Viscosity Solutions to Parabolic Master Equations and McKean-Vlasov SDEs with Closed-loop Controls

Wu, Cong; Zhang, Jianfeng

doi:10.48550/arxiv.1805.02639

Cited by 9 publications

(9 citation statements)

References 36 publications

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“…Since ∇ 2 h is bounded, b and σ are Lipschitz, we deduce, taking into account (30) and assumption (Hc)-(ii) that, for some constant C ≥ 0 independent of k and r,…”

Section: Warm-upmentioning

confidence: 88%

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Forward and Backward Stochastic Differential Equations with normal constraint in law

Briand,

Cardaliaguet,

de Raynal

et al. 2019

Preprint

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In this paper we investigate the well-posedness of backward or forward stochastic differential equations whose law is constrained to live in an a priori given (smooth enough) set and which is reflected along the corresponding "normal" vector. We also study the associated interacting particle system reflected in mean field and asymptotically described by such equations. The case of particles submitted to a common noise as well as the asymptotic system is studied in the forward case. Eventually, we connect the forward and backward stochastic differential equations with normal constraints in law with partial differential equations stated on the Wasserstein space and involving a Neumann condition in the forward case and an obstacle in the backward one.

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“…Since ∇ 2 h is bounded, b and σ are Lipschitz, we deduce, taking into account (30) and assumption (Hc)-(ii) that, for some constant C ≥ 0 independent of k and r,…”

Section: Warm-upmentioning

confidence: 88%

“…Coming back to (33), since sup k≥1 |ψ k | ∞ ≤ r and ∇h has at most a linear growth, taking into account (30), we derive, from BDG and Cauchy-Schwarz inequalities, the estimate…”

Section: Warm-upmentioning

confidence: 99%

Forward and Backward Stochastic Differential Equations with normal constraint in law

Briand,

Cardaliaguet,

de Raynal

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…As explained above, the difference between the HJB equations for V B,• S and V • S comes mainly from the fact that the control process α γ , for γ ∈ Γ • S (t, ν), depends on the initial random variable condition, which in turn modifies the Hamiltonian function which appears in the PDE. Finally, we also refer to Wu and Zhang [79] for a discussion of the McKean-Vlasov control problem in a non-Markovian framework without common noise.…”

Section: Hjb Equation For the General Strong Formulationmentioning

confidence: 99%

McKean-Vlasov optimal control: the dynamic programming principle

Djete¹,

Possamaï²,

Tan³

2019

Preprint

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We study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions. * Mao Fabrice Djete gratefully acknowledges support from the région Île-de-France. This work also benefited from support of the ANR project PACMAN ANR-16-CE05-0027.

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“…the book [12] for an account of the theory, and the papers [11,13,40] for some recent results. Concerning instead the finite-dimensional path-dependent case one can see the paper [45] for some results on viscosity solutions of the HJB equations. Finally, up to now, concerning mean-field games in infinite dimension, we only know the linear quadratic model of [21].…”

Section: Introductionmentioning

confidence: 99%

Optimal portfolio choice with path dependent benchmarked labor income: a mean field model

Djehiche¹,

Gozzi²,

Zanco³

et al. 2020

Preprint

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We consider the life-cycle optimal portfolio choice problem faced by an agent receiving labor income and allocating her wealth to risky assets and a riskless bond subject to a borrowing constraint. In this paper, to reflect a realistic economic setting, we propose a model where the dynamics of the labor income has two main features. First, labor income adjust slowly to financial market shocks, a feature already considered in Biffis et al. (2015) [8]. Second, the labor income yi of an agent i is benchmarked against the labor incomes of a population y n := (y1, y2, . . . , yn) of n agents with comparable tasks and/or ranks. This last feature has not been considered yet in the literature and is faced taking the limit when n → +∞ so that the problem falls into the family of optimal control of infinite dimensional McKean-Vlasov Dynamics, which is a completely new and challenging research field.We study the problem in a simplified case where, adding a suitable new variable, we are able to find explicitly the solution of the associated HJB equation and find the optimal feedback controls. The techniques are a careful and nontrivial extension of the ones introduced in the previous papers of Biffis et al.,[8,7].

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Viscosity Solutions to Parabolic Master Equations and McKean-Vlasov SDEs with Closed-loop Controls

Cited by 9 publications

References 36 publications

Forward and Backward Stochastic Differential Equations with normal constraint in law

Forward and Backward Stochastic Differential Equations with normal constraint in law

McKean-Vlasov optimal control: the dynamic programming principle

Optimal portfolio choice with path dependent benchmarked labor income: a mean field model

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