In my work, I have many times encountered dynamic stochastic systems which are a submartingale (increasing in expectation) below a certain value of the variable, $X^*$ and a supermartingale (decreasing in expectation) above it. So suppose that $X_{t}\in[0,1]$ for all $t$ and satisfies $E[X_{t+1}|X_{t}]\ge X_{t}$ is $X_{t}\le X^{∗}$ and $E[X_{t+1}\vert X_{t}]\le X_{t}$ is $X_{t}\ge X^{∗}$. What can I say about the limit properties of $X_{t}$?
2026-04-04 15:54:14.1775318054
Can I show that a process which a supermartingale above a certain value and a submartingale below it converges?
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This is not true. It doesn't necessarily converge.
Say that your process is the deterministic sequence: $0,1,0,1,0,1,0,1 ...$
It is a supermartingale when $X_t \ge 0.5$ and a submartingale when $X_t \le 0.5$