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Stochastic Integral Equations in Banach Spaces

Chapter
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 73)

Abstract

In this chapter, we first study the solutions of stochastic differential equations with non-Markovian Lipschitz condition and growth condition. In this case the drift and noise coefficients \(a(t,Z)\) and \(f(t,x,Z)\), being non-anticipating, depend on the path of the solution \(Z\) \(=(Z_t)_{t\in [0,T]}\). This is done by defining the equation on a probability space with \(\Omega :=\) \(D(\mathbb {R}_+,E)\), the space of càdlàg functions on \(\mathbb {R}_+ \rightarrow E\), and with the \(\sigma \)-algebra generated by the cylinder sets of \(D(\mathbb {R}_+,E)\), where \(E\) is a separable Banach space.

Keywords

Banach Space Probability Space Stochastic Differential Equation Lipschitz Condition Markov Property 
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.

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA
  2. 2.Department of Mathematics and InformaticsUniversity of WuppertalWuppertalGermany

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