Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 38 / 2 / a

Use the standard global convention that ${\mathcal{M}}_c^2$ consists of L2-bounded continuous martingales, here starting at zero. This matters for arbitrary unbounded stopping times in part (iv); square-integrability at each separate deterministic time alone is weaker. Such martingales have terminal $L^2$ limits and satisfy the optional sampling theorem at arbitrary finite stopping times. Infinite stopping times can also be included by evaluating at the terminal limit. The weakly orthogonal continuous martingales condition compares cross-time expectations. The following four parts prove its equivalence with the equal-time, bracket-mean and forward-stopping tests.

If $M,N$ are weakly orthogonal continuous martingales, taking the two times equal gives $\mathbb{E}[M_s{N}_s]=0$. Conversely suppose this equal-time condition holds. For $s\le t$, the martingale property and conditional expectation giveFor $t\le s$, condition the other factor at time $t$ instead. Products are integrable by the Cauchy-Schwarz inequality. Thus the equal-time condition implies the full cross-time definition, proving (i) equivalent to (ii).

The equal-time condition is the criterion proved equivalent to weak orthogonality in solution. Its equivalence with the quadratic-covariation condition follows from the covariance identity for continuous square-integrable martingales in solution; its equivalence with the forward-stopping condition follows from the optional sampling theorem in solution. These combine to give all four equivalences.

The polarization identity for quadratic covariation, together with $\mathbb{E}[M\pm N{]}_s=\mathbb{E}(M_s\pm N_s{)}^2$, yieldsThe bracket identity for each zero-starting square-integrable martingale is the Itô isometry for integration of the constant integrand against that martingale. Polarizing both identities gives the displayed covariance identity for continuous square-integrable martingales. The bracket is integrable: its total variation is at most $\sqrt{[M{]}_s[N{]}_s}$ by the Kunita-Watanabe inequality, whose expectation is finite by the Cauchy-Schwarz inequality. Therefore condition (iii) is equivalent to (ii).

For a stopping time $T\ge s$, the optional sampling theorem for the L2-bounded continuous martingale $M$ gives $\mathbb{E}[M_T\mid {\mathcal{F}}_s]=M_s$. ConsequentlyIf $N_s$ is unbounded, first multiply by a bounded truncation and then remove it using $L^2$ bounds and the Cauchy-Schwarz inequality. Thus (ii) implies (iv). Taking $T=s$ proves the converse. For $T=\infty$, the same calculation uses the $L^2$ terminal value.