Is an orthogonal-increment process submartingale?

Question

If I remember correctly, a stochastic process is said to be orthogonal-increment, if it is a second-order process, and the increments over disjoint intervals are uncorrelated.

I wonder if an orthogonal-increment process is a submartingale? Thanks!

Answer 1

I don't think so, at least not in the time-discrete case:

Let $(\xi_j)_{j \in \mathbb{N}}$ identically distributed independent random variables such that $\xi_j \in L^2$, $\mathbb{E}\xi_j < 0$. Define a (time-discrete) process $M_n$ bei $$M_n := \sum_{j=1}^n \xi_j$$ Then $(M_n)_n$ has orthogonal increments (in the sense of your definition), but it's not a submartingale since $$\mathbb{E}M_n = n \cdot \mathbb{E}\xi_1$$ is decreasing in $n$.