= Time-dependent test functions for a diffusion martingale problem
{title2=$M_t^f=f(t,X_t)-f(0,X_0)-\int_0^t(\partial_s+L)f(s,X_s)ds$}
For an <L-diffusion> with bounded coefficients and $f\in C_b^{1,2}$, $M^f$ is a continuous <martingale>. Freeze the time argument along a deterministic partition, apply the spatial <martingale problem> on each interval, and pass to the limit by the <dominated convergence theorem>. The same proof works if the absolute drift coefficients and the trace of the diffusivity have integrable time integrals along the path on every finite horizon. Local hypotheses alone give a <local martingale> conclusion.
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