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46
Speed scaling to manage energy and temperature
 Journal of the ACM
"... We first consider online speed scaling algorithms to minimize the energy used subject to the constraint that every job finishes by its deadline. We assume that the power required to run at speed s is P s s. We provide a tight bound on the competitive ratio of the previously proposed Optimal A ..."
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Cited by 169 (17 self)
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We first consider online speed scaling algorithms to minimize the energy used subject to the constraint that every job finishes by its deadline. We assume that the power required to run at speed s is P s s. We provide a tight bound on the competitive ratio of the previously proposed Optimal Available algorithm. This improves the best known competitive ratio by a factor of . We then introduce a new online algorithm, and show that this algorithm’s competitive ratio is at most e. This competitive ratio is significantly better and is approximately e for large . Our result is essentially tight for large . In particular, as approaches infinity, we show that any algorithm must have competitive ratio e (up to lower order terms). We then turn to the problem of dynamic speed scaling to minimize the maximum temperature that the device ever reaches, again subject to the constraint that all jobs finish by their deadlines. We assume that the device cools according to Fourier’s law. We show how to solve this problem in polynomial time, within any error bound, using the Ellipsoid algorithm. 1.
Designing options given the risk: The optimal Skorokhodembedding problem. Stochastic Process
 Appl
, 1999
"... Motivated by applications in option pricing theory [9] we formulate and solve the following problem. Given a standard Brownian motion B = (Bt)t 0 and a centered probability measure on IR having the distribution function F with a strictly positive density F0 Z satisfying: 1 x log x (dx) < 1 ..."
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Cited by 12 (3 self)
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Motivated by applications in option pricing theory [9] we formulate and solve the following problem. Given a standard Brownian motion B = (Bt)t 0 and a centered probability measure on IR having the distribution function F with a strictly positive density F0 Z satisfying: 1 x log x (dx) < 1
Sum rules and spectral measures of Schrödinger operators with L² potentials
, 2006
"... Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a halfline Schrödinger operator with square integrable potential. ..."
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Cited by 10 (1 self)
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Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a halfline Schrödinger operator with square integrable potential.
Analytical Evaluation of Economic Risk Capital for Portfolios of
 Gamma Risks, ASTIN Bulletin, Vol 31., No
, 2001
"... Based on the notions of valueatrisk and expected shortfall, we consider two functionals, abbreviated VaR and RaC, which represent the economic risk capital of a risky business over some time period required to cover losses with a high probability. These functionals are consistent with the risk pr ..."
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Cited by 9 (3 self)
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Based on the notions of valueatrisk and expected shortfall, we consider two functionals, abbreviated VaR and RaC, which represent the economic risk capital of a risky business over some time period required to cover losses with a high probability. These functionals are consistent with the risk preferences of profitseeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stoploss order). Quantitatively, RaC is equal to VaR plus an additional stoploss dependent term, which takes into account the average amount at loss. Furthermore, RaC is additive for comonotonic risks, which is an important extremal situation encountered in the modeling of dependencies in multivariate risk portfolios. Numerical illustrations for portfolios of gamma distributed risks follow. As a result of independent interest, new analytical expressions for the exact probability density of sums of independent gamma random variables are included, which are similar but different to previous expressions by Provost (1989) and Sim (1992).
On Azéma–Yor processes, their optimal properties and the Bachelier–Drawdown equation
, 2009
"... We study the class of Azéma–Yor processes defined from a general semimartingale with a continuous running supremum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the Drawdown equation. Solutions of the lat ..."
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Cited by 8 (0 self)
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We study the class of Azéma–Yor processes defined from a general semimartingale with a continuous running supremum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the Drawdown equation. Solutions of the latter have the drawdown property: they always stay above a given function of their past supremum. We then show that any process which satisfies the drawdown property is in fact an Azéma–Yor process. The proofs exploit group structure of the set of Azéma–Yor processes, indexed by functions, which we introduce. Secondly we study in detail Azéma–Yor martingales defined from a nonnegative local martingale converging to zero at infinity. We establish relations between
Holomorphic spaces: a brief and selective survey
 HOLOMORPHIC SPACES
, 1998
"... This article traces several prominent trends in the development of the subject of holomorphic spaces, with emphasis on operatortheoretic aspects. ..."
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Cited by 7 (0 self)
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This article traces several prominent trends in the development of the subject of holomorphic spaces, with emphasis on operatortheoretic aspects.
The AzémaYor Embedding in Brownian Motion with Drift
"... Let B = (Bt)t 0 be standard Brownian motion started at zero, let> 0 be given and fixed, and let be a probability measure on IR having a strictly positive density F 0. Then there exists a stopping time 3 of B such that (B 3 + 3) if and only if the following condition is satisfied: D:=Z Setting in ..."
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Cited by 4 (0 self)
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Let B = (Bt)t 0 be standard Brownian motion started at zero, let> 0 be given and fixed, and let be a probability measure on IR having a strictly positive density F 0. Then there exists a stopping time 3 of B such that (B 3 + 3) if and only if the following condition is satisfied: D:=Z Setting in this case C
Maximal operators and differentiation theorems for sparse sets
, 2009
"... We study maximal averages associated with singular measures on R. Our main result is a construction of singular Cantortype measures supported on sets of Hausdorff dimension 1 − ǫ, 0 ≤ ǫ < 1 3 for which the corresponding maximal operators are bounded on L p (R) for p> (1 + ǫ)/(1 − ǫ). As a con ..."
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Cited by 4 (0 self)
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We study maximal averages associated with singular measures on R. Our main result is a construction of singular Cantortype measures supported on sets of Hausdorff dimension 1 − ǫ, 0 ≤ ǫ < 1 3 for which the corresponding maximal operators are bounded on L p (R) for p> (1 + ǫ)/(1 − ǫ). As a consequence, we are able to answer a question of Aversa and Preiss on density and differentiation theorems in one dimension. Our proof combines probabilistic techniques with the methods developed in multidimensional Euclidean harmonic analysis, in particular there are strong similarities to Bourgain’s proof of the circular maximal theorem in two dimensions.
Complete lattices of probability measures with applications to martingale theory
, 1993
"... The set of probability measures on R with the stochastic order and the set of righttail integrable probability measures on R with the convex order form complete lattices. Connections of these lattice structures to martingale theory and to the HardyLittlewood maximal function are exhibited. ..."
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Cited by 4 (1 self)
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The set of probability measures on R with the stochastic order and the set of righttail integrable probability measures on R with the convex order form complete lattices. Connections of these lattice structures to martingale theory and to the HardyLittlewood maximal function are exhibited.
HyperbolicConcave Functions and HardyLittlewood Maximal Functions
, 2003
"... A class of generalized convex functions, the hyperbolicconcave functions, is defined, and used to characterize the collection of Hardy Littlewood maximal functions. These maximal functions and the probability measures associated with these maximal functions, the maximal probability measures, are u ..."
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Cited by 3 (0 self)
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A class of generalized convex functions, the hyperbolicconcave functions, is defined, and used to characterize the collection of Hardy Littlewood maximal functions. These maximal functions and the probability measures associated with these maximal functions, the maximal probability measures, are used in representations and inequalities within martingale theory. A related collection of minimal probability measures is also characterized, through a class of hyperbolicconcave envelopes.