Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / A / 3 / b

Integrating the preceding differential gives $M(h_j{)}_T=1+{\int }_0^T{h}_j(s)M(h_j{)}_{s}d{B}_s$. Consequently $\mathbb{E}F={\sum }_j{c}_j$, and the representation isThe process is continuous and adapted, hence progressively measurable, and the finite-sum inequality together with the previous second-moment estimate gives $\mathbb{E}{\int }_0^T|f_s{|}^{2}ds<\infty$.

If $g$ is another square-integrable predictable integrand with the same representation, Itô isometry givesThus uniqueness is up to $ds\otimes d\mathbb{P}$ null sets in the square-integrable integrand class. Pointwise uniqueness is impossible: changing a value at one deterministic time changes no Itô integral. Square integrability is the natural class for this $L^2$ representation and is needed for this uniqueness argument; the mere word adapted does not by itself specify an integrable stochastic integral.