Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 1 / c / Solution

Write $N_t=M_t-M_0$ and $V_t=⟨M{⟩}_t=[N{]}_t$. The Dambis-Dubins-Schwarz theorem supplies a standard Brownian motion $W$ such thatOn the event $V_{\infty }=\infty$, continuity and monotonicity imply that the range of $t\mapsto V_t$ is all of $[0,\infty )$.

For completeness, Brownian motion hits every positive integer almost surely. The Brownian reflection principle gives, for $a>0$,Intersect over integer $a$. The times of these increasingly large values must escape every compact interval because each Brownian path is continuous there. Therefore ${\mathrm{limsup}}_{u\to \infty }{W}_u=\infty$. The unbounded clock transfers these values to arbitrarily late times for $N$, and adding the finite initial value $M_0$ preserves unboundedness. HenceThis is an instance of infinite quadratic variation forces unbounded oscillations; applying the same argument to $-W$ also gives negative oscillations.