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

Let $A_t={\int }_0^{t}h(s)d{B}_s$ and $v_t={\int }_0^{t}h(s{)}^{2}ds$. Itô's lemma for the exponential givesTo justify that this local martingale is true, note that a deterministic Itô integral is Gaussian: step-function approximations are centered Gaussian sums with variances tending to $v_t$, and Itô isometry gives their convergence. Thus $A_t\sim N(0,v_t)$ andContinuity of $h$ on the compact interval bounds $h$ and $v_t$, soThe differential above is therefore a square-integrable stochastic integral, proving the martingale property on $[0,T]$, with initial value one.

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.

The semimartingale integration by parts givesHere $d⟨X,M{⟩}_t=h(t)M_{t}dt$. Substitution therefore gives cancellation of the drift:The continuous adapted integrand is locally square integrable. Stopping when $|X|+M$ first exceeds an integer produces true stochastic-integral martingales, hence $Y$ is a local martingale. No unproved global integrability of the product is needed.