By Andrey Itkin
We study a local stochastic volatility (LSV) model in which the variance factor follows a generalized Langevin equation (GLE) with colored noise, and past returns act on variance through a memory kernel with state-dependent amplitudes. For kernels given by finite sums of exponentials, we prove existence, uniqueness, and non-explosion under a growth condition linking the volatility map to the potential. In a one-mode family with a polynomial potential, feedback through log returns can cause explosion exactly when the squared volatility factor grows faster than the restoring force. Under structural and nondegeneracy conditions, an exact reduced Markovian state has a smooth density solving a deterministic forward equation. The Dupire conditional expectation therefore requires no stochastic partial differential equation. Under a bounded Sharpe ratio, exponential utility selects the minimal entropy martingale measure, whose premium solves a quadratic backward stochastic differential equation. Under a regularity hypothesis, which we verify in special cases with and without leverage, the premium depends only on the current state, and the squared VIX equals the expected average future local stochastic variance. Finally, a regularized calibration problem with bid–ask constraints admits a minimizer whenever a feasible point has finite objective value.
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