The Malliavin gradient method for the calibration of stochastic dynamical models

摘要

We consider a diffusion (Xt) satisfying the stochastic differential equation dXt = β(Xt, u)dt + σ(Xt, v)dWt where u and v are parameters and consider the problem of minimizing certain functionals of the form ≔ in u and v where ti ∈ [0, T] are not necessarily distinct time points. For this we combine classical gradient methods with techniques from Malliavin calculus. The proposed technique has a particular advantage to classical techniques in the case when the functions hi are not continuous or have singularities. This is the case when the functions hi represent certain quantiles, i.e. ≔ and the problem is to choose the parameters u, v in a way that the stochastic model fits the quantiles best.