Abstract
We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian motion that drives the system. This kind of problems is motivated by limiting behavior of interacting particles systems with applications in, for example, agricultural crop management. We establish a version of the geometric dynamic programming principle for the associated reachability sets and prove that the corresponding value function is a viscosity solution of a geometric partial differential equation. This provides a characterization of the initial masses that can be almost surely transported toward a given target, along the paths of a stochastic differential equation. Our results extend those of Soner and Touzi, Journal of the European Mathematical Society (2002) to our setting.
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Notes
This means that \(({{\tilde{X}}},{{\tilde{a}}})(\omega ^{\circ },\cdot )\), defined on \({{\tilde{\varOmega }}}^{\i }\), has the same law as \((X,a)(\omega ^{\circ },\cdot )\), defined on \(\varOmega ^{\i }\), for a.e. \(\omega ^{\circ }\in \varOmega ^{\circ }\).
Being \({{{\mathcal {C}}}}^{1,2}_{b}\) for the function w is not a sufficient condition for the lift W to be twice Fréchet differentiable as shown in [23, Example 2.3].
We leave the study of more precise examples to future research.
Note that, even for general stochastic target problems set on \({\mathbb {R}}^{d}\), no general comparison theorem has been established so far. This is done on a case-by-case basis, and we therefore do not enter into this issue in the abstract setting of this paper, but rather leave this to the future study of particular situations.
One could relax the constraint by just asking for \({\mathbb {P}}[{\mathbb {E}}_{B}[Y^{t,y,\nu }_{T}]\ge 0]\ge m\) for some \(m\in ]0,1[\); see [5].
The state space being increased to \({\mathbb {R}}^{d+1}\).
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Acknowledgements
The research of the first author was partially supported by the ANR project CAESARS (ANR-15-CE05-0024). Financial support of the second author from the Swedish Research Council (VR) Grant no. 2016-04086 is also gratefully acknowledged.
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Communicated by Giuseppe Buttazzo.
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Bouchard, B., Djehiche, B. & Kharroubi, I. Quenched Mass Transport of Particles Toward a Target. J Optim Theory Appl 186, 345–374 (2020). https://doi.org/10.1007/s10957-020-01704-y
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DOI: https://doi.org/10.1007/s10957-020-01704-y