 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

A process is a <supermartingale> relative to the given <filtration> if $X_n$ is $\mathcal F_n$-measurable and integrable for each $n$, and
$$
\boxed{\mathbb E(X_{n+1}\mid\mathcal F_n)\leq X_n
\quad\text{almost surely for every }n\geq0.}
$$
Iteration gives the equivalent inequality $\mathbb E(X_m\mid\mathcal F_n)\leq X_n$ for every $m\geq n$. <adapted process> and integrability are part of the definition, not consequences of the displayed inequality alone.

 Back to article page