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## LARGE DEVIATION RATE AND MOMENT EXPLOSION CONJECTURES INSPIRED BY HESTON MODEL FOR FRACTIONAL SQUARE ROOT MODEL

When solving the Heston model using affine reduction to a system of ODE, which is established practice (AndersonPiterbarg-MomentExplosions2007, SmallTimeSmileTermStructureImpliedVolHeston-2012) with notation

$dS_t = S_t\sqrt{V_t} dW^1_t$

$dV_t = -\lambda(V_t - \theta) dt + \eta \sqrt{V_t} dW^2_t$ (*)

$\langle dW^1_t,dW^2_t \rangle= \rho dt$

One notes that the the parabolic equation determined by the model can be solved using the ansatz $f(t,v,u) = \exp(\phi(t,u) + v \psi(t,u))$ by a system of ordinary differential equations

$d/dt \phi(t,u) = F(u,\psi(t,u))$

$d/dt \psi(t,u) = R(u,\psi(t,u))$

with $F(u,w) = \lambda \theta w$ and $R(u,w) = w^2\eta^2/2 + (\rho \eta u - \lambda)w + (1/2)(u^2-u)$

and the latter Riccati equation can have explosion in finite time.  So what happens when we replace $dW^2_t$ in (*) with a fractional Brownian motion?  Essentially  and not at all rigorously for the moment, $\eta \rightarrow \eta g(t)$ and $\rho \rightarrow \rho g(t)$ where $g(t) = t^{H-1/2}$.

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