In this paper, we shall focus on the optimal control for the initial value V 0 of the
LQCMKV problem with random coefficients, but following the approach developed in [32], we emphasize that time consistency can be actually restored for pre-commitment strategies, provided that one considers as state variable the conditional law of the state process instead of the state itself, therefore making possible the use of the dynamic programming method. We show that the dynamic version of the LQCMKV control problem defined by a random field value function, has a quadratic structure with respect to the conditional law of the state process, leading to a characterization of the optimal control in terms of a decoupled system of backward **stochastic** Riccati equations (BSREs) whose existence and uniqueness are obtained in connection with a standard LQ control problem. The main ingredient for such derivation is an Itˆo’s formula along a flow of conditional measures and a suitable notion of differentiability with respect to probability measures. We illustrate our results with several financial applications. We first revisit the optimal trading and benchmark tracking problem with price impact for general price and target **processes**, and obtain closed-form solutions extending some known results in the literature. We next solve a variation of the mean-variance portfolio selection problem in an incomplete market with random factor. Our last example considers an interbank systemic risk model with random factor in a common noise environment.

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of the adjoint equation is not provided by standard results on Backward **Stochastic** Differential Equa- tions (BSDEs) as the distributions of the solution **processes** (more precisely their joint distributions with the control and state **processes** α and X) appear in the coefficients of the equation. However, a slight modification of the original existence and uniqueness result of Pardoux and Peng [13] shows that existence and uniqueness still hold in our more general setting. The main lines of the proof are given in [4], Proposition 3.1 and Lemma 3.1. However, Lemma 3.1 in [4] doesn’t apply directly since the coefficients (∂ µ b(t, ˜ X t , P X t , ˜ α t )(X t ) ˜ Y t ) 0≤t≤T and (∂ µ σ(t, ˜ X t , P X t , ˜ α t )(X t ) ˜ Z t ) 0≤t≤T

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The plan of the paper is the following: In Section 2 we state our main results and compare them with the above mentioned Keller-Segel literature. In Section 3 we present our regularization procedure and we obtain density estimates of the regularized **processes** independent of the regularization parameter. These estimates enable us to prove in Section 4 the existence of a solution to the NLMP corresponding to ( 2 ). Then, in the same section we prove the global existence for the Keller-Segel system in d = 2. Section 5 is devoted to uniqueness. Finally, in Appendix 6 we prove the well-posedness of a smoothed version of ( 2 ).

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t , its net consumption on the electrical public grid is P grid ,u
t . The (deterministic) committed profile load is the curve (P grid t ,com. : 0 ≤ t ≤ T). Optimal control of a single micro-grid has already been considered in the literature, without the optimal committed load profile. A popular yet without theoretical optimality guarantee is Model Predic- tive Control [SSM16]. In discrete-time settings, **Stochastic** Dynamic Programming [IMM14; Wu+16] and **Stochastic** Dual Dynamic Programming [Pac+18] are popular approaches to get theoretical optimality guarantees. Long-term aging of the battery equipping a micro-grid is taken into account by two-time scales time decomposition in [Car+19]. Continuous time optimal control problems are considered in [Hey+15] in a deterministic setting, and in [Hey+16] in a **stochastic** environment. By jointly optimizing with the profile P grid ,com. , we change the nature of the **stochastic** control problem, compared to these works. We shall consider general filtrations with **processes** possibly exhibiting jumps, to account for sudden variations of solar irradiance or consumption for instance.

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To the best of our knowledge, this is the first time that **McKean**-**Vlasov** limits are considered where the underlying driving martingale measures are only present in the limit system, but not at the level of the N particle system. We refer however to Chevallier and Ost ( 2020 ) who work in the averaging regime and study the fluctuations of a **stochastic** system, associated to spatially structured Hawkes **processes**, around its mean field limit, and where particles in the mean field limit are still independent.

then it is clear that the composition ˜ α ε (ξ) belongs to A, that is ˜ α ε (ξ) is an F-progressively measurable process (as a matter of fact, for every i, both α ε i and the indicator function 1 {ξ∈B i } are F-progressively measurable **processes**). For a general Borel measurable function, the result follows by an approximation argument. ✷ Remark 4.1 The extension of Proposition 4.1 to the case of two-player zero-sum **stochastic** differential games presents some difficulties, as we now explain. Consider a standard two-player zero-sum **stochastic** differential game, firstly studied in the seminal paper [15], where all coeffi- cients depend only on the state and controls, but not on their probability laws:

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differentiable with bounded derivatives. The mapping φ is an at least Lipschitz function from R d to R whose precise regularity is given below.
**McKean** **Vlasov** **processes** may be regarded as a limit approximation for interacting systems with large number of particles. They appeared initially in statistical mechanics, but are now used in many fields because of the wide range of applications requiring large populations interactions. For example, they are used in finance, as factor **stochastic** volatility models [ Ber09 ] or uncertain volatility models [ GHL11 ]; in economics, in the theory of “mean field games” recently developed by J.M. Lasry and P.L. Lions in a series of papers [ LL06a , LL06b , LL07a , LL07b ] (see also [ CDL12 , CD12a , CD12b ] for the probabilistic counterpart) and also in physics, neuroscience, biology, etc. In section 5 , we present a class of control problems in which equation ( 1.1 ) explicitly appears.

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Chapter I introduces an alternative construction, by smooth approximations, of the particle system defined by Konarovskyi and von Renesse, hereinafter designed by coalesc- ing model. The coalescing model is a random process with values in the Wasserstein space, following an Itô-like formula on that space and whose short-time deviations are governed by the Wasserstein metric, by analogy with the short-time deviations of the standard Brownian motion governed by the Euclidean metric. The regular approximation con- structed in this thesis shares those diffusive properties and is obtained by smoothing the coefficients of the **stochastic** differential equation satisfied by the coalescing model. The main benefit of this variant is that it satisfies uniqueness results which are still open for the coalescing model. Moreover, up to small modifications of its structure, that smooth diffusion owns regularizing properties: this is precisely the object of study of chapters II to IV. In chapter II, an ill-posed **McKean**-**Vlasov** equation is perturbed by one of those smooth versions of the coalescing model, in order to restore uniqueness. A connection is made with recent results (Jourdain, Mishura-Veretennikov, Chaudru de Raynal-Frikha, Lacker, Röckner-Zhang) where uniqueness of a solution is proved when the noise is finite dimensional and the drift coefficient is Lipschitz-continuous in total variation distance in its measure argument. In our case, the diffusion on the Wasserstein space allows to mollify the velocity field in its measure argument and so to handle with drift functions having low regularity in both space and measure variables. Lastly, chapters III and IV are dedicated to the study, for a diffusion defined on the Wasserstein space of the circle, of the smoothing properties of the associated semi-group. Applying in chapter III the differential calculus on the Wasserstein space introduced by Lions, a Bismut-Elworthy inequality is obtained, controlling the gradient of the semi-group at those points of the space of probability measures that have a sufficiently smooth density. In chapter IV, a better explosion rate when time tends to zero is established under additional regularity conditions. This leads to a priori estimates for a PDE defined on the Wasserstein space and governed by the diffusion on the torus mentioned above, in the homogeneous case (chapter III) and in the case of a non-trivial source term (chapter IV).

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∂α (α, x s (α)) 2
ds. Up to our knowledge, these statistical results are also new for classical **stochastic** differential equations. Indeed, for ergodic diffusion **processes** with fixed diffusion term ε, the rate of con- vergence for α is √ T as T tends to infinity, while on a fixed time interval [0, T ], as ε tends to 0, the rate of estimation for α is ε −1 . With the double asymptotics ε → 0 and T → +∞, it is unexpected to obtain a rate of convergence for α which is either ε −1 √ T or ε −1 . This distinction depends on the fact that the fixed point x ∗ (α) of the ODE depends on α or not.

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measure on P 2 ([0, 1]). Interestingly, the dynamics of (µ t ) t∈[0,T ] are similar to the dynamics of a
standard Brownian motion, in the sense that the large deviations in small time are given by the Wasserstein distance W 2 and the martingale term that arises when expanding any smooth func- tion ϕ of the measure argument along the process has exactly the square norm of the Wasserstein gradient of ϕ as local quadratic variation. **Stochastic** **processes** owning those diffusive features are various and several were studied in recent years. We decide in this paper to construct a diffusion inspired by the nice model of coalescing particles called Modified Massive Arratia flow: in [Kon11, Kon17b], Konarovskyi introduces a diffusion model on P 2 (R) consisting in a mod- ification of Arratia’s system of coalescing particles on the real line. To wit, in Konarovskyi’s model, each particle carries a mass determining its quadratic variation and moves independently of the other particles as long as it does not collide with another. To make it clear, at each col- lision between two particles, both particles stick together and form a unique new particle with a mass equal to the sum of the masses of both incident particles. At each time, the quadratic variation increment of a particle is given by the inverse of its mass. That model satisfies in- teresting properties, studied by Konarovskyi and von Renesse in [Kon17b, Kon17a, KvR18], including an Itô-like formula and a Varadhan-like formula, with the Wasserstein distance W 2 playing the analogous role of the Euclidean metric for the standard Brownian motion. Moreover, those dynamics have a canonical representation as a process of quantile functions (or increasing rearrangement functions) (y t ) t∈[0,T ] : ∀u ∈ [0, 1], y t (u) := sup{x ∈ R : µ t ((−∞, x]) 6 u}.

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Decisiveness. Similarly to GSMP, STA are not decisive in general. Adapting an example from [9], one can indeed exhibit an STA which is not ID w.r.t. a given region-closed set. Still, under some fairness property, one can build an abstraction of the STA, that is sound for the almost-sure model checking of LTL properties and the-like [9]. It turns out that this fairness assumption ensures that the same abstraction is also sound for decisiveness. As for GSMPs, we consider the natural region abstraction, and we define a finite-state Markov chain Y A , in which the states are regions, and there is a transition from one region r to another r 0 as soon as there exists a configuration γ in r from which the probability to reach r 0 in one step in X A is positive. As mentioned earlier, as such, the abstraction Y A is not sound in general. Yet, it preserves almost-sure satisfaction of LTL properties when the **stochastic** timed automaton is almost-surely fair [9]. Here fairness refers as the following property, which depends on

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the representative bank, and α is the rate of borrowing/lending to a central bank.
In the literature **McKean**-**Vlasov** control problem is tackled by two different approaches: On the one hand, the **stochastic** Pontryagin maximum principle allows one to characterize solutions to the controlled **McKean**-**Vlasov** systems in terms of an adjoint backward **stochastic** differential equation (BSDE) coupled with a forward SDE: see [1], [8] in which the state dynamics depend upon moments of the distribution, and [13] for a deep investigation in a more general setting. On the other hand, the dynamic programming (DP) method (also called Bellman principle), which is known to be a powerful tool for standard Markovian **stochastic** control problem and does not require any convexity assumption usually imposed in Pontryagin principle, was first used in [24] and [5] for a specific **McKean**-**Vlasov** SDE and cost functional, depending only upon statistics like the mean of the distribution of the state variable. These papers assume a priori that the state variables marginals at all times have a density. Recently, [26] managed to drop the density assumption, but still restricted the admissible controls to be of closed-loop (a.k.a. feedback) type, i.e., deterministic and Lipschitz functions of the current value of the state, which is somewhat restrictive. This feedback form on the class of controls allows one to reformulate the **McKean**- **Vlasov** control problem (1.4) as a deterministic control problem in an infinite dimensional space with the marginal distribution as the state variable. In this paper we will consider the most general case and allow the controls to be open-loop. In this case reformulation mentioned above is no more possible. We will instead work with a proper disintegration of the value function, which we described in (1.4). The disintegration formula (1.5) was pointed out heuristically in [12], see their formulae (40) and (41), but the value function V was not identified. The idea of formulating the **McKean**-**Vlasov** control problem as in (1.3) (rather than as in (1.4)) is inspired by [9], where the uncontrolled case is addressed. We will then generalize the randomization approach developed by [21] to the **McKean**-**Vlasov** control problem corresponding to V .

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Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - [r]

Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - [r]

As already said, Equation (1.2) can be thought of as of **McKean**-**Vlasov**-type, since the process (X t ) t > 0 depends on the distribution of the solution itself. However, it is
highly non-standard, since it actually depends on the distribution of the first hitting times of the threshold by the solution. This renders the traditional approaches to **McKean**-**Vlasov** equations and propagation of chaos, such as those presented by Sznitman in [20], inapplicable, because we have no a priori smoothness on the law of the first hitting times. Thus our results are also new in this context.

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observations are attained by static observers (i.e., the agents are residing on a fixed underlying graph) and that the measured target is external to the system. In contrast, here we consider a self-organizing system, with mobile agents that measure a target (center of mass) that is a function of their positions.
The study of flocking was originally based on computer simulations, rather than on rigorous analysis [7, 2, 27]. In recent years, more attention has been given to such self-organizing **processes** by control theoreticians [20, 18], physicists [28], and computer scientists [5]. Instead of considering all components of flocking (typically assumed to be attraction, repulsion, and alignment), here we focus on the alignment component, and the ability to reach cohesion while avoiding excessive communication.

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choose as mimicking process ξ itself (or at least a copy of ξ with the same law as ξ). However, some of these constructions fail to have this property, so they cannot be qualified as a ’projection’ on the set of Markov **processes**.
Finally, many of these ad-hoc constructions fail to conserve some simple qualitative properties of ξ. For example, if ξ is a continuous process, one expects to be able to mimic it with a continuous Markov process X (preser- vation of continuity). Also, in reference to the martingale construction above, one can show that it is possible to mimic a (local) martingale ξ with a Marko- vian (local) martingale X (preservation of the martingale property). These properties are more naturally viewed in terms of the local characteristics of the process ξ rather than properties of the marginal distributions. This sug- gests a more natural, and more general approach for constructing X from the local characteristics of ξ, when ξ is a semimartingale.

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2. Write the equations that relate the mean function, the autocorrelation function, and the power spectral density function of {U c (t), t ∈ R } to those of {W (x), x ∈ R } .
3. Express dynamical equilibrium and deduce the **stochastic** differential equation that governs the displacement {U(t), t ∈ R } of the vehicle body.

2 SPATIO-TEMPORAL DEGRADATION MODEL
We look for modeling the degradation process by a convenient spatio-temporal random field to take into account its aleatory evolution with time and space. A separable model is one simple spatio-tem- poral model obtained through the tensorial product between a merely **stochastic** process ( X t t ≥0 ) and a spatial random field Z(z), where z is the spatial vari- able. This class of separable random field is exten- sively used even in situations in which they are not always physically justifiable, since separability gives important computational and mathematic benefits.

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