The Malliavin gradient method for the calibration of stochastic dynamical models

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 L(u,v)≔∑i=1k(E(hi(Xti))-qi)2 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. hi(x)≔1{x⩽pi} and the problem is to choose the parameters u, v in a way that the stochastic model fits the quantiles best.