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The Second Fundamental Theorem of Asset Pricing

, 1998
"... This paper presents a resolution of the paradox proposed by the example of an economy with complete markets and a multiplicity of martingale measures constructed by Artzner and Heath (1995). The resolution lies in noting that completeness is with respect to a topology on the space of cash flows and ..."
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Cited by 13 (4 self)
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This paper presents a resolution of the paradox proposed by the example of an economy with complete markets and a multiplicity of martingale measures constructed by Artzner and Heath (1995). The resolution lies in noting that completeness is with respect to a topology on the space of cash flows and is connected with uniqueness of the price functional in the topological dual space. Uniqueness may be lost outside the dual and this is what occurs in the counterexample of Artzner and Heath.
Fundamental Solutions of Real Homogeneous Cubic Operators of Principal Type in Three Dimensions
, 1998
"... . { Certain fundamental solutions E a of the partial dierential operators @ 3 1 + @ 3 2 + @ 3 3 + 3a@ 1 @ 2 @ 3 ; a 2 R n f1g; are represented by elliptic integrals of the rst kind. These operators are (apart from a linear change of variables) the most general homogeneous operators of real princ ..."
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Cited by 6 (3 self)
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. { Certain fundamental solutions E a of the partial dierential operators @ 3 1 + @ 3 2 + @ 3 3 + 3a@ 1 @ 2 @ 3 ; a 2 R n f1g; are represented by elliptic integrals of the rst kind. These operators are (apart from a linear change of variables) the most general homogeneous operators of real principal type in three variables and irreducible of degree three. The fundamental solutions E a show interesting nonconvex lacunas, and their remaining level sets are algebraic surfaces of degree 6, explicitly calculated. 1. INTRODUCTION 1.1. The operator @ 3 1 +@ 3 2 +@ 3 3 was consideredto my knowledgefor the rst time in 1913 in N. Zeilon's article [20], wherein he generalizes I. Fredholm's method of construction of fundamental solutions (see [5]) from homogeneous elliptic equations to arbitrary homogeneous equations in three variables with a realvalued symbol (cf. [20, II, pp. 14{ 22], [6, Ch. 11, pp. 146148]). An explicit formula for a fundamental solution was given in [19]. The o...
Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity
, 2008
"... ..."
PROJECTIVE MULTIRESOLUTION ANALYSIS AND GENERALIZED SAMPLING
, 1994
"... A standard technique in constructing wavelet bases for Hilbert spaces is the use of a Multiresolution Analysis (MRA), a sequence of closed subspaces with certain properties providing the means for obtaining wavelets. Examples for the space of integrable functions on the real line show that this appr ..."
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Cited by 5 (2 self)
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A standard technique in constructing wavelet bases for Hilbert spaces is the use of a Multiresolution Analysis (MRA), a sequence of closed subspaces with certain properties providing the means for obtaining wavelets. Examples for the space of integrable functions on the real line show that this approach is not necessarily successful in nonHilbert spaces. Therefore we introduce the concept of a projective Multiresolution Analysis, a sequence of continuous projections with properties corresponding directly to those of the subspaces forming an MRA and ensuring the existence of wavelets. We exhibit conditions under which the two concepts are equivalent, and we also show why the MRA does not yield the desired results in other cases. A projective MRA is a special type of a double projective approximation, a concept which, under reasonable assumptions on the topology of the space, is equivalent to a projective decomposition, which is the key to the construction of wavelet bases. We then study projective operators of the type used in a projective MRA. They consist of the compositions of generalized sampling operators with theiradjoints, the reconstruction operators. Understanding the action of these operators on the Fourier side requires Poisson’s Summation Formula, whose validity in different spaces we therefore examine first. Finally we apply the obtained results to the space of integrable functions. We shed some more light on the wellknown result that there are no wavelet bases for this space, but we also show that we can obtain wavelet bases for the maximal ideal of integrable functions with zero mean.
A Survey on Explicit Representation Formulae for Fundamental Solutions of Linear Partial Differential Operators
 Acta Appl. Math
, 1996
"... this paper, all partial dierential operators are linear and contain constant complex coecients, and we adopt for them the usual multiindex notation. Specically, let us consider such an operator on R ..."
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Cited by 3 (1 self)
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this paper, all partial dierential operators are linear and contain constant complex coecients, and we adopt for them the usual multiindex notation. Specically, let us consider such an operator on R
Functions and Distributions in Spaces with Thick Points
"... Euler’s vision of a generalized concept of function was a forerunner of the modern concept of distribution, and his efforts to give meaning to divergent series eventually led to the concepts of asymptotic series, summability, and distributional convergence. The introduction of such suitable abstract ..."
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Cited by 2 (2 self)
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Euler’s vision of a generalized concept of function was a forerunner of the modern concept of distribution, and his efforts to give meaning to divergent series eventually led to the concepts of asymptotic series, summability, and distributional convergence. The introduction of such suitable abstract concepts does not automatically prevent mistakes or inconsistencies resulting from careless formal reasoning. We deal with a cluster of such issues associated with the occurrence of a distributional singularity on the boundary of a domain of integration. Apparent paradoxes are resolved by introducing new classes of test functions and distributions adapted to the problems at hand; one can regard the construction as attributing internal structure to boundary points.
On the Fundamental Solutions of a Class of Elliptic Quartic Operators in Dimension 3
, 2002
"... The (uniquely determined) even and homogeneous fundamental solutions of the linear elliptic partial differential operators with constant coefficients of the form are represented by elliptic integrals of the first kind. Thereby we generalize Fredholm's example , which, up to n ..."
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Cited by 2 (0 self)
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The (uniquely determined) even and homogeneous fundamental solutions of the linear elliptic partial differential operators with constant coefficients of the form are represented by elliptic integrals of the first kind. Thereby we generalize Fredholm's example , which, up to now, was the only irreducible homogeneous quartic operator in 3 variables the fundamental solution of which was known to be expressible by tabulated functions.
A Short Proof of the MalgrangeEhrenpreis Theorem
 DOMAŃSKI (EDS.), FUNCTIONAL ANALYSIS, PROC. 1ST INT. WORKSHOP
, 1994
"... This note presents first a survey on the different methods of proof of the MalgrangeEhrenpreis theorem, then a short, new proof by means of an explicit formula, and finally some illustrative examples, wherein this formula is applied. ..."
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Cited by 2 (1 self)
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This note presents first a survey on the different methods of proof of the MalgrangeEhrenpreis theorem, then a short, new proof by means of an explicit formula, and finally some illustrative examples, wherein this formula is applied.
SMOOTH ∗ALGEBRAS
 PROGRESS OF THEORETICAL PHYSICS SUPPLEMENT
, 2001
"... Looking for the universal covering of the smooth noncommutative torus leads to a curve of associative multiplications on the space O ′ M (R2n) ∼ = OC(R 2n) of Laurent Schwartz which is smooth in the deformation parameter �. The Taylor expansion in � leads to the formal Moyal star product. The no ..."
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Cited by 2 (0 self)
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Looking for the universal covering of the smooth noncommutative torus leads to a curve of associative multiplications on the space O ′ M (R2n) ∼ = OC(R 2n) of Laurent Schwartz which is smooth in the deformation parameter �. The Taylor expansion in � leads to the formal Moyal star product. The noncommutative torus and this version of the Heisenberg plane are examples of smooth *algebras: smooth in the sense of having many derivations. A tentative definition of this concept is given.