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Under Heston LSV (HLSV) dynamics, Gatheral's equality is: $$ \sigma_{LV}^{HLSV}(S_t,t) = \sqrt{E^{HSLV}\left[V_tL(S_t,t)^2 | S_t \right]} = L(S_t,t)\sqrt{E^{HSLV}\left[V_t | S_t \right]}, $$ as $L(S_t,t)$ is $\sigma(S_t)$-measurable, where superscript $HSLV$ is meant to remind us what is our dynamics we started with (in particular the joint probability ...


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Yes it should preserve positivity. However due to numerical noise you may observe very small negative values on the edges of the lattice, that you can truncate to zero. If you solve using Fokker-Planck you may want to start from $t=\delta t$ using a gaussian approximation for the density on the first step, so as to start from a smooth density. An alternative ...


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