Theorem (Solution to SDE)

Let $(Z_{t})_{t\ge 0}$ be $\mathbb{R}^{r}$-valued continuous semimartingale and let $F:\mathbb{R}^{+}\times \mathbb{R}^{d}\to\mathscr{M}_{d,r}(\mathbb{R})$ be s.t. $\exists k>0$ s.t. $$\lVert F(t,x)-F(t,x') \rVert \le K\lVert x-x' \rVert \quad\forall t\in\mathbb{R}^{+},\forall x,x'\in\mathbb{R}^{d}$$and $t\mapsto F(t,x)$ is locally bounded $\forall x\in\mathbb{R}^{d}$. Consider the SDE $$\tag{*}dX_{t}=F(t,X_{t})dZ_{t},\quad X_{0}=x\in\mathbb{R}^{d}$$Assume the filtration $(\mathcal{F}_{t})_{t\ge 0}$ satisfies the Usual conditions. Then: $\forall x\in\mathbb{R}$, $\exists !$ up to indistinguishability a continuous $(\mathcal{F}_{t}^{Z})_{t\ge 0}$-adapted process $(X_{t})_{t\ge 0}$ satisfying $(*)$.