Electronic Communications in Probability

Optional decomposition for continuous semimartingales under arbitrary filtrations

Ioannis Karatzas and Constantinos Kardaras

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We present an elementary treatment of the Optional Decomposition Theorem  for continuous semimartingales and general filtrations. This treatment does not assume the existence of   equivalent local martingale measure(s), only that of strictly positive local martingale deflator(s).

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 59, 10 pp.

Accepted: 14 August 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 91B28

Semimartingales optional decomposition local martingale deflators

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Karatzas, Ioannis; Kardaras, Constantinos. Optional decomposition for continuous semimartingales under arbitrary filtrations. Electron. Commun. Probab. 20 (2015), paper no. 59, 10 pp. doi:10.1214/ECP.v20-4090. https://projecteuclid.org/euclid.ecp/1465320986

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  • Dellacherie, Claude. Capacites et processus stochastiques. (French) Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 67. Springer-Verlag, Berlin-New York, 1972. ix+155 pp.
  • Delbaen, Freddy; Schachermayer, Walter. A general version of the fundamental theorem of asset pricing. Math. Ann. 300 (1994), no. 3, 463–520.
  • Davis, M. H. A.; Varaiya, P. Dynamic programming conditions for partially observable stochastic systems. SIAM J. Control 11 (1973), 226–261.
  • El Karoui, Nicole; Quenez, Marie-Claire. Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 (1995), no. 1, 29–66.
  • Follmer, H.; Kramkov, D. Optional decompositions under constraints. Probab. Theory Related Fields 109 (1997), no. 1, 1–25.
  • Follmer, H.; Kabanov, Yu. M. Optional decomposition and Lagrange multipliers. Finance Stoch. 2 (1998), no. 1, 69–81.
  • Jacka, Saul. A simple proof of Kramkov's result on uniform supermartingale decompositions. Stochastics 84 (2012), no. 5-6, 599–602.
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3
  • Kardaras, Constantinos. Finitely additive probabilities and the fundamental theorem of asset pricing. Contemporary quantitative finance, 19–34, Springer, Berlin, 2010.
  • Karatzas, Ioannis; Lehoczky, John P.; Shreve, Steven E.; Xu, Gan-Lin. Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29 (1991), no. 3, 702–730.
  • Kramkov, D. O. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Related Fields 105 (1996), no. 4, 459–479.
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
  • Karatzas, Ioannis; Shreve, Steven E. Methods of mathematical finance. Applications of Mathematics (New York), 39. Springer-Verlag, New York, 1998. xvi+407 pp. ISBN: 0-387-94839-2
  • Larsen, Kasper; Žitković, Gordan. Stability of utility-maximization in incomplete markets. Stochastic Process. Appl. 117 (2007), no. 11, 1642–1662.
  • Schweizer, Martin. On the minimal martingale measure and the Follmer-Schweizer decomposition. Stochastic Anal. Appl. 13 (1995), no. 5, 573–599.
  • Stricker, C.; Yan, J. A. Some remarks on the optional decomposition theorem. Seminaire de Probabilites, XXXII, 56–66, Lecture Notes in Math., 1686, Springer, Berlin, 1998.