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280 O. L. MANQASARIAN For the case when the set / is empty, the above proof is modified by deleting (2.19), (2.22) and references thereto. For the case when / = (1, 2, • • • , n) the above proof is modified by deleting that part of the proof between (2.19) and (2.22) and references thereto. It should be noted here that the above theorem is indeed a generalization of Arrow and Enthoven's result [1, Theorem 1]. Every case covered there is covered by the above theorem, but not conversely. An example of a case not covered by Arrow and Enthoven is the following one: min j-c"' 2 | - x g 0). Another application of pseudo-convex functions may be found in duality theory. Consider the primal problem (PP) mm[0(x)\q(x) g 0], where 8(x) is a scalar function on E'" and g(x) is an ;i X 1 vector function on If". For the above problem Wolfe [10] has defined the dual problem as (DP) max \f(x, y)\ VMx, y) =0,}2 0], where +(x, y) s 6(x) + y'g(x). Under appropriate conditions Wolfe has shown [10, Theorem 2] that if x° solves (PP), then x" and some y° solve (DP). Conversely, under somewhat stronger conditions, Huard [4, Theorem 2] showed that if (x°, y") solves (DP), then x° solves (PP). Both Wolfe and Huard required, among other tilings, that 6(x) and the components of g(x) be convex. We will now show that Huard's theorem can be extended to the case where 0(x) is pseudo-convex and the components of g(x) are quasi-convex, and that Wolfe's theorem is not amenable to such an extension. THEOREM 2. (Strict converse duality theorem) . Let 6{x) be a pseudo-convex function on E" and let the components of g(x) be differentiahle quasi-convex functions on E'". (a) If (x°, y°) solves (DP) and \)/(x, 2/°) is twice continuously differentiahle with respect to x in a neighborhood of x°, and if the Hessian of 4/(x, y") with respect to x is nonzero at x", then x" solves PP. (b) Let x° solve (PP) and let g(x) S 0 satisfy the Kuhn-Tucker constraint qualification [7\. It does not necessarily follow that x" and some y solve (DP). Proof, (a) The assumption that the Hessian of
sary conditions for some v" d E": PSEUDO-CONVEX FUNCTIONS 287 V*1>(x, y") + V/Vid 1 , V) = 0, VvUx, v") + V/'v^(x°, j, 0 ) S 0, y'vMx\ y) + /v/'v^K* 0 ,,/) = o, /SO, v*° = 0. Hence the above necessary conditions become: v>(x", y) = 0, V^(/, y") = g(x a ) g 0, y°'v^(x", if) = ifg(x°) = 0, /go. But from Theorem 1, with C = E m , these conditions are sufficient for x to be a solution of (PI'). (b) This part of the theorem will be established by means of the following counter-example: (PP1) min (-
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STOCHASTIC OPTIMIZATION MODELS IN F
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STOCHASTIC OPTIMIZATION MODELS IN F
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Dedicated to the memory of my fathe
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PART II. QUALITATIVE ECONOMIC RESUL
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2. Risk Aversion over Time Implies
Page 14 and 15: PREFACE AND BRIEF NOTES TO THE 2006
Page 16 and 17: and shows that current research in
Page 18 and 19: 1990. This area presages the CVaR l
Page 20 and 21: transaction cost band. Most such mo
Page 22 and 23: through targets and are less sensit
Page 24 and 25: ing dates, the papers by Breiman, H
Page 26 and 27: Carino, D. and W.T. Ziemba (1998).
Page 28 and 29: Kallberg, J.G. and W.T. Ziemba (198
Page 30 and 31: Stone, D. and W.T. Ziemba (1993). L
Page 32 and 33: PREFACE IN 1975 EDITION There is no
Page 34: particular results are generally st
Page 37: Part I Mathematical Tools
Page 40 and 41: wealth, then his utility function m
Page 42 and 43: cases in Exercise CR-12. The relati
Page 44 and 45: However, some extensions to infinit
Page 47 and 48: 1. EXPECTED UTILITY THEORY The Anna
Page 49 and 50: SUBJECTIVE PROBABILITIES ANT EXPECT
Page 51 and 52: SUBJECTIVE PROBABILITIES AND EXPECT
Page 57: SUBJECTIVE PROBABILITIES AND EXPECT
Page 60 and 61: 282 O. L. MANGA3ARIAN" for every X
Page 62 and 63: 284 O. L. MANGASARIAN We have from
Page 66 and 67: 288 O. L. MANGASAMAN Proof. Conside
Page 68 and 69: 290 O. L. MANGASARIAN Second Berkel
Page 70 and 71: denotes the /w-dimensional vector o
Page 72 and 73: Proof (I) We shall prove this part
Page 74 and 75: (Ill) have not, to the author's kno
Page 76 and 77: for the case (3); and for the case
Page 79 and 80: 3. DYNAMIC PROGRAMMING Introduction
Page 81 and 82: INTRODUCTION TO DYNAMIC PROGRAMMING
Page 93 and 94: COMPUTATIONAL AND REVIEW EXERCISES
Page 95 and 96: (d) xxx2 on E+ 2 = {x\xt = 0, x2 S
Page 97 and 98: Show that the Kuhn-Tucker condition
Page 99 and 100: (a) Show that/, (b, c) = gt (Z>, c)
Page 101: (c) Show that the monotonicity assu
Page 104 and 105: An individual's preferences are sai
Page 106 and 107: 12. (Hessians, bordered Hessians, c
Page 108 and 109: The problem in (b)-(c) hints that t
Page 110 and 111: (1) Suppose/is twice continuously d
Page 112 and 113: (f) Assume that R(i,a) g 0 and the
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(j) Show that an optimal policy Va
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INTRODUCTION In the second part of
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andom variables when the utility fu
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detailed analysis of the qualitativ
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separation in a local sense (for al
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1. STOCHASTIC DOMINANCE G. Hanoch a
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But now, EFFICIENCY ANALYSIS OF CHO
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EFFICIENCY ANALYSIS OF CHOICES INVO
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A Unified Approach to Stochastic Do
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A UNIFIED APPROACH TO STOCHASTIC DO
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RISK AVERSION 123 a given risk the
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RISK AVERSION 125 If z is actuarial
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RISK AVERSION 127 4. CONCAVITY The
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To show that (a) implies (d), note
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RISK AVERSION 131 (b') The risk pre
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9.2. Example 2. If (30) u'(x)=(x°
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RISK AVERSION 135 12. INCREASING AN
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ADDENDUM In retrospect, 1 wish foot
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14 THE REVIEW OF ECONOMICS AND STAT
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Separation in Portfolio Analysis R.
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SEPARATION IN PORTFOLIO ANALYSIS an
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SEPARATION IN PORTFOLIO ANALYSIS II
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SEPARATION IN PORTFOLIO ANALYSIS Th
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SEPARATION IN PORTFOLIO ANALYSIS Ca
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SEPARATION IN PORTFOLIO ANALYSIS It
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SEPARATION IN PORTFOLIO ANALYSIS th
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COMPUTATIONAL AND REVIEW EXERCISES
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7. Consider an investor having the
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Then eliminate p from (2) to obtain
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(a) Show that an indifference curve
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(d) Develop a similar result to (c)
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Exercise Source Notes Exercise 1 wa
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(f) Show that k > 0. [Note (Exercis
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(c) Interpret (b). (d) Illustrate s
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9. Let the partial relative risk-av
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Condition (iv) is called the "no-ea
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Then * ( 8xj \ V - s u = \-r-\ -ax,
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exhibiting decreasing absolute and
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(a) Determine necessary and suffici
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Let the random variables Og^Sl and
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Part III Static Portfolio Selection
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given parameter. Exercise ME-30 dev
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utility function. However, many con
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the characteristic exponent is 2 (i
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proportions to maximize expected ut
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investment returns are always nonne
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case to a case of one riskless and
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538 REVIEW OF ECONOMIC STUDIES Howe
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540 REVIEW OF ECONOMIC STUDIES wher
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542 REVIEW OF ECONOMIC STUDIES expa
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JAMES A. OHLSON that quadratic util
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JAMES A. OHLSON Assumptions A1-A3 a
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JAMES A. OHLSON (i) snpl/ieSi\U (3)
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JAMES A. OHLSON A number of interes
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JAMES A. OHLSON for all y,teSx'3C{Q
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Since Z"=o^( w »0 = 0 and/(«,«)
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JAMES A. OHLSON analysis here is "o
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Choosing Investment Portfolios When
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CHOOSING INVESTMENT PORTFOLIOS univ
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CHOOSING INVESTMENT PORTFOLIOS The/
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CHOOSING INVESTMENT PORTFOLIOS Tobi
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CHOOSING INVESTMENT PORTFOLIOS a-di
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CHOOSING INVESTMENT PORTFOLIOS Henc
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CHOOSING INVESTMENT PORTFOLIOS Lemm
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CHOOSING INVESTMENT PORTFOLIOS ID.
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CHOOSING INVESTMENT PORTFOLIOS Proo
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CHOOSING INVESTMENT PORTFOLIOS is c
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CHOOSING INVESTMENT PORTFOLIOS Proo
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CHOOSING INVESTMENT PORTFOLIOS 3. C
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2. EXISTENCE AND DIVERSIFICATION OF
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ON THE EXISTENCE OF OPTIMAL POLICIE
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Reprinted from Journal of Financial
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Remarks: Differentiability assumpti
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distributed from the rest. Then an
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x * "22 "12 1 " (o22- a12) + (0n -
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to decline — as if having more mo
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solved for as a function 8.(y,0"),
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FOOTNOTES 1. H. Makower and J. Mars
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264 QUARTERLY JOURNAL OF ECONOMICS
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274 QUARTERLY JOURNAL OP ECONOMICS
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280 QVABTBRLY JOURNAL OF ECONOMICS
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Reprinted from THE REVIEW OF ECONOM
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(1.1) and (1.2) give SOME EFFECTS O
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SOME EFFECTS OF TAXES ON RISK-TAKIN
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REFERENCES SOME EFFECTS OF TAXES ON
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COMPUTATIONAL AND REVIEW EXERCISES
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and a gamble is offered that return
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among two risky assets and a riskle
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Consider problem (1) and suppose th
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21. Let pi,...,p„ be joint-normal
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Consider the following four utility
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4. In most stochastic optimizing mo
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(g) Discuss the use of semivariance
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(a) Suppose that the investor alloc
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The linear programming subproblems
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(f) Show that (j> is proportional t
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(e) Show that X has mean and varian
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(i) Show that for all Aru ...,km, t
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Some properties of the expected val
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(i) Suppose that V0 is positive def
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where V is the gradient operator an
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and where Z, Ylt...,Ym are independ
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INTRODUCTION This part of the book
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some qualitative characteristics of
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function of wealth. Hence the optim
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INVESTMENT ANALYSIS UNDER UNCERTAIN
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P if and only if INVESTMENT ANALYSI
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INVESTMENT ANALYSIS UNDEB UNCERTAIN
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2. RISK AVERSION OVER TIME IMPLIES
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ables at t, and the primed variable
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ehavior in choosing among timeless
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is to asume that the consumer behav
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The function t/,(C,_,, wt|a,, ft) i
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sumption, given state j3,+i at r+l,
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3. MYOPIC PORTFOLIO POLICIES Reprin
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326 THE JOURNAL OF BUSINESS this op
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328 THE JOURNAL OF BUSINESS with th
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330 THE JOURNAL OF BUSINESS In sum
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332 THE JOURNAL OF BUSINESS subject
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334 THE JOURNAL OF BUSINESS myopic
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where and k" 1 f (Mt-Ui) 2 4k {Oi 1
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MIND-EXPANDING EXERCISES 1. Conside
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(i.e., partial separability) and #,
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(c) Discuss the solution properties
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(d) By differentiating (cl) and (c2
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Let U be the utility function for t
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INTRODUCTION I. Two-Period Consumpt
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asked to develop explicit solutions
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that the property of nonincreasing
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The Hakansson paper studies a gener
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In the Samuelson-Hakansson additive
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techniques. The article by Breiman
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enously). These requirements are to
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in general, a monotonic function of
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ergodic theory, the book by Breiman
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or less than u; either do nothing o
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ME-24 the reader is invited to cons
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In discrete time, a nonnegative lin
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R. G. VICKSON This must not be inte
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This gives with R. G. VICKSON t(t)
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TWO-PERIOD CONSUMPTION MODELS AND P
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310 DRfeZE AND MODIGLIANI second or
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312 DREZE AND MODIGLIANI Had the sa
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314 DRfezE AND MODIGLIANI (— U^U^
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316 DREZE AND MODIGLIANI ure for te
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318 DREZE AND MODIGLIANI income sid
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320 DREZE AND MODIGLIANI (3.5) The
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322 DREZE AND MODIGLIANI Consumptio
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324 DREZE AND MODIGLIANI in (cj, c2
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326 DREZE AND MODIGLIANI Since d^Uj
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328 DREZE AND MODIGLIANI On the rig
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330 DRfeZE AND MODIGLIANI and its p
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332 DREZE AND M0DIGLIAN1 LEMMA C.2.
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334 DRfeZE AND MOD1GLIANI if the sm
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MANAGEMENT SCIENCE Vol. 19, No. 2,
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A DYNAMIC MODEL FOR BOND PORTFOLIO
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Maximize A DYNAMIC MODEL FOE BOND P
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MULTIPERIOD RISK PREFERENCE 41 mode
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MULTIPERIOD RISK PREFERENCE 43 The
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MULT1PERI0D RISK PREFERENCE between
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MULTIPERIOD RISK PREFERENCE 47 (ii)
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MULTIPERIOD RISK PREFERENCE 49 that
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or if s"(s' MULTIPERIOD RISK PREFER
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MULTIPERIOD RISK PREFERENCE 53 5. C
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Reprinted from The Review of Econom
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the risky asset, with 1 — w, goin
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w*t = glW,;Z,-1,...,Z0] = gT_i[W>]
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Since proof of the theorem is strai
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E C O N O M E T R I C A VOLUME 38 S
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OPTIMAL INVESTMENT 589 M: the numbe
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OPTIMAL INVESTMENT 591 For comparis
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(ii) in the case of Model III, 1 1
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Let Then OPTIMAL INVESTMENT 595 *.=
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where OPTIMAL INVESTMENT 597 (41) T
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where (54) T(x) = sup \ log c + log
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where b(v*) is the greatest lower b
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OPTIMAL INVESTMENT 603 noncapital i
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OPTIMAL INVESTMENT 605 Let us now t
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OPTIMAL INVESTMENT 607 REFERENCES [
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present value. Let ( be time, r the
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constant or decrease at a constant
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Denote the vector in the parenthese
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112 HOWARD M. TAYLOB easy to pictur
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114 HOWARD M. TAYLOR market and ign
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116 HOWARD M. TAYLOR the investor w
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118 HOWARD M. TAYLOR option which a
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120 HOWARD M. TAYLOR 3. BR.VDA, J.
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172 BASIL A. KALYMON more general p
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174 BASIL A. KALYMON The three case
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176 BASIL A. KALYMON an additional
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178 BASIL A. KALYMON (24) d\ = r -
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180 BASIL A. KALYMON The above resu
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182 BASIL A. KALYMON PROOF. First w
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MANAGEMENT SCIENCE Vol. 17, No. 7,
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M1NIMAX POLICIES FOB SELLING AN ASS
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MINIMAX POLICIES FOB SELLING AN ASS
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MINIMAX POLICIES FOR SELLING AN ASS
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MINIMAX POLICIES TOR SELLING AN ASS
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MINIMAX POLICIES FOB SELLING AN ASS
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648 L. BREIMAN We have discussed th
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650 L. BREIMAN Hence, it is suffici
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(a.s.) on the set lim XN < oo > u <
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EDWARD O. THORP method to this arti
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EDWARD O. THORP drastic simplificat
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EDWARD O. THORP us is that the prop
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EDWARD O. THORP Let the range of Rj
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EDWARD O. THORP is approximately on
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EDWARD O. THORP choosing P in repea
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EDWARD O. THORP return from the hed
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EDWARD O. THORP it is plausible to
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EDWARD O. THORP is maximization of
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EDWARD O. THORP 6. BREIMAN, L., "Op
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Reprinted from JOURNAL OF ECONOMIC
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CONSUMPTION AND PORTFOLIO RULES 375
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and CONSUMPTION AND PORTFOLIO RULES
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CONSUMPTION AND PORTFOLIO RULES giv
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Yin CONSUMPTION AND PORTFOLIO RULES
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Y(t) Y CONSUMPTION AND PORTFOLIO RU
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(d) Suppose n > r2. Formulate the o
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(j) Show that the absolute risk-ave
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The optimal first-period decision a
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(b) Show that the optimal policy in
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the game is stationary and favorabl
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(v) UHEV-E^CV-Af- 1 )^ w, V&0, w>0,
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MIND-EXPANDING EXERCISES 1. Conside
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Note the following distinction betw
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(e) Discuss the computational aspec
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(a) Show that U obeys the functiona
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(g) If ;ij(0) > 0 (1 g; g N), show
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Note that problem P2 has 3(N- 1)+1
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Note that (f) and (g) imply the exi
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P = (Pu), where pu = P[X„+l =j\ X
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(a) Show that {X„} is a stationar
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(b) Show that T T u = a + b Y[ c\'
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(b) Interpret this assumption. Is i
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(d) Show that (c) may be written as
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Bibliography Journal Abbreviations
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BHARUCHA-REID, A. T. (1960). Elemen
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FIACCO, A. V., and MCCORMICK, G. P.
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IGLEHART, D. (1965). "Capital accum
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MANGASARIAN, O. L. (1966). "Suffici
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PORTEUS, E. L. (1972). "Equivalent
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THOMASION, A. J. (1969). The Struct
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A Absolute risk aversion, 84, 85, 9
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Logarithmic utility functions, see
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V Value of information, 462-467, 68
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