Let $(X^1_t)_{t\ge 0}$ be an Ito process driven by a Brownian motion $(B_t)_{t\ge 0}$.
Let $N\ge 2$ and $Q_{nm} : \mathbb{R}\to\mathbb{R}^{N\times N}$ be jump intensities depending on $X^1$ and consider a continuous time Markov jump process $(Y^1_t)_{t\ge 0}$ taking values in $\{1,\dots,N\}$ such that $$ \mathbb{P}(Y^1_{t+dt} = n \mid X^1_t, Y^1_t) = Q_{Y^1_tn}(X^1_t)dt + o(dt),\ n\neq Y_t.$$
My first question is if such process $Y^1$ is unique and how do we construct it?
Now let $(X^2_t)_{t\ge 0}$ be an other Ito process driven by the same Brownian motion $(B_t)_{t\ge 0}$, and $(Y^2_t)_{t\ge 0}$ be continuous Markov jump process with values in $\{1,\dots,N\}$ such that $$ \mathbb{P}(Y^2_{t+dt} = n \mid X^2_t, Y^2_t) = Q_{Y^2_tn}(X^2_t)dt + o(dt),\ n\neq Y_t.$$
My second question what is the dependence of $Y^1$ and $Y^2$ ?
My last question is if there a simple integration formula to obtain an estimate like $$ \mathbb{E}((Y^1_T - Y^2_T)^2)\le \mathbb{E}((Y^1_0 - Y^2_0)^2) + C\int_0^T\mathbb{E}((X^1_t-X^2_t)^2) + \mathbb{E}((Y^1_t-Y^2_t)^2) dt.$$
Thank you.