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Gradient and Divergence Operators

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1982)

Abstract

In this chapter we construct an abstract framework for stochastic analysis in continuous time with respect to a normal martingale (Mt)t?R+, using the construction of stochastic calculus presented in Section 2. In particular we identify some minimal properties that should be satisfied in order to connect a gradient and a divergence operator to stochastic integration, and to apply them to the predictable representation of random variables. Some applica- tions, such as logarithmic Sobolev and deviation inequalities, are formulated in this general setting. In the next chapters we will examine concrete exam- ples of operators that can be included in this framework, in particular when (Mt)t?R+ is a Brownian motion or a compensated Poisson process

Keywords

  • Divergence Operator
  • Sobolev Inequality
  • Stochastic Integral
  • Logarithmic Sobolev Inequality
  • Jump Time

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Nicolas Privault .

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© 2009 Springer-Verlag Berlin Heidelberg

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Privault, N. (2009). Gradient and Divergence Operators. In: Stochastic Analysis in Discrete and Continuous Settings. Lecture Notes in Mathematics(), vol 1982. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02380-4_4

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