5

I would like to ask for references that may help me in tackling some of the advanced stochastic analysis books. I am interested in a variety of different areas, namely (1) Malliavin Calculus, (2) Stochastic Differential Geometry, (3) Stochastic Differential Equations, and these are in order of interest. So, I would like to know what books I can read to better comprehend the standard literature in those areas. You can suppose I have a background in real analysis from Folland, probability theory from Williams, stochastic calculus from LeGall/Kuo (which is more or less the first 3 chapters of Karatzas & Shreve, i.e. elementary stochastics knowledge).

Now, from the research I have done by myself, I concluded that for (1) I would simply need to go through a text in functional analysis. Something on the lines of "Sobolev Spaces" by Adams or Brezis' "Functional Analysis, Sobolev Spaces and Partial Differential Equations". Is that all that I should read before tackling the standard texts like Nualart?

Now, for (2), I concluded that I would simply require a deeper understanding of riemannian geometry. A text like Lee's "Riemannian Geometry" would suffice. Am I safe in assuming this? Given that I remember very little differential geometry, can I 'skim' over manifold theory and riemannian manifolds instead of rigorously tackling the classic differential geometry texts, considering that I do not want to spend/focus my time on differential geometry and would very much like to focus on pure probability theory.

For (3), from what I can tell it is an extremely wide area. If so, what would you recommend as solid foundation for tackling the basic texts in the area? Would the same books in (1) suffice?

Thank you for your time.

J. W. Tanner
  • 63,683
  • 4
  • 43
  • 88
Mundane_Matters
  • 61
  • Independent of literature recommendations, you will neither understand Malliavin calculus nor stochastic differential geometry if you don't posses a fairy good background in SDEs or stochastic analysis in general. So in any case, you should start by digging into your topic (3) before looking into the other two quite advanced topics. – Tobsn Mar 12 '21 at 18:02
  • @Tobsn Thank you so much for the reply. I have elementary knowledge in stochastic analysis from some of the easier books. I have covered stuff like semigroups, diffusions and kolmogorov equations, etc. in their most basic form. I am currently reading through Nualart's 'Introduction to Malliavin Calculus', which, from Nualart's perspective, is meant as a background for his more comprehensive book, 'The Malliavin Calculus and Related Topics'. So far I understand the material and looking through the book it looks like it wont be that difficult to go through. – Mundane_Matters Mar 13 '21 at 10:36
  • @Tobsn Do you think the leap would be too great for me? Regardless, what books for stochastic analysis and SDEs would you recommend me to read for a sufficient enough background? – Mundane_Matters Mar 13 '21 at 10:36

0 Answers0