Appendix: Proofs
In the following, we provide the proofs of our main results.
1.1 Proof of Proposition 1
As Y
j
(t, T) is assumed to have
an exponential moment, F
j
(t, T)
is integrable. We have
$$
\displaystyle\begin{array}{rcl}
& & F_{j}(t,T) = F_{j}(s,T)\exp \left (\int _{A_{j}^{s}(t)}\mu _{j}(T;u,\xi )\mathit{du}d\xi +\int _{A_{j}^{s}(t)}g_{j}(T;u,\xi )\sigma _{j}(u,\xi )W_{j}(\mathit{du},d\xi )\right.
{}\\
& & \qquad \qquad \left.+\int _{A_{j}^{s}(t)}\int _{\mathbb{R}}\mathit{zh}_{j}(T;u,\xi )\tilde{J}_{L_{j}}(\mathit{dz},\mathit{du},d\xi )\right ), {}\\
\end{array}
$$
for s ≤ t.
Hence
$$\displaystyle{
\mathbb{E}\left [F_{j}(t,T)\,\vert \,\mathcal{F}_{s}\right ] = F_{j}(s,T)\mathbb{E}\left [\exp (Z_{j}(s,t,T))\,\vert \,\mathcal{F}_{s}\right ]\,,
}$$
with
$$
\displaystyle\begin{array}{rcl}
& & Z_{j}(s,t,T) =\int _{A_{j}^{s}(t)}\mu _{j}(T;u,\xi )\mathit{dud}\xi +\int _{A_{j}^{s}(t)}g_{j}(T;u,\xi )\sigma _{j}(u,\xi )W_{j}(\mathit{du},d\xi )
{}\\
& & \qquad \qquad \quad +\int _{A_{j}^{s}(t)}\int _{\mathbb{R}}\mathit{zh}_{j}(T;u,\xi )\tilde{J}_{L_{j}}(\mathit{dz},\mathit{du},d\xi ). {}\\
\end{array}
$$
The martingale property follows if \(\mathbb{E}[\exp (Z_{j}(s,t,T))\,\vert \,\mathcal{F}_{s}] = 1\)
for all s ≤ t ≤ T. But by double
conditioning using \(\mathcal{F}^{\sigma _{j}} \vee \mathcal{F}^{\mu _{j}} \vee \mathcal{F}_{s}\)
we have, by independence of the two stochastic integrals,
$$
\displaystyle\begin{array}{rcl}
& & \mathbb{E}\left [\exp (Z_{j}(s,t,T))\,\vert \,\mathcal{F}_{s}\right ]
{}\\
& & = \mathbb{E}\left [\exp \left (\int _{A_{j}^{s}(t)}\mu _{j}(T;u,\xi )\mathit{dud}\xi + \frac{1}
{2}\int _{A_{j}^{s}(t)}g_{j}^{2}(T;u,\xi )\sigma _{
j}^{2}(u,\xi )\mathit{dud}\xi \right.\right.
{}\\
& & \quad \left.\left.\left.+\int _{A_{j}^{s}(t)}\int _{\mathbb{R}}e^{zh_{j}(T;u,\xi )} - 1 - zh_{
j}(T;u,\xi )\nu _{L_{j}}(\mathit{dz})\mathit{dud}\xi \right )\,\right \vert \,\mathcal{F}_{s}\right ]. {}\\
\end{array}
$$
But this is equal to one by assumption. Then the result follows.
1.2 Proof of Proposition 2
Using Itô’s formula, we get
$$\displaystyle\begin{array}{rcl} F_{j}(t,T)& =& F_{j}(0,T) +\int _{ 0}^{t}F_{ j}(s-,T)d_{s}Y _{j}(s,T) + \frac{1} {2}\int _{0}^{t}F_{ j}(s-,T)d_{s}[Y ]_{j}^{c}(s,T) {}\\ & & +\sum _{0\leq s\leq t}\varDelta _{s}F_{j}(s,T) - F_{j}(s-,T)\varDelta _{s}Y _{j}(s,T) {}\\ & =& F_{j}(0,T) +\int _{ 0}^{t}F_{ j}(s-,T)d_{s}Y _{j}(s,T) + \frac{1} {2}\int _{0}^{t}F_{ j}(s-,T)d_{s}[Y ]_{j}^{c}(s,T) {}\\ & & +\sum _{0\leq s\leq t}F_{j}(s-,T)\left (\exp (\varDelta _{s}Y _{j}(s,T)) - 1 -\varDelta _{s}Y _{j}(s,T)\right ) {}\\ & =& F_{j}(0,T) +\int _{A_{j}^{0}(t)}F_{j}(s-,T)g_{j}(T;s,\xi )\sigma _{j}(s,\xi )W_{j}(\mathit{ds},d\xi ) {}\\ & & +\int _{A_{j}^{0}(t)}\int _{\mathbb{R}}F_{j}(s-,T)\left (e^{\mathit{zh}_{j} (T;s,\xi )} - 1\right )\tilde{J}_{ L_{j}}(\mathit{dz},\mathit{ds},d\xi ). {}\\ \end{array}$$
Note that we have applied the Itô formula to the process \((F_{j}(t,T))_{0\leq t\leq T}\) for fixed T. From Proposition 1 we deduce that this process is indeed a martingale and hence we can apply the classical Itô formula when we fix the parameter T. The corresponding results on the quadratic variation of Y are direct consequences of the Walsh-integration theory.
1.3 Proof of Proposition 8
Formula (16) is a direct consequence of [21, Theorem 2.2] and the functional form of \(\hat{f}\) has been derived in [21, Example 5.1]. Finally, we need to compute the extended characteristic function \(\mathbb{E}_{0}^{\tilde{\mathbb{P}}}\left [\exp (i\theta (R,u)Z(t))\right ]\), where we set \(\theta =\theta (R,u) \in \mathbb{C}\) in the proof to simplify the exposition.
For fixed t define the two random variables
$$\displaystyle\begin{array}{rcl} Z_{1}(t)&:=& \int _{A_{1}^{0}(t)}(-g_{1}(T_{1};s,\xi ) + g_{1}(T_{2};s,\xi ))\sigma _{1}(s,\xi )\hat{W}_{1}(\mathit{ds},d\xi ) {}\\ & & -\frac{1} {2}\int _{A_{1}^{0}(t)}(-g_{1}(T_{1};s,\xi ) + g_{1}(T_{2};s,\xi ))^{2}\sigma _{ 1}^{2}(s,\xi )\mathit{dsd}\xi {}\\ Z_{2}(t)&:=& \int _{A_{1}^{0}(t)}\int _{\mathbb{R}}(z_{1}h_{1}(T_{2};s,\xi ) - z_{1}h_{1}(T_{1};s,\xi ))\hat{N}_{1}(\mathit{dz}_{1},\mathit{ds},d\xi ) {}\\ & & -\int _{A_{1}^{0}(t)}\int _{\mathbb{R}}(z_{1}h_{1}(T_{2};s,\xi ) - z_{1}h_{1}(T_{1};s,\xi )) + 1 -\exp (z_{1}h_{1}(T_{2};s,\xi ) {}\\ & & -z_{1}h_{1}(T_{1};s,\xi ))\nu _{L_{1},\tilde{\mathbb{P}}}(\mathit{dz}_{1})\mathit{dsd}\xi. {}\\ \end{array}$$
Note that the σ-algebra σ(Z
2(t)) is independent of \(\sigma (\mathcal{F}_{0},Z_{1}(t))\) (assuming a suitable choice of the underlying filtration \((\mathcal{F}_{t})\)). Hence
$$\displaystyle\begin{array}{rcl} \mathbb{E}_{0}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z(t))\right ] = \mathbb{E}_{ 0}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z_{ 1}(t))\right ]\mathbb{E}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z_{ 2}(t))\right ],& & {}\\ \end{array}$$
where
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}_{0}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z_{ 1}(t))\right ] {}\\ & & = \mathbb{E}_{0}^{\tilde{\mathbb{P}}}\left [\exp \left (\left (-\frac{\theta ^{2}} {2} - \frac{i\theta } {2}\right )\int _{A_{1}^{0}(t)}(-g_{1}(T_{1};s,\xi ) + g_{1}(T_{2};s,\xi ))^{2}\sigma _{ 1}^{2}(s,\xi )\mathit{dsd}\xi \right )\right ], {}\\ \end{array}$$
where we conditioned on \(\mathcal{F}^{\sigma _{1}}\). For the jump part, we have
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z_{ 2}(t))\right ] {}\\ & & =\exp \left ((1 - i\theta )\int _{A_{1}^{0}(t)}\int _{\mathbb{R}}\left (e^{z_{1}(h_{1}(T_{2};s,\xi )-h_{1}(T_{1};s,\xi ))} - 1\right )\nu _{ L_{1},\tilde{\mathbb{P}}}(dz_{1})dsd\xi \right ). {}\\ \end{array}$$
1.4 Proof of Lemma 3
This follows directly from the martingale condition derived in Proposition 1 and a straightforward extension of the Girsanov theorem for Itô-Lévy processes, see e.g. [29, Theorem 1.33]. For the Poisson random measure, note that
$$\displaystyle\begin{array}{rcl} & & \hat{N}_{1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & =\tilde{ J}_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) + (1 - e^{z_{1}h_{1}(T;s,\xi )})\nu _{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi {}\\ & & = J_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) -\nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi {}\\ & & +(1 - e^{z_{1}h_{1}(T;s,\xi )})\nu _{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi {}\\ & & = J_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) - e^{z_{1}h_{1}(T;s,\xi )}\nu _{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi, {}\\ \end{array}$$
denotes the compensated Poisson random measure under \(\tilde{\mathbb{P}}\).
1.5 Proof of Lemma 4
First of all, we apply the Itô formula for higher dimensions and get
$$\displaystyle\begin{array}{rcl} \frac{d\left (\frac{S_{2}(t)} {S_{1}(t)}\right )} {S_{2}(t-)/S_{1}(t-)}& =& -\frac{\mathit{dS}_{1}(t)} {S_{1}(t-)} + \frac{\mathit{dS}_{2}(t)} {S_{2}(t-)} + \frac{d[S_{1}]^{c}(t)} {S_{1}^{2}(t-)} - \frac{1} {S_{1}(t-)S_{2}(t-)}d[S_{1},S_{2}]^{c}(t) {}\\ & & + \frac{S_{1}(t-)} {S_{2}(t-)}\left (\frac{S_{2}(t)} {S_{1}(t)} -\frac{S_{2}(t-)} {S_{1}(t-)} -\varDelta S_{1}(t)\left (-\frac{S_{2}(t-)} {S_{1}^{2}(t-)}\right )\right. {}\\ & & \left.-\varDelta S_{2}(t)\left ( \frac{1} {S_{1}(t-)}\right )\right ) {}\\ & =& -\frac{\mathit{dS}_{1}(t)} {S_{1}(t-)} + \frac{\mathit{dS}_{2}(t)} {S_{2}(t-)} + \frac{d[S_{1}]^{c}(t)} {S_{1}^{2}(t-)} - \frac{1} {S_{1}(t-)S_{2}(t-)}d[S_{1},S_{2}]^{c}(t) {}\\ & & + \frac{S_{1}(t-)} {S_{2}(t-)}\varDelta \left (\frac{S_{2}(t)} {S_{1}(t)}\right ) -\varDelta S_{1}(t)\left (-\frac{S_{2}(t-)} {S_{1}^{2}(t-)}\right )\frac{S_{1}(t-)} {S_{2}(t-)} {}\\ & & -\varDelta S_{2}(t)\left ( \frac{1} {S_{1}(t-)}\right )\frac{S_{1}(t-)} {S_{2}(t-)} {}\\ &=& -\frac{\mathit{dS}_{1}(t)} {S_{1}(t-)} + \frac{\mathit{dS}_{2}(t)} {S_{2}(t-)} + \frac{d[S_{1}]^{c}(t)} {S_{1}^{2}(t-)} - \frac{1} {S_{1}(t-)S_{2}(t-)}d[S_{1},S_{2}]^{c}(t) {}\\ & & + \frac{1} {S_{2}(t-)/S_{1}(t-)}\varDelta \left (\frac{S_{2}(t)} {S_{1}(t)}\right ) + \frac{\varDelta S_{1}(t)} {S_{1}(t-)} - \frac{\varDelta S_{2}(t)} {S_{2}(t-)}. {}\\ \end{array}$$
Since
$$\displaystyle{ \frac{d[S_{1}]^{c}(t)} {S_{1}^{2}(t-)} =\int _{ 0}^{\infty }g_{ 1}^{2}(T;t,\xi )\sigma _{ 1}^{2}(t,\xi )\mathit{dtd}\xi, }$$
and
$$\displaystyle{ \frac{d[S_{1},S_{2}]^{c}(t)} {S_{1}(t-)S_{2}(t-)} =\int _{ 0}^{\infty }g_{ 1}(T;t,\xi )g_{2}(T;t,\xi )\sigma _{1}(t,\xi )\sigma _{2}(t,\xi )\rho \mathit{dtd}\xi, }$$
we have
$$\displaystyle\begin{array}{rcl} & & \frac{d\left (\frac{S_{2}(t)} {S_{1}(t)}\right )} {S_{2}(t-)/S_{1}(t-)} {}\\ & & = -\int _{0}^{\infty }g_{ 1}(T;t,\xi )\sigma _{1}(t,\xi )W_{1}(\mathit{dt},d\xi ) -\int _{0}^{\infty }\int _{ \mathbb{R}}\left (e^{\mathit{zh}_{1} (T;t,\xi )} - 1\right )\tilde{J}_{ L_{1}}(\mathit{dz},\mathit{dt},d\xi ) {}\\ & & \quad +\int _{ 0}^{\infty }g_{ 2}(T;t,\xi )\sigma _{2}(t,\xi )W_{2}(\mathit{dt},d\xi ) +\int _{ 0}^{\infty }\int _{ \mathbb{R}}\left (e^{\mathit{zh}_{2} (T;t,\xi )} - 1\right )\tilde{J}_{ L_{2}}(\mathit{dz},\mathit{dt},d\xi ) {}\\ & & \quad +\int _{ 0}^{\infty }g_{ 1}^{2}(T;t,\xi )\sigma _{ 1}^{2}(s,\xi )\mathit{dt}d\xi -\rho \int _{ 0}^{\infty }g_{ 1}(T;t,\xi )g_{2}(T;t,\xi )\sigma _{1}(t,\xi )\sigma _{2}(t,\xi )\mathit{dtd}\xi {}\\ & & \quad + \frac{1} {S_{2}(t-)/S_{1}(t-)}\varDelta \left (\frac{S_{2}(t)} {S_{1}(t)}\right ) +\int _{ 0}^{\infty }\int _{ \mathbb{R}}\left (e^{\mathit{zh}_{1} (T;t,\xi )} - 1\right )\tilde{J}_{ L_{1}}(\mathit{dz},\mathit{dt},d\xi ) {}\\ & & \quad -\int _{0}^{\infty }\int _{ \mathbb{R}}\left (e^{\mathit{zh}_{2} (T;t,\xi )} - 1\right )\tilde{J}_{ L_{2}}(\mathit{dz},\mathit{dt},d\xi ). {}\\ \end{array}$$
In the case of jumps of finite variation, we can further simplify and get
$$\displaystyle\begin{array}{rcl} & & \frac{d\left (\frac{S_{2}(t)} {S_{1}(t)}\right )} {S_{2}(t-)/S_{1}(t-)} {}\\ & & = -\int _{0}^{\infty }g_{ 1}(T;t,\xi )\sigma _{1}(t,\xi )W_{1}(\mathit{dt},d\xi ) +\int _{ 0}^{\infty }g_{ 2}(T;t,\xi )\sigma _{2}(t,\xi )W_{2}(\mathit{dt},d\xi ) {}\\ & & \quad +\int _{ 0}^{\infty }g_{ 1}^{2}(T;t,\xi )\sigma _{ 1}^{2}(t,\xi )\mathit{dtd}\xi -\rho \int _{ 0}^{\infty }g_{ 1}(T;t,\xi )g_{2}(T;t,\xi )\sigma _{1}(t,\xi )\sigma _{2}(t,\xi )\mathit{dtd}\xi {}\\ & & \quad + \frac{1} {S_{2}(t-)/S_{1}(t-)}\varDelta \left (\frac{S_{2}(t)} {S_{1}(t)}\right ). {}\\ \end{array}$$
In the general case—writing formally—we have
$$\displaystyle\begin{array}{rcl} & & \frac{1} {S_{2}(t-)/S_{1}(t-)}\varDelta \left (\frac{S_{2}(t)} {S_{1}(t)}\right ) {}\\ & & = \frac{S_{1}(t-)} {S_{2}(t-)}\varDelta \left (\frac{S_{2}(t)} {S_{1}(t)}\right ) = \frac{\exp (Y _{1}(t-,T)} {\exp (Y _{2}(t-,T))}\varDelta \left (\frac{\exp (Y _{2}(t,T)} {\exp (Y _{1}(t,T)}\right ) {}\\ & & = \frac{\exp (Y _{1}(t-,T)} {\exp (Y _{2}(t-,T))}\left (\frac{\exp (Y _{2}(t,T)} {\exp (Y _{1}(t,T)} -\frac{\exp (Y _{2}(t-,T)} {\exp (Y _{1}(t-,T)}\right ) {}\\ & & =\exp (Y _{1}(t-,T) - Y _{2}(t-,T))\left (\exp (Y _{2}(t,T) - Y _{1}(t,T))\right. {}\\ & & \left.-\exp (Y _{2}(t-,T) - Y _{1}(t-,T))\right ) {}\\ & & =\exp (\varDelta Y _{2}(t) -\varDelta Y _{1}(t)) - 1. {}\\ \end{array}$$
So altogether the finite variation jump term, has the form
$$\displaystyle{ \frac{1} {S_{2}(t-)/S_{1}(t-)}\varDelta \left (\frac{S_{2}(t)} {S_{1}(t)}\right )+ \frac{\varDelta S_{1}(t)} {S_{1}(t-)}- \frac{\varDelta S_{2}(t)} {S_{2}(t-)}=e^{\varDelta Y _{2}(t)-\varDelta Y _{1}(t)}+e^{\varDelta Y _{1}(t)}-e^{\varDelta Y _{2}(t)}-1. }$$
Summing up and using the joint Poisson random measure, we have
$$\displaystyle\begin{array}{rcl} & & \sum _{0\leq s\leq t}\left ( \frac{1} {S_{2}(s-)/S_{1}(s-)}\varDelta \left (\frac{S_{2}(s)} {S_{1}(s)}\right ) + \frac{\varDelta S_{1}(s)} {S_{1}(s-)} - \frac{\varDelta S_{2}(s)} {S_{2}(s-)}\right ) {}\\ & & =\int _{A_{j}(t)}\int _{\mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;s,\xi )-z_{1}h_{1}(T;s,\xi )} + e^{z_{1}h_{1}(T;s,\xi )} - e^{z_{2}h_{2}(T;s,\xi )} - 1\right ) {}\\ & & \qquad \quad \cdot J_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ). {}\\ \end{array}$$
Also,
$$\displaystyle\begin{array}{rcl} & & -\int _{A_{j}(t)}\int _{\mathbb{R}}\left (e^{z_{1}h_{1}(T;s,\xi )} - 1\right )\tilde{J}_{ L_{1}}(\mathit{dz}_{1},\mathit{ds},d\xi ) {}\\ & & \qquad +\int _{A_{j}(t)}\int _{\mathbb{R}}\left (e^{z_{2}h_{2}(T;s,\xi )} - 1\right )\tilde{J}_{ L_{2}}(\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & \qquad \qquad =\int _{A_{j}(t)}\int _{\mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;s,\xi )} - e^{z_{1}h_{1}(T;s,\xi )}\right )\tilde{J}_{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ). {}\\ \end{array}$$
Adding the jump terms, we get
$$\displaystyle\begin{array}{rcl} & & \int _{A_{j}(t)}\int _{\mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;s,\xi )} - e^{z_{1}h_{1}(T;s,\xi )}\right )\tilde{J}_{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & \quad +\int _{A_{j}(t)}\int _{\mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;s,\xi )-z_{1}h_{1}(T;s,\xi )} + e^{z_{1}h_{1}(T;s,\xi )} - e^{z_{2}h_{2}(T;s,\xi )} - 1\right ) {}\\ & & \qquad \quad \cdot J_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & =\int _{A_{j}(t)}\int _{\mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;s,\xi )-z_{1}h_{1}(T;s,\xi )} - 1\right )\tilde{J}_{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & \quad +\int _{A_{j}(t)}\int _{\mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;s,\xi )-z_{1}h_{1}(T;s,\xi )} - 1 + e^{z_{1}h_{1}(T;s,\xi )} - e^{z_{2}h_{2}(T;s,\xi )}\right ) {}\\ & & \qquad \quad \cdot \nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi. {}\\ \end{array}$$
1.6 Proof of Proposition 9
This is a direct consequence of the preceding two lemmas. Note in particular, that for the Gaussian part, we have that W
1 and W
2 are Gaussian bases with correlation coefficient ρ, meaning that there exists a homogeneous, factorisable Gaussian basis W
3 (under \(\mathbb{P}\)) such that \(W_{2} =\rho W_{1} + \sqrt{1 -\rho ^{2}}W_{3}\). Under the new measure \(\tilde{\mathbb{P}}\), we have that W
3 does not change and
$$\displaystyle\begin{array}{rcl} \hat{W}_{1}(\mathit{ds},d\xi )& =& W_{1}(\mathit{ds},d\xi ) - g_{1}(T;s,\xi )\sigma (s,\xi )\mathit{dsd}\xi, {}\\ \hat{W}_{3}(\mathit{ds},d\xi )& =& W_{3}(\mathit{ds},d\xi ), {}\\ \hat{W}_{2}(\mathit{ds},d\xi )& =& \rho \hat{W}_{1}(\mathit{ds},d\xi ) + \sqrt{1 -\rho ^{2}}\hat{W}_{3}(\mathit{ds},d\xi ) {}\\ & =& W_{2}(\mathit{ds},d\xi ) -\rho g_{1}(T;s,\xi )\sigma (s,\xi )\mathit{dsd}\xi, {}\\ \end{array}$$
are homogeneous, factorisable Gaussian bases under \(\tilde{\mathbb{P}}\), where \(\hat{W}_{1}\) and \(\hat{W}_{3}\) are independent and \(\hat{W}_{1}\) and \(\hat{W}_{2}\) have correlation coefficient ρ. So for the mixed Gaussian part, we have
$$\displaystyle\begin{array}{rcl} & & -\int _{0}^{\infty }g_{ 1}(T;t,\xi )\sigma _{1}(t,\xi )W_{1}(\mathit{dt},d\xi ) +\int _{ 0}^{\infty }g_{ 2}(T;t,\xi )\sigma _{2}(t,\xi )W_{2}(\mathit{dt},d\xi ) {}\\ & & \quad +\int _{ 0}^{\infty }g_{ 1}^{2}(T;t,\xi )\sigma _{ 1}^{2}(t,\xi )\mathit{dtd}\xi -\rho \int _{ 0}^{\infty }g_{ 1}(T;t,\xi )g_{2}(T;t,\xi )\sigma _{1}(t,\xi )\sigma _{2}(t,\xi )\mathit{dtd}\xi {}\\ & & = -\int _{0}^{\infty }g_{ 1}(T;t,\xi )\sigma _{1}(t,\xi )(W_{1}(\mathit{dt},d\xi ) - g_{1}(T;t,\xi )\sigma _{1}(t,\xi )\mathit{dtd}\xi ) {}\\ & & \quad +\int _{ 0}^{\infty }g_{ 2}(T;t,\xi )\sigma _{2}(t,\xi )(W_{2}(\mathit{dt},d\xi ) -\rho g_{1}(T;t,\xi )\sigma _{1}(t,\xi )\mathit{dtd}\xi ) {}\\ & & = -\int _{0}^{\infty }g_{ 1}(T;t,\xi )\sigma _{1}(t,\xi )\hat{W}_{1}(\mathit{dt},d\xi ) +\int _{ 0}^{\infty }g_{ 2}(T;t,\xi )\sigma _{2}(t,\xi )\hat{W}_{2}(\mathit{dt},d\xi ). {}\\ \end{array}$$
For the jump part, we have
$$\displaystyle\begin{array}{rcl} & & \int _{0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1\right )\tilde{J}_{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ) {}\\ & & \quad +\int _{ 0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1 + e^{z_{1}h_{1}(T;t,\xi )} - e^{z_{2}h_{2}(T;t,\xi )}\right ) {}\\ & & \quad \qquad \cdot \nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi {}\\ & & =\int _{ 0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1\right )(\hat{N}_{ 1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ) + (e^{z_{1}h_{1}(T;t,\xi )} - 1) {}\\ & & \quad \qquad \cdot \nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi ) {}\\ & & \quad +\int _{ 0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1 + e^{z_{1}h_{1}(T;t,\xi )} - e^{z_{2}h_{2}(T;t,\xi )}\right ) {}\\ & & \quad \qquad \cdot \nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi {}\\ & & =\int _{ 0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1\right )\hat{N}_{ 1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ) {}\\ & & \quad +\int _{ 0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1\right )(e^{z_{1}h_{1}(T;t,\xi )} - 1)\nu _{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi {}\\ & & \quad +\int _{ 0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1 + e^{z_{1}h_{1}(T;t,\xi )} - e^{z_{2}h_{2}(T;t,\xi )}\right ) {}\\ & & \quad \qquad \cdot \nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi {}\\ & & =\int _{ 0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1\right )\hat{N}_{ 1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ). {}\\ \end{array}$$
Note in particular, that
$$\displaystyle\begin{array}{rcl} & & \hat{N}_{1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ) {}\\ & & =\tilde{ J}_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ) + (1 - e^{z_{1}h_{1}(T;t,\xi )})\nu _{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi {}\\ & & = J_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ) -\nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi + (1 - e^{z_{1}h_{1}(T;t,\xi )}) {}\\ & & \qquad \cdot \nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi {}\\ & & = J_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ) - e^{z_{1}h_{1}(T;t,\xi )}\nu _{ (L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dtd}\xi, {}\\ \end{array}$$
i.e. \(\nu _{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\) has become \(e^{z_{1}h_{1}(T;t,\xi )}\nu _{(L_{ 1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2})\) under the new measure \(\tilde{\mathbb{P}}\). Overall, we get
$$\displaystyle{ \frac{d\left (\frac{S_{2}(t)} {S_{1}(t)}\right )} {S_{2}(t-)/S_{1}(t-)} = d\varXi (t), }$$
with
$$\displaystyle\begin{array}{rcl} & & d\varXi (t) = -\int _{0}^{\infty }g_{ 1}(T;t,\xi )\sigma _{1}(t,\xi )\hat{W}_{1}(\mathit{dt},d\xi ) +\int _{ 0}^{\infty }g_{ 2}(T;t,\xi )\sigma _{2}(t,\xi )\hat{W}_{2}(\mathit{dt},d\xi ) {}\\ & & \qquad \qquad +\int _{ 0}^{\infty }\int _{ \mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;t,\xi )-z_{1}h_{1}(T;t,\xi )} - 1\right )\hat{N}_{ 1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{dt},d\xi ). {}\\ \end{array}$$
Note that this differential notation should be understood in the sense that we need to consider stochastic integration over the ambit sets \(A_{j}^{0}(t) = [0,t] \times [0,\infty )\). This implies that Ξ(0) = 0. Also, the initial value is given by \(\frac{S_{2}(0)} {S_{1}(0)} =\exp (Y _{2}(0,T) - Y _{1}(0,T))\), which depends on the corresponding stochastic integrals when we integrate over the range (−∞, 0] × [0, ∞). We can solve this stochastic differential equation using the stochastic exponential. More precisely, we have
$$\displaystyle\begin{array}{rcl} \frac{S_{2}(t)} {S_{1}(t)}& =& \frac{S_{2}(0)} {S_{1}(0)}\exp \left (\varXi (t) -\varXi (0) -\frac{1} {2}[\varXi ](t)\right ) {}\\ & & \qquad \cdot \prod _{0\leq s\leq t}(1 +\varDelta \varXi _{s})\exp \left (-\varDelta \varXi _{s} + \frac{1} {2}(\varDelta \varXi _{s})^{2}\right ) {}\\ & = &\frac{S_{2}(0)} {S_{1}(0)}\exp \left (\varXi (t) -\varXi (0) -\frac{1} {2}\langle \varXi \rangle ^{c}(t)\right )\prod _{ 0\leq s\leq t}(1 +\varDelta \varXi _{s})\exp \left (-\varDelta \varXi _{s}\right ). {}\\ \end{array}$$
Note that
$$\displaystyle\begin{array}{rcl} & & \log \left (\prod _{0\leq s\leq t}(1 +\varDelta \varXi _{s})\exp \left (-\varDelta \varXi _{s}\right )\right ) =\sum _{0\leq s\leq t}(\log (1 +\varDelta \varXi _{s}) -\varDelta \varXi _{s}) {}\\ & & =\int _{A_{j}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi )) + 1 -\exp (z_{2}h_{2}(T;s,\xi ) {}\\ & & \quad - z_{1}h_{1}(T;s,\xi ))J_{(L_{1},L_{2})}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & =\int _{A_{j}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi )) + 1 -\exp (z_{2}h_{2}(T;s,\xi ) {}\\ & & \quad - z_{1}h_{1}(T;s,\xi ))\hat{N}_{1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & \quad +\int _{A_{j}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi )) + 1 -\exp (z_{2}h_{2}(T;s,\xi ) {}\\ & & \quad - z_{1}h_{1}(T;s,\xi ))\hat{\nu }_{(L_{1},L_{2}),\tilde{\mathbb{P}}}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi. {}\\ \end{array}$$
From the jump terms we get the following overall contribution:
$$\displaystyle\begin{array}{rcl} & & \varXi (t)^{d} +\log \left (\prod _{ 0\leq s\leq t}(1 +\varDelta \varXi _{s})\exp \left (-\varDelta \varXi _{s}\right )\right ) {}\\ & & =\int _{A_{j}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi ))\hat{N}_{1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & \quad +\int _{A_{j}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi )) + 1 -\exp (z_{2}h_{2}(T;s,\xi ) {}\\ & & \quad - z_{1}h_{1}(T;s,\xi ))\hat{\nu }_{(L_{1},L_{2}),\tilde{\mathbb{P}}}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi. {}\\ \end{array}$$
Overall we have
$$\displaystyle{ \frac{S_{2}(t)} {S_{1}(t)} = \frac{S_{2}(0)} {S_{1}(0)}\exp (Z(t)), }$$
where
$$\displaystyle\begin{array}{rcl} Z(t)& =& -\int _{A_{j}^{0}(t)}g_{1}(T;s,\xi )\sigma _{1}(s,\xi )\hat{W}_{1}(\mathit{ds},d\xi ) +\int _{A_{j}^{0}(t)}g_{2}(T;s,\xi )\sigma _{2}(s,\xi )\hat{W}_{2}(\mathit{ds},d\xi ) {}\\ & & -\frac{1} {2}\int _{A_{j}^{0}(t)}g_{1}^{2}(T;s,\xi )\sigma _{ 1}^{2}(s,\xi )\mathit{dsd}\xi -\frac{1} {2}\int _{A_{j}^{0}(t)}g_{2}^{2}(T;s,\xi )\sigma _{ 2}^{2}(s,\xi )\mathit{dsd}\xi {}\\ & & +\rho \int _{A_{j}^{0}(t)}g_{1}(T;s,\xi )g_{2}(T;s,\xi )\sigma _{1}(s,\xi )\sigma _{2}(s,\xi )\mathit{dsd}\xi {}\\ & & +\int _{A_{j}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi ))\hat{N}_{1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi ) {}\\ & & +\int _{A_{j}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi )) + 1 -\exp (z_{2}h_{2}(T;s,\xi ) {}\\ & & -z_{1}h_{1}(T;s,\xi ))\nu _{(L_{1},L_{2}),\tilde{\mathbb{P}}}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi. {}\\ \end{array}$$
1.7 Proof of Proposition 12
The first part of the proof is analogue to the proof of Proposition 8. Hence we only need to compute the extended characteristic function \(\mathbb{E}^{\tilde{\mathbb{P}}}\left [\exp (i\theta (R,u)Z(t))\right ]\), where we again set \(\theta =\theta (R,u) \in \mathbb{C}\). Then
$$\displaystyle{ \mathbb{E}_{0}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z(t))\right ] = \mathbb{E}_{ 0}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z_{ 1}(t))\right ]\mathbb{E}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z_{ 2}(t))\right ], }$$
where
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}_{0}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z_{ 1}(t))\right ] {}\\ & &:= \mathbb{E}_{0}^{\tilde{\mathbb{P}}}\left [\exp \left (-i\theta \int _{ A_{j}^{0}(t)}g_{1}(T;s,\xi )\sigma _{1}(s,\xi )\hat{W}_{1}(\mathit{ds},d\xi )\right.\right. {}\\ & & \quad + i\theta \int _{A_{j}^{0}(t)}g_{2}(T;s,\xi )\sigma _{2}(s,\xi )\hat{W}_{2}(\mathit{ds},d\xi ) {}\\ & & \quad -\frac{1} {2}i\theta \int _{A_{j}^{0}(t)}g_{1}^{2}(T;s,\xi )\sigma _{ 1}^{2}(s,\xi )\mathit{dsd}\xi -\frac{1} {2}i\theta \int _{A_{j}^{0}(t)}g_{2}^{2}(T;s,\xi )\sigma _{ 2}^{2}(s,\xi )\mathit{dsd}\xi {}\\ & & \quad \left.\left.+\rho i\theta \int _{A_{j}^{0}(t)}g_{1}(T;s,\xi )g_{2}(T;s,\xi )\sigma _{1}(s,\xi )\sigma _{2}(s,\xi )\mathit{dsd}\xi \right )\right ] {}\\ & & = \mathbb{E}_{0}^{\tilde{\mathbb{P}}}\left [\exp \left (\left (-\frac{\theta ^{2}} {2} - \frac{i\theta } {2}\right )\int _{A_{1}^{0}(t)}(g_{1}^{2}(T;s,\xi )\sigma _{ 1}^{2}(s,\xi )\right.\right. {}\\ & & \qquad \left.\left.-2\rho g_{1}(T;s,\xi )g_{2}(T;s,\xi )\sigma _{1}(s,\xi )\sigma _{2}(s,\xi ) + g_{2}^{2}(T;s,\xi )\sigma _{ 2}^{2}(s,\xi ))dsd\xi \right )\right ], {}\\ \end{array}$$
where we conditioned on \(\mathcal{F}^{\sigma _{1}} \vee \mathcal{F}^{\sigma _{2}}\). For the jump part, we have
$$\displaystyle\begin{array}{rcl} & & \mathbb{E}^{\tilde{\mathbb{P}}}\left [\exp (i\theta Z_{ 2}(t))\right ] {}\\ & & \quad:= \mathbb{E}^{\tilde{\mathbb{P}}}\left [\exp \left (i\theta \int _{ A_{1}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi ))\hat{N}_{1,2}(\mathit{dz}_{1},\mathit{dz}_{2},\mathit{ds},d\xi )\right.\right. {}\\ & & \qquad + i\theta \int _{A_{1}^{0}(t)}\int _{\mathbb{R}^{2}}(z_{2}h_{2}(T;s,\xi ) - z_{1}h_{1}(T;s,\xi )) + 1 -\exp (z_{2}h_{2}(T;s,\xi ) {}\\ & & \qquad \left.\left.-z_{1}h_{1}(T;s,\xi ))\nu _{(L_{1},L_{2}),\tilde{\mathbb{P}}}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi \right )\right ] {}\\ & & =\exp \left ((1 - i\theta )\int _{A_{1}^{0}(t)}\int _{\mathbb{R}^{2}}\left (e^{z_{2}h_{2}(T;s,\xi )-z_{1}h_{1}(T;s,\xi )} - 1\right )\nu _{ (L_{1},L_{2}),\tilde{\mathbb{P}}}(\mathit{dz}_{1},\mathit{dz}_{2})\mathit{dsd}\xi \right ). {}\\ \end{array}$$
