Geodesic
Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches
Louis-Pierre Chaintron1 ; Álvaro Mateos González2 ; Laurent Mertz3 ; Philippe Moireau4
1 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
ESAIM. Proceedings, Tome 73 (2023), pp. 130-157.

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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.
DOI : 10.1051/proc/202373130

Louis-Pierre Chaintron 1 ; Álvaro Mateos González 2 ; Laurent Mertz 3 ; Philippe Moireau 4

1 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 
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     title = {Mortensen observer for a class of variational inequalities {\textendash} lost equivalence with stochastic filtering approaches},
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Louis-Pierre Chaintron; Álvaro Mateos González; Laurent Mertz; Philippe Moireau. Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches. ESAIM. Proceedings, Tome 73 (2023), pp. 130-157. doi : 10.1051/proc/202373130. http://geodesic.mathdoc.fr/articles/10.1051/proc/202373130/

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