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

In this work, we consider a one-dimensional Itô diffusion process Xt with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by the observation of the expectation of the process during a small time...

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Bibliographic Details
Published in:Inverse problems 2017-09, Vol.33 (9), p.95006
Main Authors: Cristofol, Michel, Roques, Lionel
Format: Article
Language:English
Subjects:
Analysis of PDEs
inverse problem
Itô diffusion process
Mathematics
maximum principle
parabolic equation
pointwise measurements
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Summary:In this work, we consider a one-dimensional Itô diffusion process Xt with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by the observation of the expectation of the process during a small time interval, and starting from any value X0 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 any value X0 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.
ISSN:0266-5611
1361-6420
DOI:10.1088/1361-6420/aa7a1c