I have a function, lets call it $f$
$$ f(a,b) = e^{-ak} \frac{\Gamma(-a) \Gamma (-b)}{\Gamma{(-a-b+2)}} \cdot \\ \mathbb{E} \left[ e^{A(\tau) -aX_1 - bX_2 + B_1(\tau)V_1 + B_2(\tau)V_2 + B_m(\tau) V_m}| (X_1(0), X_2(0), V_1(0), V_2(0), V_m(0)) = (x_1, x_2, v_1, v_2, v_m)\right] $$ where $a<0$ and $b<0$, and conditioned on start values. My main wish is to prove that $f$ is convex. Edit
$\textit{Added information about the stochastic variables}$
The $X_i, V_i$ are stochastic processes, where $X_i$ is thought to be the $\log$ of a stock, and $V_i$ are volatilites.
\begin{align} dX_1 = & \ \left(r- \frac{V_1}{2} - V_m \frac{\sigma_{1m}^2}{2}\right) dt + \left( \sqrt{V_1} dW_{11} + \sigma _{1m} \sqrt{V_m} dW_{12} \right) \nonumber \\ dX_2 = & \ \left(r- \frac{V_2}{2} - V_m \frac{\sigma_{2m}^2}{2}\right) dt + \left( \sqrt{V_2} dW_{21} + \sigma _{2m} \sqrt{V_m} dW_{22} \right) \nonumber \\ dV_1 = & \ \kappa _1 (\theta _1 - V_1) dt + \sqrt{V_1} \sigma _1 dW_{V_1} \label{eq:21} \\ dV_2 = & \ \kappa _2 (\theta _2 - V_2) dt + \sqrt{V_2} \sigma _2 dW_{V_2} \nonumber \\ dV_m = & \ \kappa _m (\theta _m - V_m) dt + \sqrt{V_m} \sigma _m dW_{V_m}. \nonumber \end{align}
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We have $$ B_1(\tau) = \left(\frac{\rho_{11} \sigma_1 a + \kappa_1}{\sigma_1^2}\right) + \\ \sqrt{\frac{a(a+1)}{\sigma_1^2} - \left(\frac{\rho_{11}\sigma_1 a + \kappa_1}{\sigma_1 ^2}\right)^2} \cdot \tan \left( \sqrt{\left(\sigma_1 ^2 a(a+1) - (\rho_{11} \sigma_1 a + \kappa_1)^2 \right)} \left(\frac{\tau + C_1}{2}\right) \right) $$ and $$ C_1 = \frac{2\arctan \left(\frac{-\rho_{11} \sigma_1 a - \kappa_1}{\sqrt{-\sigma_1^2 a(-a-1) - \left(-\rho_{11}\sigma_1 a - \kappa_1\right)^2}}\right)} { \sqrt{\left( -\sigma_1 ^2a (-a-1) - (-\rho_{11} \sigma_1 a - \kappa_1)^2 \right)}} $$ and \begin{equation*} \begin{gathered} B_2 (\tau) = - \frac{-\rho_{22} \sigma_2 b - \kappa_2}{\sigma_2^2} + \\ \sqrt{\frac{-b(-b-1)}{\sigma_2^2} - \left(\frac{-\rho_{22}\sigma_2 b - \kappa_2}{\sigma_2 ^2}\right)^2} \cdot \tan \left( \sqrt{\left(- \sigma_2 ^2 b(-b-1) - (-\rho_{22} \sigma_2 b - \kappa_2)^2 \right)} \left(\frac{\tau + C_2}{2}\right) \right) \\ \text{and}\\ C_2 = \frac{2\arctan \left(\frac{-\rho_{22} \sigma_2 b - \kappa_2}{\sqrt{-\sigma_2^2b(-b-1) - \left(-\rho_{22}\sigma_2 b - \kappa_2\right)^2}}\right)} { \sqrt{\left(- \sigma_2 ^2 b(-b-1) - (-\rho_{22} \sigma_2 b - \kappa_2)^2 \right)}} \end{gathered} \end{equation*} Followed by \begin{equation*} \begin{gathered} B_m = - \frac{-a \sigma_{1m} \sigma_{m} \rho_{1m} - b \sigma_{2m} \sigma_{m} \rho_{2m} - \kappa_m}{\sigma_m^2} + \\ \sqrt{\frac{\sigma_{1m}^2(a^2 + a) + \sigma_{2m}^2(b^2 + b) + \sigma_{1m} \sigma_{2m} \rho_{12m} ab}{\sigma_m^2} - \left( \frac{-a \sigma_{1m} \sigma_{m}\rho_{1m} - b \sigma_{2m} \sigma_{m} \rho_{2m} - \kappa_m}{\sigma_m^2} \right)^2} \\ \cdot \\ \tan \Bigg( \sqrt{\sigma_m^2( \sigma_{1m}^2a^2 + a) + \sigma_{2m}^2 (b^2 + b) + \sigma_{1m} \sigma_{2m} \rho_{12m} ab ) - (-a \sigma_{1m} \sigma_m \rho_{1m} - b \sigma_{2m} \sigma_m \rho_{2m} - \kappa_m)^2} \\ \left( \frac{\tau + C_m}{2} \right)\Bigg) \end{gathered} \end{equation*} With \begin{equation*} \begin{gathered} C_m = \frac{2 \arctan \left(\frac{-a \sigma_{1m} \sigma_{m} \rho_{1m} - b \sigma_{2m} \sigma_{m} \rho_{2m} - \kappa_m}{\sigma_m^2} \right) }{\sqrt{\sigma_m^2( \sigma_{1m}^2(a^2 + a) + \sigma_{2m}^2 (b^2 + b) + \sigma_{1m} \sigma_{2m} \rho_{12m} ab ) - (-a \sigma_{1m} \sigma_m \rho_{1m} - a \sigma_{2m} \sigma_m \rho_{2m} - \kappa_m)^2}} \end{gathered} \end{equation*} And then the most awful one \begin{equation*} \begin{gathered} A(\tau) = C_A - \tau r (a + b) + \int \kappa_1 \theta_1 B_1(\tau) + \kappa_2 \theta_2 B_2 (\tau) + \kappa _m \theta_m B_m (\tau) d\tau \end{gathered} \end{equation*} Where \begin{equation*} \begin{gathered} C_A = \\ \kappa_1 \theta_1 \frac{c_{2,1} \log \left(\cos(c_{3,1} C_1)\right)}{c_{3,1}} + \kappa_2 \theta_2 \frac{c_{2,2} \log \left(\cos(c_{3,2} C_2)\right)}{c_{3,2}} + \kappa_m \theta_m \frac{c_{2,m} \log \left(\cos(c_{3,m} C_m)\right)}{c_{3,m}} \\ \end{gathered} \end{equation*} With \begin{align*} -&c_{1,1} = - \frac{-\rho_{11} \sigma_1 a - \kappa_1}{\sigma_1^2} & -&c_{1,2} = - \frac{-\rho_{22} \sigma_2 b - \kappa_2}{\sigma_2^2}& \\ &c_{2,1} = \sqrt{\frac{-a(-a-1)}{\sigma_1^2} - \left(\frac{-\rho_{11}\sigma_1 a - \kappa_1}{\sigma_1 ^2}\right)^2}& &c_{2,2} = \sqrt{\frac{-b(-b-1)}{\sigma_2^2} - \left(\frac{-\rho_{22}\sigma_2 b - \kappa_2}{\sigma_2 ^2}\right)^2}& \\ &c_{3,1} = \sqrt{\left(- \sigma_1 ^2 a(-a-1) - (-\rho_{11} \sigma_1 a - \kappa_1)^2 \right)}& &c_{3,2} = \sqrt{\left(-b \sigma_2 ^2 b(-b-1) - (-\rho_{22} \sigma_2 b - \kappa_2)^2 \right)}& \end{align*}
and
\begin{equation*} \begin{gathered} -c_{1,m} = - \frac{-a \sigma_{1m} \sigma_{m} \rho_{1m} - b \sigma_{2m} \sigma_{m} \rho_{2m} - \kappa_m}{\sigma_m^2} \\ c_{2,m} = \sqrt{\frac{\sigma_{1m}^2(a^2 + a) + \sigma_{2m}^2(b^2 + b) + \sigma_{1m} \sigma_{2m} \rho_{12m} ab}{\sigma_m^2} - \left( \frac{-a \sigma_{1m} \sigma_{m}\rho_{1m} - b \sigma_{2m} \sigma_{m} \rho_{2m} - \kappa_m}{\sigma_m^2} \right)^2} \\ c_{3,m} = \sqrt{\sigma_m^2( \sigma_{1m}^2(a^2 + a) + \sigma_{2m}^2 (b^2 + b) + \sigma_{1m} \sigma_{2m} \rho_{12m} ab ) - (-a \sigma_{1m} \sigma_m \rho_{1m} - b \sigma_{2m} \sigma_m \rho_{2m} - \kappa_m)^2} \end{gathered} \end{equation*}
I dont even know where to start and attack this problem. Any suggestion or strategy for getting started is appreciated.
Update The $\rho$:s are correlations so they are between $-1$ and $1$. Other constants e.g. $\sigma$ $\kappa$ can be assumed to $>0$. By correspondance with my professor he is under the impression that the $B_i$s themself doesnt need to be convex, but since the $V_i$:s can be arbitrarily close to $0$ $A$ must be convex.
It is not only convex, but even logarithmically convex.
We can write, denoting $x = -a$, $y=-b$, $g(x,y) = f(-x,-y)$ that $$ g(x,y) = e^{kx} \frac{\mathrm{B}(x+1,y+1)}{xy}\mathbb{E}^* \left[e^{xX_1+yX_2}\right], $$ where $\mathrm{B}$ is the beta function, $$ \frac{d\mathbb P^*}{d\mathbb P} = e^{A(\tau) + B_1(\tau)V_1 + B_2(\tau)V_2 + B_m(\tau) V_m}. $$
Then $$ \log g(x,y) = \log \mathrm{B}(x+1,y+1) + kx -\log x - \log y + \log \mathbb{E}^* \left[e^{xX_1+yX_2}\right]. $$ The beta function is log-convex (see e.g. Theorem 6 here), $kx-\log x$ and $-\log y$ are convex, and $\log \mathbb{E}^* \left[e^{xX_1+yX_2}\right]$ is the cumulant generating function for $(X_1,X_2)$, so also convex. Hence the claim.