Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A stochastic maximum principle for partially observed general mean-field
  control problems with only weak solution   In this paper we focus on a general type of mean-field stochastic control
problem with partial observation, in which the coefficients depend in a
non-linear way not only on the state process $X_t$ and its control $u_t$ but
also on the conditional law $E[X_t|\mathcal{F}_t^Y]$ of the state process
conditioned with respect to the past of observation process $Y$. We first
deduce the well-posedness of the controlled system by showing weak existence
and uniqueness in law. Neither supposing convexity of the control state space
nor differentiability of the coefficients with respect to the control variable,
we study Peng's stochastic maximum principle for our control problem. The
novelty and the difficulty of our work stem from the fact that, given an
admissible control $u$, the solution of the associated control problem is only
a weak one. This has as consequence that also the probability measure in the
solution $P^{u}=L^{u}_TQ$ depends on $u$ and has a density $L^{u}_T$ with
respect to a reference measure $Q$. So characterizing an optimal control leads
to the differentiation of non-linear functions
$f(P^{u}\circ\{E^{P^{u}}[X_t|\mathcal{F}_t^Y]\}^{-1})$ with respect to
$(L^{u}_T,X_t)$. This has as consequence for the study of Peng's maximum
principle that we get a new type of first and second order variational
equations and adjoint backward stochastic differential equations, all with new
mean-field terms and with coefficients which are not Lipschitz. For their
estimates and for those for the Taylor expansion new techniques have had to be
introduced and rather technical results have had to be established. The
necessary optimality condition we get extends Peng's one with new, non-trivial
terms.

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The categories of this paper are: astro-ph
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The categories of this paper are: stat
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an