Choose one of the following questions for further reading and write a brief report/essay of a few (2–5) pages with single-spaced lines. Writing more will not be penalised, but sheer length does not necessarily lead to a higher mark: quality is more important.
LATEX(with the standard document class article) is recommended because it is the system of choice for typesetting mathematic and scientific texts, for which it gives the best results, but alternatively you may resort to an editor like Word, Pages, LibreOffice,etc., or to handwriting. Please upload a PDF in Part 1 and the source code (.tex, .bib,.docx, etc.) in Part 2.
The marking criteria are similar to those for the dissertation on your summer project, that are described in detail on the Moodle pages of COMP0076/77.
1. Stochastic integral
At the end of the module we have briefly discussed the stochastic integral
where both X(t) and Y (t) are stochastic processes, in particular for the special case where X(t) and Y (t) are continuous-time random walks, and looked at Sections II and III ofGuido Germano, Mauro Politi, Enrico Scalas, Ren´e L. Schilling, “Stochastic calculus for uncoupled continuous-time random walks”, Physical Review E 79 (6), 066102, 2009, DOI 10.1103/PhysRevE.79.066102, available on the Moodle page of COMP0045.
At least, report what you have understood from there; if possible, extend to the more general case by looking up a standard book, e.g. the beginning of Chapter 4 of
Crispin W. Gardiner, Stochastic Methods — A Handbook for the Natural and Social Sciences, 4th edition, Springer, Berlin, 2009.