Appendix
Derivations of the explicit expressions of \(f(\phi _1,\phi _2)\) and \(g(\phi _3,\phi _4,\phi _5)\):
Here we first derive the explicit expression of \(f(\phi _1,\phi _2)\) defined in (10). (5) and (6) lead to
$$\begin{aligned} \ln S_i(T)= & {} \ln S_i(0)+rT+\vartheta _i\sigma _iB_i^Q(t)-\frac{1}{2}\vartheta _i^2\sigma _i^2T+\beta _i\int _{0}^{T}a(t){\mathrm {d}}W^Q(t)-\frac{1}{2}\beta _i^2\int _{0}^{T}a(t)^2 {\mathrm {d}}t,\\ \ln V(T)= & {} \ln V(0)+rT+\vartheta _0\sigma _0B_0^Q(t)-\frac{1}{2}\vartheta _0^2\sigma _0^2T+\beta _0\int _{0}^{T}a(t){\mathrm {d}}W^Q(t)-\frac{1}{2}\beta _0^2\int _{0}^{T}a(t)^2 {\mathrm {d}}t. \end{aligned}$$
Because \(GK(T)=\prod _{i=1}^{n}S_i(T)^{\omega _i}\), we have that
$$\begin{aligned} \begin{aligned} \ln GK(T)=\ln \prod _{i=1}^{n}S_i(T)^{\omega _i}=\sum _{i=1}^{n}\omega _i\ln S_i(T). \end{aligned} \end{aligned}$$
Therefore, one gets that
$$\begin{aligned} f(\phi _1,\phi _2)= & {} E^Q[GK(T)^{\phi _1}V(T)^{\phi _2}] \nonumber \\= & {} E^Q[e^{\phi _1\ln GK(T)+\phi _2\ln V(T)}] \nonumber \\= & {} e^{H_0(\phi _1,\phi _2)}E^Q[e^{H_1(\phi _1,\phi _2)}]E^Q[e^{H_2(\phi _1,\phi _2)}], \end{aligned}$$
(17)
where
$$\begin{aligned} H_0(\phi _1,\phi _2)= & {} \phi _1\ln GK(0)+\phi _2\ln V(0)+(\phi _1+\phi _2)rT-\frac{1}{2}(\phi _1(\sum _{i=1}^{n}\omega _i\vartheta _i^2\sigma _i^2)+\phi _2\vartheta _0^2\sigma _0^2)T,\nonumber \\ H_1(\phi _1,\phi _2)= & {} \phi _1\sum _{i=1}^{n}\omega _i\vartheta _i\sigma _iB_i^Q(T)+\phi _2\vartheta _0\sigma _0B_0^Q(T),\nonumber \\ H_2(\phi _1,\phi _2)= & {} (\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\int _{0}^{T}a(t){\mathrm {d}}W^Q(t)\nonumber \\&\ \ \ \ -\frac{1}{2}(\phi _1\sum _{i=1}^{n}\omega _i\beta _i^2+\phi _2\beta _0^2)\int _{0}^{T}a(t)^2{\mathrm {d}}t. \end{aligned}$$
(18)
Note that \(B^Q_i(T)\) and \(B^Q_0(T)\) are standard Brownian motions with correlation coefficient \(\rho _i\). Meanwhile, for \(i, j=1,\ldots ,n, i\ne j\), \(B^Q_i(T)\) and \(B^Q_j(T)\) are correlated to each other with correlation coefficient \(\rho _{ij}\). Then we can obtain the following result easily,
$$\begin{aligned} E^Q[e^{ H_1(\phi _1,\phi _2)}]= & {} E^Q[e^{\phi _1\sum _{i=1}^{n}\omega _i\vartheta _i\sigma _iB_i^Q(T)+\phi _2\vartheta _0\sigma _0B_0^Q(T)}]\\= & {} e^Y\nonumber . \end{aligned}$$
(19)
where
$$\begin{aligned} Y= & {} \frac{1}{2}{\phi _1^2[\sum _{i=1}^{n}\omega _i^2\vartheta _i^2\sigma _i^2T +2\sum _{i=1}^{n}\sum _{j=1}^{n}\omega _i\omega _j\vartheta _i\vartheta _j\sigma _i\sigma _j\rho _{ij}T] +\phi _1\phi _2\vartheta _0\sigma _0\sum _{i=1}^{n}\omega _i\vartheta _i\sigma _i\rho _iT+\frac{1}{2}\phi _2^2\vartheta _0^2\sigma _0^2T}. \end{aligned}$$
Next, we will deal with the term \(E^Q[e^{ H_2(\phi _1,\phi _2)}]\). Because \(W^Q(t)\) and \(M^Q(t)\) are standard Brownian motions with correlation coefficient \(\rho\), we rewrite \(W^Q(t)\) as,
$$\begin{aligned} W^Q(t)=\rho M^Q(t)+\sqrt{1-\rho ^2} \tilde{M}^Q(t), \end{aligned}$$
(20)
where \(M^Q(t)\) and \(\tilde{M}^Q(t)\) are independent Brownian motions. Substitute this into (18) and it yields that
$$\begin{aligned} H_2(\phi _1,\phi _2)= & {} (\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\int _{0}^{T}a(t){\mathrm {d}}W^Q(t)-\frac{1}{2}(\phi _1\sum _{i=1}^{n}\omega _i\beta _i^2+\phi _2\beta _0^2)\int _{0}^{T}a(t)^2{\mathrm {d}}t\nonumber \\= & {} (\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho \int _{0}^{T}a(t){\mathrm {d}}M^Q(t)+(\phi _1\sum _{i=1}^{n}\omega _i\beta _i\nonumber \\&+\phi _2\beta _0)\sqrt{1-\rho ^2}\int _{0}^{T}a(t){\mathrm {d}}{\tilde{M}}^Q(t)\nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{1}{2}(\phi _1\sum _{i=1}^{n}\omega _i\beta _i^2+\phi _2\beta _0^2)\int _{0}^{T}a(t)^2{\mathrm {d}}t. \end{aligned}$$
(21)
From the dynamics of a(t) in (5), using Itô formula, we can obtain that
$$\begin{aligned} {\mathrm {d}}a(t)^2= & {} 2 a(t){\mathrm {d}}a(t)+ \sigma _a^2{\mathrm {d}}t\\= & {} 2 \tilde{\kappa }_a a(t) (\tilde{\theta }_a-a(t)){\mathrm {d}}t+2 \sigma _a a(t){\mathrm {d}}M^Q(t)+ \sigma _a^2{\mathrm {d}}t, \end{aligned}$$
which in turn implies that
$$\begin{aligned} \int _0^T a(t){\mathrm {d}}M^Q(t)=\frac{1}{2 \sigma _a}\Big (a(T)^2-a(0)^2- 2 \tilde{\kappa }_a \int _0^T a(t) (\tilde{\theta }_a-a(t)){\mathrm {d}}t -\sigma _a^2 T\Big ). \end{aligned}$$
With the above equality, we can continue to rewrite \(H_2(\phi _1,\phi _2)\) in (21) as follows,
$$\begin{gathered} H_{2} (\phi _{1} ,\phi _{2} ) = \left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} } \right)\rho \int_{0}^{T} a (t){\text{d}}M^{Q} (t) + \left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} } \right)\sqrt {1 - \rho ^{2} } \int_{0}^{T} a (t){\text{d}}\tilde{M}^{Q} (t) - \frac{1}{2}\left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i}^{2} + \phi _{2} \beta _{0}^{2} } \right)\int_{0}^{T} a (t)^{2} {\text{d}}t \hfill \\ = \frac{{\left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} } \right)\rho }}{{2\sigma _{a} }}\left( {(a(T)^{2} - a(0)^{2} - 2\tilde{\kappa }_{a} \int_{0}^{T} a (t)\left( {\tilde{\theta }_{a} - a(t))dt - \sigma _{a}^{2} T} \right)} \right)\;\; \hfill \\ + (\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} )\sqrt {1 - \rho ^{2} } \int_{0}^{T} a (t)d\tilde{M}^{Q} (t) - \frac{1}{2}\left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i}^{2} + \phi _{2} \beta _{0}^{2} } \right)\int_{0}^{T} a (t)^{2} {\text{d}}t \hfill \\ = - \frac{{\left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} } \right)\rho }}{{2\sigma _{a} }}(a(0)^{2} + \sigma _{a}^{2} T) + \frac{{\left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} } \right)\rho }}{{2\sigma _{a} }}a(T)^{2} \;\; - \frac{{\tilde{\kappa }_{a} \tilde{\theta }_{a} (\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} )\rho }}{{\sigma _{a} }}\int_{0}^{T} a (t)dt \hfill \\ + \left( {\frac{{\tilde{\kappa }_{a} \left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} } \right)\rho }}{{\sigma _{a} }}\;\; - \frac{1}{2}\left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i}^{2} + \phi _{2} \beta _{0}^{2} } \right)} \right)\int_{0}^{T} a (t)^{2} {\text{d}}t \hfill \\ + \left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} } \right)\sqrt {1 - \rho ^{2} } \int_{0}^{T} a (t){\text{d}}\tilde{M}^{Q} (t). \hfill \\ \end{gathered}$$
Note that a(t) is driven by \(M^Q(t)\), which is independent of \(\tilde{M}^Q(t)\). Denote by \(\sigma ( a(t))_{\{0\le t\le T\}}\) the \(\sigma\)-field generated by a(t) and one can easily obtain the following result,
$$\begin{aligned} E^{Q} [e^{{H_{2} (\phi _{1} ,\phi _{2} )}} ] = E^{Q} \left[ {\exp \left\{ { - \frac{{(\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} )\rho }}{{2\sigma _{a} }}(a(0)^{2} + \sigma _{a}^{2} T) + \frac{{(\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} )\rho }}{{2\sigma _{a} }}a(T)^{2} - \frac{{\tilde{\kappa }_{a} \tilde{\theta }_{a} (\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} )\rho }}{{\sigma _{a} }}\int_{0}^{T} a (t){\rm{d}}t + \left( {\frac{{\tilde{\kappa }_{a} (\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} )\rho }}{{\sigma _{a} }} - \frac{1}{2}\left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i}^{2} + \phi _{2} \beta _{0}^{2} } \right))} \right)\int_{0}^{T} a (t)^{2} {\rm{d}}t + \left( {\phi _{1} \sum\limits_{{i = 1}}^{n} {\omega _{i} } \beta _{i} + \phi _{2} \beta _{0} } \right)\sqrt {1 - \rho ^{2} } \int_{0}^{T} a (t){\rm{d}}\tilde{M}^{Q} (t)} \right\}} \right] \nonumber \\= & {} E^Q\Big [\exp \Big \{-\frac{(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{2\sigma _a}(a(0)^2+\sigma _a^2T)\nonumber \\&+\frac{(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{2\sigma _a}a(T)^2 -\frac{\tilde{\kappa }_a\tilde{\theta }_a(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{\sigma _a}\int _{0}^{T}a(t){\mathrm {d}}t\nonumber \\&+\Big (\frac{\tilde{\kappa }_a(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{\sigma _a} -\frac{1}{2}\left(\phi _1\sum _{i=1}^{n}\omega _i\beta _i^2+\phi _2\beta _0^2\right)\Big )\int _0^T a(t)^2{\mathrm {d}}t\Big \}\nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times E^Q\Big [e^{(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\sqrt{1-\rho ^2}\int _0^T a(t){\mathrm {d}}\tilde{M}^Q(t)}\Big |\sigma (a(t))_{\{0\le t\le T\}}\Big ]\Big ]\nonumber \\= & {} E^Q\Big [\exp \Big \{-\frac{(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{2\sigma _a}\Big (a(0)^2+\sigma _a^2T\Big )\nonumber \\&+\frac{(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{2\sigma _a}a(T)^2\nonumber \\&\ \ \ -\frac{\tilde{\kappa }_a\tilde{\theta }_a\left(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0\right)\rho }{\sigma _a}\int _{0}^{T}a(t){\mathrm {d}}t\nonumber \\&+\Big (\frac{\tilde{\kappa }_a\left(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0\right)\rho }{\sigma _a}\nonumber \\&\ \ \ -\frac{1}{2}\left(\phi _1\sum _{i=1}^{n}\omega _i\beta _i^2+\phi _2\beta _0^2\right)+\frac{1}{2}\left(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0\right)^2(1-\rho ^2)\Big )\int _0^T a(t)^2{\mathrm {d}}t\Big \}\Big ]\nonumber \\= & {} e^{-\frac{\left(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0\right)\rho }{2\sigma _a}(a(0)^2+\sigma _a^2T)}E^Q\Big [\exp \Big \{\frac{(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{2\sigma _a}a(T)^2\nonumber \\&\ \ \ -\frac{\tilde{\kappa }_a\tilde{\theta }_a(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{\sigma _a}\int _0^T a(t) {\mathrm {d}}t\nonumber \\&\ \ \ +\Big (\frac{\tilde{\kappa }_a(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)\rho }{\sigma _a}-\frac{1}{2}(\phi _1\sum _{i=1}^{n}\omega _i\beta _i^2+\phi _2\beta _0^2)\\&\ \ \ +\frac{1}{2}(\phi _1\sum _{i=1}^{n}\omega _i\beta _i+\phi _2\beta _0)^2(1-\rho ^2))\int _0^T a(t)^2{\mathrm {d}}t\Big \}\Big ]\nonumber . \end{aligned}$$
(22)
The expectation in the above equality has the form of \(E^Q\Big [\exp \{\omega _1\int _0^T a(t)^2{\mathrm {d}}t + \omega _2\int _0^T a(t){\mathrm {d}}t +\omega _3a(T)^2\}]\), which is the same as (A7) and (A11) in Feng et al. (2014). Since the dynamics of a(t) in (5) are Gaussian Ornstein-Uhlenbeck processes in essence, the closed-form expressions can be derived. Here we choose to omit them and refer to Feng et al. (2014) for more details. By far, we have obtained the explicit expression of \(f(\phi _1,\phi _2)\).
Now we turn to derive the expression of \(g(\phi _3,\phi _4,\phi _5)\). Using the method we just used, we have that
$$\begin{gathered} g(\phi _{3} ,\phi _{4} ,\phi _{5} ) = E^{Q} [(\prod\limits_{{i = 1}}^{m} {S_{i} } (T)^{{q_{i} }} )^{{\phi _{3} }} (\prod\limits_{{i = m + 1}}^{n} {S_{i} } (T)^{{p_{i} }} )^{{\phi _{4} }} V(T)^{{\phi _{5} }} ] \hfill \\ \quad \quad \quad \quad \;\; = E^{Q} \left[ {e^{{\phi _{3} \ln (\prod\limits_{{i = 1}}^{m} {S_{i} } (T)^{{q_{i} }} ) + \phi _{4} \ln (\prod\limits_{{i = m + 1}}^{n} {S_{i} } (T)^{{p_{i} }} ) + \phi _{5} \ln V(T)}} } \right] \hfill \\ \quad \quad \quad \quad \;\; = e^{{H_{3} (\phi _{3} ,\phi _{4} ,\phi _{5} )}} E^{Q} [e^{{H_{4} (\phi _{3} ,\phi _{4} ,\phi _{5} )}} ]E^{Q} [e^{{H_{5} (\phi _{3} ,\phi _{4} ,\phi _{5} )}} ], \hfill \\ \end{gathered}$$
(23)
where
$$\begin{aligned} H_{3} (\phi _{3} ,\phi _{4} ,\phi _{5} ) = & \phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \ln S_{i} (0) + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \ln S_{i} (0) + \phi _{5} \ln V(0) + \left( {\phi _{3} + \phi _{4} + \phi _{5} } \right)rT \\ - & \frac{1}{2}\left( {\phi _{3} \left( {\sum\limits_{{i = 1}}^{m} {q_{i} } \vartheta _{i}^{2} \sigma _{i}^{2} } \right) + \phi _{4} \left( {\sum\limits_{{i = m + 1}}^{n} {p_{i} } \vartheta _{i}^{2} \sigma _{i}^{2} } \right) + \phi _{5} \vartheta _{0}^{2} \sigma _{0}^{2} } \right)T,H_{4} (\phi _{3} ,\phi _{4} ,\phi _{5} ) \\ = & \phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \vartheta _{i} \sigma _{i} B_{i}^{Q} (T) + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \vartheta _{i} \sigma _{i} B_{i}^{Q} (T) + \phi _{5} \vartheta _{0} \sigma _{0} B_{0}^{Q} (T),H_{5} (\phi _{3} ,\phi _{4} ,\phi _{5} ) \\ = & \left( {\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \beta _{i} + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \beta _{i} + \phi _{5} \beta _{0} } \right)\int_{0}^{T} a (t){\text{d}}W^{Q} (t) \\ - & \frac{1}{2}\left( {\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \beta _{i}^{2} + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \beta _{i}^{2} + \phi _{5} \beta _{0}^{2} } \right)\int_{0}^{T} a (t)^{2} {\text{d}}t. \\ \end{aligned}$$
(24)
Similarly, we have that
$$E^{Q} [e^{{H_{4} (\phi _{3} ,\phi _{4} ,\phi _{5} )}} ] = \;E^{Q} \left[ {e^{{\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \vartheta _{i} \sigma _{i} B_{i}^{Q} (T) + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \vartheta _{i} \sigma _{i} B_{i}^{Q} (T) + \phi _{5} \vartheta _{0} \sigma _{0} B_{0}^{Q} (T)}} } \right] = {\text{ }}e^{{Y_{1} }} ,$$
(25)
where
$$\begin{aligned} Y_1= & {} \frac{1}{2}\Big (\phi _3^2[\sum _{i=1}^{m}q_i^2\vartheta _i^2\sigma _i^2T+\sum _{i=1}^{m}\sum _{j=1}^{m}q_iq_j\vartheta _i\vartheta _j\sigma _i\sigma _j\rho _{ij}T]+2\phi _3\phi _4\sum _{i=1}^{m}\sum _{k=m+1}^{n}q_ip_j\vartheta _i\vartheta _k\sigma _i\sigma _k\rho _{ik}T\\&+2\phi _3\phi _5\vartheta _0\sigma _0\sum _{i=1}^{m}q_i\vartheta _i\sigma _i\rho _iT+\phi _4^2[\sum _{i=m+1}^{n}p_i^2\vartheta _i^2\sigma _i^2T+\sum _{i=m+1}^{n}\sum _{j=m+1}^{n}p_ip_j\vartheta _i\vartheta _j\sigma _i\sigma _j\rho _{ij}T]\\&+2\phi _4\phi _5\vartheta _0\sigma _0\sum _{i=m+1}^{n}p_i\vartheta _i\sigma _i\rho _iT+\phi _5^2\vartheta _0^2\sigma _0^2T\Big ). \end{aligned}$$
Additionally, one can get that
$$\begin{aligned} E^{Q} [e^{{H_{5} (\phi _{3} ,\phi _{4} ,\phi _{5} )}} ] = & e^{{ - \frac{{\left( {\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \beta _{i} + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \beta _{i} + \phi _{5} \beta _{0} } \right)\rho }}{{2\sigma _{a} }}\left( {a(0)^{2} + \sigma _{a}^{2} T} \right)}} E^{Q}\Big [\exp \{ \frac{{\left( {\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \beta _{i} + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \beta _{i} + \phi _{5} \beta _{0} } \right)\rho }}{{2\sigma _{a} }}a(T)^{2} \\ & - \frac{{\tilde{\kappa }_{a} \tilde{\theta }_{a} \left( {\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \beta _{i} + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \beta _{i} + \phi _{5} \beta _{0} } \right)\rho }}{{\sigma _{a} }}\int_{0}^{T} a (t){\text{d}}t \\ & + \Big (\frac{{\tilde{\kappa }_{a} \left( {\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \beta _{i} + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \beta _{i} + \phi _{5} \beta _{0} } \right)\rho }}{{\sigma _{a} }} \\ & - \frac{1}{2}\left( {\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \beta _{i}^{2} + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \beta _{i}^{2} + \phi _{5} \beta _{0}^{2} } \right) \\ & + \frac{1}{2}\left( {\phi _{3} \sum\limits_{{i = 1}}^{m} {q_{i} } \beta _{i} + \phi _{4} \sum\limits_{{i = m + 1}}^{n} {p_{i} } \beta _{i} + \phi _{5} \beta _{0} } \right)^{2} (1 - \rho ^{2} ))\int_{0}^{T} a (t)^{2} {\text{d}}t\Big )\Big \}\Big ]. \\ \end{aligned}$$
Proof of Proposition 2.1:
In order to derive the explicit expression of \(C_0\) in (7), we first rewrite it in the following form,
$$\begin{aligned} C_0= & {} e^{-rT}E^Q\Big [\max \Big \{GK(T)-K^*,0\Big \}\Big (\mathbf {1}(V(T)\ge D)+\frac{\alpha V(T)}{D}\mathbf {1}(V(T)< D)\Big )\Big ]\nonumber \\= & {} e^{-rT}\Big (A_1-K^**A_2+\frac{\alpha }{D}*(A_3 -K^**A_4)\Big ), \end{aligned}$$
(26)
where
$$\begin{aligned} A_1= & {} E^Q\Big [GK(T)I(GK(T)\ge K^*,V(T)\ge D)\Big ], \end{aligned}$$
(27)
$$\begin{aligned} A_2= & {} E^Q\Big [I(GK(T)\ge K^*,V(T)\ge D)\Big ], \end{aligned}$$
(28)
$$\begin{aligned} A_3= & {} E^Q\Big [GK(T)V(T)I(GK(T)\ge K^*,V(T)< D)\Big ], \end{aligned}$$
(29)
$$\begin{aligned} A_4= & {} E^Q\Big [V(T)I(GK(T)\ge K^*,V(T)< D)\Big ]. \end{aligned}$$
(30)
In what follows, we apply the measure change technique to get the pricing formula (see, e.g., Heston and Nandi, 2000; Wang, 2021).
Actually, \(A_2\) can be obtained directly by using standard probability theory (see, e.g., Shephard, 1991) by noting that \(f(i\phi _1,i\phi _2)\) is the joint characteristic function of \(\ln GK(T)\) and \(\ln V(T)\) under Q, i.e.,
$$\begin{aligned} A_2= & {} E^Q\Big [I(GK(T)\ge K^*,V(T)\ge D)\Big ]\nonumber \\= & {} Q\Bigg (\ln GK(T)\ge \ln K^*,\ln V(T)\ge \ln D\Big )\nonumber \\= & {} \frac{1}{4}+\frac{1}{2\pi }\int _{0}^{\infty } \text {Re}\Big [\frac{e^{-i\phi _1\ln K^*}f(i\phi _1,0)}{i\phi _1}\Big ]{\mathrm {d}}\phi _1+\frac{1}{2\pi }\int _{0}^{\infty }\text {Re}\Big [\frac{e^{-i\phi _2\ln D}f(0,i\phi _2)}{i\phi _2}\Big ]{\mathrm {d}}\phi _2\nonumber \\&- \frac{1}{{2\pi ^{2} }}\int_{0}^{\infty } {\int_{0}^{\infty } {\left( {{\text{Re}}\left[ {\frac{{e^{{ - i\phi _{1} \ln K^{*} - i\phi _{2} \ln D}} f(i\phi _{1} ,i\phi _{2} )}}{{\phi _{1} \phi _{2} }}} \right]} \right.} } \hfill \\& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - {\text{Re}}\left[ {\frac{{e^{{ - i\phi _{1} \ln K^{*} + i\phi _{2} \ln D}} f(i\phi _{1} , - i\phi _{2} )}}{{\phi _{1} \phi _{2} }}} \right]} \right){\text{d}}\phi _{1} {\text{d}}\phi _{2} . \hfill \\ \end{aligned}$$
(31)
To obtain \(A_1\), we introduce a new probability measure \(Q_1\) which is defined by
$$\begin{aligned} Q_1(\Gamma )=\frac{E^Q\Big [e^{\ln (GK(T))}I(\Gamma )\Big ]}{E^Q\Big [e^{\ln GK(T)}\Big ]} \end{aligned}$$
for any events \(\Gamma \in \mathscr {F}_{T}\). Again using standard probability theory, we have that
$$\begin{aligned} A_{1} & = E^{Q} \left[ {GK\left( T \right)I\left( {GK\left( T \right) \ge K^{*} ,V\left( T \right) \ge D} \right)} \right] \\ & = f\left( {1,0} \right)E^{{Q_{1} }} \left[ {I\left( {GK\left( T \right) \ge K^{*} ,V\left( T \right) \ge D} \right)} \right] \\ & = f(1,0)Q_{1} \left( {\ln GK\left( T \right) \ge \ln K^{*} ,\ln V\left( T \right) \ge \ln D} \right) \\ & = \frac{1}{4}f\left( {1,0} \right) + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} \left[ {\frac{{e^{{ - i\phi _{1} \ln K^{*} }} f(i\phi _{1} + 1,0)}}{{i\phi _{1} }}} \right]{\text{d}}\phi _{1} + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} \left[ {\frac{{e^{{ - i\phi _{2} \ln D}} f(1,i\phi _{2} )}}{{i\phi _{2} }}} \right]{\text{d}}\phi _{2} \\ & \quad - \frac{1}{{2\pi ^{2} }}\int_{0}^{\infty } {\int_{0}^{\infty } {\left( \begin{gathered} {\text{Re}}\left[ {\frac{{e^{{ - i\phi _{1} \ln K^{*} - i\phi _{2} \ln D}} f(i\phi _{1} + 1,i\phi _{2} )}}{{\phi _{1} \phi _{2} }}} \right] \hfill \\ - {\text{Re}}\left[ {\frac{{e^{{ - i\phi _{1} \ln K^{*} + i\phi _{2} \ln D}} f(i\phi _{1} + 1, - i\phi _{2} )}}{{\phi _{1} \phi _{2} }}} \right] \hfill \\ \end{gathered} \right)} } {\text{d}}\phi _{1} {\text{d}}\phi _{2} , \\ \end{aligned}$$
(32)
where we have used the fact that the joint characteristic function of \(\ln GK(T)\) and \(\ln V(T)\) under \(Q_1\) is given by
$$\begin{aligned} E^{Q_1}\Big [e^{i\phi _1\ln GK(T)+i\phi _2\ln V(T)}\Big ]= & {} \frac{f(0,i\phi _1+1,i\phi _2)}{f(1,0)}. \end{aligned}$$
Then we calculate \(A_3\) and \(A_4\) similarly, and the results are listed below,
$$\begin{aligned} A_3= & {} E^Q\Big [GK(T)V(T)I(GK(T)\ge K^*,V(T)< D)\Big ]\nonumber \\= & {} E^Q\Big [GK(T)V(T)I(\ln GK(T)\ge \ln K^*,-\ln V(T)>-\ln D)\Big ]\nonumber \\= & {} \frac{1}{4}f(1,1)+\frac{1}{2\pi }\int _{0}^\infty \text {Re}\Big [\frac{e^{-i \phi _1 \ln K^*}f(i\phi _1+1,1)}{ i\phi _1 }\Big ]{\mathrm {d}}\phi _1\nonumber \\&+\frac{1}{2\pi }\int _{0}^\infty \text {Re}\Big [\frac{e^{i \phi _2 \ln D}f(1,-i\phi _2+1)}{ i\phi _2 }\Big ]{\mathrm {d}}\phi _2\nonumber \\&-\frac{1}{2\pi ^2}\int _{0}^\infty \int _0^\infty \Big (\text {Re}\Big [\frac{e^{-i \phi _1 \ln K^*+i \phi _2 \ln D}f(i\phi _1+1,-i\phi _2+1)}{ \phi _1 \phi _2}\Big ]\nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\text {Re}\Big [\frac{e^{-i \phi _1 \ln K^*-i \phi _2 \ln D}f(i\phi _1+1,i\phi _2+1)}{ \phi _1 \phi _2}\Big ]\Big ){\mathrm {d}}\phi _1{\mathrm {d}}\phi _2, \end{aligned}$$
(33)
and
$$\begin{aligned} A_4= & {} E^Q\Big [V(T)I(GK(T)\ge K^*,V(T)< D)\Big ]\nonumber \\= & {} E^Q\Big [V(T)I(\ln GK(T)\ge \ln K^*,-\ln V(T)>-\ln D)\Big ]\nonumber \\= & {} \frac{1}{4}f(0,1)+\frac{1}{2\pi }\int _{0}^\infty \text {Re}\Big [\frac{e^{-i \phi _1 \ln K^*}f(i\phi _1,1)}{ i\phi _1 }\Big ]{\mathrm {d}}\phi _1\nonumber \\&+\frac{1}{2\pi }\int _{0}^\infty \text {Re}\Big [\frac{e^{i \phi _2 \ln D}f(0,-i\phi _2+1)}{ i\phi _2 }\Big ]{\mathrm {d}}\phi _2\nonumber \\&-\frac{1}{2\pi ^2}\int _{0}^\infty \int _0^\infty \Big (\text {Re}\Big [\frac{e^{-i \phi _1 \ln K^*+i \phi _2 \ln D}f(i\phi _1,-i\phi _2+1)}{ \phi _1 \phi _2}\Big ]\nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\text {Re}\Big [\frac{e^{-i \phi _1 \ln K^*-i \phi _2 \ln D}f(i\phi _1,i\phi _2+1)}{ \phi _1 \phi _2}\Big ]\Big ){\mathrm {d}}\phi _1{\mathrm {d}}\phi _2. \end{aligned}$$
(34)
This completes the proof of Proposition 2.1. \(\square\)
Proof of Proposition 2.2:
Let us rewrite the event \(\Xi\) we have defined before,
$$\begin{aligned} \Xi =\Big \{w:\frac{\prod _{i=1}^{m}S_i(T)^{q_i}}{\Big (\prod _{i=m+1}^{n}S_i(T)^{p_i}\Big )^\gamma }>\frac{e^u}{E^Q\Big [\Big (\prod _{i=m+1}^{n}S_i(T)^{p_i}\Big )^\gamma \Big ]}\Big \}. \end{aligned}$$
Then we can have that
$$\begin{aligned} \Xi= & {} \Big \{w:\frac{\prod _{i=1}^{m}S_i(T)^{q_i}}{\Big (\prod _{i=m+1}^{n}S_i(T)^{p_i}\Big )^\gamma }>\frac{e^u}{E^Q\Big [\Big (\prod _{i=m+1}^{n}S_i(T)^{p_i}\Big )^\gamma \Big ]}\Big \}\\= & {} \Big \{w:\frac{e^{q_i\sum _{i=1}^{m}\ln S_i(T)}}{\Big (e^{\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)}\Big )}>\frac{e^u}{f(0,\gamma ,0)}\Big \}\\= & {} \Big \{w:e^{q_i\sum _{i=1}^{m}\ln S_i(T)-\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)}>\frac{e^u}{f(0,\gamma ,0)}\Big \}. \end{aligned}$$
Set \(R=\frac{e^u}{f(0,\gamma p_i,0)}\), and \(\Xi\) can be reformed as
$$\begin{aligned} \Xi= & {} \Big \{w:q_i\sum _{i=1}^{m}\ln S_i(T)-\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)>\ln R\Big \}. \end{aligned}$$
Then rewrite the price of basket spread options in the following form,
$$\begin{aligned} C_0'= & {} e^{-rT}E^Q\Big [\mathop {\max }\Big \{\prod _{i=1}^{m}S_i(T)^{q_i}-\prod _{i=m+1}^{n}S_i(T)^{p_i}-K',0\Big \}\times \mathbf {1}(\Xi )\nonumber \\&\Big (\mathbf {1}(V(T)\ge D)+\frac{\alpha V(T)}{D}\mathbf {1}(V(T)<D)\Big )\Big ]. \end{aligned}$$
(35)
With
$$\begin{aligned} G_1= & {} E^Q\Big [\prod _{i=1}^{m}S_i(T)^{q_i}\times \mathbf {1}(\Xi )\Big ],\\ G_2= & {} E^Q\Big [\prod _{i=m+1}^{n}S_i(T)^{p_i}\times \mathbf {1}(\Xi )\Big ],\\ G_3= & {} E^Q\Big [\mathbf {1}(\Xi )\Big ], \end{aligned}$$
we can rewrite \(C_0'\) as follows,
$$\begin{aligned} C_0'=e^{-rT}\Big (G_1-G_2-K'G_3\Big )E^Q\Big [\Big (\mathbf {1}(V(T)\ge D)+\frac{\alpha V(T)}{D}\mathbf {1}(V(T)<D)\Big )\Big ]. \end{aligned}$$
According to the above equation, we can set
$$\begin{aligned} G_4= & {} E^Q\Big [\prod _{i=1}^{m}S_i(T)^{q_i}\mathbf {1}(e^{t_i\sum _{i=1}^{m}\ln S_i(T)-\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)}\ge R,V(T)\ge D)\Big ],\\ G_5= & {} E^Q\Big [\prod _{i=1}^{m}S_i(T)^{q_i}V(T)\mathbf {1}(e^{t_i\sum _{i=1}^{m}\ln S_i(T)-\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)}\ge R,V(T)< D)\Big ],\\ G_6= & {} E^Q\Big [\prod _{i=m+1}^{n}S_i(T)^{p_i}\mathbf {1}(e^{t_i\sum _{i=1}^{m}\ln S_i(T)-\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)}\ge R,V(T)\ge D)\Big ],\\ G_7= & {} E^Q\Big [\prod _{i=m+1}^{n}S_i(T)^{p_i}V(T)\mathbf {1}(e^{t_i\sum _{i=1}^{m}\ln S_i(T)-\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)}\ge R,V(T)< D)\Big ],\\ G_8= & {} E^Q\Big [\mathbf {1}(e^{q_i\sum _{i=1}^{m}\ln S_i(T)-\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)}\ge R,V(T)\ge D)\Big ],\\ G_9= & {} E^Q\Big [V(T)\mathbf {1}(e^{q_i\sum _{i=1}^{m}\ln S_i(T)-\gamma p_i\sum _{i=m+1}^{n}\ln S_i(T)}\ge R,V(T)< D)\Big ]. \end{aligned}$$
Then we define probability measures \(Q_2\), \(Q_3\) and \(Q_4\) as:
$$\begin{aligned} Q_2(O)= & {} \frac{E^Q[\mathbf {1}(O)e^{\ln \prod _{i=1}^{m}S_i(T)^{q_i}}]}{E^Q[e^{\ln \prod _{i=1}^{m}S_i(T)^{q_i}}]},\\ Q_3(O)= & {} \frac{E^Q[\mathbf {1}(O)e^{\ln \prod _{i=m+1}^{n}S_i(T)^{p_i}}]}{E^Q[e^{\ln \prod _{i=m+1}^{n}S_i(T)^{p_i}}]},\\ Q_4(O)= & {} \frac{E^Q[\mathbf {1}(O)e^{\ln V(T)}]}{E^Q[e^{\ln V(T)}]}. \end{aligned}$$
We start with \(G_8(T)\) and \(G_9(T)\). From the same method above and the probability measure \(Q_4\), we can have that
$$\begin{aligned} G_{8} &= E^{Q} \left[ {1(e^{{q_{i} \sum\limits_{{i = 1}}^{m} {\ln } \;S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } \;S_{i} (T)}} \ge R,V(T) \ge D)} \right] \\ &= Q\left( {q_{i} \sum\limits_{{i = 1}}^{m} {\ln } \;S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln \;} S_{i} (T) > \ln R,\ln V(T) \ge \ln D} \right) \\ &= \frac{1}{4} + \frac{1}{{2\pi }}\int_{0}^{\infty } {\text{Re} } \left[ {\frac{{e^{{ - i\phi _{3} \ln R}} g(i\phi _{3} , - \gamma i\phi _{3} ,0)}}{{i\phi _{3} }}} \right]d\phi _{3} + \frac{1}{{2\pi }}\int_{0}^{\infty } {\text{Re} } \left[ {\frac{{e^{{ - i\phi _{5} \ln D}} g(0,0,i\phi _{5} )}}{{i\phi _{5} }}} \right]d\phi _{5} \\ &\quad - \frac{1}{{2\pi ^{2} }}\int_{0}^{\infty } {\int_{0}^{\infty } {\text{Re} \left( {\left[ {\frac{{e^{{ - i\phi _{3} \ln R - i\phi _{5} \ln D}} g(i\phi _{3} , - \gamma i\phi _{3} ,i\phi _{5} )}}{{\phi _{3} \phi _{5} }}} \right]\; - \text{Re} \left[ {\frac{{e^{{ - i\phi _{3} \ln R + i\phi _{5} \ln D}} g(i\phi _{3} , - \gamma i\phi _{3} , - i\phi _{5} )}}{{\phi _{3} \phi _{5} }}} \right]} \right)} } d\phi _{3} d\phi _{5} , \\ \end{aligned}$$
(36)
and
$$\begin{aligned}G_{8} = {\text{ }}amp;E^{Q} [{\mathbf{1}}(e^{{q_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T)}} \ge R,V(T) \ge D)] = {\text{ }}amp;Q(q_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T)\,gt;\ln R,\ln V(T) \ge \ln D) = {\text{ }}amp;\frac{1}{4} + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} [\frac{{e^{{ - i\phi _{3} \ln R}} g(i\phi _{3} , - \gamma i\phi _{3} ,0)}}{{i\phi _{3} }}]{\text{d}}\phi _{3} + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} [\frac{{e^{{ - i\phi _{5} \ln D}} g(0,0,i\phi _{5} )}}{{i\phi _{5} }}]{\text{d}}\phi _{5} \; - \frac{1}{{2\pi ^{2} }}\int_{0}^{\infty } {\int_{0}^{\infty } {\left( {{\text{Re}}[\frac{{e^{{ - i\phi _{3} \ln R - i\phi _{5} \ln D}} g(i\phi _{3} , - \gamma i\phi _{3} ,i\phi _{5} )}}{{\phi _{3} \phi _{5} }}]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {\text{Re}}\Big [\frac{{e^{{ - i\phi _{3} \ln R + i\phi _{5} \ln D}} g(i\phi _{3} , - \gamma i\phi _{3} , - i\phi _{5} )}}{{\phi _{3} \phi _{5} }}\Big ]} \right)} } {\text{d}}\phi _{3} {\text{d}}\phi _{5} , \end{aligned}$$
(37)
Then using the probability measures \(Q_2\), we obtain that
$$\begin{aligned} G_{4} & = E^{Q} \left[ {\prod\limits_{{i = 1}}^{m} {S_{i} } (T)^{{q_{i} }} 1\left( {e^{{t_{i} \sum\limits_{{i = 1}}^{m} {\ln \;} S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T)}} \ge R,V(T) \ge D} \right)} \right] \\ & = E^{Q} \prod\limits_{{i = 1}}^{m} {S_{i} } (T)^{{q_{i} }} 1q_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T)\ln R,\ln V(T) \ge \ln D \\ & = \frac{1}{4}f(1,0,0) + \frac{1}{{2\pi }}\int_{0}^{\infty } {\text{Re} } \left[ {\frac{{e^{{ - i\phi _{3} \ln R}} g(i\phi _{3} + 1, - \gamma i\phi _{3} ,0)}}{{i\phi _{3} }}} \right]d\phi _{3} \\ & \quad + \frac{1}{{2\pi }}\int_{0}^{\infty } {\text{Re} } \left[ {\frac{{e^{{ - i\phi _{5} \ln D}} g(1,0,i\phi _{5} )}}{{i\phi _{5} }}} \right]d\phi _{5} \\ & \quad - \frac{1}{{2\pi ^{2} }}\int_{0}^{\infty } {\int_{0}^{\infty } {\left( \begin{gathered} \text{Re} \left[ {\frac{{e^{{ - i\phi _{3} \ln R - i\phi _{5} \ln D}} g(i\phi _{3} + 1, - \gamma i\phi _{3} ,i\phi _{5} )}}{{\phi _{3} \phi _{5} }}} \right] \hfill \\ - \text{Re} \left[ {\frac{{e^{{ - i\phi _{3} \ln R + i\phi _{5} \ln D}} g(i\phi _{3} + 1, - \gamma i\phi _{3} , - i\phi _{5} )}}{{\phi _{3} \phi _{5} }}} \right] \hfill \\ \end{gathered} \right)} } d\phi _{3} d\phi _{5} , \\ \end{aligned}$$
(38)
and
$$\begin{gathered} G_{5} = E^{Q} \left[ {\prod\limits_{{i = 1}}^{m} {S_{i} } (T)^{{q_{i} }} V(T){\mathbf{1}}\left( {e^{{t_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T)}} \ge R,V(T) < D} \right)} \right] \hfill \\ = E^{Q} \left[ {\prod\limits_{{i = 1}}^{m} {S_{i} } (T)^{{q_{i} }} V(T){\mathbf{1}}\left( {t_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T) > \ln R, - \ln V(T) > - \ln D} \right)} \right] \hfill \\ = \frac{1}{4}f(1,0,1) + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} \left[ {\frac{{e^{{ - i\phi _{3} \ln R}} g(i\phi _{3} + 1, - \gamma i\phi _{3} ,1)}}{{i\phi _{3} }}} \right]{\text{d}}\phi _{3} + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} \left[ {\frac{{e^{{i\phi _{5} \ln D}} g(1,0, - i\phi _{5} + 1)}}{{i\phi _{5} }}}
\right]{\text{d}}\phi _{5} \; \hfill \\ - \frac{1}{{2\pi ^{2} }}\int_{0}^{\infty } {\int_{0}^{\infty } {\left( {{\text{Re}}\left[ {\frac{{e^{{ - i\phi _{3} \ln R + i\phi _{5} \ln D}} g(i\phi _{3} + 1, - \gamma i\phi _{3} , - i\phi _{5} + 1)}}{{\phi _{3} \phi _{5} }}} \right] - {\text{Re}}\left[ {\frac{{e^{{ - i\phi _{3} \ln R - i\phi _{5} \ln D}} g(i\phi _{3} + 1, - \gamma i\phi _{3} ,i\phi _{5} + 1)}}{{\phi _{3} \phi _{5} }}} \right]} \right)} } {\text{d}}\phi _{3} {\text{d}}\phi _{5} . \hfill \\ \end{gathered}$$
(39)
From the
probability measure \(Q_3\), we have the following results,
$$\begin{aligned} G_{6} & = E^{Q} \left[ {\prod\limits_{{i = m + 1}}^{n} {S_{i} } (T)^{{p_{i} }} {\mathbf{1}}\left( {e^{{q_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T)}} \ge R,V(T) \ge D} \right)} \right] \\ & = E^{Q} \left[ {\prod\limits_{{i = m + 1}}^{n} {S_{i} } (T)^{{p_{i} }} {\mathbf{1}}\left( {q_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T)\ln R,\ln V(T) \ge \ln D} \right)} \right] \\ & = \frac{1}{4}f(0,1,0) + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} \left[ {\frac{{e^{{ - i\phi _{3} \ln R}} g(i\phi _{3} , - \gamma i\phi _{3} + 1,0)}}{{i\phi _{3} }}} \right]{\text{d}}\phi _{3} \\ & \quad + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} \left[ {\frac{{e^{{ - i\phi _{5} \ln D}} g(0,1,i\phi _{5} )}}{{i\phi _{5} }}} \right]{\text{d}}\phi _{5} \\ & \quad - \frac{1}{{2\pi ^{2} }}\int_{0}^{\infty } {\int_{0}^{\infty } {\left( \begin{gathered} {\text{Re}}\left[ {\frac{{e^{{ - i\phi _{3} \ln R - i\phi _{5} \ln D}} g(i\phi _{3} , - \gamma i\phi _{3} + 1,i\phi _{5} )}}{{\phi _{3} \phi _{5} }}} \right] \hfill \\ - {\text{Re}}\left[ {\frac{{e^{{ - i\phi _{3} \ln R + i\phi _{5} \ln D}} g(i\phi _{3} , - \gamma i\phi _{3} + 1, - i\phi _{5} )}}{{\phi _{3} \phi _{5} }}} \right] \hfill \\ \end{gathered} \right)} } {\text{d}}\phi _{3} {\text{d}}\phi _{5} , \\ \end{aligned}$$
(40)
and
$$\begin{aligned} G_{7} = & E^{Q} \left[ {\prod\limits_{{i = m + 1}}^{n} {S_{i} } (T)^{{p_{i} }} V(T){\mathbf{1}}\left( {e^{{q_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T)}} \ge R,V(T);D} \right)} \right] \\ = & E^{Q} \left[ {\prod\limits_{{i = m + 1}}^{n} {S_{i} } (T)^{{p_{i} }} V(T){\mathbf{1}}\left( {q_{i} \sum\limits_{{i = 1}}^{m} {\ln } S_{i} (T) - \gamma p_{i} \sum\limits_{{i = m + 1}}^{n} {\ln } S_{i} (T) > \ln R - \ln V(T) - \ln D} \right)} \right] \\ = & \frac{1}{4}f(0,1,1) + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} \left[ {\frac{{e^{{ - i\phi _{3} \ln K^{*} }} g(i\phi _{3} , - \gamma i\phi _{3} + 1,1)}}{{i\phi _{3} }}} \right]{\text{d}}\phi _{3} + \frac{1}{{2\pi }}\int_{0}^{\infty } {{\text{Re}}} \left[ {\frac{{e^{{i\phi _{5} \ln D}} g(0,1, - i\phi _{5} + 1)}}{{i\phi _{5} }}} \right]{\text{d}}\phi _{5} \; \\ - & \frac{1}{{2\pi ^{2} }}\int_{0}^{\infty } {\int_{0}^{\infty } {\left( {{\text{Re}}\left[ {\frac{{e^{{ - i\phi _{3} \ln R + i\phi _{5} \ln D}} g(i\phi _{3} , - \gamma i\phi _{3} + 1, - i\phi _{5} + 1)}}{{\phi _{3} \phi _{5} }}} \right] - {\text{Re}}\left[ {\frac{{e^{{ - i\phi _{3} \ln R - i\phi _{5} \ln D}} g(i\phi _{3} , - \gamma i\phi _{3} + 1,i\phi _{5} + 1)}}{{\phi _{3} \phi _{5} }}} \right]} \right)} } {\text{d}}\phi _{3} {\text{d}}\phi _{5} . \\ \end{aligned}$$
(41)
This completes the proof of Proposition 2.2. \(\square\)
