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

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 $M$.