Existence of a calibrated Regime Switching Local Volatility model

International audienceBy Gyongy's theorem, a local and stochastic volatility model is calibrated tothe market prices of all call options with positive maturities and strikes ifits local volatility function is equal to the ratio of the Dupire localvolatility function over the root conditional mean square of the stochasticvolatility factor given the spot value. This leads to a SDE nonlinear in thesense of McKean. Particle methods based on a kernel approximation of theconditional expectation, as presented by Guyon and Henry-Labord\`ere (2011),provide an efficient calibration procedure even if some calibration errors mayappear when the range of the stochastic volatility factor is very large. But sofar, no existence result is available for the SDE nonlinear in the sense ofMcKean. In the particular case where the local volatility function is equal tothe inverse of the root conditional mean square of the stochastic volatilityfactor multiplied by the spot value given this value and the interest rate iszero, the solution to the SDE is a fake Brownian motion. When the stochasticvolatility factor is a constant (over time) random variable taking finitelymany values and the range of its square is not too large, we prove existence tothe associated Fokker-Planck equation. Thanks to Figalli (2008), we then deduceexistence of a new class of fake Brownian motions. We then extend these resultsto the special case of the LSV model called Regime Switching Local Volatility,where the stochastic volatility factor is a jump process taking finitely manyvalues and with jump intensities depending on the spot level. Under the samecondition on the range of its square, we prove existence to the associatedFokker-Planck PDE. We then deduce existence of the calibrated model byextending the results in Figalli (2008)