= Initial-value integrability in normalized semimartingale decompositions
{title2=$X=M+A,\quad A_0=0\ \Longrightarrow\ M_0=X_0$}
With the <local martingale> definition requiring each stopped process itself to be a <martingale>, the initial value is integrable. A normalized <continuous semimartingale decomposition> $X=M+A$ with $A_0=0$ can therefore have a local-martingale part only if $X_0$ is integrable. A smooth transformation need not preserve this condition: take $M=0$, $A_t=G$ for an initially known standard <normal random variable>, and $f(x,a)=e^{a^2}$. Then $f(M_0,A_0)$ is finite almost surely but has infinite expectation. The always-valid decomposition is instead $X=X_0+N+V$ with a zero-starting <continuous local martingale> $N$ and a continuous <finite-variation process> $V$ starting at zero.
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