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A stochastic variational inequality is proposed to model a white noise excited elasto-plastic oscillator. The solution of this inequality is essentially a continuous diffusion process for which a governing diffusion equation is obtained to study the evolution in time of its probability distribution. The diffusion equation is degenerate, but using the fact that the degeneracy occurs on a bounded region we are able to show the existence of a unique solution satisfying the desired properties. We prove the ergodic properties of the process and characterize the invariant measure. Our approach relies on extending Khasminskii’s method (Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980), which in the present context leads to the study of degenerate Dirichlet problems with nonlocal boundary conditions.

We consider a newsvendor problem with partially observed Markovian demand. Demand is observed if it is less than the inventory. Otherwise, only the event that it is larger than or equal to the inventory is observed. These observations are used to update the demand distribution from one period to the next. The state of the resulting dynamic programming equation is the current demand distribution, which is generally infinite dimensional. We use unnormalized probabilities to convert the nonlinear state transition equation to a linear one. This helps in proving the existence of an optimal feedback ordering policy. So as to learn more about the demand, the optimal order is set to exceed the myopic optimal order. The optimal cost decreases as the demand distribution decreases in the hazard rate order. In a special case with finitely many demand values, we characterize a near-optimal solution by establishing that the value function is piecewise linear.

In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to nonobservation of demand, spoilage, misplacement, or theft of inventory. We study a partially observed inventory system where the demand is not observed, inventory level is noticed when it reaches zero, the unmet demand is lost, and replenishment orders must be decided so as to minimize the total discounted costs over an infinite horizon. This problem has an infinite-dimensional state space, and for it we establish the existence of a feedback policy when single-period costs are bounded or when the discount factor is sufficiently small. We also provide an approximately optimal feedback policy that uses a finite state representation.

We initiate the analysis of the response of computer owners to various offers of defence systems against a cyber-hacker (for instance, a botnet attack), as a stochastic game of a large number of interacting agents. We introduce a simple mean-field game that models their behavior. It takes into account both the random process of the propagation of the infection (controlled by the botner herder) and the decision making process of customers. Its stationary version turns out to be exactly solvable (but not at all trivial) under an additional natural assumption that the execution time of the decisions of the customers (say, switch on or out the defence system) is much faster that the infection rates.

International audience; There is a concern that the continuing trend on miniaturization (Moore’s law) in IC design and fabrication might have a negative impact on the device reliability. To understand and to possibly quantify the physics underlying this concern and phenomenon, it is natural to proceed from the experimental bathtub curve (BTC) – reliability “passport” of the device. This curve reflects the combined effect of two major irreversible governing processes: statistics-related massproduction process that results in a decreasing failure rate with time, and reliability-physics-related degradation (aging) process that leads to an increasing failure rate. It is the latter process that is of major concern of a device designer and manufacturer. The statistical process can be evaluated theoretically, using a rather simple predictive model. Owing to that and assuming that the two processes of interest are statistically independent one can assess the failure rates associated with the aging process from the BTC data by simply subtracting the predicted ordinates of the statistical failure rates (SFR) from the BTC ordinates. The objective of this analysis is to show how this could be done. The suggested methodology proceeds from the concepts that the actual (“instantaneous”) SFR is a random variable with a known (assumed, established) probability distribution, that the experimental BTC can be represented by its infant mortality and the wear-out portions only (the steady-state portion in this case is simply the boundary between the infant mortality and wear-out portions) and that the two BTC portions considered can be approximated analytically. The cases, when the “instantaneous” SFR is distributed normally and in accordance with the Rayleigh law are used as suitable illustrations of the general concept. The developed methodology can be employed when there is a need to better understand the relative roles of the statisticsrelated and physics-of-failure-related processes in reliability evaluations of electronic products. The methodology can be used also beyond the field of IC engineering, when there is a need to understand and, hence, to separate the roles of the two irreversible processes in question. One of the major challenges of the future work is to determine the probability distributions of the actual (“instantaneous”) SFRs for particular products and applications.

International audience; The establishment of European suppliers for DFB Laser Modules at 1.55 µm is considered to be essential in the context of future European space programs, where availability, cost and schedule are of primary concerns. Also, in order to minimize the risk, associated with such a development, the supplier will be requested to use components which have already been evaluated and/or validated and/or qualified for space applications. The Arrhenius model is an empirical equation able to model temperature acceleration failure modes and failure mechanisms. The Eyring model is a general representation of Arrhenius equation which takes into account additional stresses than temperature. The present paper suggests to take advantage of these existing theories and derives a unified multiple stress reliability model for electronic devices in order to quantify and predict their reliability figures when operating under multiple stress in harsh environment as for Aerospace, Space, Nuclear, Submarine, Transport or Ground. Application to DFB laser diode module technologies is analyzed and discussed based on evaluation test program under implementation.

International audience; The recently suggested probabilistic design for reliability (PDfR) concept of electronics and photonics (EP) products is based on 1) highly focused and highly cost-effective failure oriented accelerated testing (FOAT), aimed at understanding the physics of the anticipated failures and at quantifying, on the probabilistic basis, the outcome of FOAT conducted for the most vulnerable element(s) of the product of interest, for the most likely applications and for the most likely and meaningful combination of possible stressors (stimuli); 2) simple and physically meaningful predictive modeling (PM), both analytical and computer-aided, aimed at bridging the gap between the obtained FOAT data and the most likely actual operation conditions; and 3) subsequent FOAT-and-PM-based sensitivity analysis (SA) using the methodologies and algorithms developed as important by-products at the two previous steps. The PDfR concept proceeds from the recognition that nothing is perfect, and that the difference between a highly reliable and an insufficiently reliable product is “merely” in the level of the probability of its field failure. If this probability (evaluated for the anticipated loading conditions and the given time in operation) is not acceptable, then a SA can be effectively employed to determine what could/should be changed to improve the situation. The PDfR analysis enables one also to check if the product is not "over-engineered", i.e., is not superfluously robust. If it is, it might be too costly. The operational reliability cannot be low, but it does not have to be higher than necessary either. It has to be adequate for the given product and application. When reliability and cost-effectiveness are imperative, ability to optimize reliability is a must, and no optimization is possible if reliability is not quantified. We show that optimization of the total cost associated with creating a product with an adequate (high enough) reliability and acceptable (low- enough) cost can be interpreted in terms of an adequate level of the availability criterion. The major PDfR concepts are illustrated by practical examples. Although some advanced PDfR predictive modeling techniques have been recently developed, mostly for aerospace applications, the practical examples addressed in this talk employ more or less elementary analytical models. In this connection we elaborate on the roles and interaction of analytical (mathematical) and computer-aided (simulation) modeling. We show also how the recently suggested powerful and flexible Boltzmann-Arrhenius-Zhurkov (BAZ) model and particularly its multi-parametric extension could be successfully employed to predict, quantify and assure operational reliability. The model can be effectively used to analyze and design EP products with the predicted, quantified, assured, and, if appropriate and cost-effective, even maintained and specified probability of operational failure. It is concluded that these concepts and methodologies can be accepted as an effective means for the evaluation of the operational reliability of EP materials and products, and that the next generation of qualification testing (QT) specifications and practices for such products could be viewed and conducted as a quasi-FOAT, an early stage of FOAT that adequately replicates the initial non-destructive segment of the previously conducted comprehensive “full-scale” FOAT.

International audience; This book provides a perspective on a number of approaches to financial modelling and risk management. It examines both theoretical and practical issues. Theoretically, financial risks models are models of a real and a financial “uncertainty”, based on both common and private information and economic theories defining the rules that financial markets comply to. Financial models are thus challenged by their definitions and by a changing financial system fueled by globalization, technology growth, complexity, regulation and the many factors that contribute to rendering financial processes to be continuously questioned and re-assessed. The underlying mathematical foundations of financial risks models provide future guidelines for risk modeling. The book’s chapters provide selective insights and developments that can contribute to better understand the complexity of financial modelling and its ability to bridge financial theories and their practice.Future Perspectives in Risk Models and Finance begins with an extensive outline by Alain Bensoussan et al. of GLM estimation techniques combined with proofs of fundamental results. Applications to static and dynamic models provide a unified approach to the estimation of nonlinear risk models.A second section is concerned with the definition of risks and their management. In particular, Guegan and Hassani review a number of risk models definition emphasizing the importance of bi-modal distributions for financial regulation. An additional chapter provides a review of stress testing and their implications. Nassim Taleb and Sandis provide an anti-fragility approach based on “skin in the game”. To conclude, Raphael Douady discusses the noncyclical CAR (Capital Adequacy Rule) and their effects of aversion of systemic risks.A third section emphasizes analytic financial modelling approaches and techniques. Tapiero and Vallois provide an overview of mathematical systems and their use in financial modeling. These systems span the fundamental Arrow-Debreu framework underlying financial models of complete markets and subsequently, mathematical systems departing from this framework but yet generalizing their approach to dynamic financial models. Explicitly, models based on fractional calculus, on persistence (short memory) and on entropy-based non-extensiveness. Applications of these models are used to define a modeling approach to incomplete financial models and their potential use as a “measure of incompleteness”. Subsequently Bianchi and Pianese provide an extensive overview of multi-fractional models and their important applications to Asset price modeling. Finally, Tapiero and Jinquyi consider the binomial pricing model by discussing the effects of memory on the pricing of asset prices.

International audience

International audience; The industries of transportation as the space industry are faced with a strong global economic competition which sets economic constraints on the cost of the functions. The use of COTS (Commercial Off-The-Shelf) components in embedded systems is more and more necessary to shorten the development cycles and reduce manufacturing costs. The application of electronic components comes overwhelmingly from public sectors whose requirement is to provide, in short development cycles, technological innovations including risk and cost mitigation. These development cycles must incorporate the specific constraints of embedded systems in terms of reliability, dependability, and availability, held in harsh environment and life. Due to the low volume of components supplying the market of embedded systems, component manufacturers are unlikely to provide information necessary to supporting folders for certification or qualification. It is therefore necessary for the Space, Aeronautics and Automotive industries to characterize the performance and robustness of these COTS components in the operational and environmental conditions of their applications. This paper presents the objectives and main challenges of a sponsored project dedicated to characterize and model COTS reliability.

Information delays exist when the most recent inventory information available to the inventory manager (IM) is dated; namely, the IM observes only the inventory level of an earlier period. We introduce information delays into the standard multiperiod stochastic inventory problem with backorders. We consider two types of information delays: (i) a constant delay and (ii) a random delay. We define an appropriate reference inventory position, which is a sufficient statistics for finding the optimal order quantity. We show that the optimal ordering policy is of base-stock type with respect to the reference inventory position and is of (s, S) type if there is also a fixed cost of ordering.

The purpose of this Note is to show that models used in the literature for the hysteresis effect of non-linear elasto-plastic oscillators submitted to random vibrations are equivalent to stochastic variational inequalities. This powerful tool allows to study the ergodic properties of the Markov process related to the displacement. We characterize completely the invariant measure by a partially degenerate elliptic partial differential equation, with new Dirichlet coupling conditions. To cite this article: A. Bensoussan, J. Turi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Linear Random Functionals have been introduced by the author [2] to develop the theory of Kalman filtering for infinite dimensional linear systems. It is reminiscent of the concept of stochastic integral, which it partly generalizes. We compare it to that of cylindrical Wiener processes, introduced by G. Da Prato- J. Zabczyk [4]. Like distributions, linearity limits the power of the tool. We can consider however some non-linear problems. We show that it is a powerful tool to deal with statistical problems in infinite dimensional spaces. For additional relevant references see [1], [6], [7], [3].

A stochastic variational inequality is proposed to model an elasto-plastic oscillator excited by a filtered white noise. We prove the ergodic properties of the process and characterize the corresponding invariant measure. This extends Bensoussan-Turi's method (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008) with a significant additional difficulty of increasing the dimension. Two points boundary value problem in dimension 1 is replaced by elliptic equations in dimension 2. In the present context, Khasminskii's method (Stochastic Stability of Differential Equations, Sijthoff and Noordhof,1980) leads to the study of degenerate Dirichlet problems with partial differential equations and nonlocal boundary conditions.

This Note introduces recent developments in the analysis of inventory systems with partial observations. The states of these systems are typically conditional distributions, which evolve in infinite dimensional spaces over time. Our analysis involves introducing unnormalized probabilities to transform nonlinear state transition equations to linear ones. With the linear equations, the existence of the optimal feedback policies are proved for two models where demand and inventory are partially observed. In a third model where the current inventory is not observed but a past inventory level is fully observed, a sufficient statistic is provided to serve as a state. The last model serves as an example where a partially observed model has a finite dimensional state. In that model, we also establish the optimality of the basestock policies, hence generalizing the corresponding classical models with full information. To cite this article: A. Bensoussan et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

We prove that an $(s, S)$ policy is optimal in a continuous-review stochastic inventory model with a fixed ordering cost when the demand is a mixture of (i) a diffusion process and a compound Poisson process with exponentially distributed jump sizes, and (ii) a constant demand and a compound Poisson process. The proof uses the theory of impulse control. The Bellman equation of dynamic programming for such a problem reduces to a set of quasi-variational inequalities (QVI). An analytical study of the QVI leads to showing the existence of an optimal policy as well as the optimality of an $(s, S)$ policy. Finally, the combination of a diffusion and a general compound Poisson demand is not completely solved. We explain the difficulties and what remains open. We also provide a numerical example for the general case.

The study of the pricing of contingent claims under constraints leads, in the case of stocks obeying lognormal distributions, to an interesting analytical result. Namely, the price satisfies the Black Scholes equation with a different initial condition. We give a mostly analytical treatment of this result, using the probabilistic interpretation of the Cauchy problem, with nonsmooth initial conditions.

A new class of one-dimensional relativistic nonlinear wave equations with a singular δ-type nonlinear term is considered. The sense of the equations is defined according to the least-action principle. The energy and momentum conservation is established. The main results are the existence of time-periodic finite-energy solutions, the existence of global solutions and soliton-type asymptotics for a class of finite-energy initial data.

The objective of this paper is to consider the theory of regularity of systems of partial differential equations with Neumann boundary conditions. It complements previous works of the authors for the Dirichlet case. This type of problem is motivated by stochastic differential games. The Neumann case corresponds to stochastic differential equations with reflection on boundary of the domain.

We consider in this article diagonal parabolic systems arising in the context of stochastic dierential games. We address the issue of nding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic dierential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear Hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Holder solution. Mathematics Subject Classication. 35XX, 49XX.

Generalized Linear Models have been introduced by (Nelder and Wedderburn 1972). See also the book (McCullagh and Nelder 1983). They describe random observations depending on unobservable variables of interest, generalizing the standard gaussian error model. Many estimation results can be obtained in this context, which generalize with some approximation procedures the gaussian case. We revisit and extend the results. In particular, we prove the cental limit theorem for the MLE, maximum likelihood estimator, in a general setting. We also provide a recursive estimator, similar to the Kalman filter. We also consider dynamic models and develop several methods, including that of (West et al. 1985).

International audience; The attributes of and challenges in the recently suggested probabilistic design for reliability (PDfR) concept, and the role of its major constituents - failure oriented accelerated testing (FOAT) and physically meaningful predictive modeling (PM) - are addressed, advanced and discussed. The emphasis is on the application of the powerful and flexible Boltzmann-Arrhenius-Zhurkov (BAZ) model, and particularly on its multi-parametric aspect. The model can be effectively used to analyze and design optoelectronic (OE) devices and systems with the predicted, quantified, assured, and, if appropriate and cost-effective, even maintained probability of failure in the field. The numerical example is carried out for an OE system subjected to the combined action of the ionizing radiation and elevated voltage as the major stimuli (stressors). The measured leakage current is used as a suitable characteristic of the degree of degradation. It is concluded that the suggested methodology can be accepted as an effective means for the evaluation of the operational reliability of the aerospace electronics and OE systems and that the next generation of qualification testing (QT) specifications and best practices for such systems could be viewed and conducted as a “quasi-FOAT,” a sort of an “initial stage of FOAT” that adequately replicates the initial non-destructive segment of the previously conducted comprehensive “full-scale” FOAT.

From optoelectronics commercial systems through aerospace applications, new technologies based equipment must be designed taking into account reliability risks. Existing qualification standards and processes have been developed from field return “lessons learned”. The qualification methods have been built-up to take into account questions and concerns observed on previous well established processes. Is this approach applicable to new products and materials? How will new optoelectronic technologies, nanotechnologies, and new packaging techniques perform when tested under conditions defined in existing standards? Are the Aerospace and Military standards adequate to validate long term application and proof of reliability for the new devices? What is missing, and what is needed? Should we modify the methodology for qualifying new parts based on these innovative technologies? What is needed for implementing efficiently and just in time High Reliability application processes? Evolving methodology may be one solution but must be constructed based on results of experiments from FOAT (Failure Oriented Accelerated Testing) and QT (Qualification Testing) for new technologies. This presentation gives an overview of general issues of microelectronics when employed in harsh environments (electrical, optical, thermal, mechanical, radiation). Understanding the need, defining goals, accumulating field failure data, developing reliability models and educating our customers is a recommended approach to develop comprehensive FOAT and QT programs.

The attributes and challenges of the PDfR concept and the role of its major constituents - failure oriented accelerated testing (FOAT) and physically meaningful predictive modeling (PM) - are addressed, advanced and discussed in detail, with an emphasis on the application of the recently suggested powerful and flexible Boltzmann-Arrhenius-Zhurkov (BAZ) model, and particularly on its multi-parametric version. The BAZ model can be effectively used to analyze and design an opto-electronic (OE) device or a system with the predicted, quantified, assured, and, if appropriate and cost-effective, even maintained probability of failure (PoF) in the field. This could be done, e.g., by employing prognostics-and-health monitoring (PHM) methods and techniques.

International audience; The aim of the article is to identify the intraday seasonality in a wind speed time series. Following the traditional approach, the marginal probability law is Weibull and, consequently, we consider seasonal Weibull law. A new estimation and decision procedure to estimate the seasonal Weibull law intraday scale parameter is presented. We will also give statistical decision-making tools to discard or not the trend parameter and to validate the seasonal model.

International audience; Wind power is an intermittent resource due to wind speed intermittency. However wind speed can be described as a stochastic process with short memory. This allows us to derive a central limit theorem for the annual or pluri-annual wind power production and then get quantiles of the wind power production for one, ten or twenty years future periods. On the one hand, the interquantile spread offers a measurement of the intrinsic uncertainties of wind power production. On the other hand, different quantiles with different periods of time are used by financial institutions to quantify the financial risk of the wind turbine. Our method is then applied to real datasets corresponding to a French wind turbine. Since confidence intervals can be enhanced by taking into account seasonality, we present some tools for change point analysis on wind series. Résumé. L'énergie éolienne est une ressource intermittente à cause de l'intermittence du vent. Ce-pendant la vitesse du vent peut être décrite comme un processus stochastique à mémoire courte. Nous pouvons alors obtenir un théorème central limite pour la production annuelle ou pluri-annuelle d'éner-gie éolienne, ce qui nous permet de calculer les différents quantiles de la production d'énergie éolienne avec des horizons de un, dix ou vingt ans. La différence des écarts interquantiles fournit, d'une part, une mesure de l'incertitude intrinsèque sur la production annuelle d'énergie éolienne. D'autre part, différents quantiles avec différents horizons de temps sont utilisés par les institutions financières afin de quantifier le risque lié à la construction d'une éolienne. Nous appliquons ensuite notre méthode à de vrais jeux de données correspondant à une turbine française. De plus, la prévision peut être améliorée par la prise en compte de la saisonnalité, nous présentons alors quelques outils pour la détection de rupture sur la moyenne de la vitesse du vent.

We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward---backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.

For stochastic Stackelberg differential games played by a leader and a follower, there are several solution concepts in terms of the players' information sets. In this paper we derive the maximum principle for the leader's global Stackelberg solution under the adapted closed-loop memoryless information structure, where the term global signifies the leader's domination over the entire game duration. As special cases, we study linear quadratic Stackelberg games under both adapted open-loop and adapted closed-loop memoryless information structures, as well as the resulting Riccati equations.

Ever since the first attempts to model capital investment decisions as options, financial economists have sought more accurate, more realistic real options models. Strategic interactions and market incompleteness are significant challenges that may render existing classical models inadequate to the task of managing the firm’s capital investments. The purpose of this paper is to address these challenges. The issue of incompleteness comes in for the valuation of payoffs due to absence of a unique martingale measure. One approach is to valuate assets by considering a rational utility-maximizing consumer/investor’s joint decisions with respect to portfolio investment strategy and consumption rule. In our situation, we add the stopping time as an additional decision. We employ variational inequalities (V.I.s) to solve the optimal stopping problems corresponding to times to invest. The regularity of the obstacle (payoffs received at the decision time) is a major element for defining the optimal strategy. Due to the lack of smoothness of the obstacle raised by the game problem, the optimal strategy is a two-interval solution, characterized by three thresholds.

In the treatment of the risk-sensitive control problem, it is known that the criterion function may not have a finite value. The risk parameter cannot be arbitrary. Conditions have been presented by the authors in previous papers to guarantee the no breakdown. In the present article, we present a framework in which the conditions can be greatly relaxed.

In this article we consider local solutions for stochastic Navier Stokes equations, based on the approach of Von Wahl, for the deterministic case. We present several approaches of the concept, depending on the smoothness available. When smoothness is available, we can in someway reduce the stochastic equation to a deterministic one with a random parameter. In the general case, we mimic the concept of local solution for stochastic differential equations.

Stochastic control problems on a finite horizon with exponential cost criteria are considered. By taking a kind of singular limit a Hamilton--Jacobi--Isaacs equation is obtained. Its solution is characterized as the lower value function of a deterministic differential game related to robust control of nonlinear systems.

Qualification testing (QT) is the major means for making a viable photovoltaic (PV) device into a reliable and marketable product. It is well known, however, that the today’s PV modules (PVM) that passed the existing QT often exhibit premature field failures. Could the existing QT specifications and testing procedures be improved to an extent that if a PV device, module or a system passed the QT, there is a quantifiable and consistent way to assure that its performance in the field will be satisfactory and that its projected lifetime will indeed take place with the given confidence? The application of the probabilistic design for reliability (PDfR) concept enables one to provide an affirmative answer to this question. The attributes and challenges of this concept and the roles of its major constituents - failure oriented accelerated testing (FOAT) and physically meaningful predictive modeling (PM) - are addressed and discussed in detail.

We present a sufficient condition for a feedback Stackelberg equilibrium of a stochastic differential game on an infinite horizon. This condition gives rise to a system of elliptic partial differential equations involving a static Stackelberg game at the level of Hamiltonian. As an example, we consider a linear quadratic problem, obtain the corresponding algebraic Riccati equation, and provide its solution in the scalar case.

Specific Single Event Effects characterization based on RF and worst case DC conditions are used to demonstrate that two European GaAs power P-HEMT MMIC processes are safe under heavy ions.

1. The genuine savings criterion and the value of population in an economy with endogenous fertility rate Kenneth J. Arrow, Alain Bensoussan, Qi Feng, and Suresh P. Sethi 2. Continuous time overlapping generations models Hippolyte d'Albis and Emmanuelle Augeraud-Veron 3. Modeling technological change in equipment replacement problems Chin Hon Tan and Joseph C. Hartman 4. Embodied technology adoption under uncertainty Bruno Cruz and Aude Pommeret 5. Viability, optimality, and sustainability in vintage models Noel Bonneuil 6. Embodied technical change and learning-by-doing in a two sector growth model with human capital accumulation Alberto Bucci, Herb Kunze and Davide La Torre 7. Age-structured optimization models in fisheries bioeconomics: a survey Olli Tahvonen 8. Age-structured models and optimal control in mathematical epidemiology: a survey David Greenhalgh 9. Optimal management of a size distributed forest with respect to timber and non-timber values Renan-Ulrich Goertz and Angels Xabadia 10. Optimal harvesting in a two-species model of size-structured population Nobuyuki Kato 11. Optimisation problems in the management of indigenous plant populations under grazing pressure Lynne McArthur, John Boland, and Fleur Tiver 12. Identification of age-structured contamination sources in ground water Sergey Lyashko, Dmitry Klyushin, Dmitry Nomirovsky, and Vladimir Semenov

Consider a probability space \((\Omega,\mathcal{A},P)\) and a filtration \({\mathcal{F}}^{t}\) generated by a n-dimensional standard Wiener process w(t). The state space is \({\mathbb{R}}^{n}\) with the generic notation x and the control space is \({\mathbb{R}}^{d}\) with generic notation v.

The mean field game and mean field type control problems introduced in Chap. 2 are both control problems for a representative agent, with mean field terms influencing both the evolution and the objective functional of this agent. The terminology game comes from the fact that the optimal feedback of the representative agent can be used as an approximation for a Nash equilibrium of a large community of agents that are identical. In Sect. 8.2 we have shown that the theory extends to a multi-class of representative agents. However, each class still has its individual control problem.

The linear quadratic model has been developed in [10]. See also [2, 20, 22]. We highlight here the results. We take $$\displaystyle\begin{array}{rcl} f(x,m,v)& =& \dfrac{1} {2}\left [{x}^{{\ast}}Qx + {v}^{{\ast}}Rv +{ \left (x - S\int \xi m(\xi )d\xi \right )}^{{\ast}}\bar{Q}\left (x - S\int \xi m(\xi )d\xi \right )\right ]{}\end{array}$$ (6.1) $$\displaystyle\begin{array}{rcl} g(x,m,v)& = Ax +\bar{ A}\int \xi m(\xi )d\xi + Bv&{}\end{array}$$ (6.2) $$\displaystyle\begin{array}{rcl} h(x,m)& = \dfrac{1} {2}\left [{x}^{{\ast}}Q_{T}x +{ \left (x - S_{T}\int \xi m(\xi )d\xi \right )}^{{\ast}}\bar{Q_{T}}\left (x - S_{T}\int \xi m(\xi )d\xi \right )\right ].&{}\end{array}$$ (6.3)

We first assume that the functions f(x, m, v), g(x, m, v), and h(x, m)—as functions of m—can be extended to Dirac measures and the sum of Dirac measures that are probabilities. Since m is no more in L p , the reference topology will be the weak * topology of measures. That will be sufficient for our purpose, but we refer to [14] for metric space topology. At any rate, the vector space property is lost.

Let us set $$\displaystyle{ a(x) = \frac{1} {2}\sigma {(x)\sigma }^{{\ast}}(x), }$$ (3.1) and introduce the second-order differential operator $$\displaystyle{ A\varphi (x) = -\text{tr }a(x){D}^{2}\varphi (x). }$$ (3.2) We define the dual operator $$\displaystyle{ {A}^{{\ast}}\varphi (x) = -\sum _{ k,l=1}^{n} \frac{{\partial }^{2}} {\partial _{x_{k}}\partial _{x_{l}}}(a_{kl}(x)\varphi (x)). }$$ (3.3) Since m(t) is the probability distribution of \(\hat{x}(t)\), it has a density with respect to the Lebesgue measure denoted by m(x, t), which is the solution of the Fokker–Planck equation $$\displaystyle\begin{array}{rcl} \frac{\partial m} {\partial t} + {A}^{{\ast}}m + \text{div }(g(x,m,\hat{v}(x))m)& =& 0, \\ m(x,0)& =& m_{0}(x).{}\end{array}$$ (3.4) We next want the feedback \(\hat{v}(x)\) to solve a standard control problem, in which m appears as a parameter. We can thus readily associate an HJB equation with this problem, parametrized by m.