Weak existence and exact calibration of local stochastic volatility models

Working manuscript — not peer reviewed. We construct weak solutions of local stochastic volatility models that retain the exact conditional-expectation quotient and have the marginals of a prescribed local diffusion. For a real-valued factor, the variance may depend on time, price and factor, be unbounded, and approach zero; the factor diffusion need not have a uniform positive lower bound. A second result covers positive square-root factors, including Heston–LSV when the Feller ratio exceeds one. Point initial conditions and joint initial laws satisfying explicit moment bounds are allowed. The price is a true martingale and the occupation density has positive-time Sobolev regularity. The proof combines scalar score estimates with a centred conditional entropy inequality, including the additional covariance caused by price-dependent variance. Finite Fourier systems preserve the prescribed marginal, and strong convergence identifies their calibrated fluxes. We also obtain flux stability and call and put price estimates from marginal defects. For denominator regularisation, a weighted entropy yields existence with a real-valued factor and price-independent variance under an explicit temporal integrability condition; the resulting vanilla price bias is linear in the regularisation parameter, uniformly in strike.