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I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory.

I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and Shreve, for example).

My PDE theory is pretty weak. I know about the Fokker-Planck equations, and that's about it. My guess is that before I begin reading, I'll need to learn a significant amount of classical PDE theory. I would appreciate any recommendations for PDE textbooks that emphasize material that will be useful in the study of SPDEs. If someone could recommend a gentle introduction to SPDEs to go with it, I would be very grateful.

Many thanks.

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4 Answers 4

up vote 7 down vote accepted

An introduction to the theory on Hilbert spaces is this:

  • Claudia Prévôt, Michael Röckner: "A concise course on stochastic partial differential equations." (see ZMATH)

This book constructs SPDE as Hilbert space valued random processes and uses the theory of linear operators Hilbert spaces mostly, no PDE theory. So if you are familiar with linear functional analysis on Hilbert spaces, this is the place to start.

Another nice reference is:

  • Helge Holden, Bernt Øksendal, Jan Ubøe, Tusheng Zhang: "Stochastic partial differential equations. A modeling, white noise functional approach. 2nd ed." (see ZMATH)

This book uses spaces of distributions, so you'll have to learn a little bit about that. But you'll have to, anyway, in order to understand SPDE. Besides that, I'd recommend that you dive in and whenever something comes up that you don't understand, you try to look it up in some nice introductory PDE text.

For linear PDE, I'd recommend

  • Francois Treves: "Basic Linear Partial Differential Equations"

but that topic is flooded with textbooks for each and every taste.

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Great answer, thanks. –  Simon Jun 15 '11 at 22:56
    
But the theory in Hilbert spaces is not that interesting, the notes I have linked to also include Banach spaced value random processes (the UMD-property is important here). There is also "Stochastic equations in infinite dimensions" by da Prato & Zabczyk. That book also includes the theory in Banach spaces. –  Jonas Teuwen Jun 16 '11 at 13:22
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You might have a look at Martin Hairer's "An Introduction to Stochastic PDEs", which is available on arXiv at http://arxiv.org/abs/0907.4178. At the very least, having a look at the topics he discusses might help you to plan out a road map of things you want to learn more about in preparation for reading Hairer's notes.

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You can check out the Internet Seminar Notes 2007/2008 bij Jan van Neerven, they are quite good and are about stochastic evolution equations.

Here you can find them: http://fa.its.tudelft.nl/~neerven/publications/papers/ISEM.pdf

The author also happens to be my supervisor so I might be biased but I'd suggest you take a look at them.

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On the flip side, how much probability theory does the notes assume? Thanks for the link, by the way. –  Willie Wong Jun 18 '11 at 21:14
    
@Willie: Not that much. I think somewhere up to martingales. –  Jonas Teuwen Jun 18 '11 at 23:22
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Rama Cont compiled a rather comprehensive resource page about SPDEs, but some of the links are broken, still this gives you an overview of what books/papers you wanna read in order to know certain aspects of SPDEs.

http://www.cmap.polytechnique.fr/~rama/spde/articles.htm

And don't worry about PDE theory too much, any graduate level PDE textbook, for example L. Evans or Gilbarg and Trudinger, has a too long technicalities introduction, read certain chapters when needed.

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