scholarly journals Optimal Strategy for the Vardi Casino with Interest Payments

2007 ◽  
Vol 44 (01) ◽  
pp. 199-211 ◽  
Author(s):  
Ilie Grigorescu ◽  
Robert Chen ◽  
Larry Shepp
Keyword(s):  
General Result ◽  
Optimal Strategy ◽  
Critical Value ◽  
Discount Factor ◽  
Optimal Probability ◽  
Special Cases ◽  
Interest Payments

A gambler starts with fortune f < 1 and plays in a Vardi casino with infinitely many tables indexed by their odds, r ≥ 0. In addition, all tables return the same expected winnings per dollar, c < 0, and a discount factor is applied after each round. We determine the optimal probability of reaching fortune 1, as well as an optimal strategy that is different from bold play for fortunes larger than a critical value depending exclusively on c and 1 + a, the discount factor. The general result is computed explicitly for some relevant special cases. The question of whether bold play is an optimal strategy is discussed for various choices of the parameters.

scholarly journals Optimal Strategy for the Vardi Casino with Interest Payments

2007 ◽  
Vol 44 (1) ◽  
pp. 199-211 ◽  
Author(s):  
Ilie Grigorescu ◽  
Robert Chen ◽  
Larry Shepp
Keyword(s):  
General Result ◽  
Optimal Strategy ◽  
Critical Value ◽  
Discount Factor ◽  
Optimal Probability ◽  
Special Cases ◽  
Interest Payments

A gambler starts with fortune f < 1 and plays in a Vardi casino with infinitely many tables indexed by their odds, r ≥ 0. In addition, all tables return the same expected winnings per dollar, c < 0, and a discount factor is applied after each round. We determine the optimal probability of reaching fortune 1, as well as an optimal strategy that is different from bold play for fortunes larger than a critical value depending exclusively on c and 1 + a, the discount factor. The general result is computed explicitly for some relevant special cases. The question of whether bold play is an optimal strategy is discussed for various choices of the parameters.

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

Raman scattering from impurities in semiconductors. II. Special cases

1978 ◽  
Vol 56 (5) ◽  
pp. 560-564
Author(s):  
Robert Barrie ◽  
H. -C. Chow
Keyword(s):  
Raman Scattering ◽  
General Result ◽  
Phonon Scattering ◽  
Phonon Coupling ◽  
Strong Electron ◽  
Impurities In Semiconductors ◽  
Electron Phonon Coupling ◽  
Special Cases ◽  
Weak Electron

Special cases of the general result for Raman scattering from an impurity in a semiconductor are discussed. For weak electron–phonon coupling the zero-phonon and one-phonon scattering intensities are derived. For strong electron–phonon coupling a comparison is made between two different approximations that have been previously used.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

Universal Hilbert subsets

1998 ◽  
Vol 124 (1) ◽  
pp. 127-134 ◽  
Author(s):  
PIERRE DÈBES ◽  
UMBERTO ZANNIER
Keyword(s):  
General Result ◽  
Special Cases

We show that the sequence 2n+n is a universal Hilbert sequence. That is, for each polynomial P(T, Y) irreducible in ℚ(T) [Y], the polynomial P(2n+n, Y) is irreducible in ℚ[Y] for all but finitely many n. This answers a question of M. Yasumoto. Other examples, like 2n+5n, are given. They are all obtained as special cases of a more general result which is proved from classical diophantine arguments.

scholarly journals On the Structure of a Swing Contract's Optimal Value and Optimal Strategy

2008 ◽  
Vol 45 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Sheldon M. Ross ◽  
Zegang Zhu
Keyword(s):  
Optimal Strategy ◽  
Value Function ◽  
Market Price ◽  
Time Interval ◽  
Optimal Contract ◽  
Future Time ◽  
Special Cases ◽  
Optimal Value ◽  
Swing Contract ◽  
Time Point

Consider a sales contract, called a swing contract, between a seller and a buyer concerning some underlying commodity, with the contract specifying that during some future time interval the buyer will purchase an amount of the commodity between some specified minimum and maximum values. The purchase price and capacity at each time point is also prespecified in the contract. Assuming a random market price process and ignoring the possibility of storage, we look for the maximal expected net gain for the buyer of such a contract, and the strategy that achieves this maximal expected net gain. We study the effects that various contract constraints and market price processes have on the optimal strategy and on the contract value. We show how we can reduce the general swing contract to a multiple exercising of American (Bermudan) style options. Also, in important special cases, we give explicit expressions for the optimal contract value function and the optimal strategy.

scholarly journals MOCK THETA DOUBLE SUMS

2016 ◽  
Vol 59 (2) ◽  
pp. 323-348 ◽  
Author(s):  
JEREMY LOVEJOY ◽  
ROBERT OSBURN
Keyword(s):  
General Result ◽  
Theta Functions ◽  
Similar Type ◽  
Mock Theta Functions ◽  
Bailey Pairs ◽  
Special Cases ◽  
Seventh Order

AbstractWe prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.

scholarly journals Cylindrical-wave diffraction by a rational wedge

1987 ◽  
Vol 411 (1841) ◽  
pp. 285-303 ◽  
Keyword(s):  
Half Plane ◽  
General Result ◽  
Wave Diffraction ◽  
Cylindrical Wave ◽  
Open Angle ◽  
Source Terms ◽  
Special Cases ◽  
Rational Multiple

In this paper, new expressions for the field produced by the diffraction of a cylindrical wave by a wedge, whose angle can be expressed as a rational multiple of π, are given. The solutions are expressed in terms of source terms and real integrals that represent the diffracted field. The general result obtained includes as special cases, Macdonald’s solution for diffraction by a half plane, a solution for Carslaw’s problem of diffraction by a wedge of open angle 2/3π, and a new representation for the solution of the problem of diffraction by a mixed soft-hard half plane.

Slender-body expressions for the wave load on offshore structures

1995 ◽  
Vol 450 (1939) ◽  
pp. 391-416 ◽  
Keyword(s):  
Cross Sections ◽  
General Result ◽  
Offshore Structures ◽  
Slender Body ◽  
Structural Members ◽  
Wave Load ◽  
Offshore Structure ◽  
Complete Structure ◽  
Special Cases ◽  
Surface Intersections

Expressions for the wave load on an offshore structure have recently been derived by several authors, by applying the classical slender-body approach to individual structural members. This paper shows that they are all special cases of a more general result, which covers a complete structure, including the effect of non-circular member cross-sections, joints between members, and surface intersections. This more general result cannot readily be derived by the methods adopted by the earlier authors; it relies on an energy argument.

The Problem of Air Flow Over Mountains: A Summary of Theoretical Studies

1948 ◽  
Vol 29 (1) ◽  
pp. 16-26 ◽  
Author(s):  
Paul Queney
Keyword(s):  
Wave Length ◽  
Potential Temperature ◽  
Vertical Plane ◽  
Critical Value ◽  
Mountain Range ◽  
Theoretical Problem ◽  
Wave System ◽  
Special Cases ◽  
Lee Waves ◽  
Horizontal Wave

The theoretical researches on the problem of the disturbance of an atmospheric current flowing over a mountain range, carried on during the last ten years, prove that most of the characteristic observed features can be explained, to a very large extent, by the hydrodynamical theory of internal, small adiabatic perturbations in a stratified rotating atmosphere. In the case of a uniform current of velocity u, in which the coefficient of vertical stability is also uniform (θ is the potential temperature, g the gravity, z the elevation), the perturbation pattern corresponding to a smooth, low, gently-sloping mountain range varies widely according to the half-width a of this mountain range:i) If a is comparable to the critical value (Ls/2π) = (u/s) ≈ 1 km, there is a system of short stationary lee waves, or gravity waves, with the wave length Ls.ii) If a is comparable to (Lf/2π) = (u/f) ≈ 100 km, where f is the Coriolis parameter, there is a complex system of gravity-inertia lee waves, with the horizontal wave length Lf at ground level, and the vertical wave length Ls in the vertical plane of the crest.iii) In the case of a westerly wind, if a is comparable to the third critical value , where β is the latitudinal variation of f, there is a system of long geostrophic lee waves, of the type studied by Rossby, with the horizontal wave length Lβ at ground level.iv) In the other cases, there are no lee waves, and the wave system is either missing (cases a ≪ (u/s) and ≫ (u/f)) or restricted to the vicinity of the vertical plane of the crest, with the wave length Ls (case (u/s) ≪ a ≪ (u/f). One of the effects of the tropopause, and of atmospheric discontinuities in general, is to produce eventually an additional system of waves on the upwind side of the mountain range, but these waves must be generally much smaller than the lee waves. Similar results are obtained for a plateau with smooth sides, and in the case of a more complex mountain profile the perturbation can be computed by superimposing elementary solutions. If the current is not uniform, if the ground deformation is steep, or if the perturbation is not adiabatic, the preceding results are not valid, and the solution of the theoretical problem has been obtained in only very special cases.

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