Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 36 / 3 / d

Interpret the otherwise undefined $b_k(t)$ here as the moment ${\beta }_k(t)$ defined above. For $k=1$, the process is $B_t$, and for $k=2$ it is $B_t^2-t$; both are martingales.

For $k\ge 3$, the Itô formula and the derivative of the Brownian moment recursion yieldThe stochastic integral is a true martingale on finite horizons, since the required $2k-2$ moment is integrable in time. The continuous finite-variation term is not identically zero. If it were, its continuous derivative would vanish at every time, whereas at a fixed $t>0$, the nondegenerate Gaussian $B_t$ does not almost surely satisfy the polynomial equation $B_t^{k-2}={\beta }_{k-2}(t)$. Uniqueness of continuous semimartingale decomposition therefore rules out even the local martingale property. HenceFor example $\mathbb{E}(B_t^3\mid {\mathcal{F}}_s)=B_s^3+3(t-s)B_s$, so merely subtracting its zero expectation cannot make the cubic power a martingale. The centered Brownian powers need not be martingales distinction is resolved by subtracting the random drift compensator $k(k-1){\int }_0^t{B}_s^{k-2}ds/2$ instead of the deterministic mean.