Mean Field Games with Singular Controls

Fu, Guanxing; Horst, Ulrich

doi:10.1137/17m1123742

Cited by 39 publications

(39 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Take the M 1 topology for the Skorokhod space D([0, ∞)) with a Wasserstein distance W 1 ( [40,19]). Fix a mean field measure {µ t } t≥0 ∈ P 1 (D([0, ∞))), with m t = xµ t (dx) and P 1 the class of all probability measures with finite moment of first order.…”

Section: )mentioning

confidence: 99%

Stochastic Games for Fuel Follower Problem: $N$ versus Mean Field Game

Guo¹,

Xu²

2019

SIAM J. Control Optim.

View full text Add to dashboard Cite

In this paper we formulate and analyze an N -player stochastic game of the classical fuel follower problem and its Mean Field Game (MFG) counterpart. For the N -player game, we obtain the Nash Equilibrium (NE) explicitly by deriving and analyzing a system of Hamilton-Jacobi-Bellman (HJB) equations, and by establishing the existence of a unique strong solution to the associated Skorokhod problem on an unbounded polyhedron with an oblique reflection. For the MFG, we derive a bang-bang type NE under some mild technical conditions and by the viscosity solution approach. We also show that this solution is an -NE to the N -player game, with = O( 1 √ N ). The N -player game and the MFG differ in that the NE for the former is state dependent while the NE for the latter is threshold-type bang-bang policy where the threshold is state independent . Our analysis shows that the NE for a stationary MFG may not be the NE for the corresponding MFG.

show abstract

Section: )mentioning

confidence: 99%

Stochastic Games for Fuel Follower Problem: $N$ versus Mean Field Game

Guo¹,

Xu²

2019

SIAM J. Control Optim.

View full text Add to dashboard Cite

show abstract

“…In the original MFG framework (regular control, without optimal stopping), the controlled martingale problem approach was first used in [39] to show the existence of a mean field game equilibrium under general assumptions. Further developments have been made in the case of mean field games with branching [19] or mean field games with singular controls [27]. Another relaxation technique used in the classical stochastic control theory is based on the linear programming formulation (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Control and optimal stopping Mean Field Games: a linear programming approach

Dumitrescu

Leutscher

Tankov

2021

Electron. J. Probab.

View full text Add to dashboard Cite

We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in earlier papers in the pure control case.

show abstract

“…Hence, we may consider Y in some space with appropriate topologies, for instance, Meyer-Zheng topology and obtain the convergence of probability measures deduced by Y involving relaxed control. For this topic, reader can also refer articles [35,33] in this direction. We point out that the related work from the technique of PDEs (see [14,15] therein).…”

Section: Dynamic Programming Principlementioning

confidence: 99%

Singular optimal controls of stochastic recursive systems and Hamilton–Jacobi–Bellman inequality

Zhang

2019

Journal of Differential Equations

View full text Add to dashboard Cite

In this paper, we study the optimal singular controls for stochastic recursive systems, in which the control has two components: the regular control, and the singular control. Under certain assumptions, we establish the dynamic programming principle for this kind of optimal singular controls problem, and prove that the value function is a unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman inequality, in a given class of bounded and continuous functions. At last, an example is given for illustration.T ] → R n×m are given deterministic functions, (W s ) s≥0 is an d-dimensional Brownian motion, (x, t) are initial time and state, v (·) : [0, T ] → R k is classical control process, and ξ (·) : [0, T ] → R m , with nondecreasing left-continuous with right limits stands for the singular control.The aim is to minimize the cost functional:where. . . m} are given deterministic functions, where l (·) represents the running cost tare of the problem and K (·) the cost rate of applying the singular control. [40], Karatzas and Shreve [43], and Menaldi and Taksar [52]. The second one is related to probabilistic methods; see Baldursson [6], Boetius [8, 9], Boetius and Kohlmann [10], El Karoui and Karatzas [29, 30], Karatzas [39], and Karatzas and Shreve [41, 42]. Third, the DPP, has been studied in a general context, for example, by Boetius [9], Haussmann and Suo [36], Fleming and Soner [31], and Zhu [66]. At last the maximum principle for optimal singular controls (see, for example, Cadenillas and Haussmann [17], Dufour and Miller [26], Dahl and Øksendal [27] see references therein). The existence for optimal singular control can be found in Haussmann and Suo [36] and Dufour and Miller [25] via different approaches. It is necessary to point out that singular control problems are largely used in diverse areas such as mathematical finance (see Baldursson and Karatzas [12], Chiarolla and Haussmann [18], Kobila [44], and Karatzas and Wang [45], Davis and Norman [24]), manufacturing systems (see, Shreve, Lehoczky, and Gaver [60]), and queuing systems (see Martins and Kushner [53]). Particularly, the application of H-J-B inequality in finance can be seen in Pagès and Possamaï [58], which is employed to investigate the bank monitoring incentives.As is well known for the classical stochastic control problems, DPP is satisfied. Moreover, if the value function has appropriate regularity, it admits a second-order nonlinear partial differential equation (H-J-B equation) (see Fleming and Rishel [32] and Lions [49] reference therein). In the frame work of singular stochastic control, the H-J-B equation becomes a second-order variational inequality (see Fleming and Soner [31], Haussmannand, Suo [35]).Wang [63] firstly introduces and studies a class of singular control problems with recursive utility, where the cost function is determined by a backward stochastic differential equation (BSDE in short). More preciously, the cost functional is defined bywhere Y t,x;v,ξ t is determined by    dX t,x;ξ s = aX t,x;ξ s + b ds + ...

show abstract

Mean Field Games with Singular Controls

Cited by 39 publications

References 45 publications

Stochastic Games for Fuel Follower Problem: $N$ versus Mean Field Game

Stochastic Games for Fuel Follower Problem: $N$ versus Mean Field Game

Control and optimal stopping Mean Field Games: a linear programming approach

Singular optimal controls of stochastic recursive systems and Hamilton–Jacobi–Bellman inequality

Contact Info

Product

Resources

About