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Stochastic Integration

  • Ernst Eberlein
  • Jan Kallsen
Chapter
  • 1.2k Downloads
Part of the Springer Finance book series (FINANCE)

Abstract

Traditional stochastic calculus is based on stochastic integration. It constitutes the basis of modern Mathematical Finance. The most important notions and results from the theory are presented in this chapter.

References

  1. 2.
    D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge Univ. Press, Cambridge, 2009)CrossRefGoogle Scholar
  2. 8.
    G. Barles, R. Buckdahn, E. Pardoux, Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60(1–2), 57–83 (1997)MathSciNetCrossRefGoogle Scholar
  3. 43.
    P. Brockwell, Lévy-driven CARMA processes. Ann. Inst. Stat. Math. 53(1), 113–124 (2001)MathSciNetCrossRefGoogle Scholar
  4. 58.
    E. Çinlar, J. Jacod, P. Protter, M. Sharpe, Semimartingales and Markov processes. Z. Wahrsch. verw. Gebiete 54(2), 161–219 (1980)MathSciNetCrossRefGoogle Scholar
  5. 60.
    R. Cont, P. Tankov, Financial Modelling with Jump Processes (Chapman & Hall/CRC, Boca Raton, 2004)Google Scholar
  6. 76.
    C. Dellacherie, P.-A. Meyer, Probabilities and Potential B: Theory of Martingales (North-Holland, Amsterdam, 1982)zbMATHGoogle Scholar
  7. 114.
    H. Föllmer, Yu. Kabanov, Optional decomposition and Lagrange multipliers. Finance Stochast. 2(1), 69–81 (1998)MathSciNetzbMATHGoogle Scholar
  8. 125.
    T. Goll, J. Kallsen, Optimal portfolios for logarithmic utility. Stoch. Process. Appl. 89(1), 31–48 (2000)MathSciNetCrossRefGoogle Scholar
  9. 152.
    J. Jacod, Calcul Stochastique et Problèmes de Martingales, volume 714 of Lecture Notes in Math (Springer, Berlin, 1979)CrossRefGoogle Scholar
  10. 154.
    J. Jacod, A. Shiryaev, Limit Theorems for Stochastic Processes, 2nd edn. (Springer, Berlin, 2003)CrossRefGoogle Scholar
  11. 163.
    J. Kallsen, Semimartingale Modelling in Finance. PhD thesis, University of Freiburg, 1998Google Scholar
  12. 168.
    J. Kallsen, σ-localization and σ-martingales. Teor. Veroyatnost. i Primenen. 48(1), 177–188 (2003)MathSciNetCrossRefGoogle Scholar
  13. 169.
    J. Kallsen, A didactic note on affine stochastic volatility models. From Stochastic Calculus to Mathematical Finance (Springer, Berlin, 2006), pp. 343–368CrossRefGoogle Scholar
  14. 182.
    J. Kallsen, A. Shiryaev, The cumulant process and Esscher’s change of measure. Finance Stochast. 6(4), 397–428 (2002)MathSciNetCrossRefGoogle Scholar
  15. 186.
    I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)zbMATHGoogle Scholar
  16. 196.
    D. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields 105(4), 459–479 (1996)MathSciNetCrossRefGoogle Scholar
  17. 204.
    D. Lamberton, B. Lapeyre, Stochastic Calculus Applied to Finance (Chapman & Hall, London, 1996)zbMATHGoogle Scholar
  18. 230.
    E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)MathSciNetCrossRefGoogle Scholar
  19. 238.
    P. Protter, Stochastic Integration and Differential Equations, 2nd edn. (Springer, Berlin, 2004)zbMATHGoogle Scholar
  20. 239.
    M.-C. Quenez, A. Sulem, BSDEs with jumps, optimization and applications to dynamic risk measures. Stoch. Process. Appl. 123(8), 3328–3357 (2013)MathSciNetCrossRefGoogle Scholar
  21. 241.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar
  22. 252.
    C. Rogers, D. Williams, Diffusions, Markov processes, and Martingales: Volume 2, Itô Calculus (Cambridge Univ. Press, Cambridge, 1994)zbMATHGoogle Scholar
  23. 259.
    K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge Univ. Press, Cambridge, 1999)zbMATHGoogle Scholar
  24. 263.
    W. Schoutens, Lévy Processes in Finance (Wiley, New York, 2003)CrossRefGoogle Scholar
  25. 284.
    S. Tang, X. Li, Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32(5), 1447–1475 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ernst Eberlein
    • 1
  • Jan Kallsen
    • 2
  1. 1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
  2. 2.Department of MathematicsKiel UniversityKielGermany

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