Title: Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations,
Abstract: In this work, we consider a one-dimensional It{ô} 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.