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Nonlinear stochastic integrals for hyperfinite Lévy processes

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Logic and Analysis

Abstract

I develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes and use it to find exact formulas for expressions which are intuitively of the form \(\sum_{s=0}^t\phi(\omega,dl_{s},s)\) and \(\prod_{s=0}^t\psi(\omega,dl_{s},s)\), where l is a Lévy process. These formulas are then applied to geometric Lévy processes, infinitesimal transformations of hyperfinite Lévy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic calculus for processes with jumps on manifolds, and the paper may be regarded as a reworking of his ideas in a different setting and with totally different techniques.

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References

  1. Albeverio S, Fenstad JE, Høegh-Krohn R and Lindstrøm T (1986). Nonstandard methods in stochastic analysis and mathematical physics. Academic, New York

    MATH  Google Scholar 

  2. Albeverio S and Herzberg FS (2006). Lifting Lévy processes to hyperfinite random walks. Bull Sci Math 130: 697–706

    Article  MathSciNet  MATH  Google Scholar 

  3. Albeverio S and Herzberg FS (2006). A combinatorial infinitesimal representation of Lévy processes and an application to incomplete markets. Stochastics 78: 301–325

    MathSciNet  MATH  Google Scholar 

  4. Applebaum D (2004). Lévy processes and stochastic calculus. Cambridge University Press, New York

    MATH  Google Scholar 

  5. Chan T (1999). Pricing contingent claims on stocks driven by Lévy processes. Ann Appl Probab 9: 504–528

    Article  MathSciNet  MATH  Google Scholar 

  6. Cohen S (1996). Géométrie différentielle stochastique avec sauts 1. Stoch Stoch Rep 56: 179–203

    MATH  Google Scholar 

  7. Cohen S (1996). Géométrie différentielle stochastique avec sauts 2: discrétisation et applications des EDS avec sauts. Stoch Stoch Rep 56: 205–225

    MATH  Google Scholar 

  8. Di Nunno G, Rozanov YA (2007) Stochastic integrals and adjoint derivatives. In: Benth FE, Di Nunno G, Lindstrom T, Øksendal B, Zhang T (eds) Stochastic analysis and applications: proceedings of the second Abel Symposium, Oslo, July 29–August 4, 2005. Abel symposia, vol 2, pp 265–307. Springer, Berlin

  9. Föllmer H and Schweizer M (1991). Hedging of contingent claims under incomplete information. In: Davis, MHA and Elliott, RJ (eds) Applied stochastic analysis. Stochastics monographs, vol 5, pp 389–414. Gordon and Breach, New York

    Google Scholar 

  10. Hoover DN and Perkins EA (1983). Nonstandard construction of the stochastic integral and applications to stochastic differential equations I–II. Trans Am Math Soc 275: 1–58

    Article  MathSciNet  MATH  Google Scholar 

  11. Ibéro M (1976). Intégrales stochastique multiplicatives et construction de diffusions sur un groupe de Lie. Bull Sci Math 100: 175–191

    MathSciNet  MATH  Google Scholar 

  12. Jacod J (1979). Calcul stochastique et problemes de martingales. Lecture notes in mathematics, vol 714. Springer, Berlin

    Google Scholar 

  13. Lindstrøm T (1980). Hyperfinite stochastic integration I–III. Math Scand 46: 265–333

    MathSciNet  MATH  Google Scholar 

  14. Lindstrøm T (1997) Internal martingales and stochastic integration. In: Arkeryd LO, Cutland NJ, Henson CW (eds) Nonstandard analysis: theory and applications. NATO science series C, vol 493, pp 209–259. Kluwer, Dordrecht

  15. Lindstrøm T (2004). Hyperfinite Lévy processes. Stoch Stoch Rep 76: 517–548

    MathSciNet  Google Scholar 

  16. McKean HP (1969). Stochastic integrals. Academic, New York

    MATH  Google Scholar 

  17. Melnikov AV, Shiryaev AN (1996) Criteria for the absence of arbitrage in the financial market. In: Shiryaev AN, Mel’nikov AV, Niemi K, Valkejla E (eds) Frontiers in pure and applied probability II: proceedings of the Fourth Russian-Finnish Symposium on Probability Theory and Mathematical Statistics, pp 121–134. TVP, Moscow

  18. Ng S-A (2007) A nonstandard Lévy–Khintchine formula and Lévy processes. Acta Math Sin 23 (in press)

  19. Øksendal B and Sulem A (2005). Applied stochastic control of jump diffusions. Springer, Berlin

    Google Scholar 

  20. Protter P (2004). Stochastic integration and differential equations, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  21. Schweizer M (1995). Variance-optimal hedging in discrete time. Math Oper Res 20: 1–32

    Article  MathSciNet  MATH  Google Scholar 

  22. Schweizer M (1995). On the minimal martingale measure and the Föllmer–Schweizer decomposition. Stoch Anal Appl 13: 573–599

    Article  MathSciNet  MATH  Google Scholar 

  23. Skorokhod AV (1965). Studies in the theory of random processes. Addison-Wesley, Reading

    MATH  Google Scholar 

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Authors and Affiliations

  1. Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316, Oslo, Norway

    Tom Lindstrøm

  2. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316, Oslo, Norway

    Tom Lindstrøm

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  1. Tom Lindstrøm
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Correspondence to Tom Lindstrøm.

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Lindstrøm, T. Nonlinear stochastic integrals for hyperfinite Lévy processes. Log Anal 1, 91–129 (2008). https://doi.org/10.1007/s11813-007-0004-7

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  • DOI: https://doi.org/10.1007/s11813-007-0004-7

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