Finding an example of a discrete-time strict local martingale.

Find an example of a discrete-time local martingale that is not a true martingale.

I was thinking hard for some time about this fun problem. I know that $\mathbb{E}[|M|_t]=\infty \text{ for some } t\geq0$ should hold. Moreover any non-negative local martingale in discrete time is a true martingale, so this restricts my choice even more. I played around with Cauchy distribution, doubling strategy.

Solution 1:

Let $X$ be a random variable with finite mean and infinite variance. Let $B$ be $1$ with probability half and $−1$ with probability half, independent of $X$. Fix a filtration $\mathcal{F}$ by $\mathcal{F}_0 = \sigma(X)$ and $\mathcal{F}_i = \sigma(X,B)$ for every $i\geq 1$.

Let $M_0=X$ and $M_i=M_0+BM_0^2$ for every $i\geq1$. Then $(M_i)$ is not a true martingale, since $M_i$ is not integrable when $i\geq1$. For every $n$, set $T_n=\inf\{k:|M_k|\ge n\}$.

Fix an $n$. Then $\mathbb{E}[|M^{T_n}_0|]=\mathbb{E}[|X|]<\infty$ and, for every $i\geq1$,

$\begin{align} \mathbb{E}[|M^{T_n}_i|] &= \mathbb{E}[|M^{T_n}_1|\mathbf{1}(T_n=0)]+\mathbb{E}[|M^{T_n}_1|\mathbf{1}(T_n>0)]\\ &= \mathbb{E}[|M_0|\mathbf{1}(T_n=0)]+\mathbb{E}[|M_1|\mathbf{1}(T_n>0)]\\ &\leq \mathbb{E}[|M_0|]+\mathbb{E}[|M_1|\mathbf{1}(M_0\leq n)]\\ &\leq \mathbb{E}[|M_0|] + n+n^2 <\infty \end{align}$

So $M^{T_n}$ is integrable. We may also check that $\mathbb{E}[M^{T_n}_1\mid X]=M^{T_n}_0$, so $(T_n)$ localizes $M$. So $M$ is indeed a local martingale.