This note is about Doob decomposition and the basics of Square integrable martingales Contents 1 Doob-Meyer Decomposition 1 2 Square Integrable Martingales 4 Brownian Motion and Stochastic Calculus Continuout Time Submartingales Usually it’s su¢ ce to only discuss submartingales by symmetry in de…nition and techniques are the same. 1 Doob-Meyer Decomposition Doob-meyer decomposition clears the ...

We study Doob’s martingale convergence theorem for computable continuous time martingales on Brownian motion, in the context of algorithmic randomness. A characterization of the class of sample points for which the theorem holds is given. Such points are given the name of Doob random points. It is shown that a point is Doob random if its tail is computably random in a certain sense. Moreover, D...

We prove a version of the Doob Meyer decomposition for supermartingales with a linearly ordered index set. Key words Doob Meyer decomposition, natural increasing processes, potentials, supermartingales. Mathematics Subject Classi…cation (2000): : 28A12, 60G07, 60G20.

Joseph Leonard Doob was born in Cincinnati, Ohio, USA on 27 February 1910. In an interview Joe gave for Statistical Science (see Snell (1997)), he said that in grammar school he built a crystal set and in high school operated a radio transmitter. This led him to think he would major in mathematical physics when he entered Harvard in 1926. However, after taking two physics courses, Doob decided ...

We consider nonconservative diffusion processes xt on the unit interval, so with absorbing barriers. Using Doob-transformation techniques involving superharmonic functions, we modify the original process to form a new diffusion process x̃t presenting an additional killing rate part d > 0. We limit ourselves to situations for which x̃t is itself nonconservative with upper bounded killing rate. For...

We show that the Doob and Hamming graphs are hyper-self-dual. We then show that although the Doob graphs are formally dual to certain Hamming graphs, they are not hyper-dual to them. We do so by showing that Bose-Mesner subalgebras and Kronecker products of Bose-Mesner algebras inherit hyper-duality.

Both the Doob-Meyer and the Graversen-Rao decomposition theorems can be proved following an approach based on predictable compensators of discretizations and weak-L technique, which was developed by K.M. Rao. It is shown that any decomposition obtained by Rao’s method gives predictability of compensators without additional assumptions (like submartingality in the original Doob-Meyer theorem or ...