Some exercises for MOP. The separate documents may overlap occasionally but more important will be a supply of solutions. These are "in preparation". Exercise 1 ,....... Mop exercises ,....... Stopping Times Introductory Exercises ,......... 2011 Assessed Coursework 2,...... Supplementary Questions and Comments .........

Course notes for MOP

The notes on sigma-fields deal with Conditional Expectations, An example of a filtration and how conditional expectation works in this specific example, and a Martingale convergence result. Part 1 of the notes on sigma fields , Part 2 of the notes on sigma fields, Part 3 of the notes on sigma fields.

These notes prove a martingale representation result in a discrete setting. , A Martingale Representation Theorem ,

The files Doob_Meyer_1 and Doob_Meyer_2 give a treatment of the Doob_Meyer Decomposition for a submartingale. Along the way the idea of a Natural process is introduced. In discrete time, Natural is equivalent to Predictable in the strongest sense. In continuous time you modify your idea of equivalent for the result to remain true.