= Conditionally symmetric increments
{title2=$\mathcal L(X_t-X_s\mid\mathcal F_s)=\mathcal L(X_s-X_t\mid\mathcal F_s)$}
A process has conditionally symmetric increments if, for all deterministic $s\le t$ and bounded measurable $g$,
$$
\mathbb E[g(X_t-X_s)\mid\mathcal F_s]=\mathbb E[g(X_s-X_t)\mid\mathcal F_s].
$$
Equivalently, the <conditional characteristic function> of every increment is invariant under changing the sign of its argument. This property is stronger than unconditional symmetry and is different from <independent increments>. It supplies identities between positive and negative exponential terminal martingales.
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