Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations

Michel Cristofol (1, 2, 3, 4) , Lionel Roques (5)

1 I2M - Institut de Mathématiques de Marseille
2 CNRS - Centre National de la Recherche Scientifique
3 AMU - Aix Marseille Université
4 ECM - École Centrale de Marseille
5 BioSP - Biostatistique et Processus Spatiaux

In this work, we consider a one-dimensional Itô diffusion process X t with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by an observation of the expectation of the process during a small time interval, and starting from values X 0 in a given subset of R. With the same type of observation, and given the drift coefficient, we also show that the diffusion coefficient is uniquely determined. When both coefficients are unknown, we show that they are simultaneously uniquely determined by the observation of the expectation and variance of the process, during a small time interval, and starting again from values X 0 in a given subset of R. To derive these results, we apply the Feynman-Kac theorem which leads to a linear parabolic equation with unknown coefficients in front of the first and second order terms. We then solve the corresponding inverse problem with PDE technics which are mainly based on the strong parabolic maximum principle.

Mots clés: measurements · Maximum principle; Itô diffusion process · Parabolic equation · Inverse problem · Pointwise; 1

Domaines: Equations aux dérivées partielles [math.AP]

Michel Cristofol, Lionel Roques. Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations. Inverse Problems, 2017, 33 (9), pp.095006. ⟨hal-01471568⟩