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Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 1, Pages 177–188 (Mi tvp309)  

This article is cited in 21 scientific papers (total in 21 papers)

Short Communications

$\sigma$-localization and $\sigma$-martingales

J. Kallsen

Albert Ludwigs University of Freiburg

Abstract: This paper introduces the concept of $\sigma$-localization, which is a generalization of localization in the general theory of stochastic processes. The $\sigma$-localized class derived from the set of martingales is the class of $\sigma$-martingales, which plays an important role in mathematical finance. These processes and the corresponding $\sigma$-martingale measures are considered in detail. By extending the stochastic integral with respect to compensated random measures, a canonical representation of $\sigma$-martingales as for local martingales is derived.

Keywords: $\sigma$-localization, $\sigma$-martingale, stochastic integral, canonical representation, $\sigma$-martingale measure.

DOI: https://doi.org/10.4213/tvp309

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English version:
Theory of Probability and its Applications, 2004, 48:1, 152–163

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Received: 06.09.2002

Citation: J. Kallsen, “$\sigma$-localization and $\sigma$-martingales”, Teor. Veroyatnost. i Primenen., 48:1 (2003), 177–188; Theory Probab. Appl., 48:1 (2004), 152–163

Citation in format AMSBIB
\by J.~Kallsen
\paper $\sigma$-localization and $\sigma$-martingales
\jour Teor. Veroyatnost. i Primenen.
\yr 2003
\vol 48
\issue 1
\pages 177--188
\jour Theory Probab. Appl.
\yr 2004
\vol 48
\issue 1
\pages 152--163

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  • http://mi.mathnet.ru/eng/tvp/v48/i1/p177

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    This publication is cited in the following articles:
    1. Kallsen J., Kühn C., “Pricing derivatives of American and game type in incomplete markets”, Finance Stoch., 8:2 (2004), 261–284  crossref  mathscinet  zmath  isi  scopus
    2. Cherny A., Shiryaev A., “On stochastic integrals up to infinity and predictable criteria for integrability”, Séminaire de Probabilités XXXVIII, Lecture Notes in Math., 1857, Springer, Berlin, 2005, 165–185  crossref  mathscinet  zmath  isi
    3. Kramkov D., Sîrbu M., “On the two–times differentiability of the value functions in the problem of optimal investment in incomplete markets”, Ann. Appl. Probab., 16:3 (2006), 1352–1384  crossref  mathscinet  zmath  isi  elib  scopus
    4. Kallsen J., “A didactic note on affine stochastic volatility models”, From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, 2006, 343–368  crossref  mathscinet  zmath  isi
    5. Karatzas I., Kardaras C., “The numeraire portfolio in semimartingale financial models”, Finance Stoch., 11:4 (2007), 447–493  crossref  mathscinet  zmath  isi  elib  scopus
    6. Černý A., Kallsen J., “On the structure of general mean–variance hedging strategies”, Ann. Probab., 35:4 (2007), 1479–1531  crossref  mathscinet  zmath  isi  scopus
    7. Kallsen J., Vierthauer R., “Quadratic hedging in affine stochastic volatility models”, Review of Derivatives Research, 12:1 (2009), 3–27  crossref  mathscinet  zmath  isi  scopus
    8. Kallsen J., Muhle-Karbe J., “Exponentially affine martingales, affine measure changes and exponential moments of affine processes”, Stochastic Process. Appl., 120:2 (2010), 163–181  crossref  mathscinet  zmath  isi  scopus
    9. Kallsen J., Muhle-Karbe J., “Utility Maximization in Models with Conditionally Independent Increments”, Ann Appl Probab, 20:6 (2010), 2162–2177  crossref  mathscinet  zmath  isi  scopus
    10. Mayerhofer E., Muhle-Karbe J., Smirnov A.G., “A characterization of the martingale property of exponentially affine processes”, Stochastic Process Appl, 121:3 (2011), 568–582  crossref  mathscinet  zmath  isi  elib  scopus
    11. Biagini S., Cerny A., “Admissible Strategies in Semimartingale Portfolio Selection”, SIAM J Control Optim, 49:1 (2011), 42–72  crossref  mathscinet  zmath  isi  elib  scopus
    12. Nutz M., “The Bellman Equation for Power Utility Maximization with Semimartingales”, Ann Appl Probab, 22:1 (2012), 363–406  crossref  mathscinet  zmath  isi  elib  scopus
    13. Kardaras C., “Market Viability via Absence of Arbitrage of the First Kind”, Financ. Stoch., 16:4 (2012), 651–667  crossref  mathscinet  zmath  isi  elib  scopus
    14. Czichowsky Ch., “Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time”, Financ. Stoch., 17:2 (2013), 227–271  crossref  mathscinet  zmath  isi  scopus
    15. Pulido S., “The Fundamental Theorem of Asset Pricing, the Hedging Problem and Maximal Claims in Financial Markets with Short Sales Prohibitions”, Ann. Appl. Probab., 24:1 (2014), 54–75  crossref  mathscinet  zmath  isi  scopus
    16. Kardaras C., “On the Stochastic Behaviour of Optional Processes Up To Random Times”, Ann. Appl. Probab., 25:2 (2015), 429–464  crossref  mathscinet  zmath  isi  scopus
    17. Kallsen J., Kruehner P., “on a Heath-Jarrow-Morton Approach For Stock Options”, Financ. Stoch., 19:3 (2015), 583–615  crossref  mathscinet  zmath  isi  scopus
    18. Fontana C., “Weak and Strong No-Arbitrage Conditions For Continuous Financial Markets”, Int. J. Theor. Appl. Financ., 18:1 (2015), 1550005  crossref  mathscinet  zmath  isi  elib  scopus
    19. Choulli T. Schweizer M., “Locally Phi-Integrable SIGMA-Martingale Densitiesfor General Semimartingales”, Stochastics, 88:2 (2016), 191–266  crossref  mathscinet  zmath  isi  scopus
    20. Criens D., “Structure-Preserving Equivalent Martingale Measures For H-Sii Models”, J. Appl. Probab., 55:1 (2018), 1–14  crossref  mathscinet  isi  scopus
    21. Biagini S., Cerny A., “Convex Duality and Orlicz Spaces in Expected Utility Maximization”, Math. Financ., 30:1 (2020), 85–127  crossref  mathscinet  isi
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