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:


  • $X_t$ 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.

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