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The following articles are about Rough path theory which is developed by T. Lyons from Oxford University:

  1.  Lyons-Victoir extension
  2. An introduction to Rough paths

For more reference on Rough paths, one could visit Rough paths.


The following is collections of articles published on subject of Tanaka’s formula:

  1.  Tanaka’s formula for symmetric levy processe

The following articles are a collection of local time related publications:

  1. Continuity of local time for Levy processes 
  2. Continuity of local time for markov processes
  3. Joint continuity of local time for markov process
  4. Large deviation of local time for levy processes
  5. Limit theorem and variation properties for fractional derivatives of the local time of a stable process
  6. Local time of markov processes approximated by a generalized iterated brownian motion
  7. Local time for Markov processes
  8. local time and related properties of multidimensional iterated brownian motion
  9. Occupation densities
  10. On the barlow yor inequality of local time
  11. Semi-martingale inequality and local time
  12. Unbounded local time
  13. Two parameter p,q variation path and integration of local time
  14. Generalized Itǒ Formulae and Space-Time Lebesgue–Stieltjes Integrals of Local Times

The following article I wrote is based on Bertoin’s Levy Processes. It reviews on how potential theory has been used to construct Occupation densities . It is well known that there are different constructions and definitions of local times corresponding to different classes of stochastic processes. For a large panorama of such definitions, refer to Occupation densities 

This book is designed for postgraduate course in stochastic analysis. It require readers familiar with measure-theoretic probability and discrete-time processes, it continues to explore stochastic process in a continuous time set-up. The stochastic process chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. 

The theory of stochastic integration and stochastic calculus is developed in this book.  The power of Stochastic calculus is illustrated by results concerning the martingales representation and change of measure on Wiener space, and these development furthered application in financial economics, such as option pricing and consumption optimization. This book also contains a detailed discussion on weak and strong solutions of SdE and a study of local time for semimartingale, with special emphasis on the theory of Brownian local time. 

This is one of the jokes that i know of on a mathematical theorem.Q. Why did the mathematician name his dog Cauchy?A. Because it left a residue at every pole. 

February 2008
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Stochastic analysis