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Supposse $X=(X_t)$ is a cadlag martingale taking values in $\mathbb{R}$. If $f:\mathbb{R}\to\mathbb{R}$ is a convex function, then we have Tanaka Formulae. Now let $g: \mathbb{R}_+\times\mathbb{R}\to\mathbb{R}$ is function s.t.

  1. For every $x\in\mathbb{R}$, $g(\cdot,x): \mathbb{R}_+\to\mathbb{R}$ is decreasing;

  2. For every $t\in\mathbb{R}_+$, $g(t,\cdot): \mathbb{R}\to\mathbb{R}$ is regular enough and the related derivatives $\partial_xg(t,x)$, $\partial_{xx}g(t,x)$ are uniformly bouned on $\mathbb{R}_+\times\mathbb{R}$.

My question is whether we have a formulae as Tanaka Formulae for $g(t,X_t)$? Many thanks for the reply and related reference!

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