135–161 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Pal, S., Protter, P.: Analysis of continuous strict local martingales via h-transforms. Stoch. Process. Appl. 120, 1424–1443 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Parthasarathy, K.R.: Probability Measures on Metric Spaces. Probability and Mathematical Statistics, vol. 3. Academic Press, New York (1967)

    Google Scholar 

  42. 42.

    Perkowski, N.: Studies of robustness in stochastic analysis and mathematical finance. Ph.D. thesis, Humboldt-Universität zu Berlin (2014). Available online http://edoc.hu-berlin.de/dissertationen/perkowski-nicolas-simon-2013-12-13/PDF/perkowski.pdf

  43. 43.

    Perkowski, N., Ruf, J.: Supermartingales as Radon–Nikodým densities and related measure extensions. Ann. Probab. (2015, to appear). arXiv preprint arXiv:1309.4623

  44. 44.

    Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004)

    Google Scholar 

  45. 45.

    Rokhlin, D.B.: On the existence of an equivalent supermartingale density for a fork-convex family of random processes. Math. Notes 87, 556–563 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    Ross, S.A.: A simple approach to the valuation of risky streams. J. Bus. 51, 453–475 (1978)

    Article  Google Scholar 

  47. 47.

    Ruf, J.: Hedging under arbitrage. Math. Finance 23, 297–317 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Schweizer, M.: On the minimal martingale measure and the Föllmer–Schweizer decomposition. Stoch. Anal. Appl. 13, 573–599 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

    Song, S.: An alternative proof of a result of Takaoka (2013). arXiv preprint arXiv:1306.1062

  50. 50.

    Takaoka, K., Schweizer, M.: On the condition of no unbounded profit with bounded risk. Finance Stoch. 18, 393–405 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  51. 51.

    von Weizsäcker, H., Winkler, G.: Stochastic Integrals: An Introduction. Vieweg, Wiesbaden (1990)

    Google Scholar 

  52. 52.

    Yan, J.-A.: Caractérisation d’une classe d’ensembles convexes de \(L^{1}\) ou \(H^{1}\). In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités XIV. Lecture Notes in Mathematics, vol. 784, pp. 220–222. Springer, Berlin (1980)

    Google Scholar 

  53. 53.

    Chantha, Y.: Théorème de Girsanov généralisé et grossissement d’une filtration. In: Jeulin, Th., Yor, M. (eds.) Grossissements de Filtrations: Exemples et Applications. Lecture Notes in Mathematics, vol. 1118, pp. 172–196. Springer, Berlin (1985)

    Google Scholar 

  54. 54.

    Yor, M., Meyer, P.-A.: Sur l’extension d’un théorème de Doob à un noyau \(\sigma\)-fini d’après G. Mokobodzki. In: Dellacherie, C., et al. (eds.) Séminaire de Probabilités, XII, Univ. Strasbourg, 1976/1977. Lecture Notes in Math., vol. 649, pp. 482–488. Springer, Berlin (1978)

    Google Scholar 

  55. 55.

    Žitković, G.: A filtered version of the bipolar theorem of Brannath and Schachermayer. J. Theor. Probab. 15, 41–61 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  56. 56.

    Žitković, G.: Convex compactness and its applications. Math. Financ. Econ. 3, 1–12 (2010)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

We wish to express our gratitude to the anonymous referees for their careful reading of the manuscript and for their detailed comments which helped to correct a mistake in Sect. 5 and to improve the presentation. N.P. thanks Asgar Jamneshan for the introduction to filtration enlargements. We are grateful to Stefan Ankirchner, Kostas Kardaras and Johannes Ruf for their helpful comments on earlier versions of this paper. We thank Alexander Gushchin for pointing out the reference [45]. Part of the research was carried out during a 2011 visit at the University of Illinois at Urbana-Champaign. We are grateful for the hospitality at UIUC.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Peter Imkeller.

Additional information

N.P. was supported by a Ph.D. scholarship of the Berlin Mathematical School, by the Fondation Sciences Mathématiques de Paris (FSMP), and by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098). N.P. acknowledges generous support from Université Paris Dauphine, where a part of this work was completed.

Appendix: Incomplete filtrations

Appendix: Incomplete filtrations

Here we collect some classical observations which allow transferring to our setting results of other authors that were obtained under complete filtrations. There are at least three important monographs which avoid the use of complete filtrations as far as possible, namely Jacod [25], Jacod and Shiryaev [27] and von Weizsäcker and Winkler [51]. Here we follow [27].

Let \((\varOmega, \mathcal {F}, (\mathcal {F}_{t})_{t \ge0}, P)\) be a filtered probability space equipped with a right-continuous filtration \((\mathcal {F}_{t})\). Write \(\mathcal {F}^{P}\) for the \(P\)-completion of ℱ and \(\mathcal{N}^{P}\) for the \(P\)-nullsets of \(\mathcal {F}^{P}\). Then \(\mathcal {F}_{t}^{P} = \mathcal {F}_{t} \vee\mathcal{N}^{P}\), \(t\ge0\), satisfies the usual conditions. It is well known and easy to show that for every random variable \(X\) on \((\varOmega, \mathcal {F}^{P})\), there exists a random variable \(Y\) on \((\varOmega, \mathcal {F})\) with \(P[X=Y]=1\).

Recall that the optional \(\sigma\)-algebra over \((\mathcal {F}_{t})\) is the \(\sigma \)-algebra on \(\varOmega\times[0,\infty)\) that is generated by all processes of the form \(X_{t}(\omega) = 1_{A}(\omega) 1_{[r,s)}(t)\) for some \(0\le r < s < \infty\) and \(A \in \mathcal {F}_{r}\), and the predictable \(\sigma\)-algebra is generated by all processes of the form \(X_{t}(\omega) = 1_{A}(\omega) 1_{\{0\}}(t) + 1_{B}(\omega) 1_{(r,s]}(t)\) for some \(0\le r < s < \infty\), \(A \in \mathcal {F}_{0}\) and \(B \in \mathcal {F}_{r}\). Similarly, we define the predictable and optional \(\sigma\)-algebras over \((\mathcal {F}^{P}_{t})\).

The first result relates stopping times under \((\mathcal {F}_{t})\) and under \((\mathcal {F}^{P}_{t})\).

Lemma A.1

(Lemma I.1.19 of [27])

Any stopping time on the completion \((\varOmega, (\mathcal {F}^{P}_{t}))\) is almost surely equal to a stopping time on \((\varOmega, (\mathcal {F}_{t}))\).

We also have a comparable result at the level of processes.

Lemma A.2

Any predictable (optional) process on the completion \((\varOmega, (\mathcal {F}^{P}_{t}))\) is indistinguishable from a predictable (optional) process on \((\varOmega, (\mathcal {F}_{t}))\).

Proof

The predictable case is Lemma I.2.17 of [27]. The proof of the optional case works exactly in the same way: the claim is trivial for the generating processes described above, and we can use the monotone class theorem to pass to indicator functions of general optional sets. Then we use monotone convergence to pass to general optional processes. □

This allows us to deduce a similar result for càdlàg processes.

Lemma A.3

Let \(S\) be an \((\mathcal {F}^{P}_{t})\)-adapted process that is almost surely càdlàg. Then \(S\) is indistinguishable from an \((\mathcal {F}_{t})\)-adapted process (which is then, of course, almost surely càdlàg as well).

Proof

Since \((\mathcal {F}^{P}_{t})\) is complete, \(S\) admits an indistinguishable version \(\widetilde{S}\) which is \((\mathcal {F}_{t}^{P})\)-adapted and càdlàg for every \(\omega\in\varOmega\). This \(\widetilde {S}\) is optional; so now the result follows from Lemma A.2. □

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Imkeller, P., Perkowski, N. The existence of dominating local martingale measures. Finance Stoch 19, 685–717 (2015). https://doi.org/10.1007/s00780-015-0264-0

Download citation

Keywords

  • Dominating local martingale measure
  • Arbitrage of the first kind
  • Fundamental theorem of asset pricing
  • Supermartingale densities
  • Föllmer’s measure
  • Enlargement of filtration
  • Jacod’s criterion

Mathematics Subject Classification

  • 60G44
  • 60G48
  • 91B70
  • 46N10

JEL Classification

  • G10