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1.
The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest
rates are constant and the exchange rate follows a log-normal diffusion process.
In this paper we consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process
with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency
risk-free interest rates may be arbitrary continuous functions of time. First non-trivial problem we encounter in time-dependent
case is the continuity in time argument of the value function of the American put option and the regularity properties of
the optimal exercise boundary. We establish these properties based on systematic use of the monotonicity in volatility for
the value functions of the American as well as European options with convex payoffs together with the Dynamic Programming
Principle and we obtain certain type of comparison result for the value functions and corresponding exercise boundaries for
the American puts with different strikes, maturities and volatilities.
Starting from the latter fact that the optimal exercise boundary curve is left continuous with right-hand limits we give a
mathematically rigorous and transparent derivation of the significant early exercise premium representation for the value
function of the American foreign exchange put option as the sum of the European put option value function and the early exercise
premium.
The proof essentially relies on the particular property of the stochastic integral with respect to arbitrary continuous semimartingale
over the predictable subsets of its zeros. We derive from the latter the nonlinear integral equation for the optimal exercise
boundary which can be studied by numerical methods. 相似文献
2.
Constantin N. Beli 《Journal of Number Theory》2003,102(1):125-182
The spinor norms of integral rotations of an arbitrary quadratic lattice over an arbitrary dyadic local field are determined. The results are given in terms of BONGs, short for “bases of norm generators”. This approach provides a new way to describe lattices over dyadic local fields. 相似文献
3.
We construct lattices with quadratic structure over the integers in quadratic number fields having the property that the rank
of the quadratic structure is constant and equal to the rank of the lattice in all reductions modulo maximal ideals. We characterize
the case in which such lattices are free. The construction gives a representative of the genus of such lattices as an orthogonal
sum of “standard” pieces of ranks 1–4 and covers the case of the discriminant of the real quadratic number field congruent
to 1 modulo 8 for which a general construction was not known.
相似文献
4.
We describe algorithms which address two classical problems in lattice
geometry: the lattice covering and the simultaneous lattice
packing-covering problem. Theoretically our algorithms solve the two
problems in any fixed dimension d in the sense that they approximate
optimal covering lattices and optimal packing-covering lattices within
any desired accuracy. Both algorithms involve semidefinite
programming and are based on Voronoi's reduction theory for positive
definite quadratic forms, which describes all possible Delone
triangulations of ℤd. In practice, our implementations reproduce known results in dimensions d ≤ 5 and in particular solve the two problems in
these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are
closely related to the lattice E*6. For d = 7,8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort
to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices
are locally optimal. 相似文献
5.
The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic methods and algebraic geometric methods. We prove two conjectures which were proposed recently by the above authors in a previous article (2022). Here we obtain an asymptotic formula for arbitrary power means of generalized quadratic Gauss sums and one corresponding power moment of a character sum. 相似文献
6.
7.
We establish that each lattice bimorphism from the Cartesian product of two vector lattices into a universally complete vector lattice is representable as the product of two lattice homomorphisms defined on the factors. This fact makes it possible to reduce the problem to the linear case and obtain some results on representation of an order bounded disjointness preserving bilinear operator as a strongly disjoint sum of weighted shift or multiplicative operators. 相似文献
8.
Saptarshi Das Indranil Pan Kaushik Halder Shantanu Das Amitava Gupta 《Applied Mathematical Modelling》2013
The continuous and discrete time Linear Quadratic Regulator (LQR) theory has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are formulated as the optimal state-feedback gains, corresponding to the standard quadratic cost function involving the state variables and the controller effort. A real coded Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator design while minimizing another time domain integral performance index, comprising of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. The proposed methodology is extended for a new kind of fractional order (FO) integral performance indices. The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control. Guidelines for the choice of integral order of the performance index are given depending on the characteristics of the process, to be controlled. 相似文献
9.
Yu. G. Teterin 《Journal of Mathematical Sciences》1997,83(5):664-672
It is proved that the group of spinor norms of autometries of a generalized quadratic lattice ℒ over the ring of integral
elements v
p
of a local field k
p
, in the case wherep∤2 and ℒ is a generalized translation, is generated by the spinor norms of symmetries contained in the group of autometries
of ℒ. As a corollary, an extension to the case of generalized quadratic lattices is given for known sufficient conditions
of coincidence of the genus and the spinor genus of a quadratic lattice. Bibliography: 9 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 161–173.
Translated by Yu. G. Teterin. 相似文献
10.
Bosbach and Rie?an states on residuated lattices both are generalizations of probability measures on Boolean algebras. Just from the observation that both of them can be defined by using the canonical structure of the standard MV-algebra on the unit interval [0, 1], generalized Rie?an states and two types of generalized Bosbach states on residuated lattices were recently introduced by Georgescu and Mure?an through replacing the standard MV-algebra with arbitrary residuated lattices as codomains. In the present paper, the Glivenko theorem is first extended to residuated lattices with a nucleus, which gives several necessary and sufficient conditions for the underlying nucleus to be a residuated lattice homomorphism. Then it is proved that every generalized Bosbach state (of type I, or of type II) compatible with the nucleus on a nucleus-based-Glivenko residuated lattice is uniquely determined by its restriction on the nucleus image of the underlying residuated lattice, and every relatively generalized Rie?an state compatible with the double relative negation on an arbitrary residuated lattice is uniquely determined by its restriction on the double relative negation image of the residuated lattice. Our results indicate that many-valued probability theory compatible with nuclei on residuated lattices reduces in essence to probability theory on algebras of fixpoints of the underlying nuclei. 相似文献
11.
This paper is to provide some new generalizations of the Pick Theorem. We first derive
a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron. Then, we
use the idea of a weight function of [2] to obtain a weighted version. Other Pick type theorems
known to the author for the integral lattice
Z2 are reduced to some special cases of this generalization.
Finally, using an idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of
the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to
higher dimensions.AMS Subject Classification: 52C05, 11H06, 57N05, 57N15, 57N35. 相似文献
12.
13.
This is a survey of our papers [3, 4]. We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, translation, real linear transformation and functional differentiation. The invariance under translation and orthogonal transformation, the interchange of the order with Riemann-Stieltjes integrals and some limits, the semiclassical approximation, the integration by parts and the Taylor expansion formula with respect to functional differentiation, and the fundamental theorem of calculus hold in Feynman path integral. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
《Quaestiones Mathematicae》2013,36(4):481-508
Abstract This paper offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frame T of truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and we demonstrate its usefulness by employing it to derive a number of results, some old and some new, concerning these partially ordered sets. In particular, we prove they are complete lattices, algebraic or continuous, depending on whether T is algebraic or continuous, respectively (Proposition 3); they satisfy the same lattice identities for arbitrary T that hold in the case T = 2 (Corollary of Proposition 4); and they are coherent frames for any coherent T whenever this is the case for T = 2 (Proposition 6). In addition we show, generalizing a result by Makamba and Murali [10], that the familiar classical situations where the congruences of an algebra correspond to certain other entities, such as the normal subgroups of a group or the ideals of a ring, extend to the fuzzy case by proving that the corresponding propositional theories are equivalent (Proposition 2). Further, we obtain the result of Gupta and Kantroo [5] that the fuzzy radical ideals of a commutative ring with unit are the meets of fuzzy prime ideals for arbitrary continuous T in place of the unit interval, using basic facts concerning continuous frames (Proposition 7). 相似文献
15.
We use the residue theorem to derive an expression for the number of lattice points in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart polynomial. We show that it is a polynomial in t, where t is the integral dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these tetrahedra, relating the Ehrhart polynomials of the interior and the closure of the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient for codimension 2. Finally, we show how our ideas can be used to compute the Ehrhart polynomial for an arbitrary convex lattice polytope. 相似文献
16.
A. Z. Asekov 《Moscow University Mathematics Bulletin》2017,72(5):210-212
An optimal control problem of minimization of the sum of a linear-quadratic integral functional and a terminal of arbitrary form is considered. It is proved that in this case a control can be also constructed in an explicit analytic form. 相似文献
17.
C. Tudor 《Mathematische Nachrichten》1990,147(1):205-218
The infinite dimensional version of the linear quadratic cost control problem is studied by Curtain and Pritchard [2], Gibson [5] by using Riccati integral equations, instead of differential equations. In the present paper the corresponding stochastic case over a finite horizon is considered. The stochastic perturbations are given by Hilbert valued square integrable martingales and it is shown that the deterministic optimal feedback control is also optimal in the stochastic case. Sufficient conditions are given for the convergence of approximate solutions of optimal control problems. 相似文献
18.
Larry J Gerstein 《Journal of Number Theory》1973,5(5):332-338
Orthogonal splitting for lattices on quadratic spaces over algebraic number fields is studied. It is seen that if the rank of a lattice is sufficiently large, then its spinor genus must contain a decomposable lattice. Also, splitting theory is used to obtain a lower bound for the class number of a lattice (in the definite case) in terms of its rank, via the partition function. 相似文献
19.
Peter M. Gruber 《Proceedings of the Steklov Institute of Mathematics》2012,276(1):103-124
A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this
article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using
an idea of Voronoĭ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function
is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein
zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoĭ type notions such as versions
of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are
extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm. 相似文献
20.
《Journal of Applied Mathematics and Mechanics》2001,65(1):33-46
A mechanical system, consisting of a non-variable rigid body (a carrier) and a subsystem, the configuration and composition of which may vary with time (the motion of its elements with respect to the carrier is specified), is considered. The system moves in a central force field at a distance from its centre which considerably exceeds the dimensions of the system. The effect of the system motion about the centre of mass on the motion of the centre of mass, which is assumed to be known, is ignored (the analogue of the limited problem [1] for a rigid body). The necessary and sufficient conditions for a quadratic integral of the motion around the centre of mass to exist are obtained in the case when there is no dynamic symmetry. It is shown that, for a quadratic integral to exist, it is necessary that the trajectory of the motion of the centre of mass should be on the surface of a certain circular cone, fixed in inertial space, with its vertex at the centre of the force field. If the trajectory does not lie on the generatrix of the cone, only one non-trivial quadratic integral can exist and the initial system, in the presence of this quadratic integral, reduces to autonomous form. For the motion of the centre of mass along the generatrix or the motion of the system around a fixed centre of mass, the necessary and sufficient conditions for a non-trivial quadratic integral to exist are obtained, which are generalizations of the energy integral, the de Brun integral [2] and the integral of the projection of the kinetic moment. When three non-trivial quadratic integrals exist, the condition for reduction to an autonomous system describing the rotation of the rigid body around the centre of mass and integrable in quadratures are indicated [3, 4]. 相似文献