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ity criterium. More precisely, it consists in dening reduced approximation spaces V M
and S M such that they verify simultaneously **equations** (69) and (71). One then shows
that reduced basis are solution **of** an invariant subspace problem. This problem can be assimilated to an eigenproblem whose dominant eigenspace leads to **the** researched reduced basis functions. **The** GSD method has been initially introduced **for** **solving** a particular class **of** linear elliptic **stochastic** partial dierential **equations** [89]. In this context, **the** method appears as a natural extension **of** Hilbert Karhunen-Loève decom- position (see appendix B). Dedicated algorithms, inspired from classical algorithms **for** **solving** eigenproblems, have been proposed **for** **the** construction **of** reduced basis func- tions. **The** main advantage **of** these algorithms is that they only require **the** resolution **of** a few deterministic problems, with a well mastered mathematical structure, and **of** a few **stochastic** algebraic **equations**. Computational costs are then drastically reduced. Moreover, **stochastic** **equations** and deterministic problems being uncoupled, **the** GSD method allows **for** recovering a part **of** non intrusivity **for** Galerkin spectral approaches. In [90], **the** method has been used **for** **solving** a nonlinear **stochastic** elliptic problem **for** which a classical **global** nonlinear solver led to **the** resolution **of** successive linear **stochastic** problems. Each linear **stochastic** problem were solved by GSD algorithms proposed in [90], with a re-use and an enrichment **of** **the** reduced basis **of** deterministic functions at each iteration **of** **the** nonlinear solver. **The** GSD method has been extended to a wider class **of** linear problems in [91], where it has been also proposed some new ecient algorithms **for** building **the** generalized decomposition. More recently, a natural extension to **the** non-linear context has been proposed in [94].

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Fig. 15. Example 1. Integration precision: convergence **of** **global** **error** indicator ε k with respect to k **for** isotropic and anisotropic procedures.
7.1.3 Quality **of** **the** X-SFEM solution
Some examples **of** post-processing will now illustrate **the** quality **of** **the** solution obtained by X-SFEM. **The** solution being explicit in terms **of** basic random variables, post-processing can be performed at a very low cost. **The** following results have been obtained with **the** mesh presented on figure 13 and with a **stochastic** integration based on a **stochastic** partition **of** order k = 2. Figure 16 presents **the** response surfaces **of** **the** horizontal displacement **for** two points: x 1 = (1, 0.5) which is surely inside **the** random domain and x2 = (1, 0.95) which is possibly inside or outside **the** random domain. **For** both points, we observe that **the** approximation matches very well **the** exact physical solution. **For** point x2, we observe that **the** non-physical part **of** **the** solution is **the** unique possible prolongation (in **the** approximation space) **of** **the** physical part **of** **the** solution.

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is directly sought as a polynomial function **of** a Gaussian process (G t ) t∈[0,T ]
and **the** coefficients **of** **the** **expansion** are obtained by **solving** ordinary differen- tial **equations** (ODE). **The** macroscopic quantities (mean, variance,...) can be computed directly from **the** coefficients **of** **the** decomposition. Thus, two major drawbacks **of** competing methods can be avoided: **the** curse **of** dimensionality **for** Polynomial chaos expansions and **the** resort to Monte-Carlo simulations, with slow convergence, **for** Euler-like or exact sampling techniques. Assuming that **the** ODEs can be solved with high degree **of** accuracy, then **the** **error** **of** our method mainly comes from **the** interpolation **error**.

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In this paper, we combine different ideas from **the** mathematics (**numerical** probability) and **the** computer science (reinforcement learning) communities to propose and compare several algo- rithms based on dynamic programming (DP), and deep neural networks (DNN) **for** **the** approxima- tion/learning **of** (i) **the** optimal policy, and then **of** (ii) **the** value function. Notice that this differs from **the** classical approach in DP recalled above, where we first approximate **the** Q-optimal state/control value function, and then approximate **the** optimal control. Our learning **of** **the** optimal policy is achieved in **the** spirit **of** [ HE16 ] by DNN, but sequentially in time though DP instead **of** a **global** learning over **the** whole period 0, . . . , N − 1. Once we get an approximation **of** **the** optimal policy, and recalling **the** martingale property ( 5.1.3 ), we approximate **the** value function by Monte-Carlo (MC) regression based on simulations **of** **the** forward process with **the** approximated optimal control. In particular, we avoid **the** issue **of** a priori endogenous simulation **of** **the** controlled process in **the** classical Q-approach. **The** MC regressions **for** **the** approximation **of** **the** optimal policy and/or value function, are performed according to different features leading to algorithmic variants: Performance iteration (PI) or hybrid iteration (HI), and regress now or regress later/quantization in **the** spirit **of** [ LS01 ] or [ GY04 ]. **Numerical** results on several applications are devoted to a companion paper [ BHLP18 ]. **The** theoretical contribution **of** **the** current paper is to provide a detailed convergence analysis **of** our three proposed algorithms: Theorem 5.4.1 **for** **the** NNContPI Algo based on control learning by performance iteration with DNN, Theorem 5.4.2 **for** **the** Hybrid-Now Algo based on control learning by DNN and then value function learning by regress-now method, and Theorem 5.4.3 **for** **the** Hybrid-LaterQ Algo based on control learning by DNN and then value function learning by regress later method combined with quantization. We rely mainly on arguments from statistical learning and non parametric regression as developed notably in **the** book [ GKKW02 ], **for** giving estimates **of** approximated control and value function in terms **of** **the** universal approximation **error** **of** **the** neural networks, and **of** **the** statistical **error** in **the** estimation **of** network functions.

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If another phenomenon acts on **the** same constituent,
**the** corresponding interaction can only act before or after **the** previous one. If this last phenomenon is modeled by PDEs with its own **numerical** scheme, **the** resulting **numerical** resolution **of** both phenomena within asynchronous iterations can be viewed as a sort **of** **stochastic** splitting method applied to both **schemes**. Another way to solve this problem could be to write a new single scheme valid **for** **the** PDE system obtained by both phenomena; this is not very accurate in terms **of** structural coupling, thus moves away from **the** MIS principles **for** complex system modeling. Anyway, if one **of** **the** phenomena involved in **the** complex sys- tem and acting on such a constituent is not modeled by PDEs, its intrication with PDE-based models won’t be possible **for** time intervals smaller than **the** time step

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Our initial motivation **for** **the** study **of** **the** **error** ε n (f ) comes from **the** second point **of** view
(**numerical** probabilities) but all **the** results **of** this paper are significant from both perspectives. Let us mention that **the** difficulty **of** **the** analysis and **the** interest **of** **the** result depend on **the** regularity **of** **the** test function f. It turns out that if f is a smooth function, then **the** analysis **of** **the** **error** is rather simple, using a Taylor type **expansion** in short time first, and a concatenation argument after. However, **the** study is much more subtle if f is simply a bounded and measurable test function - this is **the** so called convergence in total variation distance. A lot **of** work has been done in this direction in **the** case **of** **the** CLT. In particular, Bhattacharya and Rao [15] obtained **the** convergence when f (x) = 1 A (x) where A is a measurable set that belongs to a large class (including convex sets). From that point, one would hope to get such results **for** every measurable set A and consequently **for** every measurable and bounded test function f. Eventually, **the** seminal result **of** Prokhorov [62] clarified this point: He proved that **the** convergence in total variation in **the** CLT may not be obtained without some regularity assumptions on **the** law **of** Z k . Essentially, one has to assume that **the** law **of** Z k has an absolute

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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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Title: Weak **error** analysis **for** time and particle discretizations **of** some **stochastic** **differential** **equations** non linear in **the** sense **of** McKean
Summary: This thesis is dedicated to **the** theoretical and **numerical** study **of** **the** weak **error** **for** time and particle discretizations **of** some **Stochastic** **Differential** **Equations** non linear in **the** sense **of** McK- ean. In **the** first part, we address **the** weak **error** analysis **for** **the** time discretization **of** standard SDEs. More specifically, we study **the** convergence in total variation **of** **the** Euler-Maruyama scheme applied to d-dimensional SDEs with additive noise and a measurable drift coefficient. We prove weak convergence with order 1/2 when assuming boundedness on **the** drift coefficient. By adding more regularity to **the** drift, namely **the** drift has a spatial divergence in **the** sense **of** distributions with ρ-th power integrable with respect to **the** Lebesgue measure in space uniformly in time **for** some ρ ≥ d, **the** order **of** convergence at **the** terminal time improves to 1 up to some logarithmic factor. In dimension d = 1, this result is pre- served when **the** spatial derivative **of** **the** drift is a measure in space with total mass bounded uniformly in time. In **the** second part **of** **the** thesis, we analyze **the** weak **error** **for** both time and particle discretizations **of** two classes **of** nonlinear SDEs in **the** sense **of** McKean. **The** first class consists in multi-dimensional SDEs with regular drift and diffusion coefficients in which **the** dependence in law intervenes through moments. **The** second class consists in one-dimensional SDEs with a constant diffusion coefficient and a singular drift coefficient where **the** dependence in law intervenes through **the** cumulative distribution function. We approximate **the** SDEs by **the** Euler-Maruyama **schemes** **of** **the** associated particle systems and obtain **for** both classes a weak order **of** convergence equal to 1 in time and particles. We also prove, **for** **the** second class, a trajectorial propagation **of** chaos result with optimal order 1/2 in particles as well as a strong order **of** convergence equal to 1 in time and 1/2 in particles. All our theoretical results are illustrated by **numerical** experiments.

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In a population context, there exist four estimation methods: Hermite **expansion** **of** **the** marginal likelihood, **the** combination **of** FOCE with **the** extended Kalman filtering and **the** use **of** a **stochastic** version **of** **the** EM algorithm and Bayesian approach.
Note that Hermite **expansion** can not be directly extended when **the** diffusion is observed with noise whereas Kalman filter, Extended Kalman filter and FOCE methods are strictly designed **for** models with observation noise. Methods relying on **the** EM algorithm can be applied to models with or without noise, provided **the** complete likelihood belongs to **the** exponential family. Bayesian approach is finally **the** most adaptable method since it can be applied to any model (one or several trajectories, with or without noise observation) as soon as **the** transition density is explicit. If not, a Euler approximation has to be performed and **the** introduction **of** set **of** auxiliary latent data

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Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE` S NANCY Unite´ de recherche INRIA Rennes, Ir[r]

Abstract
In this paper, we study existence and uniqueness to multidimensional Reflected Backward **Stochastic** **Differential** **Equations** in an open convex domain, allowing **for** oblique directions **of** reflection. In a Markovian framework, combining a priori estimates **for** penalised **equations** and compactness arguments, we obtain existence results under quite weak assumptions on **the** driver **of** **the** BSDEs and **the** direction **of** reflection, which is allowed to depend on both Y and Z. In a non Markovian framework, we obtain existence and uniqueness result **for** direction **of** reflection depending on time and Y . We make use in this case **of** stability estimates that require some smoothness conditions on **the** domain and **the** direction **of** reflection.

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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 Futurs : Parc Club Orsay Université - ZAC des Vi[r]

m b+1 −x b+1 , and conditionally on M y→x **the** time is takes to do **the** down-crossing scales like λ −2 and in
particular its law converges to that **of** **the** zero random variable. We shall not give **the** details because this is intuitive given **the** second computation. Let us note however that making this rigorous would not be difficult: we could describe **the** down-crossing from y to x with maximum m as **the** concatenation **of** two independent processes, **the** first is X started at y and conditioned to reach m before x, and **the** second is X stated at m and conditioned to reach x before touching m again. Both pieces are described as diffusions via Girsanov’s theorem, **the** additional drift term due to conditioning being explicit in terms **of** P [x,m] (y). **For** both pieces **the** trajectories

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Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Ir[r]

ically with a wide class **of** PDEs which admit steady states. **The** method is based on an underlying **numerical** discretization and is capable to preserve **the** steady states by keeping **the** accuracy **of** **the** underlying method. Therefore, **the** approach can be used also in conjunction with spectral techniques **for** which **the** direct construction **of** a steady state preserving approach is often impossible. Thanks to its structural simplicity it can be applied to very different PDEs, examples ranging from linear and nonlinear diffusion **equations** to Boltzmann **equations** and hyperbolic balance laws have been presented and show **the** effectiveness **of** **the** residual equilibrium approach. In principle **the** methods admits several improvement when applied to a specific kind **of** PDE, **for** example in order to preserve some peculiar property **of** **the** **differential**

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Additional information is available at **the** end **of** **the** chapter http://dx.doi.org/10.5772/intechopen.73183
Abstract
In this chapter, we consider a periodic SDE in **the** dimension n ≥ 2, and we study **the** existence **of** periodic solutions **for** this type **of** **equations** using **the** Massera principle. On **the** other hand, we prove an analogous result **of** **the** Massera’s theorem **for** **the** SDE considered. Keywords: **stochastic** **differential** **equations**, periodic solution, Markov process, Massera theorem

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