Institute of Mathematical Statistics Collections

No Arbitrage and General Semimartingales

Philip Protter and Kazuhiro Shimbo

Full-text: Open access


No free lunch with vanishing risk (NFLVR) is known to be equivalent to the existence of an equivalent martingale measure for the price process semimartingale. We give necessary conditions for such a semimartingale to have the property NFLVR. We also extend Novikov’s criterion for the stochastic exponential of a local martingale to be a martingale to the general case; that is, the case where the paths need not be continuous.

Chapter information

Stewart N. Ethier, Jin Feng and Richard H. Stockbridge, eds., Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 267-283

First available in Project Euclid: 28 January 2009

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Copyright © 2008, Institute of Mathematical Statistics


Protter, Philip; Shimbo, Kazuhiro. No Arbitrage and General Semimartingales. Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, 267--283, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000426.

Export citation


  • [1] Ansel, J.-P. and Stricker, C. (1992). Lois de martingale, densités et décomposition de Föllmer–Schweizer. Ann. Inst. H. Poincaré Probab. Statist. 28 375–392.
  • [2] Beneš, V. E. (1971). Existence of optimal stochastic control laws. SIAM J. Control 9 446–472.
  • [3] Cheridito, P., Filipovic, D. and Yor, M. (2005). Equivalent and absolutely continuous measure changes for jump-diffusion processes. Ann. Applied Probability 15 1713–1732.
  • [4] Cherney, A. S. and Shiryaev, A. N. (2001). On criteria for the uniform integrability of Brownian stochastic exponentials. Optimal Control and Partial Differential Equations, in honour of Alain Bensoussan’s 60th birthday, 80–92. IOS Press.
  • [5] Delbaen, F. and Schachermayer, W. (1994). A general version of the fun- damental theorem of asset pricing. Math. Ann. 300 463–520.
  • [6] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem for unbounded stochastic processes. Math. Ann. 312 215–250.
  • [7] Delbaen, F. and Schachermayer, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Applied Probability 5 926–945.
  • [8] Dellacherie, C. and Meyer, P. A. (1978). Probabilities and Potential B. North-Holland Mathematics Studies 72. North-Holland, Amsterdam.
  • [9] Eberlein, E. and Jacod, J. (1997). On the ranges of options prices. Finance Stochast. 1 131–140.
  • [10] Jacod, J. and Shiryaev, A. N. (2002). Limit Theorems for Stochastic Processes, second edition. Springer–Verlag, Heidelberg.
  • [11] Jarrow, R. and Protter, P. (2005). Large traders, hidden arbitrage and complete markets. Journal of Banking and Finance 29 2803–2820.
  • [12] Jarrow, R. and Protter, P. (2008). A partial introduction to financial asset pricing theory. In Handbook in OR & MS: Financial Engineering 15 (J. R. Birge and V. Linetsky, eds.) 1–59. Elsevier.
  • [13] Kabanov, Yuri (1997). On the FTAP of Kreps–Delbaen–Schachermayer. In The Liptser Festschrift. Papers from the Steklov Seminar held in Moscow, 1995–1996 (Yu. M. Kabanov, B. L. Rozovskii and A. N. Shiryaev, eds.) 191–203. World Scientific Publishing Co., Inc., River Edge, NJ.
  • [14] Kallsen, J. and Shiryaev, A. N. (2002). The cumulant process and Esscher’s change of measure. Finance Stoch. 6 (4) 397–428.
  • [15] Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semi- martingale financial models. Preprint.
  • [16] Kazamaki, N. (1977). On a problem of Girsanov. Tôhoku Math. J. 29 (4) 597–600.
  • [17] Kazamaki, N. (1978). Correction: On a problem of Girsanov (Tôhoku Math. J. (2) 29 (4) (1977) 597–600). Tôhoku Math. J. (2) 30 (1) 175.
  • [18] Kazamaki, N. (1994). Continuous exponential martingales and BMO. Lecture Notes in Mathematics 1579. Springer-Verlag, Berlin.
  • [19] Lépingle, Dominique and Mémin, Jean. (1978). Sur l’intégrabilité uniforme des martingales exponentielles. Z. Wahrsch. Verw. Gebiete 42 (3) 175–203.
  • [20] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of random processes. I, expanded edition. Applications of Mathematics 5. Springer-Verlag, Berlin. General theory, translated from the 1974 Russian original by A. B. Aries, Stochastic Modeling and Applied Probability.
  • [21] Mémin, J. (1978). Décompositions multiplicatives de semimartingales exponentielles et applications. Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977). Lecture Notes in Math. 649 35–46. Springer, Berlin.
  • [22] Novikov, A.A. (1980). On conditions for uniform integrability for continuous exponential martingales. In Stochastic differential systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978). Lecture Notes in Control and Information Sci. 25 304–310. Springer, Berlin.
  • [23] Protter, P. (2005). Stochastic Integration and Differential Equations, second edition, Version 2.1. Springer–Verlag, Heidelberg.
  • [24] Revuz, D. and Yor, M. (1999). Continuous martingales and Brownian mo- tion, third edition. Grundlehren der Mathematischen Wissenschaften 293. Springer-Verlag, Berlin.
  • [25] Sato, H. (1990). Uniform integrability of an additive martingale and its exponential. Stochastics Stochastics Rep. 30 163–169.
  • [26] Schweizer, M. (1994). Approximating random variables by stochastic integrals. Ann. Probability 22 1536–1575.
  • [27] Schweizer, M. (1995). On the minimal martingale measure and the Föllmer–Schweizer decomposition. Stochastic Analysis and Applications 13 573–599.
  • [28] Strasser, Eva (2005). Characterization of arbitrage-free markets. Ann. Appl. Probability 15 116–124.
  • [29] Stroock, D. W. (2003). Markov processes from K. Itô’s perspective. Annals of Mathematics Studies 155. Princeton University Press, Princeton, NJ.