ABSTRACT
As will be explained in Chapters 11 and 12, the calibration of local stochastic volatility models and local correlation models to market smiles leads to the so-called McKean nonlinear stochastic differential equations (SDEs). In such nonlinear SDEs, in contrast with classical Itoˆ SDEs, the drift and volatility coefficients depend on the (unknown) marginal law of the process. The Fokker-Planck equation associated to this SDE is therefore nonlinear, hence the denomination. In this chapter, we review some basic properties of McKean SDEs and introduce the particle method, an elegant stochastic simulation of such processes, which will prove to be extremely efficient for calibration purposes (see Chapters 11 and 12).
