Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 2 / b

Apply the realized absolute covariation theorem to each dyadic partition of an interval. More explicitly, use the continuous increasing clock $C=[M]+[N]$ and the Radon-Nikodym theorem to writeFor each time, let $(U,V)$ have the centered bivariate normal distribution with covariance matrix $\left(\begin{array}{cc}a& c\\ c& b\end{array}\right)$, and defineThis process is continuous and increasing. To prove convergence, localize $M,N$, represent the pair as stochastic integrals against a two-dimensional Brownian motion, and approximate the integrands in $L^2(dC)$ by bounded step previsible processes. For step integrands, the result is the weak law of large numbers applied on each block to independent Gaussian random variables. The Burkholder-Davis-Gundy inequality and the Cauchy-Schwarz inequality make the error uniform on each compact interval in probability. Consequentlyin the sense of uniform convergence on compacts in probability.

With the notation from part (i), the total-variation process of $[M,N]$ isFor the centered bivariate normal distribution $(U,V)$ used there, $c=\mathbb{E}[UV]$. The integral triangle inequality givesIntegrating this pointwise inequality against $dC$ proves $V_t\le {\tilde{V}}_t$ for every $t\ge 0$.

The Kunita-Watanabe inequality applied to the continuous local martingales $M,N$ gives directlyEquivalently, with the clock and densities from part (i), positivity of the covariance matrix gives $|c|\le \sqrt{ab}$, and the Cauchy-Schwarz inequality gives