Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 5

For every real $\lambda$, positivity of quadratic variation givesThe discriminant of this quadratic is nonpositive, soContinuity lets the almost-sure assertion hold simultaneously for every $t$.

For a partition $\pi$ of $[0,t]$, apply the first inequality to each increment of the covariation and then the Cauchy-Schwarz inequality:Taking the supremum over partitions proves the Kunita-Watanabe inequality

Let ${\tau }_n$ localize the nonnegative local martingale $M$. For $s\le t$,Conditional Fatou lemma and nonnegativity giveThus $M$ is a supermartingale, recovering the general fact about a nonnegative local martingale.

A strong solution of a stochastic differential equation is adapted to the completed filtration of a prescribed Brownian motion $B$ on a prescribed probability space and satisfiesalmost surely. A weak solution of a stochastic differential equation may choose the filtered probability space, Brownian motion, and adapted process as part of the solution; only the displayed integral equation and the prescribed initial law are required.

Apply the Itô formula to $f(t,x)=x_0{e}^{\sigma x+(\mu -{\sigma }^2/2)t}$ and the semimartingale vector $(t,B_t)$. Its derivatives giveThereforeThis is adapted to the given Brownian filtration and is consequently a strong solution; it is geometric Brownian motion.