Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 2 / a

Let $0=t_0<t_1<\cdots <t_m<\infty$ and let ${\xi }_j$ be bounded ${\mathcal{F}}_{t_j}$-measurable random variables. For the bounded simple predictable processthe Itô integral against a continuous martingale $M\in {\mathcal{H}}^{2,c}$ is defined by the increment sumThe same definition applies to elementary predictable processes with stopping-time intervals, replacing $t_j$ by ordered stopping times $T_j$ and taking bounded ${\mathcal{F}}_{T_j}$-measurable coefficients. Refinement of the intervals leaves the increment sum unchanged. The initial value of the integral is zero, and its martingale property follows from the conditional expectation identity for subsequent increments of $M$.

For a bounded simple predictable process $H$ and a continuous martingale $M\in {\mathcal{H}}^{2,c}$, the Itô isometry statesHere $⟨M⟩$ is the predictable quadratic variation, equal to $[M]$ for a continuous martingale. On the whole half-line, the terminal form isThe Itô integral starts at zero even when $M_0\ne 0$. For Brownian motion, $d⟨B{⟩}_s=ds$.

For completeness, cross terms of the finite increment sum have expectation zero by conditioning on the earlier endpoint of the later increment. Also $M^2-⟨M⟩$ is a martingale, so the remaining terms satisfyThis proves the displayed identity without a change-of-variables formula. Completing bounded simple predictable processes in the norm $(\mathbb{E}\int H^{2}d⟨M⟩{)}^{1/2}$ defines the Itô integral of every square-integrable stochastic integrand. This completion will justify the unbounded Brownian integrands below.

Fix $t>0$, a deterministic partition of an interval $\pi$, and write $\eta (s)=t_j$ on $[t_j,t_{j+1})$. Its mesh is $|\pi |={\max }_j(t_{j+1}-t_j)$. The Brownian increment variance givesThus the left-endpoint sums ${\sum }_j{B}_{t_j}\Delta B_j$ converge in L2 space to the Itô integral. Although these step coefficients are unbounded, clipping them at finite levels first makes them bounded simple predictable processes; the Itô isometry and their finite second moments remove the clipping.

The elementary identity $B_{t_{j+1}}^2-B_{t_j}^2=2B_{t_j}\Delta B_j+(\Delta B_j{)}^2$ givesThe squared Brownian increments are independent, with means $\Delta t_j$ and variances $2(\Delta t_j{)}^2$, since a centered normal random variable of variance $v$ has fourth moment $3v^2$. ThereforePassing to the convergence in L2 limits yieldsThis holds for each fixed $t$; continuous sample paths of both sides extend equality from rational times to all $t$ on one event of probability one.