Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches
ESAIM. Proceedings, Tome 73 (2023), pp. 130-157
Cet article a éte moissonné depuis la source EDP Sciences
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.
Affiliations des auteurs :
Louis-Pierre Chaintron 1 ; Álvaro Mateos González 2 ; Laurent Mertz 3 ; Philippe Moireau 4
@article{EP_2023_73_a7,
author = {Louis-Pierre Chaintron and \'Alvaro Mateos Gonz\'alez and Laurent Mertz and Philippe Moireau},
title = {Mortensen observer for a class of variational inequalities {\textendash} lost equivalence with stochastic filtering approaches},
journal = {ESAIM. Proceedings},
pages = {130--157},
year = {2023},
volume = {73},
doi = {10.1051/proc/202373130},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/202373130/}
}
TY - JOUR AU - Louis-Pierre Chaintron AU - Álvaro Mateos González AU - Laurent Mertz AU - Philippe Moireau TI - Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches JO - ESAIM. Proceedings PY - 2023 SP - 130 EP - 157 VL - 73 UR - http://geodesic.mathdoc.fr/articles/10.1051/proc/202373130/ DO - 10.1051/proc/202373130 LA - en ID - EP_2023_73_a7 ER -
%0 Journal Article %A Louis-Pierre Chaintron %A Álvaro Mateos González %A Laurent Mertz %A Philippe Moireau %T Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches %J ESAIM. Proceedings %D 2023 %P 130-157 %V 73 %U http://geodesic.mathdoc.fr/articles/10.1051/proc/202373130/ %R 10.1051/proc/202373130 %G en %F EP_2023_73_a7
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
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