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BISMUT-ELWORTHY-LI FORMULA FOR TIME-FRACTIONAL DIFFUSIONS

The Bismut-Elworthy-Li formula (elworthy-li-formula)

$d(P_t f)(x_0)(v_0) = \frac{1}{t} Ef(x_t) \int_0^t \langle Y(x_s), dB_s \rangle$

for the process satisfying $dx_t = X(x_t) dB_s + Z(x_t)dt$ can be modified for time-fractional SDE by replacing $dB_s with S_{\alpha}(s) dB_s$.  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 $S_{\alpha}(t) = kappa t^{\alpha-1} E_{\alpha,\alpha}(-kappa t^\alpha)$ with $E_{\alpha,\beta}$ the Mittag-Leffler function.

The appropriate formula is:

$\frac{\partial}{\partial z_k} E[f(x_T)] = \frac{1}{G(T)} E[ f(x_T) \int_0^T S_{\alpha}(s) R(s,x_s) \frac{\partial x_s}{\partial z_k} ds]$

where $R(s,x_s)$ is the right-inverse of $X(s,x_s)$ (see Cass-Fritz for notation).  The new feature here is
$G(T) = \int_0^T S_{\alpha}(t)^2 a(t) dt$.  We follow the Cass-Fritz argument

Note that the Ito isometry extends to

$E[ \int_0^t X_s S_\alpha(s) dw(s) \int_0^t Y_s S_{\alpha}(s) dw(s) ] = E[ \int_0^t X_s Y_s S_{\alpha}(s)^2 ds]$

without problems because it depends only on the identity $ab = \frac{1}{4}((a+b)^2 - (a-b)^2)$ and one can apply the standard Ito isometry.