Analytical Derivatives for Haar Wavelet Variance

White Noise: \[\begin{aligned} {\nu ^2}\left( \tau \right) &= \frac{{\sigma _0^2}}{\tau } \\ \frac{\partial }{{\partial \sigma _0^2}}{\nu ^2}\left( \tau \right) &= \frac{1}{\tau } \\ \frac{\partial ^2 }{{\partial \sigma _0^4}}{\nu ^2}\left( \tau \right) &= 0 \\ \end{aligned}\]

Random Walk: \[\begin{aligned} {\nu ^2}\left( \tau \right) &= \frac{{\left( {2{\tau ^2} + 1} \right)\gamma _0^2}}{{24\tau }} \\ \frac{\partial }{{\partial \gamma _0^2}}{\nu ^2}\left( \tau \right) &= \frac{{\left( {2{\tau ^2} + 1} \right)}}{{24\tau }} \\ \frac{\partial ^2 }{{\partial \gamma _0^4}}{\nu ^2}\left( \tau \right) &= 0 \\ \end{aligned}\]

Drift Process: \[\begin{aligned} {\nu ^2}\left( \tau \right) &= \frac{{{\tau ^2}\omega _0^2}}{2} \\ \frac{\partial }{{\partial {\omega _0}}}{\nu ^2}\left( \tau \right) &= {\tau ^2}{\omega _0} \\ \frac{{{\partial ^2}}}{{\partial \omega _0^2}}{\nu ^2}\left( \tau \right) &= {\tau ^2} \\ \end{aligned}\]

Quantisation Noise (QN): \[\begin{aligned} {\nu ^2}\left( \tau \right) &= \frac{{3Q_0^2}}{{2{\tau ^2}}} \\ \frac{\partial }{{\partial Q_0^2}}{\nu ^2}\left( \tau \right) &= \frac{3}{{2{\tau ^2}}} \\ \frac{{{\partial ^2}}}{{\partial Q_0^4}}{\nu ^2}\left( \tau \right) &= 0 \\ \end{aligned}\]

AR 1 Process: \[\begin{aligned} {\nu ^2}\left( \tau \right) &= \frac{{\left( {\frac{\tau }{2} - 3{\rho _0} - \frac{{\tau \rho _0^2}}{2} + 4\rho _0^{\frac{\tau }{2} + 1} - \rho _0^{\tau + 1}} \right)\nu _0^2}}{{\frac{{{\tau ^2}}}{8}{{\left( {1 - {\rho _0}} \right)}^2}\left( {1 - \rho _0^2} \right)}} \\ \\ &{\text{Derivatives w}}{\text{.r}}{\text{.t to }}\nu _0^2 \\ \frac{\partial }{{\partial \nu _0^2}}{\nu ^2}\left( \tau \right) &= \frac{{\left( {\frac{\tau }{2} - 3{\rho _0} - \frac{{\tau \rho _0^2}}{2} + 4\rho _0^{\frac{\tau }{2} + 1} - \rho _0^{\tau + 1}} \right)}}{{\frac{{{\tau ^2}}}{8}{{\left( {1 - {\rho _0}} \right)}^2}\left( {1 - \rho _0^2} \right)}} \\ &= \frac{{4\left( {{\rho _0}\left( { - 8\rho _0^{\tau /2} + 2\rho _0^\tau + {\rho _0}\tau + 6} \right) - \tau } \right)}}{{{{\left( {{\rho _0} - 1} \right)}^3}\left( {{\rho _0} + 1} \right){\tau ^2}}} \\ \frac{{{\partial ^2}}}{{\partial \nu _0^4}}{\nu ^2}\left( \tau \right) &= 0 \\ \\ &{\text{Derivatives w}}{\text{.r}}{\text{.t to }}{\rho _0} \\ \frac{\partial }{{\partial {\rho _0}}}{\nu ^2}\left( \tau \right) &= \frac{{8\nu _0^2\left( {4\left( {\frac{\tau }{2} + 1} \right)\rho _0^{\tau /2} - (\tau + 1)\rho _0^\tau - {\rho _0}\tau - 3} \right)}}{{{{\left( {1 - {\rho _0}} \right)}^2}\left( {1 - \rho _0^2} \right){\tau ^2}}} \\ &+ \frac{{16\nu _0^2\left( {4\rho _0^{\frac{\tau }{2} + 1} - \rho _0^{\tau + 1} - \frac{{\rho _0^2\tau }}{2} - 3{\rho _0} + \frac{\tau }{2}} \right)}}{{{{\left( {1 - {\rho _0}} \right)}^3}\left( {1 - \rho _0^2} \right){\tau ^2}}} \\ &+ \frac{{16\nu _0^2{\rho _0}\left( {4\rho _0^{\frac{\tau }{2} + 1} - \rho _0^{\tau + 1} - \frac{{\rho _0^2\tau }}{2} - 3{\rho _0} + \frac{\tau }{2}} \right)}}{{{{\left( {1 - {\rho _0}} \right)}^2}{{\left( {1 - \rho _0^2} \right)}^2}{\tau ^2}}} \\ &= \frac{{8\nu _0^2}}{{{{\left( {{\rho _0} - 1} \right)}^4}{{\left( {{\rho _0} + 1} \right)}^2}{\tau ^2}}}\left( \begin{gathered} - 2\left( {{\rho _0}\left( {{\rho _0}(\tau - 6) - 4} \right) - \tau - 2} \right)\rho _0^{\tau /2} \\ + \left( {{\rho _0}\left( {{\rho _0}(\tau - 3) - 2} \right) - \tau - 1} \right)\rho _0^\tau \\ - {\rho _0}\left( {{\rho _0}\left( {{\rho _0}\tau + \tau + 9} \right) - \tau + 6} \right) \\ + \tau - 3 \\ \end{gathered} \right) \\ \frac{{{\partial ^2}}}{{\partial \rho _0^2}}{\nu ^2}\left( \tau \right) &= \frac{{8\nu _0^2}}{{{\tau ^2}}}\left( \begin{gathered} \frac{{2\left( {\frac{\tau }{2} + 1} \right)\tau \rho _0^{\frac{\tau }{2} - 1} - \tau (\tau + 1)\rho _0^{\tau - 1} - \tau }}{{{{\left( {1 - {\rho _0}} \right)}^2}\left( {1 - \rho _0^2} \right)}} \\ + \left( {\frac{{8{\rho _0}}}{{{{\left( {1 - {\rho _0}} \right)}^3}{{\left( {1 - \rho _0^2} \right)}^2}}} + \frac{{\frac{{8\rho _0^2}}{{{{\left( {1 - \rho _0^2} \right)}^3}}} + \frac{2}{{{{\left( {1 - \rho _0^2} \right)}^2}}}}}{{{{\left( {1 - {\rho _0}} \right)}^2}}} + \frac{6}{{{{\left( {1 - {\rho _0}} \right)}^4}\left( {1 - \rho _0^2} \right)}}} \right)\left( {4\rho _0^{\frac{\tau }{2} + 1} - \rho _0^{\tau + 1} - \frac{{\rho _0^2\tau }}{2} - 3{\rho _0} + \frac{\tau }{2}} \right) \\ + 2\left( {\frac{{2{\rho _0}}}{{{{\left( {1 - {\rho _0}} \right)}^2}{{\left( {1 - \rho _0^2} \right)}^2}}} + \frac{2}{{{{\left( {1 - {\rho _0}} \right)}^3}\left( {1 - \rho _0^2} \right)}}} \right)\left( {4\left( {\frac{\tau }{2} + 1} \right)\rho _0^{\tau /2} - (\tau + 1)\rho _0^\tau - {\rho _0}\tau - 3} \right) \\ \end{gathered} \right) \\ &= \frac{{8\nu _0^2}}{{{{\left( {{\rho _0} - 1} \right)}^5}{\rho _0}{{\left( {{\rho _0} + 1} \right)}^3}{\tau ^2}}}\left( \begin{gathered} - \left( {{\rho _0}\left( {{\rho _0}\left( {{\rho _0}(\tau - 8)\left( {{\rho _0}(\tau - 6) - 8} \right) - 2(\tau - 6)\tau + 64} \right) + 8(\tau + 2)} \right) + \tau (\tau + 2)} \right)\rho _0^{\tau /2} \\ + \left( {{\rho _0}\left( {{\rho _0}\left( {{\rho _0}(\tau - 4)\left( {{\rho _0}(\tau - 3) - 4} \right) - 2(\tau - 3)\tau + 16} \right) + 4(\tau + 1)} \right) + \tau (\tau + 1)} \right)\rho _0^\tau \\ + 3{\rho _0}\left( {{\rho _0}\left( {{\rho _0}\left( {\rho _0^2\tau + 2{\rho _0}(\tau + 6) + 16} \right) - 2(\tau - 8)} \right) - \tau + 4} \right) \\ \end{gathered} \right) \end{aligned}\]