On Wald-Type Optimal Stopping for Brownian Motion
|
|
- Vanessa Dalton
- 6 years ago
- Views:
Transcription
1 J Al Probab Vol 34, No 1, 1997, (66-73) Prerint Ser No 1, 1994, Math Inst Aarhus On Wald-Tye Otimal Stoing for Brownian Motion S RAVRSN and PSKIR The solution is resented to all otimal stoing roblems of the form: where B (B t ) t su jb j c is standard Brownian motion and the suremum is taken over all stoing times for B with finite exectation, while the ma : R +! R satisfies jxj 1 cjxj + d for some d R with c > being given and fixed The otimal stoing time is shown to be the hitting time by the reflecting Brownian motion jbj (jb t j) t of the set of all ( aroximate ) imum oints of the ma x 7! (jxj) cx The method of roof relies uon Wald s identity for Brownian motion and simle real analysis arguments A simle roof of the Dubins-Jacka-Schwarz-She-Shiryaev ( square root of two ) imal inequality for randomly stoed Brownian motion is given as an alication 1 Introduction The aim of this aer is to resent the solution to a class of Wald s tye otimal stoing roblems for Brownian motion, and from this deduce some shar inequalities which give bounds for the exectation of functionals of randomly stoed Brownian motion in terms of the exectation of the stoing time More recisely, let B (B t ) t be standard Brownian motion defined on the robability sace (; F ; P ) Then in this aer we find the solution to all otimal stoing roblems of the following form Maximize the exectation: (11) jb j over all stoing times for B with finite exectation, where the measurable ma : R +! R satisfies jxj 1 cjxj + d for some d R with c > being given and fixed It is shown that the otimal stoing time is the hitting time by the reflecting Brownian motion jbj (jb t j) t of the set of all ( aroximate ) imum oints of the ma x 7! (jxj) cx The result just indicated will be resented in more detail in Section, as well as extended to all continuous local martingales by using the standard time change method To conclude the introduction we should like to say that our main emhasis in this aer is on simlicity of our solution to the roblem under consideration Nevertheless, we will see in Section 3 that our c AMS 198 subject classifications Primary 64, 6J65 Secondary 64, 644, 6H5 Key words and hrases: Brownian motion (Wiener rocess), otimal stoing (time), Wald s identity for Brownian motion, Burkholder-undy s inequality, Doob s otional samling theorem, the concave conjugate, continuous (local) martingale, time change, the square root of two imal inequality goran@imfaudk 1
2 method is flexible enough to rovide a simle roof of the Dubins-Jacka-Schwarz-She-Shiryaev inequality for Brownian motion which was firstly found in [] (and indeendently in [4]), and then roved by an entirely different method in [3] Wald s otimal stoing for Brownian motion 1 In this section we resent the solution to the otimal stoing roblem (11) For simlicity, we shall only consider the case where (jxj) jxj for <, and it will be clear from our roof below that the case of general ma ( satisfying the boundedness condition ) could be treated by exactly the same method Thus, if B (Bt)t is standard Brownian motion, then the roblem under consideration in this section is the following Maximize the exectation: B (1) c over all stoing times for B with finite exectation, where < and c > are given and fixed First, it should be noted that in the case, we find by Wald s identity (see [5]) for Brownian motion ( jb j ( ) ) that the exression in (1) equals (1c)( ) Thus taking n or for n 1, deending on whether < c < 1 or 1 < c < 1, we see that the suremum equals +1 or resectively If c 1, then the suremum equals, and any stoing time for B is otimal These facts solve the roblem (1) in the case The solution in the general case < < is formulated in the following theorem Theorem 1 ( Wald s otimal stoing for Brownian motion ) Let B (Bt)t be standard Brownian motion, and let < < and c > be given and fixed Consider the otimal stoing roblem: () su B c where the suremum is taken over all stoing times for B with finite exectation Then the otimal stoing time (at which the suremum is attained) in () may be defined as follows: n (3) ;c 3 1 o 1() inf t > : jbtj c Moreover, for all stoing times for B with finite exectation we have: () (4) jb j c The uer bound in (4) is the best ossible c Proof iven < < and c >, denote: (5) V(; c) jb j c whenever is a stoing time for B Then by Wald s identity for Brownian motion we find
3 out that the exression in (5) may be equivalently written in the following form: jxj cx (6) V (; c) Z 1 1 dp B (x) whenever is a stoing time for B with () < 1 Our next ste is to imize the ma x 7! D(x) jxj cx over R For this, note that D(x) D(x) for all x R, and therefore it is enough to consider D(x) for x > We have D (x) x 1 cx for x >, and hence we see that D attains its imal value at the oint 6(c) 1() Thus it is clear from (6) that the otimal stoing time in () might be defined by (3) This comletes the first art of the roof Finally, inserting 3 3 ;c from (3) into (6), we easily find: V 3(; c) D (c) 1() () c This establishes (4) with the last statement of the theorem, and the roof is comlete Remark The receding roof shows that the solution to the roblem (11) in the case of general ma ( satisfying the boundedness condition ) could be found by using exactly the same method: The otimal stoing time is the hitting time by the reflecting Brownian motion jbj (jb t j) t of the set of all (aroximate) imum oints of the ma x 7! D(x) (jxj) cx ( Here aroximate stands to cover the case ( in an obvious manner ) when D does not attain its least uer bound on the real line) In the remaining art of this section we will exlore some consequences of the inequality (4) in more detail For this, let a stoing time for B with ( ) < 1 and < < be given and fixed Then from (4) we get: (7) jb j inf c> c( ) + c () It is elementary to comute that this infimum equals ( ) In this way we obtain: (8) jb j 1 ( < ) with the constant 1 being the best ossible in all of the inequalities ( Observe that this also follows by Wald s identity and Jensen s inequality in a straightforward way) Next consider the case < < 1 Thus we shall look at V (; c) instead of V (; c) in (5) and (6) By the same argument as for (6) we obtain: c jb j V (; c) Z 1 1 cx jxj dp B (x) where < < 1 The same calculation as in the roof of Theorem 1 shows that the ma x 7! D(x) cx jxj attains its imal value over R at the oint 6(c) 1() Thus as in the roof of Theorem 1 we find: 3
4 From this inequality we get: su c> c jb j c( ) + () c () c jb j The same calculation as for the roof of (8) shows that this suremum equals ( ) Thus as above for (8) we obtain: (9) 1 jb j ( < 1 ) with the constant 1 being the best ossible in all of the inequalities ( Observe that this also follows by Wald s identity and Jensen s inequality in a straightforward way) 3 The revious calculations together with conclusions (8) and (9) indicate that the inequality (4)+(7) rovide shar estimates which are otherwise obtainable by a different method that relies uon convexity and Jensen s inequality ( see Remark 4 below ) This leads recisely to the main oint of our observation: The revious rocedure can be reeated for any measurable ma satisfying the boundedness condition In this way we obtain a shar estimate of the form: jb j 1 ( ) where is a ma to be found ( by imizing and minimizing certain real valued functions of real variable ) We formulate this more recisely in the next corollary Corollary 3 Let B (B t ) t Then for any stoing time for B the inequality holds: (1) be standard Brownian motion, and let : R!R be a measurable ma jb j 1 inf c> c( ) + su xr jxj 1 cx and is shar whenever the right-hand side is finite Similarly, if H : R!R is a measurable ma, then for any stoing time for B with finite exectation the inequality holds: (11) su c> c( ) + inf xr H and is shar whenever the left-hand side is finite jxj 1 cx H jb j 1 Proof It follows from the roof of Theorem 1 as indicated in Remark and the lines above following it (or just straightforward by using Wald s identity) It should be noted that the boundedness condition on the mas and H is contained in the non-triviality of the conclusions Remark 4 As noted by a referee, if we set H(x) ( x) 1 cx su xr jxj inf x for x, then we have: cx H(x) ~ H(c) 4
5 where ~ H denotes the concave conjugate of H Similarly, we have: inf c> Thus (1) reads as follows: (1) H c( ) + su xr jb j 1 jxj 1 cx H ( ~ ) inf c( ) ~ H(c) H ~ ( )1 Moreover, since ~H is the (smallest) concave function which dominates H, it is clear from a simle comarison that (1) also follows by Jensen s inequality This rovides an alternative way of looking at (1) and (11) and clarifies (7)+(8) ( A similar remark might be directed to (11) with (9)) Note that (1) gets the following form: jb j 1 ( ) whenever x 7! ( x) is concave on R + Remark 5 By using the standard time change method, we can generalize and extend the inequalities (1) and (11) to cover the case of all continuous local martingales Let M (M t ) t be a continuous local martingale with the quadratic variation rocess [M ] ([M ] t ) t such that M, and let, H : R +! R be measurable functions Then for any t > for which ([M ] t ) < 1 the inequalities hold: (13) (14) su c> 1 jmt j inf c> c ([M ] t ) + inf xr c ([M ] t ) + su xr H jxj 1 cx jxj 1 cx H 1 jm t j and are shar whenever the right-hand side in (13) and the left-hand side in (14) are finite To rove the sharness of (13) and (14) for every given and fixed t >, consider M t B t+ with > and being the hitting time of some > by the reflecting Brownian motion jbj (jb t j) t Letting! 1 and using ( integrability ) roerties of ( in the context of Corollary 3 ), by Burkholder-undy s inequalities (see [1]) and uniform integrability arguments we (eventually) finish with the inequalities (1) and (11) for otimal, at least in the case when allows the limiting rocedures which are required ( the case of general could then follow by aroximation ) Thus the sharness of (13)+(14) follows from the sharness of (1)+(11) 3 Alications In this section, as an alication of our method and results obtained, we shall resent a simle roof of the Dubins-Jacka-Schwarz-She-Shiryaev (square root of two) imal inequality for randomly stoed Brownian motion which was firstly found in [] ( and indeendently in [4] ), and then roved by an entirely different method in [3] The method of attack in the roof of (31) and (3) is based uon the trick of icking u the two martingales (33) and (35) with the roerties 5
6 desired We shall begin by stating the two inequalities to be roved Let B (B t ) t be standard Brownian motion, and let be a stoing time for B with finite exectation Then the following inequalities are shar: (31) (3) t jb t j t B t 1 We shall first deduce these inequalities by our method, and then show their sharness by icking u the otimal stoing times ( for which the equalities are attained ) Our aroach to the roblem of establishing (31) is motivated by the fact that the rocess ( st B s B t ) t is equally distributed as the reflecting Brownian motion rocess (jb t j) t for which we have found otimal bound (4) ( from where by (7) we get (8) with 1 ), while (B ) whenever ( ) < 1 These observations clearly lead us to (31), at least for some stoing times To extend this to all stoing times, we shall use a simle martingale argument Proof of (31): Set S t st B s for t Since (B t) t t is a martingale, and (S t B t ) t is equally distributed as (jb t j) t, we see that: St 1 (33) Z t c B t t + 14c is a martingale ( with resect to the natural filtration which is known to be the same as the natural filtration of B ) Using (B ), by Doob s otional samling theorem and the elementary inequality xct c(x t) + 14c, we find: for any bounded stoing time S c 1 S B c 1 Z ) (Z ) 14c Hence we get: S 1 inf c> c( )+14c for any bounded stoing time Passing to the limit, we obtain (31) for all stoing times with finite exectation This comletes the roof of (31) Next we extend (31) to any continuous local martingale M (M t ) t with M For this, note that by the time change and (31) we obtain: (34) M s B [M ] s B s q 1 [M ] t st st s[m ] t for all t > 3 In the next ste we will aly (34) to the continuous martingale M defined by: (35) M t jb j jb j 1 F t^ for t In this way we get: 6
7 (36) t<1 jb j 1 jb j Ft^ We now ass to the roof of the square root of two inequality Proof of (3): Since A x + x A for < x < t r r jb t j t<1 jb j t<1 jb t^ j jb j 1 Ft^ 1 1 jb j jb j + jb j ( ) r jb j jb j 1 A, by (36) we find: t<1 1 + jb j 1 jb j + 1 jb j ( ) This establishes (3) and comletes the first art of the roof 4 To rove the sharness of (31) one may take the stoing time: inf 8 t > : jb t j a jb j Ft^ for any a > Then the equality in (31) is attained It follows by Wald s identity Note that for any a > the stoing time 3 1 could be equivalently (in distribution) defined by: 3 1 inf t > : B s B t a st 5 To rove the sharness of (3) one may take the stoing time: 3 inf t > : jb s jjb t j a st for any a > Then it is easily verified that t 3 jb t j 1 a and ( 3 ) a ( see [3] ) Thus the equality in (3) is attained, and the roof of the sharness is comlete RFRNCS [1] BURKHOLDR, D: L: and UNDY, R: F: (197): xtraolation and interolation of quasilinear oerators on martingales: Acta Math: 14 (49-34) [] DUBINS, L: : and SCHWARZ, : (1988): A shar inequality for sub-martingales and stoing times: Astérisque (19-145) [3] DUBINS, L: B: SHPP, L: A: and SHIRYAV, A: N: (1993): Otimal stoing rules and imal inequalities for Bessel rocesses: Theory Probab: Al: 38 (6-61) 7
8 [4] JACKA, S: D: (1991): Otimal stoing and best constants for Doob-like inequalities I: The case 1 : Ann: Probab: 19 ( ) [5] WALD, A: (1945): Sequential tests of statistical hyotheses: Ann: Math: Statist: 16 ( ) Svend rik raversen Deartment of Mathematical Sciences University of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus matseg@imfaudk oran Peskir Deartment of Mathematical Sciences University of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus homeimfaudk/goran goran@imfaudk 8
On Doob s Maximal Inequality for Brownian Motion
Stochastic Process. Al. Vol. 69, No., 997, (-5) Research Reort No. 337, 995, Det. Theoret. Statist. Aarhus On Doob s Maximal Inequality for Brownian Motion S. E. GRAVERSEN and G. PESKIR If B = (B t ) t
More informationt 0 Xt sup X t p c p inf t 0
SHARP MAXIMAL L -ESTIMATES FOR MARTINGALES RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI ABSTRACT. Let X be a suermartingale starting from 0 which has only nonnegative jums. For each 0 < < we determine the best
More informationRandomly Weighted Series of Contractions in Hilbert Spaces
Math. Scand. Vol. 79, o. 2, 996, (263-282) Prerint Ser. o. 5, 994, Math. Inst. Aarhus. Introduction Randomly Weighted Series of Contractions in Hilbert Saces G. PESKIR, D. SCHEIDER, M. WEBER Conditions
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Al. 44 (3) 3 38 Contents lists available at SciVerse ScienceDirect Journal of Mathematical Analysis and Alications journal homeage: www.elsevier.com/locate/jmaa Maximal surface area of a
More informationConvex Optimization methods for Computing Channel Capacity
Convex Otimization methods for Comuting Channel Caacity Abhishek Sinha Laboratory for Information and Decision Systems (LIDS), MIT sinhaa@mit.edu May 15, 2014 We consider a classical comutational roblem
More informationLECTURE 7 NOTES. x n. d x if. E [g(x n )] E [g(x)]
LECTURE 7 NOTES 1. Convergence of random variables. Before delving into the large samle roerties of the MLE, we review some concets from large samle theory. 1. Convergence in robability: x n x if, for
More informationSums of independent random variables
3 Sums of indeendent random variables This lecture collects a number of estimates for sums of indeendent random variables with values in a Banach sace E. We concentrate on sums of the form N γ nx n, where
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationWALD S EQUATION AND ASYMPTOTIC BIAS OF RANDOMLY STOPPED U-STATISTICS
PROCEEDINGS OF HE AMERICAN MAHEMAICAL SOCIEY Volume 125, Number 3, March 1997, Pages 917 925 S 0002-9939(97)03574-0 WALD S EQUAION AND ASYMPOIC BIAS OF RANDOMLY SOPPED U-SAISICS VICOR H. DE LA PEÑA AND
More informationHAUSDORFF MEASURE OF p-cantor SETS
Real Analysis Exchange Vol. 302), 2004/2005,. 20 C. Cabrelli, U. Molter, Deartamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and CONICET, Pabellón I - Ciudad Universitaria,
More informationLocation of solutions for quasi-linear elliptic equations with general gradient dependence
Electronic Journal of Qualitative Theory of Differential Equations 217, No. 87, 1 1; htts://doi.org/1.14232/ejqtde.217.1.87 www.math.u-szeged.hu/ejqtde/ Location of solutions for quasi-linear ellitic equations
More informationOn a Markov Game with Incomplete Information
On a Markov Game with Incomlete Information Johannes Hörner, Dinah Rosenberg y, Eilon Solan z and Nicolas Vieille x{ January 24, 26 Abstract We consider an examle of a Markov game with lack of information
More informationStochastic integration II: the Itô integral
13 Stochastic integration II: the Itô integral We have seen in Lecture 6 how to integrate functions Φ : (, ) L (H, E) with resect to an H-cylindrical Brownian motion W H. In this lecture we address the
More informationSTRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R 2
STRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R 2 ANGELES ALFONSECA Abstract In this aer we rove an almost-orthogonality rincile for
More informationLEIBNIZ SEMINORMS IN PROBABILITY SPACES
LEIBNIZ SEMINORMS IN PROBABILITY SPACES ÁDÁM BESENYEI AND ZOLTÁN LÉKA Abstract. In this aer we study the (strong) Leibniz roerty of centered moments of bounded random variables. We shall answer a question
More informationImproved Bounds on Bell Numbers and on Moments of Sums of Random Variables
Imroved Bounds on Bell Numbers and on Moments of Sums of Random Variables Daniel Berend Tamir Tassa Abstract We rovide bounds for moments of sums of sequences of indeendent random variables. Concentrating
More informationAn Estimate For Heilbronn s Exponential Sum
An Estimate For Heilbronn s Exonential Sum D.R. Heath-Brown Magdalen College, Oxford For Heini Halberstam, on his retirement Let be a rime, and set e(x) = ex(2πix). Heilbronn s exonential sum is defined
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationTranspose of the Weighted Mean Matrix on Weighted Sequence Spaces
Transose of the Weighted Mean Matri on Weighted Sequence Saces Rahmatollah Lashkariour Deartment of Mathematics, Faculty of Sciences, Sistan and Baluchestan University, Zahedan, Iran Lashkari@hamoon.usb.ac.ir,
More informationEstimation of the large covariance matrix with two-step monotone missing data
Estimation of the large covariance matrix with two-ste monotone missing data Masashi Hyodo, Nobumichi Shutoh 2, Takashi Seo, and Tatjana Pavlenko 3 Deartment of Mathematical Information Science, Tokyo
More informationApproximating min-max k-clustering
Aroximating min-max k-clustering Asaf Levin July 24, 2007 Abstract We consider the roblems of set artitioning into k clusters with minimum total cost and minimum of the maximum cost of a cluster. The cost
More information16.2. Infinite Series. Introduction. Prerequisites. Learning Outcomes
Infinite Series 6.2 Introduction We extend the concet of a finite series, met in Section 6., to the situation in which the number of terms increase without bound. We define what is meant by an infinite
More informationCombining Logistic Regression with Kriging for Mapping the Risk of Occurrence of Unexploded Ordnance (UXO)
Combining Logistic Regression with Kriging for Maing the Risk of Occurrence of Unexloded Ordnance (UXO) H. Saito (), P. Goovaerts (), S. A. McKenna (2) Environmental and Water Resources Engineering, Deartment
More information4. Score normalization technical details We now discuss the technical details of the score normalization method.
SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules
More informationIMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES
IMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES OHAD GILADI AND ASSAF NAOR Abstract. It is shown that if (, ) is a Banach sace with Rademacher tye 1 then for every n N there exists
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationOn Isoperimetric Functions of Probability Measures Having Log-Concave Densities with Respect to the Standard Normal Law
On Isoerimetric Functions of Probability Measures Having Log-Concave Densities with Resect to the Standard Normal Law Sergey G. Bobkov Abstract Isoerimetric inequalities are discussed for one-dimensional
More informationThe Azéma-Yor Embedding in Non-Singular Diffusions
Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let
More informationON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE. 1. Introduction
ON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE GUSTAVO GARRIGÓS ANDREAS SEEGER TINO ULLRICH Abstract We give an alternative roof and a wavelet analog of recent results
More informationConvex Analysis and Economic Theory Winter 2018
Division of the Humanities and Social Sciences Ec 181 KC Border Conve Analysis and Economic Theory Winter 2018 Toic 16: Fenchel conjugates 16.1 Conjugate functions Recall from Proosition 14.1.1 that is
More informationsubstantial literature on emirical likelihood indicating that it is widely viewed as a desirable and natural aroach to statistical inference in a vari
Condence tubes for multile quantile lots via emirical likelihood John H.J. Einmahl Eindhoven University of Technology Ian W. McKeague Florida State University May 7, 998 Abstract The nonarametric emirical
More informationTowards understanding the Lorenz curve using the Uniform distribution. Chris J. Stephens. Newcastle City Council, Newcastle upon Tyne, UK
Towards understanding the Lorenz curve using the Uniform distribution Chris J. Stehens Newcastle City Council, Newcastle uon Tyne, UK (For the Gini-Lorenz Conference, University of Siena, Italy, May 2005)
More informationInference for Empirical Wasserstein Distances on Finite Spaces: Supplementary Material
Inference for Emirical Wasserstein Distances on Finite Saces: Sulementary Material Max Sommerfeld Axel Munk Keywords: otimal transort, Wasserstein distance, central limit theorem, directional Hadamard
More informationTRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES
Khayyam J. Math. DOI:10.22034/kjm.2019.84207 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES ISMAEL GARCÍA-BAYONA Communicated by A.M. Peralta Abstract. In this aer, we define two new Schur and
More informationApplications to stochastic PDE
15 Alications to stochastic PE In this final lecture we resent some alications of the theory develoed in this course to stochastic artial differential equations. We concentrate on two secific examles:
More informationON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX)0000-0 ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS WILLIAM D. BANKS AND ASMA HARCHARRAS
More informationRadial Basis Function Networks: Algorithms
Radial Basis Function Networks: Algorithms Introduction to Neural Networks : Lecture 13 John A. Bullinaria, 2004 1. The RBF Maing 2. The RBF Network Architecture 3. Comutational Power of RBF Networks 4.
More informationMultiplicity of weak solutions for a class of nonuniformly elliptic equations of p-laplacian type
Nonlinear Analysis 7 29 536 546 www.elsevier.com/locate/na Multilicity of weak solutions for a class of nonuniformly ellitic equations of -Lalacian tye Hoang Quoc Toan, Quô c-anh Ngô Deartment of Mathematics,
More informationAnalysis of some entrance probabilities for killed birth-death processes
Analysis of some entrance robabilities for killed birth-death rocesses Master s Thesis O.J.G. van der Velde Suervisor: Dr. F.M. Sieksma July 5, 207 Mathematical Institute, Leiden University Contents Introduction
More informationProbability Estimates for Multi-class Classification by Pairwise Coupling
Probability Estimates for Multi-class Classification by Pairwise Couling Ting-Fan Wu Chih-Jen Lin Deartment of Comuter Science National Taiwan University Taiei 06, Taiwan Ruby C. Weng Deartment of Statistics
More informationLecture 6. 2 Recurrence/transience, harmonic functions and martingales
Lecture 6 Classification of states We have shown that all states of an irreducible countable state Markov chain must of the same tye. This gives rise to the following classification. Definition. [Classification
More information16.2. Infinite Series. Introduction. Prerequisites. Learning Outcomes
Infinite Series 6. Introduction We extend the concet of a finite series, met in section, to the situation in which the number of terms increase without bound. We define what is meant by an infinite series
More information1 Riesz Potential and Enbeddings Theorems
Riesz Potential and Enbeddings Theorems Given 0 < < and a function u L loc R, the Riesz otential of u is defined by u y I u x := R x y dy, x R We begin by finding an exonent such that I u L R c u L R for
More informationHaar type and Carleson Constants
ariv:0902.955v [math.fa] Feb 2009 Haar tye and Carleson Constants Stefan Geiss October 30, 208 Abstract Paul F.. Müller For a collection E of dyadic intervals, a Banach sace, and,2] we assume the uer l
More informationSolvability and Number of Roots of Bi-Quadratic Equations over p adic Fields
Malaysian Journal of Mathematical Sciences 10(S February: 15-35 (016 Secial Issue: The 3 rd International Conference on Mathematical Alications in Engineering 014 (ICMAE 14 MALAYSIAN JOURNAL OF MATHEMATICAL
More information#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS
#A47 INTEGERS 15 (015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS Mihai Ciu Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 5,
More informationSCHUR S LEMMA AND BEST CONSTANTS IN WEIGHTED NORM INEQUALITIES. Gord Sinnamon The University of Western Ontario. December 27, 2003
SCHUR S LEMMA AND BEST CONSTANTS IN WEIGHTED NORM INEQUALITIES Gord Sinnamon The University of Western Ontario December 27, 23 Abstract. Strong forms of Schur s Lemma and its converse are roved for mas
More information#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS
#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt
More informationPell's Equation and Fundamental Units Pell's equation was first introduced to me in the number theory class at Caltech that I never comleted. It was r
Pell's Equation and Fundamental Units Kaisa Taiale University of Minnesota Summer 000 1 Pell's Equation and Fundamental Units Pell's equation was first introduced to me in the number theory class at Caltech
More information1 1 c (a) 1 (b) 1 Figure 1: (a) First ath followed by salesman in the stris method. (b) Alternative ath. 4. D = distance travelled closing the loo. Th
18.415/6.854 Advanced Algorithms ovember 7, 1996 Euclidean TSP (art I) Lecturer: Michel X. Goemans MIT These notes are based on scribe notes by Marios Paaefthymiou and Mike Klugerman. 1 Euclidean TSP Consider
More informationMaximum Process Problems in Optimal Control Theory
J. Appl. Math. Stochastic Anal. Vol. 25, No., 25, (77-88) Research Report No. 423, 2, Dept. Theoret. Statist. Aarhus (2 pp) Maximum Process Problems in Optimal Control Theory GORAN PESKIR 3 Given a standard
More informationIntroduction to Probability and Statistics
Introduction to Probability and Statistics Chater 8 Ammar M. Sarhan, asarhan@mathstat.dal.ca Deartment of Mathematics and Statistics, Dalhousie University Fall Semester 28 Chater 8 Tests of Hyotheses Based
More informationOn the Chvatál-Complexity of Knapsack Problems
R u t c o r Research R e o r t On the Chvatál-Comlexity of Knasack Problems Gergely Kovács a Béla Vizvári b RRR 5-08, October 008 RUTCOR Rutgers Center for Oerations Research Rutgers University 640 Bartholomew
More informationASYMPTOTIC BEHAVIOR FOR THE BEST LOWER BOUND OF JENSEN S FUNCTIONAL
75 Kragujevac J. Math. 25 23) 75 79. ASYMPTOTIC BEHAVIOR FOR THE BEST LOWER BOUND OF JENSEN S FUNCTIONAL Stojan Radenović and Mirjana Pavlović 2 University of Belgrade, Faculty of Mechanical Engineering,
More informationB8.1 Martingales Through Measure Theory. Concept of independence
B8.1 Martingales Through Measure Theory Concet of indeendence Motivated by the notion of indeendent events in relims robability, we have generalized the concet of indeendence to families of σ-algebras.
More informationBrownian Motion and Random Prime Factorization
Brownian Motion and Random Prime Factorization Kendrick Tang June 4, 202 Contents Introduction 2 2 Brownian Motion 2 2. Develoing Brownian Motion.................... 2 2.. Measure Saces and Borel Sigma-Algebras.........
More informationA Note on Guaranteed Sparse Recovery via l 1 -Minimization
A Note on Guaranteed Sarse Recovery via l -Minimization Simon Foucart, Université Pierre et Marie Curie Abstract It is roved that every s-sarse vector x C N can be recovered from the measurement vector
More informationOn a class of Rellich inequalities
On a class of Rellich inequalities G. Barbatis A. Tertikas Dedicated to Professor E.B. Davies on the occasion of his 60th birthday Abstract We rove Rellich and imroved Rellich inequalities that involve
More informationApplied Probability Trust (24 March 2004) Abstract
Alied Probability Trust (24 March 2004) STOPPING THE MAXIMUM OF A CORRELATED RANDOM WALK, WITH COST FOR OBSERVATION PIETER ALLAART, University of North Texas Abstract Let (S n ) n 0 be a correlated random
More informationOptimism, Delay and (In)Efficiency in a Stochastic Model of Bargaining
Otimism, Delay and In)Efficiency in a Stochastic Model of Bargaining Juan Ortner Boston University Setember 10, 2012 Abstract I study a bilateral bargaining game in which the size of the surlus follows
More informationTRIGONOMETRIC WAVELET INTERPOLATION IN BESOV SPACES. Woula Themistoclakis. 1. Introduction
FACTA UNIVERSITATIS NIŠ) Ser. Math. Inform. 4 999), 49 7 TRIGONOMETRIC WAVELET INTERPOLATION IN BESOV SPACES Woula Themistoclakis Dedicated to Prof. M.R. Occorsio for his 65th birthday Abstract. In the
More informationSOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES. 1. Introduction
SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES SEVER S. DRAGOMIR 1 AND MOHAMMAD SAL MOSLEHIAN Abstract. An oerator T is called (α, β)-normal (0 α 1 β) if α T T T T β T T. In this aer,
More informationElementary theory of L p spaces
CHAPTER 3 Elementary theory of L saces 3.1 Convexity. Jensen, Hölder, Minkowski inequality. We begin with two definitions. A set A R d is said to be convex if, for any x 0, x 1 2 A x = x 0 + (x 1 x 0 )
More informationarxiv:cond-mat/ v2 25 Sep 2002
Energy fluctuations at the multicritical oint in two-dimensional sin glasses arxiv:cond-mat/0207694 v2 25 Se 2002 1. Introduction Hidetoshi Nishimori, Cyril Falvo and Yukiyasu Ozeki Deartment of Physics,
More informationHolder Continuity of Local Minimizers. Giovanni Cupini, Nicola Fusco, and Raffaella Petti
Journal of Mathematical Analysis and Alications 35, 578597 1999 Article ID jmaa199964 available online at htt:wwwidealibrarycom on older Continuity of Local Minimizers Giovanni Cuini, icola Fusco, and
More informationCOMPARISON OF VARIOUS OPTIMIZATION TECHNIQUES FOR DESIGN FIR DIGITAL FILTERS
NCCI 1 -National Conference on Comutational Instrumentation CSIO Chandigarh, INDIA, 19- March 1 COMPARISON OF VARIOUS OPIMIZAION ECHNIQUES FOR DESIGN FIR DIGIAL FILERS Amanjeet Panghal 1, Nitin Mittal,Devender
More informationPETER J. GRABNER AND ARNOLD KNOPFMACHER
ARITHMETIC AND METRIC PROPERTIES OF -ADIC ENGEL SERIES EXPANSIONS PETER J. GRABNER AND ARNOLD KNOPFMACHER Abstract. We derive a characterization of rational numbers in terms of their unique -adic Engel
More informationUniform Law on the Unit Sphere of a Banach Space
Uniform Law on the Unit Shere of a Banach Sace by Bernard Beauzamy Société de Calcul Mathématique SA Faubourg Saint Honoré 75008 Paris France Setember 008 Abstract We investigate the construction of a
More informationChapter 7 Sampling and Sampling Distributions. Introduction. Selecting a Sample. Introduction. Sampling from a Finite Population
Chater 7 and s Selecting a Samle Point Estimation Introduction to s of Proerties of Point Estimators Other Methods Introduction An element is the entity on which data are collected. A oulation is a collection
More informationA PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL
A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL LAPLACE EQUATIONS Abstract. We establish ointwise a riori estimates for solutions in D 1, of equations of tye u = f x, u, where
More informationOn the capacity of the general trapdoor channel with feedback
On the caacity of the general tradoor channel with feedback Jui Wu and Achilleas Anastasooulos Electrical Engineering and Comuter Science Deartment University of Michigan Ann Arbor, MI, 48109-1 email:
More informationGENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS
International Journal of Analysis Alications ISSN 9-8639 Volume 5, Number (04), -9 htt://www.etamaths.com GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS ILYAS ALI, HU YANG, ABDUL SHAKOOR Abstract.
More informationCorrespondence Between Fractal-Wavelet. Transforms and Iterated Function Systems. With Grey Level Maps. F. Mendivil and E.R.
1 Corresondence Between Fractal-Wavelet Transforms and Iterated Function Systems With Grey Level Mas F. Mendivil and E.R. Vrscay Deartment of Alied Mathematics Faculty of Mathematics University of Waterloo
More informationAlmost 4000 years ago, Babylonians had discovered the following approximation to. x 2 dy 2 =1, (5.0.2)
Chater 5 Pell s Equation One of the earliest issues graled with in number theory is the fact that geometric quantities are often not rational. For instance, if we take a right triangle with two side lengths
More informationPositivity, local smoothing and Harnack inequalities for very fast diffusion equations
Positivity, local smoothing and Harnack inequalities for very fast diffusion equations Dedicated to Luis Caffarelli for his ucoming 60 th birthday Matteo Bonforte a, b and Juan Luis Vázquez a, c Abstract
More informationMATHEMATICAL MODELLING OF THE WIRELESS COMMUNICATION NETWORK
Comuter Modelling and ew Technologies, 5, Vol.9, o., 3-39 Transort and Telecommunication Institute, Lomonosov, LV-9, Riga, Latvia MATHEMATICAL MODELLIG OF THE WIRELESS COMMUICATIO ETWORK M. KOPEETSK Deartment
More information15-451/651: Design & Analysis of Algorithms October 23, 2018 Lecture #17: Prediction from Expert Advice last changed: October 25, 2018
5-45/65: Design & Analysis of Algorithms October 23, 208 Lecture #7: Prediction from Exert Advice last changed: October 25, 208 Prediction with Exert Advice Today we ll study the roblem of making redictions
More information2 K. ENTACHER 2 Generalized Haar function systems In the following we x an arbitrary integer base b 2. For the notations and denitions of generalized
BIT 38 :2 (998), 283{292. QUASI-MONTE CARLO METHODS FOR NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES II KARL ENTACHER y Deartment of Mathematics, University of Salzburg, Hellbrunnerstr. 34 A-52 Salzburg,
More informationNONLINEAR OPTIMIZATION WITH CONVEX CONSTRAINTS. The Goldstein-Levitin-Polyak algorithm
- (23) NLP - NONLINEAR OPTIMIZATION WITH CONVEX CONSTRAINTS The Goldstein-Levitin-Polya algorithm We consider an algorithm for solving the otimization roblem under convex constraints. Although the convexity
More informationThe non-stochastic multi-armed bandit problem
Submitted for journal ublication. The non-stochastic multi-armed bandit roblem Peter Auer Institute for Theoretical Comuter Science Graz University of Technology A-8010 Graz (Austria) auer@igi.tu-graz.ac.at
More informationIntrinsic Approximation on Cantor-like Sets, a Problem of Mahler
Intrinsic Aroximation on Cantor-like Sets, a Problem of Mahler Ryan Broderick, Lior Fishman, Asaf Reich and Barak Weiss July 200 Abstract In 984, Kurt Mahler osed the following fundamental question: How
More informationSolving the Poisson Disorder Problem
Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, Springer-Verlag, 22, (295-32) Research Report No. 49, 2, Dept. Theoret. Statist. Aarhus Solving the Poisson Disorder Problem
More informationON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS
#A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,
More informationBEST CONSTANT IN POINCARÉ INEQUALITIES WITH TRACES: A FREE DISCONTINUITY APPROACH
BEST CONSTANT IN POINCARÉ INEQUALITIES WITH TRACES: A FREE DISCONTINUITY APPROACH DORIN BUCUR, ALESSANDRO GIACOMINI, AND PAOLA TREBESCHI Abstract For Ω R N oen bounded and with a Lischitz boundary, and
More informationAn Analysis of Reliable Classifiers through ROC Isometrics
An Analysis of Reliable Classifiers through ROC Isometrics Stijn Vanderlooy s.vanderlooy@cs.unimaas.nl Ida G. Srinkhuizen-Kuyer kuyer@cs.unimaas.nl Evgueni N. Smirnov smirnov@cs.unimaas.nl MICC-IKAT, Universiteit
More informationKhinchine inequality for slightly dependent random variables
arxiv:170808095v1 [mathpr] 7 Aug 017 Khinchine inequality for slightly deendent random variables Susanna Sektor Abstract We rove a Khintchine tye inequality under the assumtion that the sum of Rademacher
More informationAdvanced Calculus I. Part A, for both Section 200 and Section 501
Sring 2 Instructions Please write your solutions on your own aer. These roblems should be treated as essay questions. A roblem that says give an examle requires a suorting exlanation. In all roblems, you
More informationThe inverse Goldbach problem
1 The inverse Goldbach roblem by Christian Elsholtz Submission Setember 7, 2000 (this version includes galley corrections). Aeared in Mathematika 2001. Abstract We imrove the uer and lower bounds of the
More informationA note on the random greedy triangle-packing algorithm
A note on the random greedy triangle-acking algorithm Tom Bohman Alan Frieze Eyal Lubetzky Abstract The random greedy algorithm for constructing a large artial Steiner-Trile-System is defined as follows.
More informationMulti-Operation Multi-Machine Scheduling
Multi-Oeration Multi-Machine Scheduling Weizhen Mao he College of William and Mary, Williamsburg VA 3185, USA Abstract. In the multi-oeration scheduling that arises in industrial engineering, each job
More informationTopic 7: Using identity types
Toic 7: Using identity tyes June 10, 2014 Now we would like to learn how to use identity tyes and how to do some actual mathematics with them. By now we have essentially introduced all inference rules
More informationCOMMUNICATION BETWEEN SHAREHOLDERS 1
COMMUNICATION BTWN SHARHOLDRS 1 A B. O A : A D Lemma B.1. U to µ Z r 2 σ2 Z + σ2 X 2r ω 2 an additive constant that does not deend on a or θ, the agents ayoffs can be written as: 2r rθa ω2 + θ µ Y rcov
More informationMartingale transforms and their projection operators on manifolds
artingale transforms and their rojection oerators on manifolds Rodrigo Bañuelos, Fabrice Baudoin Deartment of athematics Purdue University West Lafayette, 4796 Setember 9, Abstract We rove the boundedness
More informationOn the rate of convergence in the martingale central limit theorem
Bernoulli 192), 2013, 633 645 DOI: 10.3150/12-BEJ417 arxiv:1103.5050v2 [math.pr] 21 Mar 2013 On the rate of convergence in the martingale central limit theorem JEAN-CHRISTOPHE MOURRAT Ecole olytechnique
More informationOn split sample and randomized confidence intervals for binomial proportions
On slit samle and randomized confidence intervals for binomial roortions Måns Thulin Deartment of Mathematics, Usala University arxiv:1402.6536v1 [stat.me] 26 Feb 2014 Abstract Slit samle methods have
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLOCAL p-laplacian PROBLEMS
Electronic Journal of ifferential Equations, Vol. 2016 (2016), No. 274,. 1 9. ISSN: 1072-6691. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu EXISTENCE AN UNIQUENESS OF SOLUTIONS FOR NONLOCAL
More informationCommutators on l. D. Dosev and W. B. Johnson
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Commutators on l D. Dosev and W. B. Johnson Abstract The oerators on l which are commutators are those not of the form λi
More informationSolved Problems. (a) (b) (c) Figure P4.1 Simple Classification Problems First we draw a line between each set of dark and light data points.
Solved Problems Solved Problems P Solve the three simle classification roblems shown in Figure P by drawing a decision boundary Find weight and bias values that result in single-neuron ercetrons with the
More informationMODELING THE RELIABILITY OF C4ISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL
Technical Sciences and Alied Mathematics MODELING THE RELIABILITY OF CISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL Cezar VASILESCU Regional Deartment of Defense Resources Management
More informationOnline Appendix to Accompany AComparisonof Traditional and Open-Access Appointment Scheduling Policies
Online Aendix to Accomany AComarisonof Traditional and Oen-Access Aointment Scheduling Policies Lawrence W. Robinson Johnson Graduate School of Management Cornell University Ithaca, NY 14853-6201 lwr2@cornell.edu
More information