Aug 28 – 31, 2018
Angers - France
Europe/Paris timezone

Controlled mean-field dynamics in stochastic control problems depending on the law of state process

Aug 31, 2018, 4:00 PM
Angers - France

Angers - France


Rainer Buckdahn (Université de Bretagne Occidentale)


The starting point of the talk is a recent work with Juan Li (Shandong University,
Weihai, P.R.China) and Jin Ma (University of South California, Los Angeles, U.S.A.), “A mean-
field stochastic control problem with partial observations” (Annals of Appl.Probability, 2017: [1]) in
which we studied Pontryagin’s optimality principle for a stochastic control problem whose dynamics
are given by the stochastic differential equation (SDE)
Z t
Z t
X t = x +
b(s, X .∧s , μ s , u s (Y ))ds +
σ(s, X .∧s , μ s X|Y , u s (Y ))dB s 1 ,
R t
Y t = 0 h(s, X s )ds + B t 2 , t ∈ [0, T ], P -a.s.,
where (B 1 , B 2 ) is a P -Brownian motion, the controlled state process X is only observable through
the observation process Y and so the control process u = u(Y ) is a non anticipating functional of
the observation process Y . Moreover, unlike classical controlled dynamics, the coefficients σ and b
do not only depend on the paths of the controlled state process X and the control u(Y ) but also
on the law μ s = P ◦ [E[X s |Y r , r ≤ s]] −1 , s ∈ [0, T ]. Motivations for such a type of dynamics are
given in [1]. However, in [1] the dependence of the law is linear; the talk will study the case where
the coefficients are non linear functions of the law. Moreover, unlike in [1] the coefficients b and σ
are only supposed to be continuous in the law w.r.t. the 1-Wasserstein metric and on h we only
impose boundedness and Lipschitz continuity in the state variable. The main objective of the talk
is to prove the weak existence and the uniqueness in law for the above dynamics.

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