Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2008 / A / 1

For a partition $0=t_0<\cdots <t_n=1$, approximate the integral by $X_{\pi }={\sum }_j{t}_j(B_{t_{j+1}}-B_{t_j})$, using left endpoints. The defining independent Brownian increment of Brownian motion make this a centered Gaussian variable with variance ${\sum }_j{t}_j^2(t_{j+1}-t_j)$. As the mesh tends to zero, that variance tends to ${\int }_0^1{t}^{2}dt=1/3$.

The stochastic integral is the mean-square convergence of these sums. mean-square convergence implies convergence in distribution, while the characteristic function converge to $e^{-u^2/6}$. HenceThe probability is exactly one half because this centered nondegenerate normal distribution is symmetric and has no atom at zero.

Use a filtration with respect to which $B$ is Brownian. The process is adapted and continuous because it is a continuous function of $B_t$ and $t$. The moment-generating function givesFor $0\le s<t$, writeThe increment is independent of ${\mathcal{F}}_s$ and normal with variance $t-s$, so the conditional expectation of the second factor is one. ThusThis verifies the martingale property from the definition and proves square integrability at each finite time, as well as continuity. It does not assert uniform square integrability over an infinite time horizon.

Lévy's characterization of Brownian motion states that a real continuous local martingale $M$ with $M_0=0$ and quadratic variation $⟨M{⟩}_t=t$ is a standard Brownian motion with respect to its filtration. Equivalently, its increments are independent of the past and have centered normal distribution with variances equal to elapsed times.

Let $H_s=\mathrm{sgn}({\sup }_{r\le s}{B}_r-1)$. The running maximum is continuous and adapted, so $H$ is bounded and progressively measurable process. It has a predictable version for integration; changing its value at the single switching time does not change its stochastic integral. Thus $X=\int HdB$ is a continuous local martingale starting at zero. Since $H_s^2=1$,Lévy's characterization of Brownian motion yields $X$ is standard Brownian motion. The sign depends on the entire past through the running maximum, but the unit quadratic-variation rate is exactly what the characterization needs.