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Linear skorohod stochastic differential equations
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  • Published: June 1991

Linear skorohod stochastic differential equations

  • Rainer Buckdahn1 

Probability Theory and Related Fields volume 90, pages 223–240 (1991)Cite this article

Summary

Let σ andb be bounded processes on the Wiener space (Ω,ℱP), Ω=C([0,1]), which are possibly anticipating the Brownian motionW t (ω)=ω(t), and let η be a bounded random variable. We deduce the existence and uniqueness of a solutionX for the linear equation with Skorohod integral

$$X_t = \eta + \int\limits_0^t {\sigma _s X_s dW_s } + \int\limits_0^t {b_s X_s ds,} t \in [0,1],$$
((1))

under rather weak assumptions on σ and no additional requirement onb and η. The description of the solutionX requires to study the family {T t ,t∈[0,1]} of transformationT t of Ω into itself associated to (1) by the equation

$$T_t = \omega + \int\limits_0^{t \wedge .} {\sigma _s (T_s \omega )ds} , \omega \in \Omega , t \in [0,1]$$

.

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References

  1. Buckdahn, R.: Transformations on the Wiener space and Skorohod-type stochastic differential equations. Seminarbericht 105, Sektion Mathematik, Humboldt-Universität Berlin (1989)

  2. Buckdahn, R.: Anticipative Girsanov transformations. Preprint No. 108, Centre de Recerca Matemàtica, Inst, d'Estudis Catalans, 1990. Probab. Th. Rel. Fields (in press)

  3. Buckdahn, R., Nualart, D.: Skorohod stochastic differential equations with boundary conditions. Stochastics and Stochastic Rep. (in press)

  4. Kusuoka, S.: The non-linear transformation of Gaussian measure on Banach space and its absolute continuity (1). J. Fac. Sci., Univ. Tokyo, Sect. I A,29, 567–597 (1982)

    Google Scholar 

  5. Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Th. Rel. Fields78, 535–582 (1988)

    Google Scholar 

  6. Ocone, D., Pardoux, E.: Linear stochastic differential equations with boundary conditions. Probab. Th. Rel. Fields82, 489–526 (1989)

    Google Scholar 

  7. Ocone, D., Pardoux, E.: A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations. Ann. Inst. Henri Poincaré, Probab. Stat.25, 39–71 (1989)

    Google Scholar 

  8. Ramer, R.: On non-linear transformations of Gaussian measures. J. Funkt. Anal.15, 166–187 (1974)

    Google Scholar 

  9. Shiota, Y.: A linear stochastic integral equation containing the extended Itô integral. Math. Rep., Toyama Univ.9, 43–65 (1986)

    Google Scholar 

  10. Ustunel, A.S.: Some comments on the filtering of diffusions and the Malliavin Calculus. In: Korezlioglu, H., Ustunel, A.S. (eds.) Proc. Silivri Conf. 1986 (Lect. Notes Math., vol. 1316, pp. 247–266) Berlin Heidelberg New York: Springer 1988

    Google Scholar 

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Authors and Affiliations

  1. Unter den Linden 6, Fachbereich Mathematik der Humboldt-Universität, O-1086, Berlin, Germany

    Rainer Buckdahn

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  1. Rainer Buckdahn
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Buckdahn, R. Linear skorohod stochastic differential equations. Probab. Th. Rel. Fields 90, 223–240 (1991). https://doi.org/10.1007/BF01192163

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  • Received: 19 June 1990

  • Revised: 03 May 1991

  • Issue Date: June 1991

  • DOI: https://doi.org/10.1007/BF01192163

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Keywords

  • Differential Equation
  • Stochastic Process
  • Linear Equation
  • Probability Theory
  • Mathematical Biology
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