Modeling the **term** **structure** of interest rates has long been an important topic in economics and finance. Most of the papers about modelling of the interest rate **term** **structure** are relative to the family of the Affine **Term** **Structure** **Models** (ATSM). This modelling considers a linear relation between the log price of a zero coupon bond and its states factors. Those **models** have been first studied by Vasicek (1977) in [ 13 ] and Cox, Ingersoll and Ross (1985) in [ 4 ]. Then developed by Duffie and Kan (1996) in [ 6 ] and Dai and Singleton (2000) in [ 5 ]. A first extension of this class of model was to use regime switching model. Thus, Elliott et al. (2011) in [ 8 ] considered a discrete-time, Markov, regime-switching, affine **term** **structure** model for valuing bonds and other interest-rate securities. Recently, Goutte and Ngoupeyou (2013) in [ 10 ] obtained explicit formulas to price defaultable bond under this class of regime switching **models**. The proposed model incorporates the impact of structural changes in economic conditions on interest rate dynamics and so can capture different economics (financials) levels or trends of the economy. A second extension was to not only consider a linear model. Thus to model the **term** **structure** of interest rates with Quadratic **Term** **Structure** **Models** (QTSM). This family, first introduce by Beaglehole and Tammey (1991) in [ 2 ] are applied to price contingent claims (Lieppold and Wu (2002) in [ 12 ]) and to the credit risk pricing (Chen, Filipovic and Poor (2004) in [ 3 ]). Hence, in this paper we propose to use both of the previous extension and so a regime switching discrete-time version of quadratic **term** **structure** **models** (RS-QTSM).

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The storage theory constitutes the main basis for the elaboration of **term** **structure** **models** of commodity prices. Indeed, it brings useful conclusions to construct such a model. First, the relationship between spot and futures prices allows the identification of at least three variables influencing the futures price: the spot price, the convenience yield, and the interest rate, which is implicitly included in the financing costs. Second, convenience yield and spot price are positively correlated: both of them are an inverse function of the stock level. Third, the examination of arbitrage relationships between physical and paper markets shows that the basis has an asymmetrical behavior: in contango, its level is limited to storage costs. This is not the case for backwardation. Furthermore, the basis is stable in contango, and volatile in backwardation, since in this situation stocks cannot absorb price fluctuations. This asymmetry has implications on the dynamic of convenience yield. These implications were exploited in the context of **term** **structure** **models**.

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C EREG1
A BSTRACT : A Kalman filter can be used for the estimation of a model’s parameters, when the model relies on non observable data. In finance, this kind of problem arises for example with **term** **structure** **models** of interest rates, **term** **structure** **models** of commodity prices, and with the market portfolio in the capital asset pricing model. The Kalman filter is also an interesting method when a large volume of information must be taken into account, because it is very fast. Last but not least, when associated with an optimization procedure, the filter provides a mean to obtain the model’s parameters. In a first section, this article exposes the basic principles of the method, shows how we can use it to estimate a model’s parameters, and presents two Kalman filters. The first one is the simple filter, which accepts only linear **models**. The second one, the extended filter, allows working with non-linear **models**. The second section is devoted to the application of the Kalman filter in finance. To explain how this method can be used in this field, we apply it to a very famous **term** **structure** model of commodity prices, and we discuss practical problems usually not mentioned in the literature, regarding the implementation of the method. The third section presents and compares the performances obtained with the two filters.

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The purpose of this study is to develop a uni¯ed framework in which all exponential{ a±ne **term** **structure** **models** can be estimated with the simultaneous use of many bond yield series. We utilize a result of Du±e and Kan (1993) which establishes the su±cient and necessary conditions for the obtention of an exponential{a±ne **term** **structure** model. These conditions simply require the drift and variance functions of the underlying di®usion process to be a±ne in state variables. In this paper, we establish that the conditional mean and variance function of the state variables, over any discrete{time interval, must also be a±ne in state variables. This result for the discrete{time interval makes it possible to use the Kalman ¯lter and the prediction{error decomposition to obtain an approximate quasi{ maximum likelihood solution to the estimation problem for the entire class of exponential a±ne **term** **structure** **models**.

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1 Introduction
Multifactor affine **term** **structure** **models** (ATSMs) have become the standard in the literature on the valuation of fixed income securities, such as government bonds, corporate bonds, interest rate swaps, credit default swaps, and interest rate derivatives. Even though we have made significant progress in specifying these **models**, their implementation is still subject to substantial challenges. One of the challenges is the proper identification of the parameters governing the dynamics of the risk premia (see Dai and Singleton (2002)). It has been recognized in the literature that the use of contracts that are nonlinear in the state variables, such as interest rate derivatives, can potentially help achieve such identification. Nonlinear contracts can also enhance the ability of affine **models** to capture time variation in excess returns and conditional volatility (see Bikbov and Chernov (2009) and Almeida, Graveline and Joslin (2011)).

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These statements are important for several reasons. Firstly, such a study constitutes a useful prerequisite for the elaboration of **term** **structure** **models** of commodity prices, especially for the crude oil market. Indeed, if two factors are sufficient to explain more than 99% of prices’ volatility, even when the whole curve is taken into account, then it is relevant to retain solely two underlying factors in a **term** **structure** model. Secondly, if the steepness factor has a significant impact for long-**term** maturities, then a long-**term** analysis should retain a mean-reverting behavior for at least one of the state variables of the model. Thirdly, understanding the prices’ dynamic is important for risk management. Indeed, principal component analysis can be used in order to quantify the risks of a

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The size of the predictability and nature of projection coeﬃcients in regressions like (22) is quite puzzling and, as Bansal, Tauchen, and Zhou (2004) state, “constitutes a serious challenge to **term** **structure** **models**.” Bansal, Tauchen, and Zhou account for the predictability evidence from the perspective of latent factor **term** **structure** **models**. They show that the regime-switching model of Bansal and Zhou (2002) can empirically account for these challenging features of the data, while aﬃne speciﬁcations cannot. In this section, we ask whether the risk premiums generated by our model (based on observable factors) can also account for the tent-shaped predictability pattern. To preview the results, it is only the non-expected utility model that can do so. Both the expected utility version of the equilibrium model and the reduced-form model fail to account for these important features. An important note is that the question is not whether one can construct market prices of risk that generate the return regressions in an aﬃne model. Cochrane and Piazzesi (2005) show exactly how that can be done. As with the Campbell-Shiller regressions, the question we ask is whether any of the considered **term** **structure** **models** can generate the required risk premiums given the speciﬁc set of parameter values that correctly ﬁt the data.

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The answers given to these questions have crucial implications for financial decisions, particularly for all the hedging and valuation operations relying on the relationship between different futures prices. It is the case, for example, of the “stack and roll” hedging strategies that rely on short **term** futures contracts to protect long-**term** positions on the physical market (in 1994, Metallgesellschaft tried to build such a strategy in order to exploit the higher liquidity of the nearest contracts). The efficiency of these strategies can be affected by differences in the informational content of futures prices. It is also the case of investment decision, when the latter is based on the extrapolation from observed prices curves to value cash flows for maturities that are not available on the market 1 . All these operations rely on **term** **structure** **models** of commodity prices. Such a tool aims

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4.3 Assessing Directional Accuracy of Forecasts
Another criterion according to which we can evaluate our **models** is the extent to which, for a given forecast horizon, they can on average correctly predict the sign of forecast change. Knowing how well **models** perform according to this dimension can add further useful in- formation to the information provided by the MSFE and MAPE criteria. We therefore test the null of no directional accuracy of forecasts for selected **models** and for specific forecast horizons. For this purpose, and given that multi-period forecasts are serially correlated, we make use of the methods proposed by Pesaran and Timmermann (2009). The test is applied in the context of a linear regression where, for a given model specification, the binary vari- able consisting of signs of oil price changes relative to the previous period is regressed on a constant and on the binary variable consisting of forecasted signs of oil price changes relative to the previous period. If, for a given model, the coefficient of the regression is found to be significant, then that model is interpreted as being able to, on average, accurately forecast the sign of the actual oil price change.

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The dotted curve is the output of the QSS model including the correction (20). The two responses differ mainly by the times at which shunt reactors are tripped. This difference is due to short-**term** transients, which are not captured by the QSS model. For instance, in the FTS simulation, the voltage spike at t = 36 s resets some LTCs (the controlled voltages re- entering the deadbands transiently) and delays their reaction. Since the voltage spike is not present in the QSS response, the LTCs move earlier in the latter, which causes the voltage to drop and, hence, the second MAIS to be triggered earlier as well.

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metrics for assessing the sensitivity of the carbon cycle to climate change. A next logical step for benchmarking ter- restrial carbon cycle **models** would be to use eddy covari- ance observations of flux‐climate relationships [Reichstein et al., 2007; Ito et al., 2008], global satellite information [Running, 2006], and manipulative experiments of warming or altered precipitation at ecosystem scale [Luo et al., 2008; Gerten et al., 2008]. Further, the definition of metrics should be extended to assess ocean carbon cycle **models** against observations, using, for instance, ocean flux observations [Takahashi et al., 2009], ocean interior inventories [Sabine et al., 2004], and atmospheric tracers such as atmospheric potential oxygen [Naegler et al., 2007]. Finally, the avail- ability of observational data over longer time series is essential in view of benchmarking **models** over the longest possible periods which is necessary, even if not sufficient, to build confidence in future projections.

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3 Equilibrium with full asset liquidation
One implication of our particular market **structure** is that available assets play the role of buﬀer stocks, since they allow agents to partially oﬀset the lack of full credit and insurance markets. However, many **models** of this class imply a smooth portfolio rebalancing in equilibrium: High income agents gradually build up their asset wealth, while low income ones decumulate assets at a suﬃciently slow pace not to ever actually hit the borrowing constraint (e.g. Scheinkman and Weiss (1986), or Aiyagari (1994)). Since we focus on the implications of the liquidation risk for the bond pricing, we construct our equilibrium in such a way that agents indeed liquidate assets when a bad idiosyncratic income shock hurts (i.e., b i t,k h t , e i,t = 0 for k = 1, . . . , n if e i t = 0). All low income agents face therefore a binding credit constraint, which cannot be oﬀset through bond liquidation. Incidentally, this full liquidation of bond holdings drastically reduces the number of agent types in the economy, thereby allowing us to study bond pricing analytically for an arbitrarily large number of maturities.

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March 19, 2018
Abstract. In [4, 5, 6], we have introduced the technique of classical realizability, which permits to extend the Curry-Howard correspondence between proofs and programs, to Zermelo-Fraenkel set theory. The **models** of ZF we obtain in this way, are called realizability **models** ; this technique is an extension of the method of forcing, in which the ordered sets (sets of conditions) are replaced with more complex first order structures called realizability algebras.

3 Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, Belgium
Abstract. Porosity-based shallow water **models** for the simulation of urban
floods incorporate additional energy dissipation terms compared to the usual two-dimensional shallow water equations. These terms account for head losses stemming from building drag. They are usually modelled using turbulence-based equations of state (drag proportional to the squared velocity). However, refined numerical simulations of wave propagation in periodic urban layouts indicate that such drag **models** do not suffice to reproduce energy dissipation properly. Correct wave propagation speeds, energy dissipation rates and flow fields are obtained by incorporating a new type of source **term**, active only under transient situations involving positive waves. This source **term** does not take the form of an equation of state. It can be modelled as an artificial increase in water inertia. In this communication, an experimental validation of this source **term** model is presented by means of new dam-break flow experiments in idealized, periodic urban layouts. The experimental results validate both the existence and the proposed formulation of this new source **term**.

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Abstract
We analyse the **term** **structure** of interest rates in a general equilibrium model with incom- plete markets, borrowing constraint, and positive net supply of government bonds. Uninsured idiosyncratic shocks generate bond trades, while aggregate shocks cause ‡uctuations in the trading price of bonds. Long bonds command a “liquidation risk premium” over short bonds, because they may have to be liquidated before maturity –following a bad idiosyncratic shock– precisely when their resale value is low –due to the simultaneous occurrence of a bad aggre- gate shock. Our framework endogenously generates limited cross-sectional wealth heterogeneity among the agents (despite the presence of uninsured idiosyncratic shocks), which allows us to characterise analytically the shape of the entire yield curve, including the yields on bonds of arbitrarily long maturities. Agents’ desire to hedge the idiosyncratic risk together with their fear of having to liquidate long bonds at unfavourable terms imply that a greater bond supply raises the level of the yield curve, while an increase in the relative supply of long bonds raises its slope.

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and Tang (2010) and Baker (2014). The **models** proposed in those papers, however, do not deal with the possibility that market frictions may prevent different parts of the price curve from moving in sync.
Questions related to the information contained in a **term** **structure** of prices and the possible implications of market imperfections for segmentation date back to the works of Culbertson (1957) and Modigliani and Sutch (1966) on “preferred habitats” in bond markets. Spurred in part by interest rate behaviors during the 2008-2011 financial crisis and the so-called Great Recession, the past ten years have seen a resurgence of theoretical and empirical work on segmentation. The latter is defined as a situation in which different parts of the price curve are disconnected from each other. Gürkaynak and Wright (2012), who review this still-growing literature, conclude that “the preferred habitat approach (has) value, especially at times of unusual financial market turmoil” (p. 360). For example, D’Amico and King (2013) document the existence of a “local supply” effect in the yield curve in 2009 during the U.S. Federal Reserve’s unprecedented program to purchase $300 billion of U.S. Treasury securities.

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Our interest in uncovering the structural shocks that drive the **term** **structure** slope is motivated by the recent macro-finance literature that has taken the first step of linking simple atheoretical **term** **structure** factor **models** with macroeconomic factors. In this paper we take the next step of uncovering the fundamental shocks and **structure** that propagates these shocks between macroeco- nomic and financial variables. To do so, we adopt a novel approach in the search for a structural explanation for slope movements. Instead of postulating a particular type of shock and then analyz- ing its effects, our strategy consists of first uncovering (in a statistical sense) the main innovations of movements in the slope of the **term** **structure** and then trying to provide an economic inter- pretation of these shocks. As in existing papers, we start by combining **term** **structure** variables with prominent macroeconomic aggregates in a VAR. We then apply a methodology developed by Uhlig (2003) to extract the exogenous shocks that explain as much as possible of the Forecast Error Variance (FEV) of a target variable in the VAR, which in our case is the slope. That is, we first look for a quantitatively important shock, and then interpret it. We do so by analyzing the impulse responses of the different variables in the VAR and contrasting them with the theoretical implications of different types of macroeconomic shocks.

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Shock Propagation Across the Futures **Term** **Structure**: Evidence from Crude Oil Prices
Abstract
To what extent are futures prices interconnected across the maturity curve? Where in the **term** **structure** do price shocks originate, and which maturities do they reach? We propose a new approach, based on information theory, to study these cross-maturity linkages and the extent to which connectedness is impacted by market events. We introduce the concepts of backward and forward information flows, and propose a novel type of directed graph, to investigate the propagation of price shocks across the WTI **term** **structure**. Using daily data, we show that the mutual information shared by contracts with different maturities increases substantially starting in 2004, falls back sharply in 2011-2014, and recovers thereafter. Our findings point to a puzzling re-segmentation by maturity of the WTI market in 2012-2014. We document that, on average, short-dated futures emit more information than do backdated contracts. Importantly, however, we also show that significant amounts of information flow backwards along the maturity curve – almost always from intermediate maturities, but at times even from far-dated contracts. These backward flows are especially strong and far-reaching amid the 2007-2008 oil price boom/bust.

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is no more binding of electrons. These minimizers share the same density thanks to the strict convexity of ρ ÞÑ Dpρ, ρq .
The rHF model is a one-electron description of the quantum system (the ground state can be described by a mean–field Hamiltonian), but electrons "interact" with each other through the common electronic den- sity ρ in the **term** Dp¨, ¨q. The rHF description is therefore a simple framework which allows one to conduct rigorous mathematical analysis, while still being realistic enough to describe the physics of quantum sys- tems. Furthermore, it can be seen as a good approximation of Kohn-Sham **models** [ KS65 , DG90 , LLS19 ] presented the next section, which are commonly used in solid-state physics. Besides, most results obtained on the rHF model can be extended to the Kohn–Sham LDA model [ AC09 ], up to possibly some assumptions on the uniqueness of the ground-state density matrix or on the coercivity of the second-order derivative of the Kohn–Sham energy functional at the ground-state under consideration. Due to these reasons, in the sequel we shall extend the rHF description to infinite systems and use this framework to treat extended defects, the junction of quasi 1D materials as well as lattice dynamics respectively in Sections 1.3.2 , 1.4

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Understanding variability in response to antiretroviral treatment in HIV patients is an important challenge. HIV dynamic **models** describe the viral load decrease and the CD4 cells increase under treatment by modeling the interaction between several types of CD4 cells and virions ( Perelson et al., 1996 Perelson and Nelson, 1997 Wu and Ding, 1999 Nowak and May, 2000 Rong and Perelson, 2009 ; ; ; ; ). They are defined as nonlinear differential systems and have generally no closed-form solutions. As available data are measurements of total number of the CD4 and the total number of virions, these differential systems are partially observed, complicating the parameter estimation. Nonlinear mixed effect **models** (NLMEMs) are appropriate to estimate model parameters and their inter-patient variability. The first modeling of viral load dynamic, using standard nonlinear regression or mixed **models**, considered a short time period and assumed non-infected CD4 cells to be constant ( Perelson et al., 1996 Wu et al., 1998 Ding and Wu, 2001 ; ; ). Another simplified approach assumed inhibition of any new infection by initiated therapy. Under that unrealistic assumption, the system can be solved explicitly (Wu et al., 1998 ; Putter et al., 2002 ). Putter et al. (2002) proposed the first simultaneous estimation of the viral load and CD4 dynamics based on a differential system under this assumption, but had to focus only on the first two weeks of the dynamic after initiation of an anti-retroviral treatment. These two assumptions are unsatisfactory when studying long-**term** response to anti-retroviral treatment for which the use of complete **models** expressed as ordinary differential equations (ODEs) is mandatory.

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