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

A càdlàg process $A$ is a finite-variation process when, almost surely, for every $T<\infty$,where the supremum is over every finite partition of an interval $0=t_0<\cdots <t_n=T$.

Uniform convergence on compacts in probability of $X^n$ to $X$ means that, for every $T<\infty$ and $\epsilon >0$,

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

The martingale product identity says that $N_t{K}_t-[N,K{]}_t$ is a martingale. Passing to the terminal values of the square-integrable martingales and using $N_0=K_0=0$ givesThe quadratic covariation identity for a stochastic integral isApplying the same product identity to $H\cdot M$ and $K$ therefore gives

Let $L=N-H\cdot M$. It is a continuous square-integrable martingale, and the assumed bracket identity givesfor every continuous square-integrable martingale $K$. Choose $K=L$ and use part (i):Thus $L_{\infty }=0$ almost surely, and the conditional expectation property gives $L_t=\mathbb{E}[L_{\infty }\mid {\mathcal{F}}_t]=0$ for every $t$. Hence $N=H\cdot M$ up to indistinguishability of stochastic processes.