# p-Variation distance defines semi-martingales

Question

When, does the process $$\tilde{X}_t$$, defined path-wise by $$\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),$$ define a semi-martingale? Where the notation is:

Notation

• $$X_t$$ is an $$\mathbb{R}^d$$-valued semi-martingale defined on the filtered probability space $$(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$$,
• $$(x_t(\omega),\mathbb{X}_t(\omega))$$ is a lift of $$X_t(\omega)$$ to a (random) rough-path of regularity at most $$\frac1{2}$$,
• $$(y_t,\mathbb{Y}_t):[0,T]\rightarrow \mathbb{R}^d \oplus (\mathbb{R}^d\otimes \mathbb{R}^d)$$ is a rough path of Holder continuity atmost $$\frac1{2}$$.
• $$\rho_{\frac1{2}}$$ is the $$\frac1{2}$$-Holder rough path metric.