Examples of a continuous martingale with $E[\sup\limits_{0\leq s\leq t} |M_s|]=\infty$?

Question

A local martingale is a martingale iff it is in the class DL.

The condition: for every $t\in[0,\infty)$

$$E[\sup\limits_{0\leq s\leq t} |M_s|]<\infty\tag1$$

guarantees a local martingale $M$ is a martingale by ensuring it satisfies the condition for being in the class DL. Moreover, by Burkholder-Davis-Gundy, this means: for every $t\in[0,\infty)$,

$$E[\langle M\rangle^{1/2}_t]<\infty$$

My question is: do there exist (simple?) examples of continuous martingales where these two conditions are violated?

What I am really asking is that these are known to be sufficient conditions to guarantee local martingales are martingales, why are they not necessary?

Answer 1

There exist indeed a uniformly integrable martingale $X$ such that $\sup_{t} |X_t|$ is not integrable. Here a list of several examples:

1. A simple example in discrete time can be found here http://www.math.fsu.edu/~nichols/martingalezoo.pdf (see the last 2 lines of point 5). A related construction is found in example 4.1 of the paper by A.S. Cherny "Some particular problems of martingale theory", which produces a uniformly integrable martingale $(X_n)_{n\in \mathbb{N}}$ and a bounded process $(H_n)_{n\in \mathbb{N}}$ such that the stochastic integral $M_n:=\sum_{k=0}^n H_k (X_{k+1}-X_k)$ is a martingale which is not uniformly integrable (by the Burkholder-Davis-Gundy inequality this implies that $\sup_{k\in \mathbb{N}} |X_k|$ is not integrable)
2. A (completely different) example in continuous time is sketched in Exercise 3.15 of the 2nd Chapter of Revuz-Yor's Book "Continuous Martingales and Brownian Motion, 3rd ed"
3. An example, with time index [0,∞): see section 4 in https://arxiv.org/pdf/1508.07564.pdf. This paper also discusses in detail the relation between local martingales and uniformly integrable martingales
4. An example (in French, in discrete time) is found at pages II-58 and II-59 of

Pellaumail, Jean, Sur l'intégrale stochastique et la décomposition de Doob-Meyer., Société mathématique de France. (1973).

Answer 2

I guess the following is a counter-example: take $B$ a standard two-dimensional Brownian motion and $$ M_t = \log ( \Vert B_t \Vert ) $$ $M$ is a local martingale (compute the SDE satisfied by $M$, rembering that $\Vert B \Vert$ is a Bessel process) , but it does not hold that $$ E( \langle M \rangle_t ) < \infty $$ nor that $$ E( \sup_{0 \le s \le t} |M_s|) < \infty. $$ (to be checked; I am not 100% sure about this last statement).