# probability theory – On the higher moments of a continuous martingale

In discrete-time, it is possible to construct a martingale which is in $$L^p$$ but not $$L^q$$ for $$p. To see this, simply let $$(Omega, mathcal{F}, mathcal{F}_n, mathbb{P})$$ be a probability space for which $$(epsilon_n)$$ is an iid sequence of centred random variables in $$L^p setminus L^q$$. Now construct the random walk $$X_0=0$$,
$$X_n = sum_{k=1}^n epsilon_n$$
This is now a martingale with the desired properties.

Is this possible to construct a continuous martingale with the same property? Namely, can one have a (true) continuous martingale $$X_t$$ such that $$X_t$$ has $$p$$-th moments, but all higher moments are infinite?