The Bismut-Elworthy-Li formula (elworthy-li-formula)
for the process satisfying can be modified for time-fractional SDE by replacing . The BEL formula with jumps added is worked out in Theorem 2 of Cass-Fritz 2007 (BELjumps-Cass-Fritz-2007; the extension to long memory can be worked out following a similar modification. The applications to finance are obvious given the empirical improvements to volatility surfaces for the ZulfIII model I have undertaken in the past few months.
We set with the Mittag-Leffler function.
The appropriate formula is:
where is the right-inverse of (see Cass-Fritz for notation). The new feature here is
. We follow the Cass-Fritz argument
Note that the Ito isometry extends to
without problems because it depends only on the identity and one can apply the standard Ito isometry.