is there a characterization of submartingales in terms of stopping times similar to the martingale case in A martingale characterization.
Especially is the following true:
If $X$ is an adapted stochastic process with cadlag paths and $\mathrm{E}|X_t|<\infty \forall t\geq 0$, tfae:
- X is a submartingale
- For every bounded stopping time $\tau$ we have $\mathrm{E}X_{\tau} \geq \mathrm{E}X_0$
Thanks!
Best Answer
The following construction satisfies 2. but not 1.
Let $W$ be standard Brownian motion.
For $0 \le t\le 1$ set $X_t= W_t^2$.
For $1 \le t \le 2$ set $X_t=X_1(1-t)$. Note that $\mathbb{E}(W_t^2)=1$ so that $\mathbb{E}(X_{t})\ge 0 \text{ for } t\le 2$
For $2 \le t$ set $X_t=X_2+W_{t-2}^2$
The process $X$ is not a submartingale since $\mathbb{E}(X_2)<X_1$.