A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control

Djehiche, Boualem; Tembiné, Hamidou; Tempone, Raúl

doi:10.1109/tac.2015.2406973

Cited by 77 publications

(32 citation statements)

References 20 publications

(43 reference statements)

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“…If several producers need to match the demand, a cooperative strategy is proposed in order to reduce of cost of production and hence reduce electricity price for the consumers. Thanks to a novel stochastic maximum principle established in [8] we are able to handle mean-field-type optimization which helps to reduce not only the mean cost but also variance of the cost and risk-minimizing criteria. We have provided closed-form expression for the optimal supply and the optimal production strategy.…”

Section: Discussionmentioning

confidence: 99%

“…Note that Problem (rs), which is not a classical optimization problem since the mean-field term m is involved in l. In order to solve Problem (rs), we use recent development in mean-field-type optimization. The reader is referred to [8] for more details on the optimality equation. We introduce the risk-sensitive Hamiltonian: for θ ∈ R and (p,q,˜ ) ∈ R × R × R, H θ (t, e, m, s,p,q,˜ ) := bp + l + σ (q + θ˜ p).…”

Section: Solving the Mismatch Risk Minimization Problemmentioning

confidence: 99%

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Mean-field-type optimization for demand-supply management under operational constraints in smart grid

Tembiné

2015

Energy Syst

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The electricity business depends significantly on continuous, cost-effective sources of electricity, the efficient reaction to the demand and storage cost. But the increasing demands and alternative renewable energy sources on the power grid are causing instability and price volatility. Smart grid technology and programs are emerging to address these problems. In this paper we study demand-supply management under operational constraints and outage scheduling. We consider a model of an electricity market in which finite number of suppliers offer electricity. Each supplier has several power plants with different maximum capacity of production and different cost depending on the operational constraints. The response of the market to these offers is the quantities bought from the suppliers. The objective of the market is to satisfy the electricity demand at minimal cost. We develop a theoretical framework for cooperative production strategies between electricity producers. We formulate the problem as an optimal control problem under constraints. Using maximum principle techniques, we provide closed-form expressions of the joint production efforts between the electricity producers. Applying inf-convolution technique to the Hamiltonian we transform the multiple variable optimization problem into a single-variable optimization problem, reducing significantly the curse of dimensionality. In order to capture the random nature of most of the renewable energy sources, we introduce stochastic demand and stochastic production variabilities. We derive closed-form expressions using mean-field-type optimization techniques. The framework is shown to be flexible enough to include both risk-neutral and risk-sensitive behavior of a decision-maker when facing uncertainties. B Hamidou Tembine

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Section: Discussionmentioning

confidence: 99%

Section: Solving the Mismatch Risk Minimization Problemmentioning

confidence: 99%

Mean-field-type optimization for demand-supply management under operational constraints in smart grid

Tembiné

2015

Energy Syst

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“…The assumptions are clearly necessary and sufficient for mean-field games with quadratic controls, but it is still necessary to establish how certain assumptions (particularly Assumptions 4.1.2, 4.1.4) can be relaxed. The spike variation method has been shown to be very successful for establishing a maximum principle for non-convex cost functions, as seen in Peng (1990), Djehiche et al (2015), Buckdahn et al (2016). Given the generality of the cost functions under consideration in these papers, the use of a second order approximation for the Hamiltonian may help generalize the results in Chapter 4.…”

Section: Mean-field Games With Ambiguity Aversionmentioning

confidence: 99%

Mean-Field Games and Ambiguity Aversion

Huang

2017

SSRN Journal

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This thesis focuses on incorporating the idea of ambiguity aversion into mean-field games.Intuitively, mean-field games describes the dynamics of a system with an infinite population. It is useful in approximating systems with a large population, for which an exact result is computationally intractable. By adding in ambiguity aversion, the mean-field game reflects how players in the population should act if they wish to protect themselves from model misspecification.Two applications of mean-field games are considered through two distinct approaches.The broker execution problem is investigated in a multi-agent framework containing (i) a major agent who is liquidating a large number of shares, (ii) a number of minor agents (high-frequency traders (HFTs)) who search for statistical arbitrage strategies, and (iii) noise traders who buy and sell for exogenous reasons. All optimizing agents (the broker and HFTs) trade against noise traders as well as one another. We use a mean-field game approach to solve the problem and obtain a set of decentralized feedback trading strategies for the major and minor agents. Furthermore, the mean-field game strategies have an N -Nash equilibrium property where N → 0 as N → ∞.The second application focuses on interbank borrowing and lending, which may induce systemic risk into financial markets. A simple model of this is to assume that log-monetary reserves are coupled, and that banks can also borrow/lend from/to a central bank. When all banks optimize their cost of borrowing and lending, this leads to a stochastic game. We account for model uncertainty by recasting the problem as a robust stochastic game and succeed in providing a strategy which leads to a Nash equilibria for ii both the finite game and the mean-field game limit. We prove that an -Nash equilibrium exists and show that when firms are ambiguity-averse, default probabilities can be reduced relative to their ambiguity-neutral counterparts.Finally, this thesis develops a modified stochastic maximum principle for min-max problems, and derives new existence and uniqueness results for a mean-field game with ambiguity averse players. An -Nash equilibrium is shown to exist for this class of mean-field games, and the mean-field game equations are explicitly derived in the linearquadratic framework.iii

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“…These game problems are of practical interest, and a detailed exposition of this theory can be found in [7,12,[22][23][24][25]. The popularity of these game problems is due to practical considerations in signal processing, pattern recognition, filtering, prediction, economics, and management science [26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Linear–Quadratic Mean-Field-Type Games: A Direct Method

Duncan

Tembiné²

2018

Games

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Abstract:In this work, a multi-person mean-field-type game is formulated and solved that is described by a linear jump-diffusion system of mean-field type and a quadratic cost functional involving the second moments, the square of the expected value of the state, and the control actions of all decision-makers. We propose a direct method to solve the game, team, and bargaining problems. This solution approach does not require solving the Bellman-Kolmogorov equations or backward-forward stochastic differential equations of Pontryagin's type. The proposed method can be easily implemented by beginners and engineers who are new to the emerging field of mean-field-type game theory. The optimal strategies for decision-makers are shown to be in a state-and-mean-field feedback form. The optimal strategies are given explicitly as a sum of the well-known linear state-feedback strategy for the associated deterministic linear-quadratic game problem and a mean-field feedback term. The equilibrium cost of the decision-makers are explicitly derived using a simple direct method. Moreover, the equilibrium cost is a weighted sum of the initial variance and an integral of a weighted variance of the diffusion and the jump process. Finally, the method is used to compute global optimum strategies as well as saddle point strategies and Nash bargaining solution in state-and-mean-field feedback form.

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A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control

Cited by 77 publications

References 20 publications

Mean-field-type optimization for demand-supply management under operational constraints in smart grid

Mean-field-type optimization for demand-supply management under operational constraints in smart grid

Mean-Field Games and Ambiguity Aversion

Linear–Quadratic Mean-Field-Type Games: A Direct Method

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