The Annals of Probability

${\scr E}$-martingales and their applications in mathematical finance

Tahir Choulli, Leszek Krawczyk, and Christophe Stricker

Full-text: Open access


After introducing a new concept, the notion of $\mathscr{E}$-martingale, we extend the well-known Doob inequality (for $1 < p < +\infty)$ and the Burkholder–Davis–Gundy inequalities (for $p = 2$) to $\mathscr{E}$-martingales. By means of these inequalities, we give sufficient conditions for the closedness of a space of stochastic integrals with respect to a fixed $\mathbb{R}^d$-valued semimartingale, a question which arises naturally in the applications to financial mathematics. We also provide a necessary and sufficient condition for the existence and uniqueness of the Föllmer–Schweizer decomposition.

Article information

Ann. Probab., Volume 26, Number 2 (1998), 853-876.

First available in Project Euclid: 31 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G48: Generalizations of martingales 60H05: Stochastic integrals 90A09

Semimartingales stochastic integrals stochastic exponential reverse Hölder inequalities weighted norm inequalities Föllmer-Schweizer decompositin


Choulli, Tahir; Krawczyk, Leszek; Stricker, Christophe. ${\scr E}$-martingales and their applications in mathematical finance. Ann. Probab. 26 (1998), no. 2, 853--876. doi:10.1214/aop/1022855653.

Export citation


  • Ansel, J. P. and Stricker, C. (1992). Lois de martingale, densit´es et d´ecomposition de F¨ollmer- Schweizer. Ann. Inst. H. Poincar´e 28 375-392.
  • Bonami, A. and L´epingle, D. (1979). Fonction maximale et variation quadratique des martingales en pr´esence d'un poids. S´eminaire de Probabilit´es XIII. Lecture Notes in Math. 721 294-306. Springer, Berlin.
  • Choulli, T. and Stricker, C. (1996). Deux applications de la d´ecomposition de Galtchouk- Kunita-Watanabe. S´eminaire de Probabilit´es XXX. Lecture Notes in Math. 1626 12-23. Springer, Berlin.
  • Delbaen, F., Monat, P., Schachermayer, W., Schweizer, M. and Stricker, C. (1997). Weighted norm inequalities and hedging in incomplete markets. Finance and Stochastics 1 181- 227.
  • Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463-520.
  • Delbaen, F. and Schachermayer, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5 926-945.
  • Delbaen, F. and Schachermayer, W. (1996). The variance-optimal martingale measure for continuous processes. Bernoulli 2 81-106.
  • Delbaen, F. and Shirakawa, H. (1996). A note on the no arbitrage condition for international financial markets. Unpublished manuscript.
  • Dellacherie, C. and Meyer, P. A. (1980). Probabilit´es et Potentiel. Ch. V-VIII. Hermann, Paris.
  • Dol´eans-Dade, C. and Meyer, P. A. (1979). In´egalit´es de normes avec poids. S´eminaire de Probabilit´es XIII 313-331. Springer, Berlin.
  • Grandits, P. and Krawczyk, L. (1997). Closedness of some spaces of stochastic integrals. S´eminaire de Probabilit´es XXXII. Lecture Notes in Math. Springer, Berlin. To appear.
  • Jacod, J. (1979). Calcul stochastique et probl emes de martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, New York.
  • Jawerth, B. (1986). Weighted inequalities for maximal operators: linearization, localization and factorization. Amer. J. Math. 108 361-414.
  • Kazamaki, N. (1994). Continuous exponential martingales and BMO. Lecture Notes in Math. 1579. Springer, Berlin.
  • L´epingle, D. and M´emin, J. (1978). "Sur l'int´egrabilit´e uniforme des martingales exponentielles.Wahrsch. Verw. Gebiete 42 175-203.
  • Long, R. L. (1993). Martingale Spaces and Inequalities. Peking Univ. Press and Vieweg, Braunschweig.
  • Monat, P. and Stricker, C. (1994). Fermeture de GT et de L2 0 + GT. S´eminaire de Probabilit´es XXVIII. Lecture Notes in Math. 1583 189-194. Springer, Berlin.
  • Monat, P. and Stricker, C. (1995). F¨ollmer-Schweizer decomposition and mean-variance hedging for general claims. Ann. Probab. 23 605-628.
  • Pratelli, M. (1976). Sur certains espaces de martingales localement de carr´e int´egrable. S´eminaire de Probabilit´es X. Lecture Notes in Math. 511 401-413. Springer, Berlin.
  • Protter, P. (1990). Stochastic integration and differential equations. Appl. Math. 21.
  • Ruiz de Chavez, J. (1984). Le th´eor eme de Paul Levy pour des mesures sign´ees. S´eminaire de Probabilit´es XVIII. Lecture Notes in Math. 1059 245-255. Springer, Berlin.
  • Schweizer, M. (1994). Approximating random variables by stochastic integrals. Ann. Probab. 22 1536-1575.
  • Stricker, C. (1990). Arbitrage et lois de martingale. Ann. Inst. H. Poincar´e 26 451-460.
  • Yoeurp, C. (1982). Contribution au calcul stochastique. Th ese, Univ. Paris VI.
  • Yor, M. (1976). Sur les int´egrales stochastiques optionnelles et une suite remarquable de formules exponentielles. S´eminaire de Probabilit´es X. Lecture Notes in Math. 511 481- 500. Springer, Berlin.
  • Yor, M. (1985). In´egalit´es de martingales continues arr et´ees a un temps quelconque. Lecture Notes in Math. 1118 110-146. Springer, Berlin.