I derived the variance of an Ornstein-Uhlenbeck process filtered with a function A, but when I define A, I can't get an acceptable answer. I wonder whether my mistake is from the derivation before or after defining $A$. Hence, I would like to validate my derivation when A is not yet define. Here it is:
Given $\chi$ an OU process, I convolute $\chi$ with A (of integral 1) \begin{align} {\chi} &= A \star \tilde{\chi} = \int_{-\infty}^{\infty} A(p-t) \tilde{\chi}(p) \mathrm{d}p \end{align}
The variance is (with $\mu_{\tilde{\chi}}=\mu_{\chi}$) \begin{align} \sigma_{\chi}^2 &= \mathbb{E}(\chi^2) - \mu_{\tilde{\chi}}^2 \nonumber \\ \sigma_{\chi}^2 &= \mathbb{E}\left[\left( \int_{-\infty}^{\infty} A(p-t) \tilde{\chi}(p) \mathrm{d}p\right)^2\right] - \mu_{\tilde{\chi}}^2 \nonumber \\ \sigma_{\chi}^2 &= \mathbb{E}\left[ \int_{-\infty}^{\infty} A(p-t) \tilde{\chi}(p) \mathrm{d}p \int_{-\infty}^{\infty} A(s-t) \tilde{\chi}(s) \mathrm{d}s\right] - \mu_{\tilde{\chi}}^2 \nonumber \\ \sigma_{\chi}^2 &= \mathbb{E}\left[ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(p-t) \tilde{\chi}(p) A(s-t) \tilde{\chi}(s) \mathrm{d}p \mathrm{d}s\right] - \mu_{\tilde{\chi}}^2 \nonumber \\ \sigma_{\chi}^2 &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(p-t) A(s-t) \mathbb{E}\left[ \tilde{\chi}(p) \tilde{\chi}(s) \right] \mathrm{d}p \mathrm{d}s - \mu_{\tilde{\chi}}^2 \label{eqSI:var} \end{align}
An OU process can be solved analytical following (with zero initial value) \begin{align} \tilde{\chi}(t) = \mu_{\tilde{\chi}} (1-e^{-\frac{t}{T_{\tilde{\chi}}}}) + \left(\frac{2 \sigma_{\tilde{\chi}}^2}{T_{\tilde{\chi}}} \right)^{{1}/{2}} \int_0^t e^{-\frac{t-s}{T_{\tilde{\chi}}}} \mathrm{d} \mathcal{W}_s \end{align} The esperance $\mathbb{E}\left[ \tilde{\chi}(p) \tilde{\chi}(s) \right]$ is therefore \begin{align} \mathbb{E}\left[ \tilde{\chi}(p) \tilde{\chi}(s) \right] &= \mathbb{E}\left[ \left( \mu_{\tilde{\chi}} (1-e^{-\frac{p}{T_{\tilde{\chi}}}}) + \left(\frac{2 \sigma_{\tilde{\chi}}^2}{T_{\tilde{\chi}}} \right)^{{1}/{2}} \int_0^p e^{-\frac{p-u}{T_{\tilde{\chi}}}} \mathrm{d} \mathcal{W}_u \right) \right. \nonumber \\ &\quad \left. \left( \mu_{\tilde{\chi}} (1-e^{-\frac{s}{T_{\tilde{\chi}}}}) + \left(\frac{2 \sigma_{\tilde{\chi}}^2}{T_{\tilde{\chi}}} \right)^{{1}/{2}} \int_0^s e^{-\frac{s-u}{T_{\tilde{\chi}}}} \mathrm{d} \mathcal{W}_u \right) \right] \nonumber \\ &= \mu_{\tilde{\chi}}^2 (1-e^{-\frac{p}{T_{\tilde{\chi}}}}) (1-e^{-\frac{s}{T_{\tilde{\chi}}}}) + \frac{2 \sigma_{\tilde{\chi}}^2}{T_{\tilde{\chi}}} \mathbb{E}\left[\int_0^p e^{-\frac{p-u}{T_{\tilde{\chi}}}} \mathrm{d} \mathcal{W}_u \int_0^s e^{-\frac{s-u}{T_{\tilde{\chi}}}} \mathrm{d} \mathcal{W}_u \right] \nonumber \\ &= \mu_{\tilde{\chi}}^2 (1-e^{-\frac{p}{T_{\tilde{\chi}}}}) (1-e^{-\frac{s}{T_{\tilde{\chi}}}}) + \frac{2 \sigma_{\tilde{\chi}}^2}{T_{\tilde{\chi}}} e^{-\frac{p}{T_{\tilde{\chi}}}} e^{-\frac{s}{T_{\tilde{\chi}}}} \mathbb{E}\left[\int_0^{\min(p,s)} e^{\frac{u}{T_{\tilde{\chi}}}} \mathrm{d} \mathcal{W}_u \int_0^{\min(p,s)} e^{\frac{u}{T_{\tilde{\chi}}}} \mathrm{d} \mathcal{W}_u \right] \nonumber \\ &= \mu_{\tilde{\chi}}^2 (1-e^{-\frac{p}{T_{\tilde{\chi}}}}) (1-e^{-\frac{s}{T_{\tilde{\chi}}}}) + \frac{2 \sigma_{\tilde{\chi}}^2}{T_{\tilde{\chi}}} e^{-\frac{p}{T_{\tilde{\chi}}}} e^{-\frac{s}{T_{\tilde{\chi}}}} \int_0^{\min(p,s)} e^{2\frac{u}{T_{\tilde{\chi}}}} \mathrm{d}u \nonumber \\ &= \mu_{\tilde{\chi}}^2 (1-e^{-\frac{p}{T_{\tilde{\chi}}}}) (1-e^{-\frac{s}{T_{\tilde{\chi}}}}) + \sigma_{\tilde{\chi}}^2 e^{-\frac{p}{T_{\tilde{\chi}}}} e^{-\frac{s}{T_{\tilde{\chi}}}} \left( e^{2\frac{\min(p,s)}{T_{\tilde{\chi}}}} -1 \right) \end{align} Injecting this in the variance, one gets \begin{align} \sigma_{\chi}^2 &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(p-t) A(s-t) \mathbb{E}\left[ \tilde{\chi}(p) \tilde{\chi}(s) \right] \mathrm{d}p \mathrm{d}s - \mu_{\tilde{\chi}}^2 \nonumber \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(p-t) A(s-t) \left( \mu_{\tilde{\chi}}^2 (1-e^{-\frac{p}{T_{\tilde{\chi}}}}) (1-e^{-\frac{s}{T_{\tilde{\chi}}}}) + \sigma_{\tilde{\chi}}^2 e^{-\frac{p}{T_{\tilde{\chi}}}} e^{-\frac{s}{T_{\tilde{\chi}}}} \left( e^{2\frac{\min(p,s)}{T_{\tilde{\chi}}}} -1 \right) \right) \mathrm{d}p \mathrm{d}s - \mu_{\tilde{\chi}}^2 \nonumber \\ &= \mu_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} A(p-t)(1-e^{-\frac{p}{T_{\tilde{\chi}}}}) \mathrm{d}p \int_{-\infty}^{\infty} A(s-t)(1-e^{-\frac{s}{T_{\tilde{\chi}}}}) \mathrm{d}s - \mu_{\tilde{\chi}}^2 \nonumber \\ &\quad + \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(p-t) A(s-t)\sigma_{\tilde{\chi}}^2 e^{-\frac{p}{T_{\tilde{\chi}}}} e^{-\frac{s}{T_{\tilde{\chi}}}} \left( e^{2\frac{\min(p,s)}{T_{\tilde{\chi}}}} -1 \right) \mathrm{d}p \mathrm{d}s \end{align} Decomposing to remove the min function \begin{align} & \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(p-t) A(s-t)\sigma_{\tilde{\chi}}^2 e^{-\frac{p}{T_{\tilde{\chi}}}} e^{-\frac{s}{T_{\tilde{\chi}}}} \left( e^{2\frac{\min(p,s)}{T_{\tilde{\chi}}}} -1 \right) \mathrm{d}p \mathrm{d}s \nonumber \\ & \quad = \int_{-\infty}^{\infty} \left( \int_{-\infty}^{s} A(p-t) A(s-t)\sigma_{\tilde{\chi}}^2 e^{-\frac{p}{T_{\tilde{\chi}}}} e^{-\frac{s}{T_{\tilde{\chi}}}} \left( e^{2\frac{p}{T_{\tilde{\chi}}}} -1 \right) \mathrm{d}p \right. \nonumber \\ & \quad \left. + \int_{s}^{\infty} A(p-t) A(s-t)\sigma_{\tilde{\chi}}^2 e^{-\frac{p}{T_{\tilde{\chi}}}} e^{-\frac{s}{T_{\tilde{\chi}}}} \left( e^{2\frac{s}{T_{\tilde{\chi}}}} -1 \right) \mathrm{d}p \right) \mathrm{d}s \nonumber \\ & \quad = \sigma_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} A(s-t) e^{-\frac{s}{T_{\tilde{\chi}}}} \left[\int_{-\infty}^{s} A(p-t) \; 2\mathrm{sinh}\left(\frac{p}{T_{\tilde{\chi}}}\right) \mathrm{d}p + \left( e^{2\frac{s}{T_{\tilde{\chi}}}} -1 \right) \int_{s}^{\infty} A(p-t) e^{-\frac{p}{T_{\tilde{\chi}}}} \mathrm{d}p \right] \mathrm{d}s \nonumber \\ & \quad = \sigma_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} \left( A(s-t) e^{-\frac{s}{T_{\tilde{\chi}}}} \int_{-\infty}^{s} A(p-t) \; 2\mathrm{sinh}\left(\frac{p}{T_{\tilde{\chi}}}\right) \mathrm{d}p \right) \mathrm{d}s\nonumber \\ & \quad+ \sigma_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} \left( A(s-t) 2\mathrm{sinh}\left(\frac{s}{T_{\tilde{\chi}}}\right) \int_{s}^{\infty} A(p-t) e^{-\frac{p}{T_{\tilde{\chi}}}} \mathrm{d}p \right) \mathrm{d}s \end{align} Thus giving \begin{align} \sigma_{\chi}^2 &= \mu_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} A(p-t)(1-e^{-\frac{p}{T_{\tilde{\chi}}}}) \mathrm{d}p \int_{-\infty}^{\infty} A(s-t)(1-e^{-\frac{s}{T_{\tilde{\chi}}}}) \mathrm{d}s - \mu_{\tilde{\chi}}^2 \nonumber \\ &\quad + \sigma_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} A(s-t) e^{-\frac{s}{T_{\tilde{\chi}}}} \left[\int_{-\infty}^{s} A(p-t) \; 2\mathrm{sinh}\left(\frac{p}{T_{\tilde{\chi}}}\right) \mathrm{d}p + \left( e^{2\frac{s}{T_{\tilde{\chi}}}} -1 \right) \int_{s}^{\infty} A(p-t) e^{-\frac{p}{T_{\tilde{\chi}}}} \mathrm{d}p \right] \mathrm{d}s \nonumber \\ & =\mu_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} A(p-t)(1-e^{-\frac{p}{T_{\tilde{\chi}}}}) \mathrm{d}p \int_{-\infty}^{\infty} A(s-t)(1-e^{-\frac{s}{T_{\tilde{\chi}}}}) \mathrm{d}s - \mu_{\tilde{\chi}}^2 \nonumber \\ &\quad + \sigma_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} \left( A(s-t) e^{-\frac{s}{T_{\tilde{\chi}}}} \int_{-\infty}^{s} A(p-t) \; 2\mathrm{sinh}\left(\frac{p}{T_{\tilde{\chi}}}\right) \mathrm{d}p \right) \mathrm{d}s\nonumber \\ & \quad+ \sigma_{\tilde{\chi}}^2 \int_{-\infty}^{\infty} \left( A(s-t) 2\mathrm{sinh}\left(\frac{s}{T_{\tilde{\chi}}}\right) \int_{s}^{\infty} A(p-t) e^{-\frac{p}{T_{\tilde{\chi}}}} \mathrm{d}p \right) \mathrm{d}s \end{align}
Then one can continue the derivation by using the definition of $A$. Thanks for the help !