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April 8, 2018
Abstract
Discrete time hedging produces a residual risk, namely, the tracking error. The major problem is to get valuation/hedging policies minimizing this error. We evaluate the risk between trading dates through a function penalizing asymmetri- cally profits **and** losses. After deriving the asymptotics within a discrete time risk measurement for a large number of trading dates, we derive the optimal strategies minimizing the **asymptotic** risk in the continuous time setting. We characterize the optimality through a class of fully nonlinear Partial **Differential** **Equations** (PDE). Numerical experiments show that the optimal strategies associated with discrete **and** **asymptotic** approach coincides asymptotically.

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Swanson [38] summarized the classical results of the oscillation theory. We also nd a nice review of such theory in Kreith Oscillation theory" Springer, 1973.
The oscillation theory of non linear dierential **equations** of second order has also attracted an attention, see for instance the books of Bogolyubov **and** Mitropolski Méthodes asympto- tiques en théorie des oscillations non linéaires", 1962, of Roseau Vibrations non linéaires", 1966, **and** of Coddington **and** Levinson Theory of ODEs", 1955. We do not know exactly the rst work on the oscillation theory of non linear dierential **equations** of second order. It seems that a rst attempt to study the oscillation theory of dierential equation with delays was done by Fite in 1921.

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Already in the context of nonswitched systems several challenges arise in the study of DAEs. The difference basi- cally arises due to the presence of **algebraic** **equations** (static relations) in the description of the system because of which the state trajectories can only evolve on the sets defined by the **algebraic** **equations** of the active mode. Observer designs have been studied for (nonswitched) DAEs since 1980’s, e.g. (Dai, 1989; Fahmy **and** O’Reilly, 1989). Unlike ODEs, the ob- server design in DAEs requires additional structural assump- tions **and**, furthermore, the order of the observer may de- pend on the design method. Because of these added general- ities, observer design for nonswitched DAEs is still an active research field (Bobinyec et al., 2011; Darouach, 2012), **and** the recent survey articles summarize the development of this field (Berger **and** Reis, 2015; Bobinyec **and** Campbell, 2014). In studying switched DAEs, our modeling framework al- lows for time-varying **algebraic** relations. The changes in al- gebraic constraints due to switching introduce jumps in the state of the system, **and** because of the possibility of a higher- index DAE, these jumps may get differentiated **and** generate impulsive solutions. The notion of observability studied in this paper takes into account the additional structure due to **algebraic** constraints, **and** the added information from the outputs in case there are impulses observed in the measure- ments. This observability concept is then used to construct a mapping from the output space to the state space, which allows us to theoretically reconstruct the state. The key ele- ment of constructing this mapping is to show how the struc- ture of a linear DAE is exploited to recover the information about the state in individual subsystems. This structural de- composition is then combined with the expressions used for evolution of states in switched DAEs to accumulate all possi- ble recoverable information from past measurements about the state at one time instant. The construction then yields a systematic procedure for writing the value of the state at a time instant in terms of outputs measured over an interval for which the system is observable.

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However, loop **equations** can be generalized beyond the context of matrix models, just as a set of **algebraic** relationships among the W n s.
In [ 1 ], the authors derived loop **equations** in the case g = sl 2 (C) on the Riemann
sphere. However, the proof in [ 1 ] involved an “insertion operator”, that was hard to deﬁne rigorously in all cases, **and** involved analysis (inﬁnitesimal deformations). It was unsatisfactory because loop **equations** are **algebraic** statements, that cry for an **algebraic** proof.

k + 1 is in the generic case the multiplicity of the singularity), while in the very
general case the method is called accelero-summation (or multisummation). Based on an original idea of É. Borel from the end of the 19th century, it has been largely developed during 1970–1980’s by J. Écalle (cf. [Ec]), **and** by J.-P. Ramis (cf. [Ra3]), **and** became one of the main tools in the local study of singularities of analytic dif- ferential **equations**. In general, the solutions on different sectors do not coincide, **and** if extended to larger sectors, they may drastically change their **asymptotic** behavior due to the presence of hidden exponentially small terms. This is traditionally known as the (linear or non-linear) Stokes phenomenon. It is now understood, that the divergence of the **asymptotic** series is caused by singularities of its Borel transform, which also encode information on the geometry of the singularity.

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A few years ago, we started to resurrect Tutte’s technique in order to solve a more general problem: the Potts model on planar maps. In combinatorial terms, this means that we count all q-colourings, not necessarily proper, but with a weight ν m , where ν is an indeterminate **and** m
is the number of monochromatic edges (edges whose endpoints have the same colour). The case ν = 0 is thus the problem solved by Tutte. Moreover, we do not only study degree-constrained maps (triangulations) as Tutte did, but also general planar maps (counted by edges **and** vertices). In a first paper on this topic [ 2 ], we wrote the counterpart of ( 2 ) for each of these two problems. Then, we performed a first step, by establishing an equation with only one catalytic variable, y. But this equation holds only for values of q of the form 2 + 2 cos(jπ/m), for integers j **and** m (with q 6= 0, 4). Moreover, the size of this equation grows with m. Nevertheless, **equations** with one catalytic variable are much better understood that those with two [ 11 ], **and** we were able to prove that the generating function of q-coloured maps is **algebraic** for all such values of q, including q = 2 **and** q = 3. It is known to be transcendental in general [ 2 ].

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−→ A good rule of thumb is to give I u a value close to that of the lower
bound, which depends on the system.
−→ A guess can be made from the behaviour of the system in the first
timesteps : I u = I 1 **and** I max = 3I u ( γ a = 1 . 2 **and** γ d = 0 . 5 for ARES **and**

which is called a fundamental circuit of a matroid, induced by a certain bipartite graph. Jian-Wan et al [7] **and** Nilsson [13] consider the SAP in relation with Modelica models. A Modelica source code is first translated into a so-called ”flat model” which is a system of **equations** of type 1. In [7], the authors propose a method for analysing **and** de- tecting minimal bad-constrained subsystems. The method uses Dulmage et Mendelsohn decomposition techniques in a first step to isolate the bad-constrained subsets of equa- tions. Then for each such subsystem, a set of fictitious **equations** is formulated. These are related to the underlay- ing physical system. The resulting system of **equations** is in turn decomposed **and** so on until a minimal bad-constrained subsystem is detected. The method is applied in a recursive way until all the minimal bad-constrained components are localized. In [13], the author studies the SAP for modular systems of **equations**, that is systems which are constructed by composition of individual equation system fragments. Leitold **and** Hangos [8] consider the DAS for dynamic pro- cess models. These are DAS which are sometimes diffi- cult to solve numerically due to index problems [1]. They propose a graph-theoretical method for analysing the dif- ferential index **and** the structural solvability of these mod- els. The method is an extension of Murota’s approach [10], where a representation graph is considered for each differ- ential index.

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Since the pioneering work of [29], in the Brownian case, the relations between more general BSDEs **and** associated deterministic problems have been studied extensively, **and** innovations have been made in several directions.
In [5] the authors introduced a new kind of BSDE including a term with jumps generated by a Poisson measure, where an underlying forward process X solves a jump diﬀusion equation with Lipschitz type conditions. They associated with it an Integro-Partial Diﬀerential Equation (in short IPDE) in which some non- local operators are added to the classical partial diﬀerential maps, **and** proved that, under some continuity conditions on the coeﬃcients, the BSDE provides a viscosity solution of the IPDE. In chapter 13 of [6], under some speciﬁc con- ditions on the coeﬃcients of a Brownian BSDE, one produces a solution in the sense of distributions of the parabolic PDE. Later, the notion of mild solution of the PDE was used in [3] where the authors tackled diﬀusion operators generating symmetric Dirichlet **forms** **and** associated Markov processes thanks to the the- ory of Fukushima Dirichlet **forms**, see e.g. [21]. Those results were extended to the case of non symmetric Markov processes in [36]. Inﬁnite dimensional setups were considered for example in [20] where an inﬁnite dimensional BSDE could produce the mild solution of a PDE on a Hilbert space. Concerning the study of BSDEs driven by more general martingales than Brownian motion, we have already mentioned BSDEs driven by Poisson measures. In this respect, more recently, BSDEs driven by marked point processes were introduced in [13], see also [4]; in that case the underlying process does not contain any diﬀusion term. Brownian BSDEs involving a supplementary orthogonal term were studied in [18]. We can also mention the study of BSDEs driven by a general martin- gale in [11]. BSDEs of the same type, but with partial information have been investigated in [12]. A ﬁrst approach to face deterministic problems for those **equations** appears in [25]; that paper also contains an application to ﬁnancial hedging in incomplete markets. Finally, BSDEs in general ﬁltered space were studied in [26] as we have already mentioned.

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later case, Σ is called a realization of H(s) if
H(s) = c(sI − A) −1 b.
The above realization theory between state space form **and** transfer function form for SISO systems is easily generalized to multi-input multi-output (MIMO) linear time-invariant systems [3]; however, the corresponding generalization to nonlinear systems has been a longstanding problem. For nonlinear systems, the realization problem is finding a suitable state equation for a given system of input-output equa- tions. Recall that the success of the concept of transfer function in solving the linear realization problem is because those transfer functions could be computed either from the input-output **equations** or from the state space **equations**. However, for nonlinear systems it is impossible to define transfer functions by using the Laplace transform which results in great difficulty in the study of nonlinear realization the- ory. Furthermore, the problem of minimal realization for nonlinear systems is not adequately understood.

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operator **and** H ( p ) = jpj **and** for convex initial sets were studied using dierent methods
by Yip [Y].
(iii) **Asymptotic** problems in phase transitions
We present here an example of an **asymptotic** problem arising in phase transitions | see [BCESS] **and** [BS] for an extended discussion of such problems in the deterministic setting **and** [LS2,3] for more general problems in random environments. The problem is about a modied Allen-Cahn equation of the form

Abstract. This paper is concerned with state-constrained discontinuous or- dinary **differential** **equations** for which the corresponding vector field has a set of singularities that **forms** a stratification of the state domain. Existence of solutions **and** robustness with respect to external perturbations of the right- hand term are investigated. Moreover, notions of regularity for stratifications are discussed.

Index Terms— **Differential**-**algebraic** **equations**, Mo- ments, Optimal Control, Switched Systems
I. I NTRODUCTION
In most all areas of electrical, mechanical, or chem- ical engineering the modeling **and** control of the dy- namics of complex systems is nowadays highly modu- larized, thus allowing the efficient generation of mathe- matical models for substructures **and** link them together via constraints. **Differential**-**algebraic** **equations** (DAEs) arise in the modeling of physical systems where the state variables satisfy certain **algebraic** constraints alongside some **differential** **equations** that govern the evolution of these state variables. In complex systems naturally arise the switched control systems, which are characterized by a set of several continuous nonlinear state dynamics with a logic-based controller, which determines simultaneously a sequence of switching times **and** a sequence of modes [4], [11], [9].

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The current paper should be seen as a continuation of the program initiated in [16, 17]. In particular, in [16] solutions to the initial value problem (1.1) were not allowed to have jumps. In addition, we restricted ourselves to purely **differential** systems where I is invertible **and** purely alge- braic systems where I = 0. Furthermore, we only considered linear **equations**. By contrast, in [17] we considered smooth nonlinear **algebraic** **equations** **and** constructed local center manifolds around equilibrium solutions. Special care needed to be taken to address the intricate compatibility condi- tions that the nonlinear terms must satisfy. We continue this analysis here for (1.1) **and** construct local stable manifolds for the zero equilibrium, under less restrictive conditions on the nonlinear terms.

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• The values of state variable enclosure w.r.t. time under the form: [y(t)]; t • The values of **algebraic** variable enclosure w.r.t. time under the form: [x(t)]; t • The log reported in Tab. 5.1
With the files containing state **and** **algebraic** values w.r.t. time, we are able to plot three figures, given in Fig. 5.1. On the figure (a) **and** (b), it is apparent that even if state variable **and** **algebraic** variable evolves exponentially but quite slowly, **algebraic** variable evolves in a stiff way w.r.t. state variable (c). In general, DAEs leads to a stiff problem, which is the motivation for using RADAU IIA method. This method is known for its efficiency in front of stiff problems.

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Similarly, we study the systems of DAE encountered in the numerical modeling of HTS devices using the FEM with edge elements for a 2-D model. We give the discretization of the **equations** in space **and** identify the resulting system of DAE. Depending on the boundary conditions, the resulting system of DAE can be of index 0 or 2 in Hessenberg form. For the system of DAE of index 2, we cannot conclude if it is better to discretize it directly or to reduce its index. We note that reducing the index from 2 to 1 yields a system of ODEs for the Degrees Of Freedom (DOFs) of interest but that a matrix needs to be inverted. We verify the code develop for this project against two analytical solutions for three different problems. The strategy of direct discretization is implemented in the code through the IDAS library. There are no difficulties reported when computing the approximations with the direct discretization strategy. Therefore, this strategy works for the problems considered. We show that the code gives good approximations for all the problems implemented, except where the solution is not linear. The approximations get better when the mesh is refined.

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In the following, we give the necessary notation **and** basic definitions **and** lem- mas which will be used in this paper.
Definition 2.1: A real valued function f (t), t > 0 is said to be in the space C µ , µ ∈ R, if there exists a real number p > µ such that f (t) = t p f 1 (t), where

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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(b) Stabilized scheme, c f as in ( 15 ).
Figure 6. Non-conforming DG finite elements ⇤ 1 h ( T h ) = P r d ⇤ 1 ( T h ).
6.2. Stationary problem: Velocity with non-resolved discontinuities. The derivation of the method, as of Section 2 , relies on the assumption that the mesh resolves the possible discontinuities of the velocity field. In the following experiment we investigate an example of normal jump discontinuity in the velocity field not resolved by the mesh or any of its refinements **and** observe that the instabilities arising downstream of the discontinuity irremediably compromise the accuracy of the numerical solution **and** wreck the performance of the method. The failure of the numerical scheme in this test case may be ascribed to the fact that, since across the mesh facets the jump of the velocity vanishes, the scheme itself does not capture the discontinuity of the velocity **and** hence of the solution. Jump discontinuities are only taken into account through numerical quadrature.

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