Volume 73, 2023CEMRACS 2021 - Data Assimilation and Reduced Modeling for High Dimensional Problems
|Page(s)||130 - 157|
|Published online||30 August 2023|
Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches
DMA, École normale supérieure, Université PSL, CNRS, Paris, France
2 ECNU-NYU Institute of Mathematical Sciences, NYU Shanghai, Shanghai, China
3 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China
4 Inria – LMS, Ecole Polytechnique, CNRS – Institut Polytechnique de Paris, Palaiseau, France
We address the problem of deterministic sequential estimation for a nonsmooth dynamics governed by a variational inequality. An example of such dynamics is the Skorokhod problem with a reflective boundary condition. For smooth dynamics, Mortensen introduced in 1968 a nonlinear estimator based on likelihood maximisation. Then, starting with Hijab in 1980, several authors established a connection between Mortensen’s approach and the vanishing noise limit of the robust form of the so-called Zakai equation. In this paper, we investigate to what extent these methods can be developed for dynamics governed by a variational inequality. On the one hand, we address this problem by relaxing the inequality constraint by penalization: this yields an approximate Mortensen estimator relying on an approximating smooth dynamics. We verify that the equivalence between the deterministic and stochastic approaches holds through a vanishing noise limit. On the other hand, inspired by the smooth dynamics approach, we study the vanishing viscosity limit of the Hamilton-Jacobi equation satisfied by the Hopf-Cole transform of the solution of the robust Zakai equation. In contrast to the case of smooth dynamics, the zero-noise limit of the robust form of the Zakai equation cannot be understood in our case from the Bellman equation on the value function arising in Mortensen’s procedure. This unveils a violation of equivalence for dynamics governed by a variational inequality between the Mortensen approach and the low noise stochastic approach for nonsmooth dynamics.
© EDP Sciences, SMAI 2023
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