The main diculty about **equations** like (0.1) is the well-known fact, even in the deterministic case, that there are no global smooth solutions in general. Moreover, the fully **nonlinear** character of the **equations** seems to make them inaccessible to the classical martingale theory employed for the linear case. Finally, even when smooth solutions may exist, the **equations** can not be described in a pointwise sense, because of the everywhere lack of dierentiability of the Brownian motion. In the deterministic case the lack of regularity was overcome with the introduction by Crandall and Lions [CL] of the notion of viscosity solutions | we refer to [CIL], [B], [FS] and [BCESS] for an up-to-last year overview of the theory of viscosity solutions and their applications in the deterministic setting.

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on [0,T] to (x t (α)), solution of (10). Let us consider the problem of estimating (α, β)
from a continuous observation (X t , t ∈ [0, T ]). In classical **stochastic** **differential** **equations**
with small diffusion coefficient, all drift parameters have rate ε −1 . Here, the situation is different since we observe that the parameter β is no longer present in the limiting ODE (10). We show that α can be consistently estimated as ε tends to 0, but not β.

The extension to fully **nonlinear** PDE, motivated in particular by uncertain volatility model and more generally by **stochastic** control problem where control can affect both drift and diffusion terms of the state process, generated important recent developments. Soner, Touzi and Zhang [101] introduced the notion of second order BSDEs (2BSDEs), whose basic idea is to require that the solution verifies the equation P α a.s. for every pro- bability measure in a non dominated class of mutually singular measures. This theory is closely related to the notion of **nonlinear** and G-expectation of Peng [89]. Alternatively, Kharroubi and Pham [75], following [74], introduced the notion of BSDE with nonposi- tive jumps. The basic idea was to constrain the jumps-component solution to the BSDE driven by Brownian motion and Poisson random measure, to remain nonpositive, by ad- ding a nondecreasing process in a minimal way. A key feature of this class of BSDEs is its formulation under a single probability measure in contrast with 2BSDEs, thus avoiding technical issues in quasi-sure analysis, and its connection with fully **nonlinear** HJB equa- tion when considering a Markovian framework with a simulatable regime switching dif- fusion process, defined as a randomization of the controlled state process. This approach opens new perspectives for probabilistic scheme for fully **nonlinear** PDEs as currently investigated in [73].

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Keywords: backward doubly **stochastic** **differential** **equations**, **stochastic** partial **differential** **equations**, monotone condition, singular terminal data.
Introduction
Backward Doubly **Stochastic** **Differential** **Equations** (BDSDEs for short) have been intro- duced by Pardoux and Peng [35] to provide a non-linear Feynman-Kac formula for classical solutions of SPDE. The main idea is to introduce in the standard BSDE a second **nonlinear** term driven by an external noise representing the random perturbation of the **nonlinear** SPDE. Roughly speaking, the BSDE becomes:

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Second order BSDEs were introduced by Cheredito, Soner, Touzi and Victoir in [13]. In a Markovian framework, they show that there exists a connection between 2BSDEs and fully **nonlinear** PDEs while standard BS- DEs induce quasi-linear PDEs. However, except in the case where the PDEs admits sufficiently regular solutions, they do not provide a general existence result. In [24], Denis and Martini generalized the uncertain volatility model introduced in [1] or [59] to a family of martingale measures thanks to the quasi-sure analysis. The uncertain volatility model is directly linked to the Black-Scholes-Barrenblat equation which is fully **nonlinear**. This problem is strongly linked to the problem of G-integration theory studied mainly by Peng (see [68], [67]) for the definition of the main properties. Denis, Hu and Peng in [23] established connections between [68] and [24] while Soner, Touzi and Zhang in [83] provide a martingale representation theorem for the G-martingale which corresponds to a hedging strategy in the uncer- tain volatility model. Inspired by this quasi-sure framework, Soner, Touzi and Zhang study in [85] the second order **stochastic** target problem whose solution solves a 2BSDE and prove existence and uniqueness for general 2BSDEs in [86] with an undominated family of mutually singular martin- gale measures. Recently, Possamai and Zhou extend their results for a one dimensional 2BSDE with bounded terminal condition and continuous gen- erator with quadratic growth in the z variable (see Possamai [75], Possamai and Zhou [76]). This result allow to solve second order reflected BSDEs and utility maximization problem under volatility uncertainty as we can see in

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The existence of periodic solutions for **differential** **equations** has received a particular interest. We quote the famous results of Massera [9]. In its approach, Massera was the first to establish a relation between the existence of bounded solutions and that of a periodic solution for a **nonlinear** ODE.

CONTROLLED MCKEAN VLASOV DYNAMICS
REN ´ E CARMONA AND FRANC ¸ OIS DELARUE
A BSTRACT . The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of **nonlinear** **stochastic** dynamical systems of the McKean Vlasov type. Motivated by the recent interest in mean field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the **stochastic** maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large **stochastic** games with mean field interactions.

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1. Introduction
We are interested in finite elements approximations for Cauchy problems for **stochastic** parabolic PDEs of the form of equation (2.1) below. Such kind of **equations** arise in various fields of sciences and engineering, for example in **nonlinear** filtering of partially observed diffusion processes. Therefore these **equations** have been intensively studied in the litera- ture, and theories for their solvability and numerical methods for approximations of their solutions have been developed. Since the computational effort to get reasonably accurate numerical solutions grow rapidly with the dimension d of the state space, it is important to investigate the possibility of accelerating the convergence of spatial discretisations by Richardson extrapolation. About a century ago Lewis Fry Richardson had the idea in [18] that the speed of convergence of numerical approximations, which depend on some parameter h converging to zero, can be increased if one takes appropriate linear combina- tions of approximations corresponding to different parameters. This method to accelerate the convergence, called Richardson extrapolation, works when the approximations admit a power series expansion in h at h = 0 with a remainder term, which can be estimated by a higher power of h. In such cases, taking appropriate mixtures of approximations with different parameters, one can eliminate all other terms but the zero order term and the remainder in the expansion. In this way, the order of accuracy of the mixtures is the exponent k + 1 of the power h k+1 , that estimates the remainder. For various numerical methods applied to solving deterministic partial **differential** **equations** (PDEs) it has been proved that such expansions exist and that Richardson extrapolations can spectacularly increase the speed of convergence of the methods, see, e.g., [16], [17] and [20]. Richard- son’s idea has also been applied to numerical solutions of **stochastic** **equations**. It was shown first in [21] that by Richardson extrapolation one can accelerate the weak conver- gence of Euler approximations of **stochastic** **differential** **equations**. Further results in this direction can be found in [14], [15] and the references therein. For **stochastic** PDEs the first result on accelerated finite difference schemes appears in [7], where it is shown that

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The assumptions on A are made for simplicity and can be relaxed. Moreover, since the coefficients of the noise in (2.1) are deterministic, the question of whether we need to use Itˆ o’s or Stratonovich **stochastic** **differential** does not arise here.
In the context of (2.1), a process is stationary, if it is adapted to the filtration generated by the (B k )

allowance price equals the marginal abatement cost, and market participants implement all the abate- ment measures whose costs are not greater than the cost of compliance (i.e. the equilibrium price of an allowance).
The next section puts together the economic activities of a large number of producers and search for the existence of an equilibrium price for the emissions allowances. Such a problem leads naturally to a forward **stochastic** **differential** equation (SDE) for the aggregate emissions in the economy, and a backward **stochastic** **differential** equation (BSDE) for the allowance price. However, these equa- tions are ”coupled” since a **nonlinear** function of the price of carbon (i.e. the price of an emission allowance) appears in the forward equation giving the dynamics of the aggregate emissions. This feedback of the emission price in the dynamics of the emissions is quite natural. For the purpose of option pricing, this approach was described in[4] where it was called detailed risk neutral approach. Forward backward **stochastic** **differential** **equations** (FBSDEs) of the type considered in this sec- tion have been studied for a long time. See for example [12], or [16]. However, the FBSDEs we need to consider for the purpose of emission prices have an unusual pecularity: the terminal condition of the backward equation is given by a discontinuous function of the terminal value of the state driven by the forward equation. We use our first model to prove that this lack of continuity is not an issue when the forward dynamics are strongly elliptic, in other words when the volatility of the forward SDE is bounded from below. However, using our second equilibrium model, we also show that when the forward dynamics are degenerate (even if they are hypoelliptic), discontinuities in the terminal con- dition and lack of uniform ellipticity in the forward dynamics can conspire to produce point masses in the terminal distribution of the forward component, at the locations of the discontinuities. This implies that the terminal value of the backward component is not given by a deterministic function of the forward component, for the forward scenarios ending at the locations of jumps in the terminal condition, and justifies relaxing the definition of a solution of the FBSDE.

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such that P ≡ P µ,σ
x 0 is the probability distribution of the unique strong solution to the **stochastic** **differential** with coefficients µ and σ and initial condition x 0 .
The definition of the following Wasserstein-type distance on the set P results from the obvious but important observation that not any coupling measure of two probability distributions in P can be represented as the solution of a 2d- dimensional martingale problem. We thus modify the definition of the standard W 2 distance by restricting the set of possible coupling measures.

Another approach to strong existence and pathwise uniqueness was re- cently initiated by Le Bris and Lions in [18, 19], based on well-posedness re- sults for the backward Kolmogorov equation. The authors define the notion of almost everywhere **stochastic** flows for (1.1), which combines existence and a flow property for almost all initial conditions, and give precise results in the case where a = Id. The general case was recently studied deeply by P.-L. Lions in [21], who reduces the question to well-posedness, L 1 norms and stability properties for two backward Kolmogorov **equations**; the first one associated to the SDE (1.1) and the other one obtained by a doubling of variable technique. Note that this approach does not require assump- tions of uniform ellipticity for a. In [19], the authors also define a **stochastic** transport equation whose solutions are in correspondence with the stochas- tic flow. This approach was also used in [30], where the existence of almost everywhere **stochastic** flows was obtained for divF and ∇σ bounded, for ∇F in L log L and with some bounds of ∇(divσ), but without any assumption of uniform ellipticity for σ.

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Dans cette partie de notre travail, nous mettons en oeuvre les techniques du calcul de Malliavin combinées à la méthode de Stein aﬁn de determiner, dans un cadre gaussien, des bornes de [r]

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values in the set of orthogonal matrices and converges to C ∗ (s, x, x) in L p -norm.
The results below do not bring further information, either to the distance f W 2 ,
or to its **stochastic** control representation. However, it may be interesting for numerical purposes, notably if one is interested in simulating a simplified model by using a standard discretization method such as the Euler scheme whose con- vergence requires the coefficients are at least continuous.

If the observability inequality ( 9 ) is not satisfied, then the decay of the energy cannot be exponential, however, it may be polynomial in some cases. It may be so for instance for some weakly damped wave **equations** in the absence of geometric control condition (see [ 12 ]) but also for indirect stabilization for coupled systems, that is when certain **equations** are not directly stabilized, even though the usual geometric conditions are satisfied (see [ 2 , 3 , 4 ]). In that case, it would be of interest to establish a uniform polynomial decay rate for space and/or time semi-discrete and full discrete approximations of ( 1 ). In [ 1 ], such results are stated for second-order linear **equations** (certain examples being taken from [ 3 , 4 ]) , with appropriate viscosity terms, and under adequate spectral gap conditions.

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2 UMAB University of Mostaganem
LPAM, Faculty SEI, UMAB University of Mostaganem, Algeria
Abstract In this paper, we study a **nonlinear** coupled system of n−fractional dif- ferential **equations**. Applying Banach contraction principle and Schaefer’s fixed point theorem, new existence and uniqueness results are established. We also give some concrete examples to illustrate the possible application of the established analytical results.

where B is a Brownian motion, h is a function from R n to R, the law of the initial condition X
0 is such
that E[h(X 0 )] ≥ 0 and where, for the time being, n = d = 1 .
When focusing on this forward system, there are several ways to understand the mean reflected SDE. One striking example lies into the equation satisfied by the law of the solution X, which turns out to be a reflected Fokker-Planck equation. In other words, solving the above system translates into searching for solution to the Skorohod problem stated on Partial **Differential** Equation of Fokker-Planck type. Indeed, let (X, K) be a solution of the above system (which, according to [3], under suitable assumptions , exists, is unique and where the process K is supposed to be a deterministic increasing process starting from 0). By Itô’s formula, the law µ t = [X t ] of X t satisfies, in the sense of distributions,

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tailed. A particular care has been devoted to the numerical integration of the weak form, requiring the development of a specific quadrature technique at the **stochastic** level. For the numerical examples treated in this article, the proposed **stochastic** quadrature technique has given very good results since elementary matrices and vectors had a nice dependence on the random vari- ables. The case where these quantities have a more complex dependence on the random variables and also the case of higher **stochastic** dimension are key questions which are currently under investigation. These points will certainly require the development of simpler quadrature techniques and the use of the high degree of parallelism of the method in order to reduce computational costs.

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Our main purpose is thus to propose an efficient algorithmic estimation method of the vector of parameters θ together with theoretical convergence results. We consider an approximate statistical model, of which the regression term is the Euler-Maruyama discretized approximate diffusion process of the SDE. The parameter inference is then performed on this new model, using a **stochastic** version of the EM algorithm. Section 1 describes the setup of the problem which is considered in this paper, detailing the diffusion process and its Euler-Maruyama approximation. The estimation algorithm is presented in Section 2. This section details a tuned MCMC procedure supplying both theoretical and computational convergence properties. The error on the estimation induced by the Euler-Maruyama scheme is quantified in Section 3. In Section 4, the estimation algorithm is applied to a non-linear mixed effects model issued from pharmacokinetics. Section 5 concludes with some discussion.

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