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Stochastic Differentiation – A Generalized Approach

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Abstract

The space (D *) of Wiener distributions allows a natural Pettis-type stochastic calculus. For a certain class of generalized multiparameter processes X: R N→(D *) we prove several differentiation rules (Itô formulas); these processes can be anticipating. We then apply these rules to some examples of square integrable Wiener functionals and look at the integral versions of the resulting formulas.

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References

  1. Betounes, D. and Redfern, M.: A generalized Itô formula for N-dimensional time, In: T. Hida, H.-H. Kuo, J. Potthoff and L. Streit (eds), White Noise Analysis – Mathematics and Applications, World Scientific, Singapore, 1990, 337–343.

    Google Scholar 

  2. Betounes, D. and Redfern, M.:Wiener distributions and white noise analysis, Appl. Math. Optim. 26(1992), 63–93.

    Google Scholar 

  3. Betounes, D. and Redfern, M.: Stochastic integrals for nonprevisible, multiparameter processes, Appl. Math. Optim. 28(1993), 197–223.

    Google Scholar 

  4. Hida, T.: Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes 13 (1975).

  5. Hida, T.: Generalized multiple Wiener integrals, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), 55–58.

    Google Scholar 

  6. Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L.: White Noise, An Infinite Dimensional Calculus, Kluwer Acad. Publ., Dordrecht, 1993.

    Google Scholar 

  7. Jolis, M. and Sanz, M.: On generalized multiple stochastic integrals and multiparameter anticipative calculus, In: Lecture Notes in Math. 1444, Springer, New York, 1988, 141–182.

    Google Scholar 

  8. Kuo, H.-H.: Lectures on white noise analysis, Soochow J. Math. 18(1992), 229–300.

    Google Scholar 

  9. Kuo, H.-H.: White Noise Distribution Theory, CRC Press, Boca Raton, 1996.

    Google Scholar 

  10. Nualart, D.: The Malliavin Calculus and Related Topics, Springer, New York, 1995.

    Google Scholar 

  11. Nualart, D. and Pardoux, E.: Stochastic calculus with anticipating integrands, Probab. Theory Related Fields 78(1988), 535–581.

    Google Scholar 

  12. Simon, B.: The P.(∅) 2 Euclidean (Quantum) Field Theory, Princeton University Press, 1974.

  13. Wong, E. and Zakai, M.: Martingales and stochastic integrals for processes with a multidimensional parameter, Z. Wahrsch. Verw. Gebiete 29(1974), 109–122.

    Google Scholar 

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  1. Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS, 39406-5045, U.S.A.

    Mylan Redfern

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  1. Mylan Redfern
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Redfern, M. Stochastic Differentiation – A Generalized Approach. Acta Applicandae Mathematicae 63, 349–361 (2000). https://doi.org/10.1023/A:1010770207382

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