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Pathwise differentiability with respect to a parameter of solutions of stochastic differential equations

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Theory and Application of Random Fields

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 49))

Abstract

We consider a stochastic differential equation

$$x^u (t) = v^u (t) + \int_o^t {\sigma (u,s,x_{s^ - }^u ) dS_S } + \int_o^t f (u,s,x_{s^ - }^u ,x) q(ds,dx)$$

where S is a semimartingale and q a random measure and where the "coefficients" depend on a parameter u. We prove under suitable differentiability-conditions that the solution X u(t,ω) can be choosen for each u in such a way that the mapping u→X u(t,ω) is continuously differentiable for every (t,ω).

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Bibliography

  1. BICHTELER, S.: Stochastic integrations with stationary independant increments. to appear in Z. Wahr. verw. Geb.

    Google Scholar 

  2. BISMUT, M.: A generalized formula of Ito and some other properties of stochastic flows. Z. Wahr. verw. Geb. 55, 331–350 (1981).

    Google Scholar 

  3. BLAGOVEČENSKII, Y.M., FREIDLIN, M.I.: Some properties of diffusion processes depending on a parameter. DAN, 138 (1961) Sov. Math, 2, 633–636.

    Google Scholar 

  4. GIKHMAN, I.I.: On the theory of differential equations of random processes. Uhr. Mat. Zb. 2, 4, 37–63 (1950).

    Google Scholar 

  5. GIKHMAN, I.I., SKOROKHOD, A.V.: Stochastic differential equations. Springer Verlag, 1972.

    Google Scholar 

  6. JACOD, J.: Calcul stochastique et problèmes de martingales. Lecture Notes in Math. 714, Springer-Verlag, New York, 1979.

    Google Scholar 

  7. JACOD, J.: Equations différentielles stochastiques: continuité et dérivabilité en probabilité. Preprint.

    Google Scholar 

  8. KUNITA, H.: On the decomposition of solutions of stochastic differential equations. Proc. of the L.M.S. Symposium on Stochastic Different. Equations, Durham, Juillet 1980, Lecture Notes in Math. Springer-Verlag, 1981.

    Google Scholar 

  9. MALLIAVIN, P.: Stochastic calculus of variations and Hypoelliptic operators. Proc. of the Intern. Symposium on Stochastic Different, Equations of Kyoto, 195–263 (1976). Tokyo, Kinokuniya and New York, Wiley, 1978.

    Google Scholar 

  10. METIVIER, M.: Stability theorems for stochastic Integral equations driven by random measures and semimartingales. J. Integral Equations, 3, 109–135 (1981).

    Google Scholar 

  11. METIVIER, M., PELLAUMAIL, J.: Stochastic Integration. Acad. Press, New York, 1980.

    Google Scholar 

  12. MEYER, P.A.: Flot d'une équation différentielle stochastique. Séminaire de Probabilité XV. Lecture Notes in Math. 850, Springer-Verlag, 1981.

    Google Scholar 

  13. NEVEU, J.: Intégrales stochastiques et applications. Cours de 3e cycle. Université de Paris VI, 1971–1972.

    Google Scholar 

  14. PRIOURET, P.: Processus de diffusion et équations différentielles stochastiques. Ecole d'Eté de Probabilités de St-Flour. Lecture Notes in Math. 390, Springer-Verlag, 1974.

    Google Scholar 

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Authors and Affiliations

  1. Ecole Polytechnique, 91128, Palaiseau, France

    Michel Metivier

Authors
  1. Michel Metivier
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G. Kallianpur

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© 1983 Springer-Verlag

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Metivier, M. (1983). Pathwise differentiability with respect to a parameter of solutions of stochastic differential equations. In: Kallianpur, G. (eds) Theory and Application of Random Fields. Lecture Notes in Control and Information Sciences, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0044692

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  • DOI: https://doi.org/10.1007/BFb0044692

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12232-6

  • Online ISBN: 978-3-540-39564-5

  • eBook Packages: Springer Book Archive

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