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## JOINT MOMENTS OF LM HESTON MODEL

For the following long-memory Heston model:
$dX_t = \sigma X_t \sqrt{Y_t} dw^1_t$

$dY_t = \kappa(\theta-Y_t)dt + \nu \sqrt{Y_t} dw^2_t$

with $dw^2_t$ a fractional Brownian motion with Hurst exponent $H$ we just assume that the following Ito formula holds (say with Wick product definition of stochastic integral:

$f(w^2_t) = f(w^2_0) + \int_0^t f'(w^2_s) dw^2_s + \frac{1}{2}\int_0^t f''(w^2_s) s^{2-2H} ds$

Then

$\frac{d}{dt} E[X^a_t Y^{b}_t] =$

$\frac{a(a-1) \sigma^2}{2} E[X^a_t Y^{b+1}_t]$

$+ \kappa \theta b E[ X^a_t Y^{b-1}_t]$

$- b\kappa E[X^a_t Y^{b-1}_t] + \nu^2 \frac{b(b-1)}{2} t^{2-2H} E[ X^a_t Y^{b-2}_t]$

The recursive structure for the short memory Heston model had been exploited in Dassios-Nagaradjasarma-SquareRoot where the authors then take a Laplace transform for recursive computation of the joint moments of $(X_t,Y_t)$ the formula above simply has an additional term from their Theorem 3.1.

This follows from using Ito formulae and using $E[dw^1_t]=E[dw^2_t]=0$.

For integers $k$, the Laplace transform $L( t^k f) = (-1)^k \frac{d^k}{dt^k} Lf$, which we consider extended to fractional powers without justification for the moment and let $G_{a,b} = LM_{a,b}$, $A = a(a-1) \sigma^2/2$, $B = \kappa \theta b$, $C= -\kappa b$, $D=\nu^2 b(b-1)/2$.  Then

$s G_{a,b} - G_{a,b}(0) = A G_{a,b+1} + B G_{a,b-1} + C G_{a,b} + D (-1)^\mu D_t^\mu G_{a,b+2}$