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

To establish the Brownian weighted quadratic variation in L2 for the deterministic partition of an interval, decomposewhere $U_{\pi }={\sum }_j{B}_j((\Delta B_j{)}^2-\Delta t_j)$ and $V_{\pi }={\sum }_j{B}_j\Delta t_j-{\int }_0^t{B}_{s}ds$.

The summands in $U_{\pi }$ are martingale differences: $B_j$ is ${\mathcal{F}}_{t_j}$-measurable, and the following Brownian increment is independent of ${\mathcal{F}}_{t_j}$. Their conditional means are zero, so distinct summands are orthogonal in L2 space. Since $\mathbb{E}{B}_j^2=t_j$ and $\mathbb{E}[((\Delta B_j{)}^2-\Delta t_j{)}^2]=2(\Delta t_j{)}^2$,For the Riemann sum error, the Cauchy-Schwarz inequality givesThus the triangle inequality in L2 space provesThe time integration variable is $s$.

Apply the exact discrete product identityand sum over the deterministic partition of an interval. Since $B_0=0$,The left side converges in L2 space to ${\int }_0^{t}sd{B}_s$ by the definition and Itô isometry, sinceThe right-endpoint Riemann sum converges in L2 space to ${\int }_0^t{B}_{s}ds$: its squared error is at mostConsequently,As in (a)(iii), continuity extends fixed-time equality to equality of the continuous processes.

Summing the exact cubic increment identity givesThe first sum tends to zero in L2 space. Indeed, odd moments of centered normal random variables vanish, a centered normal random variable of variance $v$ has sixth moment $15v^3$, and independence of the Brownian increments givesThe second sum converges in L2 space to $3{\int }_0^t{B}_{s}ds$ by (i).

For the last sum, first note that $\mathbb{E}{\int }_0^t{B}_s^{4}ds={\int }_0^{t}3s^{2}ds=t^3$, so $B_s^2$ is a square-integrable stochastic integrand. If $s=t_j+u$ and $D=B_s-B_j$, then $D$ is independent of $B_j$, with $D\sim N(0,u)$. HenceIt follows thatThe Itô isometry therefore makes ${\sum }_j{B}_j^2\Delta B_j$ converge in L2 space to ${\int }_0^t{B}_s^{2}d{B}_s$, with bounded truncations supplying the initial simple predictable processes if needed. Taking limits in the cubic identity provesThe proof uses only increment algebra, moments of normal random variables and the definition of the Itô integral.