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

  • Michel Metivier
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 920)

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 differentia-bility-conditions that the solution X u (t, ω) can be choosen for each u in such a way that the mapping uX u (t, ω) is continuously differentiable for every (t, ω).

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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Michel Metivier
    • 1
  1. 1.Ecole PolytechniquePalaiseauFrance

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