Stochastic Control of Memory Mean-Field Processes

Agram, Nacira; Øksendal, Bernt

doi:10.1007/s00245-017-9425-1

Cited by 23 publications

(20 citation statements)

References 32 publications

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“…We omit the details. In the same manner, we can get the equivalence between d ds J 2 (µ, u + sπ)| s=0 = 0 and E[ ∂H 2 ∂u (t)|G (2) t ] = 0.…”

Section: Definition 11mentioning

confidence: 83%

“…In this section, we as in Agram and Øksendal [2] construct a Hilbert space M of random measures on R. It is simpler to work with than the Wasserstein metric space that has been used by many authors previously.…”

Section: A Weighted Sobolev Space Of Random Measuresmentioning

confidence: 99%

“…χ V , λ = λ(V ) for all λ ∈ M 0 . The first order condition for the optimal consumption rateρ is E[ 1 ρ(t) −p 0 (t)X(t)|G (2) t ] = 0.…”

Section: Optimal Consumption Of a Mean-field Cash Flow Under Uncertaintymentioning

confidence: 99%

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Model uncertainty stochastic mean-field control

Agram

Øksendal

2019

Stochastic Analysis and Applications

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We consider the problem of optimal control of a mean-field stochastic differential equation (SDE) under model uncertainty. The model uncertainty is represented by ambiguity about the law L(X(t)) of the state X(t) at time t. For example, it could be the law L P (X(t)) of X(t) with respect to the given, underlying probability measure P. This is the classical case when there is no model uncertainty. But it could also be the law L Q (X(t)) with respect to some other probability measure Q or, more generally, any random measure µ(t) on R with total mass 1. We represent this model uncertainty control problem as a stochastic differential game of a mean-field related type SDE with two players. The control of one of the players, representing the uncertainty of the law of the state, is a measure-valued stochastic process µ(t) and the control of the other player is a classical real-valued stochastic process u(t). This optimal control problem with respect to random probability processes µ(t) in a non-Markovian setting is a new type of stochastic control problems that has not been studied before. By constructing a new Hilbert space M of measures, we obtain a sufficient and a necessary maximum principles for Nash equilibria for such games in the general nonzero-sum case, and for saddle points in zero-sum games.As an application we find an explicit solution of the problem of optimal consumption under model uncertainty of a cash flow described by a mean-field related type SDE. MSC(2010): 60H05, 60H20, 60J75, 93E20, 91G80, 91B70.

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“…We omit the details. In the same manner, we can get the equivalence between d ds J 2 (µ, u + sπ)| s=0 = 0 and E[ ∂H 2 ∂u (t)|G (2) t ] = 0.…”

Section: Definition 11mentioning

confidence: 83%

Section: A Weighted Sobolev Space Of Random Measuresmentioning

confidence: 99%

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Model uncertainty stochastic mean-field control

Agram

Øksendal

2019

Stochastic Analysis and Applications

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“…We now define a weighted Sobolev spaces of measures. It is strongly related to the space introduced in Agram and Øksendal [5], [6], but with a different weight, which is more suitable for estimates (see e.g. Lemma 2.4 below):…”

Section: Sobolev Spaces Of Measuresmentioning

confidence: 99%

Mean-field stochastic control with elephant memory in finite and infinite time horizon

Agram

Øksendal

2019

Stochastics

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Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.• For infinite horizon, we derive sufficient and necessary maximum principles.As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.MSC (2010): 60H05, 60H20, 60J75, 93E20, 91G80,91B70.

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“…In this section, we proceed as in Agram and Øksendal [2], [3] and construct a Hilbert space M of random measures on R. It is simpler to work with than the Wasserstein metric space that has been used by many authors previously. See e.g.…”

Section: A Hilbert Space Of Random Measuresmentioning

confidence: 99%

Singular Control Optimal Stopping of Memory Mean-Field Processes

Agram¹,

Bachouch²,

Øksendal³

et al. 2019

SIAM J. Math. Anal.

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The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory, (ii) reflected advanced mean-field backward stochastic differential equations, and (iii) optimal stopping of mean-field stochastic differential equations.More specifically, we do the following:• We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations (AMBSDEs),• we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information, and• we deduce a relation between the optimal singular control of a MMSDE, and the optimal stopping of such processes.

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Stochastic Control of Memory Mean-Field Processes

Cited by 23 publications

References 32 publications

Model uncertainty stochastic mean-field control

Model uncertainty stochastic mean-field control

Mean-field stochastic control with elephant memory in finite and infinite time horizon

Singular Control Optimal Stopping of Memory Mean-Field Processes

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