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I've read a proof for existence of solutions to stochastic differential equation from a book of Ikeda and Watanabe and have a question. Is it possible to prove existence (and uniquness) by means of the Banach contraction principle, similarly like in case of ordinary differential equations? It so, could you give a reference?

Thank you for help and hints, Almost sure.

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See for example: Stochastic Calculus - A practical Introduction by R. Durrett. Theorem 2.2. in Section 5.2. It exactly the same idea as the classical Picard Iteration for ODE –  math Aug 8 '12 at 15:34
    
But this proof is basically the same as in Ikeda and Watanabe. I'm searching for another one, not by the Picard Iteration, but by the Banach contraction principle. The first thing which puzzles me is the following: what (Banach) space should I consider? –  Almost sure Aug 8 '12 at 16:47
    
Sorry, for the misunderstanding. So you want to use a fixpoint argument to use Banachs fixpoint theorem. Am I right? –  math Aug 9 '12 at 6:08
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The method of Picard iterates is a special case of the contraction mapping theorem, so what do you mean by asking for a proof by contraction mappings while not considering a method like Picard iterates to be acceptable? –  KCd Aug 9 '12 at 23:56
    
Very warm thanks for comments! To math: Yes, I want to use some fixpoint argument to prove existence and uniquness of solutions to SDE. To KCd: your comment is very interesting, thanks! I've never thought of Picard's iterates as a special case of the contraction mapping theorem. I have to think of it a little... –  Almost sure Aug 10 '12 at 16:57

1 Answer 1

This is actually the usual technique for solving BSDEs(Backward Stochastic Differential Equations). Check out the first (or second) chapter of

Yong, Jiongmin and Zhou, Xun Yu. (1999) Stochastic controls.

and the references to see how this works.

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Gerard, you're right! Thank you very much indeed! –  Almost sure Aug 10 '12 at 18:36

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