points $\xi_0, \dotsc, \xi_d$ in $\R^d$, we define the $(d+1) \times (d+1)$-matrix $$\label{eq:H_def}
H(\xi_0, \dotsc, \xi_n) := \begin{pmatrix} 1 & \xi_0^\top \\ \vdots & \vdots \\ 1 & \xi_n^\top \end{pmatrix}.$$ The matrix-valued process $\Theta_t$ is formed by inserting $\mathbf{X} = (X^0, \dotsc, X^d)$ into $H$, i.e. we set $$\label{eq:theta_def}
\Theta_t(\omega) = H(X^0, \dotsc, X^d) 
= \begin{pmatrix} 1 & X_t^0(\omega)^\top \\  \vdots & \vdots \\ 1 & X_t^d(\omega)^\top \end{pmatrix}.$$

\[lem:inversion\] Let $s > 0$ and let $X$ be an affine semimartingale satisfying the support condition \[def:full\_support\]. Then there exists $\epsilon > 0$ and a set $E \in \cF_s$with $\P\brac{E}>0$, such that the matrices $\Theta_t(\omega)$ and $\Theta_{t-}(\omega)$; are regular for all $(t,\omega) \in (s,s+\epsilon) \times E$.

Define the first hitting time $$\tau :=  \inf \set{t > s: \Theta_t \text{ singular, or } \Theta_{t-} \text{ singular}}.$$ Since the set of singular matrices is a closed subset of the vector space of $\RR^{(d+1) \times (d+1)}$-matrices, $\tau$ is a stopping time, cf. [@Protter Thm. 1.4]. Moreover, by monotone convergence, we have $$\lim_{n \to \infty} \P\Big(\Theta_t \text{ and } \Theta_{t-} \text{ regular for all $t \in (s,s+1/n)$}\Big) = \lim_{n \to \infty}\P (\tau \ge s+1/n) = \P(\tau > s).$$ If we can show that $\P(\tau > s) 