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Well-posedness of the calibrated Heston-type LSV McKean–Vlasov SDE

Establish the well-posedness (existence and uniqueness of solutions) for the calibrated Heston-type local stochastic volatility McKean–Vlasov stochastic differential equation in which the log-spot process X_t satisfies dX_t = −(1/2)·V_t·[σ_Dup(t, e^{X_t})]^2 / E[V_t | X_t]·dt + √V_t·σ_Dup(t, e^{X_t}) / √E[V_t | X_t]·dW_t^x, while the squared volatility V_t follows Cox–Ingersoll–Ross dynamics dV_t = k(θ−V_t)·dt + ξ√V_t·dW_t^v with correlation ρ between W^x and W^v, and σ_Dup is the local volatility obtained via the Dupire formula from market option prices.

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Background

The calibrated Heston-type local stochastic volatility model leads to a McKean–Vlasov SDE for the log-spot process because the diffusion coefficient depends on the conditional expectation E[V_t | X_t]. This induces a singular measure dependence that is not covered by existing well-posedness results for McKean–Vlasov equations.

The paper introduces a regularised model and proves well-posedness, propagation of chaos, and strong convergence of numerical schemes for the regularised system. However, the original (unregularised) calibrated SDE with the conditional expectation in the diffusion remains theoretically unresolved, and prior works do not address this exact financial setting.

References

As a result, the well-posedness of sde is currently an open question.

sde:

$\begin{split} &\diff X_t = -\frac{1}{2}V_t\frac{\sigma^2_{\text{Dup}}(t,e^{X_t})}{\mathbb{E}^{\mathbb{Q}}[V_t|X_{t}]}\diff t + \sqrt{V_t}\frac{\sigma_{\text{Dup}}(t,e^{X_t})}{\sqrt{\mathbb{E}^{\mathbb{Q}}[V_{t}|X_{t}]}}\diff W^x_t,\\ &\diff V_t = k(\theta-V_t)\diff t + \xi\sqrt{V_t}\diff W^v_t,\\ &\diff W^x_t \diff W^v_t = \rho \diff t, \, \rho \in (-1,1),\\ \end{split} $

— Numerical analysis of a particle system for the calibrated Heston-type local stochastic volatility model (2504.14343 - Reisinger et al., 19 Apr 2025) in Introduction; paragraph following the definition of equation (sde)