Appendix: Generator and \(\mathfrak {b}_{\beta }\) for the HSDJ Model
By conferring (7), (8), and (26), we have in fact
$$ v(x,dz)=v^{0}(dz)+x^{\top }v^{1}(dz)=\lambda _{0}\mu _{0}(z_{1})\delta _{0} (z_{2})dz_{1}dz_{2}+x_{2}\lambda _{1}\mu _{1}(z_{1})\delta _{0}(z_{2} )dz_{1}dz_{2} $$
with \(\delta _{0}\) being the Dirac delta function, that is the (singular) density of the Dirac probability measure \(\mathbb {R}\) concentrated in \(\left\{ 0\right\} .\) Thus, the generator of the HSDJ model is given by
$$\begin{aligned} Af\,(x_{1},x_{2})&=\left( -\lambda _{0}a_{0}-\left( \frac{1}{2}\alpha ^{2}+\lambda _{1}a_{1}\right) x_{2}\right) \partial _{x_{1}}f+\kappa \left( \theta -x_{2}\right) \partial _{x_{2}}f\\&+\,\frac{1}{2}\alpha ^{2}x_{2}\partial _{x_{1}x_{1}}f+\alpha \sigma \rho x_{2}\partial _{x_{1}x_{2}}f+\frac{1}{2}\sigma ^{2}x_{2}\partial _{x_{2}x_{2}}f\\&\!\!\!\!\!+\,\int _{\mathbb {R}}\left[ f(x_{1}+z_{1},x_{2})-f(x_{1} ,x_{2})-z_{1}\partial _{x_{1}}f\right] \left( \lambda _{0}\mu _{0}(z_{1} )dz_{1}+x_{2}\lambda _{1}\mu _{1}(z_{1})dz_{1}\right) . \end{aligned}$$
Since we are dealing with jump probability densities rather than infinite jump measures, as in the case of infinite activity processes, the generator may be written as
$$\begin{aligned}\begin{gathered} Af\,(x_{1},x_{2})=\left( -\lambda _{0}\left( \mathfrak {m}_{0}+a_{0}\right) -\left( \frac{1}{2}\alpha ^{2}+\lambda _{1}\left( \mathfrak {m}_{1} +a_{1}\right) \right) x_{2}\right) \partial _{x_{1}}f\\ +\,\kappa \left( \theta -x_{2}\right) \partial _{x_{2}}f+\frac{1}{2}\alpha ^{2}x_{2}\partial _{x_{1}x_{1}}f+\alpha \sigma \rho x_{2}\partial _{x_{1}x_{2} }f+\frac{1}{2}\sigma ^{2}x_{2}\partial _{x_{2}x_{2}}f\\ \!\!\!\!\!+\,\lambda _{0}\int _{\mathbb {R}}\left[ f(x_{1}+y,x_{2})-f(x_{1} ,x_{2})\right] \mu _{0}(y)dy\\ \!\!\!\!\!+\,x_{2}\lambda _{1}\int _{\mathbb {R}}\left[ f(x_{1}+y,x_{2} )-f(x_{1},x_{2})\right] \mu _{1}(y)dy, \end{gathered}\end{aligned}$$
using (29).
With \(f_{u}(x)=e^{\mathfrak {i}u^{\top }x}\) we so obtain,
$$\begin{aligned}\begin{gathered} \frac{Af_{u}(x)}{f_{u}(x)}=\left( -\lambda _{0}\left( \mathfrak {m}_{0} +a_{0}\right) -\left( \frac{1}{2}\alpha ^{2}+\lambda _{1}\left( \mathfrak {m}_{1}+a_{1}\right) \right) x_{2}\right) \mathfrak {i}u_{1}\\ +\,\kappa \left( \theta -x_{2}\right) \mathfrak {i}u_{2}-\frac{1}{2}\alpha ^{2}x_{2}u_{1}^{2}-\alpha \sigma \rho x_{2}u_{1}u_{2}-\frac{1}{2}\sigma ^{2} x_{2}u_{2}^{2}\\ +\,\lambda _{0}\psi _{0}(u_{1})+x_{2}\lambda _{1}\psi _{1}(u_{1}) \end{gathered}\end{aligned}$$
with
$$\begin{aligned} \psi _{i}(\xi ):=\int _{\mathbb {R}}\left( e^{\mathfrak {i}\xi y}-1\right) \mu _{i}(y)dy, \,\, i=0,1. \end{aligned}$$
(33)
Note that we have
$$\begin{aligned} \mathfrak {m}_{i}+a_{i}=\psi _{i}(-\mathfrak {i}), \,\, i=0,1. \end{aligned}$$
(34)
The first order derivatives w.r.t. u are,
$$\begin{aligned} \partial _{u_{1}}\frac{Af_{u}(x)}{f_{u}(x)}&=-\lambda _{0}\left( \mathfrak {m}_{0}+a_{0}\right) \mathfrak {i}-\left( \frac{1}{2}\alpha ^{2}+\lambda _{1}\left( \mathfrak {m}_{1}+a_{1}\right) \right) \mathfrak {i} x_{2}\\&-\,\alpha ^{2}x_{2}u_{1}-\alpha \sigma \rho x_{2}u_{2}+\lambda _{0} \partial _{u_{1}}\psi _{0}(u_{1})+x_{2}\lambda _{1}\partial _{u_{1}}\psi _{1} (u_{1})\\ \partial _{u_{2}}\frac{Af_{u}(x)}{f_{u}(x)}&=\kappa \left( \theta -x_{2}\right) \mathfrak {i}-\alpha \sigma \rho x_{2}u_{1}-\sigma ^{2}x_{2}u_{2}. \end{aligned}$$
For the second order derivatives we have
$$\begin{aligned} \partial _{u_{1}u_{1}}\frac{Af_{u}(x)}{f_{u}(x)}&=-\alpha ^{2}x_{2} +\lambda _{0}\partial _{u_{1}u_{1}}\psi _{0}(u_{1})+x_{2}\lambda _{1} \partial _{u_{1}u_{1}}\psi _{1}(u_{1})\\ \partial _{u_{1}u_{2}}\frac{Af_{u}(x)}{f_{u}(x)}&=-\alpha \sigma \rho x_{2}, \,\, \partial _{u_{2}u_{2}}\frac{Af_{u}(x)}{f_{u}(x)}=-\sigma ^{2}x_{2}, \end{aligned}$$
and for multi-indices \(\beta \) with \(\left| \beta \right| \ge 3,\) i.e. the higher order ones,
$$\begin{aligned} \partial _{u^{\beta }}\frac{Af_{u}(x)}{f_{u}(x)}= {\left\{ \begin{array}{ll} \lambda _{0}\partial _{u_{1}^{\left| \beta \right| }}\psi _{0} (u_{1})+x_{2}\lambda _{1}\partial _{u_{1}^{\left| \beta \right| }} \psi _{1}(u_{1}) \text { for } \beta =(\left| \beta \right| ,0),\\ 0 \text { if } \beta \ne (\left| \beta \right| ,0). \end{array}\right. } \end{aligned}$$
(35)
Hence the ingredients (14) of the recursion (15) are in multi-index notation as follows.
\(\left| \beta \right| =0:\)
$$\begin{aligned}\begin{gathered} \mathfrak {b}_{0}(x,u)=-\lambda _{0}\left( \mathfrak {m}_{0}+a_{0}\right) \mathfrak {i}u_{1}+\kappa \theta \mathfrak {i}u_{2}+\lambda _{0}\psi _{0}(u_{1})\\ +x_{2}\left( \lambda _{1}\psi _{1}(u_{1})-\left( \frac{1}{2}\alpha ^{2} +\lambda _{1}\left( \mathfrak {m}_{1}+a_{1}\right) \right) \mathfrak {i} u_{1}-\kappa \mathfrak {i}u_{2}-\frac{1}{2}\alpha ^{2}u_{1}^{2}-\alpha \sigma \rho u_{1}u_{2}-\frac{1}{2}\sigma ^{2}u_{2}^{2}\right) , \end{gathered}\end{aligned}$$
whence
$$\begin{aligned} \mathfrak {b}_{0}^{0}(u)&=-\lambda _{0}\left( \mathfrak {m}_{0} +a_{0}\right) \mathfrak {i}u_{1}+\kappa \theta \mathfrak {i}u_{2}+\lambda _{0} \psi _{0}(u_{1}),\\ \mathfrak {b}_{0,e_{1}}^{1}(u)&=0, \,\, \mathfrak {b}_{0,e_{2}} ^{1}(u)=\lambda _{1}\psi _{1}(u_{1})-\left( \frac{1}{2}\alpha ^{2}+\lambda _{1}\left( \mathfrak {m}_{1}+a_{1}\right) \right) \mathfrak {i}u_{1}\\&-\,\kappa \mathfrak {i}u_{2}-\frac{1}{2}\alpha ^{2}u_{1}^{2}-\alpha \sigma \rho u_{1}u_{2}-\frac{1}{2}\sigma ^{2}u_{2}^{2}. \end{aligned}$$
For \(\left| \beta \right| \) \(=1,\) (14) yields
$$\begin{aligned} \mathfrak {b}_{(1,0)}(x,u)&=-\lambda _{0}\left( \mathfrak {m}_{0} +a_{0}\right) -\lambda _{0}\partial _{u_{1}}\psi _{0}(u_{1})\mathfrak {i}-\left( \frac{1}{2}\alpha ^{2}+\lambda _{1}\left( \mathfrak {m}_{1}+a_{1}\right) \right) x_{2}\\&+\,\alpha ^{2}x_{2}u_{1}\mathfrak {i}+\alpha \sigma \rho x_{2}u_{2} \mathfrak {i}-x_{2}\lambda _{1}\partial _{u_{1}}\psi _{1}(u_{1})\mathfrak {i}\\ \mathfrak {b}_{(0,1)}(x,u)&=\kappa \left( \theta -x_{2}\right) +\alpha \sigma \rho x_{2}u_{1}\mathfrak {i}+\sigma ^{2}x_{2}u_{2}\mathfrak {i,} \end{aligned}$$
whence
$$\begin{aligned} \mathfrak {b}_{(1,0)}^{0}(u)&=-\lambda _{0}\left( \mathfrak {m}_{0} +a_{0}\right) -\lambda _{0}\partial _{u_{1}}\psi _{0}(u_{1})\mathfrak {i,}\,\, \mathfrak {b}_{(1,0),e_{1}}^{1}(u)=0,\\ \mathfrak {b}_{(1,0),e_{2}}^{1}(u)&=-\left( \frac{1}{2}\alpha ^{2} +\lambda _{1}\left( \mathfrak {m}_{1}+a_{1}\right) \right) +\alpha ^{2} u_{1}\mathfrak {i}+\alpha \sigma \rho u_{2}\mathfrak {i}-\lambda _{1} \partial _{u_{1}}\psi _{1}(u_{1})\mathfrak {i} \end{aligned}$$
and
$$\begin{aligned} \mathfrak {b}_{(0,1)}^{0}(u)&=\kappa \theta , \,\, \mathfrak {b} _{(0,1),e_{1}}^{1}(u)=0,\\ \mathfrak {b}_{(0,1),e_{2}}^{1}(u)&=-\kappa +\alpha \sigma \rho u_{1} \mathfrak {i}+\sigma ^{2}u_{2}\mathfrak {i.} \end{aligned}$$
Next, for \(\left| \beta \right| \) \(=2,\) (14) yields
$$\begin{aligned} \mathfrak {b}_{(2,0)}(x,u)&=\alpha ^{2}x_{2}-\lambda _{0}\partial _{u_{1} u_{1}}\psi _{0}(u_{1})-x_{2}\lambda _{1}\partial _{u_{1}u_{1}}\psi _{1}(u_{1}),\\ \mathfrak {b}_{(1,1)}(x,u)&=\alpha \sigma \rho x_{2},\\ \, \mathfrak {b}_{(0,2)}(x,u)&=\sigma ^{2}x_{2}, \end{aligned}$$
whence
$$\begin{aligned} \mathfrak {b}_{(2,0)}^{0}(u)&=-\lambda _{0}\partial _{u_{1}u_{1}}\psi _{0}(u_{1}), \,\, \mathfrak {b}_{(2,0),e_{1}}^{1}(u)=0,\\ \mathfrak {b}_{(2,0),e_{2}}^{1}(u)&=\alpha ^{2}-\lambda _{1}\partial _{u_{1}u_{1}}\psi _{1}(u_{1}),\\ \mathfrak {b}_{(1,1)}^{0}(u)&=\mathfrak {b}_{(1,1),e_{1}}^{1}(u)=0,\quad \mathfrak {b}_{(1,1),e_{2}}^{1}(u)=\alpha \sigma \rho ,\\ \mathfrak {b}_{(0,2)}^{0}(u)&=\mathfrak {b}_{(0,2),e_{1}}^{1}(u)=0,\quad \mathfrak {b}_{(0,2),e_{2}}^{1}(u)=\sigma ^{2}. \end{aligned}$$
For multi-indices \(\beta \) with \(\left| \beta \right| \ge 3\) we get
$$ \mathfrak {b}_{\beta }(x,u)= {\left\{ \begin{array}{ll} \lambda _{0}\mathfrak {i}^{-\left| \beta \right| }\partial _{u_{1}^{\left| \beta \right| }}\psi _{0}(u_{1})+x_{2}\lambda _{1}\mathfrak {i}^{-\left| \beta \right| }\partial _{u_{1}^{\left| \beta \right| }}\psi _{1}(u_{1}) \text { for } \beta =(\left| \beta \right| ,0),\\ 0 \text { if } \beta \ne (\left| \beta \right| ,0), \end{array}\right. } $$
whence
$$ \mathfrak {b}_{\beta }^{0}(u)= {\left\{ \begin{array}{ll} \lambda _{0}\mathfrak {i}^{-\left| \beta \right| }\partial _{u_{1}^{\left| \beta \right| }}\psi _{0}(u_{1}) \text { for } \beta =(\left| \beta \right| ,0),\\ 0 \text { if } \beta \ne (\left| \beta \right| ,0), \end{array}\right. } $$
and
$$\begin{aligned} \mathfrak {b}_{\beta ,e_{1}}^{1}(u)&=0,\\ \mathfrak {b}_{\beta ,e_{2}}^{1}(u)&= {\left\{ \begin{array}{ll} \lambda _{1}\mathfrak {i}^{-\left| \beta \right| }\partial _{u_{1}^{\left| \beta \right| }}\psi _{1}(u_{1}) \text { for } \beta =(\left| \beta \right| ,0),\\ 0\text { if } \beta \ne (\left| \beta \right| ,0). \end{array}\right. } \end{aligned}$$
