ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS


 Coral Fields
 1 years ago
 Views:
Transcription
1 ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga, Spain. () Deparmen of Foundaions of Economic Analysis, Universiy Compluense of Madrid, Spain. ABSTRACT: The aim of his sudy is o analyse he resoluion of Sochasic Programming Problems in which he objecive funcion depends on parameers which are coninuous random variables wih a known disribuion probabiliy. In he lieraure on hese quesions differen soluion conceps have been defined for problems of hese characerisics. These conceps are obained by applying a ransformaion crierion o he sochasic objecive which conains a saisical feaure of he objecive, implying ha for he same sochasic problem here are differen opimal soluions available which, in principle, are no comparable. Our sudy analyses and esablishes some relaions beween hese soluion conceps. KEY WORDS: Sochasic Programming. Opimal soluion conceps. We hank J.B. Readman for his linguisic revision of he ex.
2 . Inroducion When a real problem is modelled and solved by means of a mahemaical programming problem i may happen ha some of he parameers which figure in he problem are unknown, wheher i be in he objecive funcion or in he feasible se. If hese parameers of unknown value can be aken as random variables he resuling problem is a sochasic programming one. Obviously, he fac ha one of he consrain funcions or he objecive funcion is affeced by random parameers gives rise o he view ha ha funcion is also a random variable and, given ha, in general, a random variable does no admi a relaion of order, i is necessary o specify a soluion concep for hese problems. The characerisics of sochasic programming problems and he deermining of opimal soluions for hem have been widely sudied in he lieraure. Among oher sudies we should make special menion of he books of Prékopa (995), Kall and Wallace (994), Kibzun and Kan (996) or he aricles of Kall (98), Leclerq (98) and Zare and Daneshmand (995). In hese sudies we can observe how he exising soluion conceps for sochasic programming problems originae from he ransformaion of he problem ino a deerminisic one which is given he erm deerminisic equivalen. This ransformaion is achieved by aking saisical feaures of he objecive funcion or of he consrain funcions dependen on random parameers in he problem. In his sudy we focus our aenion on problems of sochasic programming in which he random parameers affec only he objecive funcion of i or in which, if hey affec he consrains, his laer has already been ransformed ino is deerminisic equivalen by one of he exising procedures in sochasic programming. In addiion, we assume ha hey are coninuous random variables,
3 so ha he sochasic programming problem which we are considering is as follows: Min %(,) xc% (.) x D z where x is he vecor of decision variables, D R n is a closed se, bounded, convex and no empy, c ~ is a random vecor defined on he se E R s, wih known probabiliy disribuion and whose componens are coninuous random variables. Furhermore, we assume ha he probabiliy disribuion of vecor c ~ is independen of he decision variables of he problem, x,..., x. n From he hypoheses esablished for he problem (.) i follows ha he objecive funcion of he problem depends on random parameers and, as previously indicaed, his implies ha he funcion is a random variable. In order o solve he problem (.), differen soluion conceps have been defined in he lieraure. Each of hese conceps proceeds from he applicaion of a crierion o ransform he sochasic objecive ino a deerminisic funcion. These crieria draw on some saisical feaure from he sochasic objecive, so ha he problem (.) is aribued various deerminisic equivalens and, in general, various possible opimal soluions. The diversiy of conceps and soluions can give rise in some way o cerain confusion when a problem is being solved and immediaely poses various quesions, such as he exising differences beween some conceps and ohers, wheher any of he conceps so far defined is beer han anoher, ec. In his sudy we consider five soluion conceps o he problem (.) which correspond o he applicaion of differen crieria o ransform he sochasic problem ino is deerminisic equivalen, all of which have been defined previously, and some relaions are esablished beween hem. We begin by defining in Secion hese five soluion conceps and esablishing some differences beween 3
4 hem. In Secion 3 some relaions beween hese conceps are esablished and in Secion 4 we pose and solve an example o illusrae he resuls obained.. Some Soluion Conceps for Sochasic Programming Problems Le us consider he sochasic programming problem (.). Le z (x) be he expeced value funcion of ~ z ( x, ~ c ) and σ ( x ) is variance, which we assume o be finie for all x D. Le us now look a five opimum conceps for his problem. As previously indicaed, each one of hese conceps derives from he applicaion of ransformaion crieria o he objecive funcion of he problem and he obaining of a deerminisic equivalen problem from he iniial problem. The ransformaion crieria which we consider are: expeced value, minimum variance, expeced value sandard deviaion efficiency, minimum risk and Kaaoka. Definiion : Expeced value soluion Poin x D is he expeced value soluion o (.) if i is he opimal soluion o he problem: Min x D z( x) (.) In oher words, he expeced value opimal soluion o he problem (.) is simply he opimal soluion resuling from he subsiuion of he sochasic objecive funcion by is expeced value, hus aking up his measuremen of cenral endency of he random variable for is opimisaion. This soluion concep has been frequenly used in he lieraure for he resoluion of sochasic programming problems, alhough his crierion canno always be considered appropriae given ha if we remain wih only he expeced value of he random variable, cerain feaures of he sochasic objecive funcion of he problem migh no be included in he deerminisic equivalen problem. In his sense, some auhors such as Kaaoka (963) or Prékopa (995) make some criical commens abou he 4
5 applicaion of he above crierion and esablish same condiions under which i can be considered appropriae. Definiion : Minimum variance soluion Poin x D is he minimum variance soluion o he problem (.) if i is he opimum of he problem: Min x D σ ( x) (.) In oher words, in order o solve he sochasic programming problem, he crierion considered is ha of minimising he variance of he sochasic objecive funcion ~ z ( x, ~ c ), choosing, herefore, he vecor x for which he random variable ~ z ( x, ~ c ) is more concenraed around is expeced value. This implies ha he opimisaion crierion is ha of he minimum, regardless of wheher he problem of opimisaion be ha of he minimum (a hypohesis which we susain in his sudy) or of he maximum, and in some ways, can be considered a crierion of low risk. Definiion 3: Expeced value sandard deviaion efficien soluion Poin x D is a expeced value sandard deviaion efficien soluion o he problem (.) if i is an efficien soluion in he Pareo sense o he following bicrieria problem: Min x D ( z ( x), σ ( x) ) (.3) which includes he expeced value and he sandard deviaion of he sochasic objecive funcion. We denoe (.3) and E Eσ as he se of efficien soluions o he problem p E Eσ as he se of properly efficien soluions o his problem. This concep of efficiency was inroduced by Markowiz (95) as a means of solving problems in he field of porfolio selecion in finance economics. Markowiz considers he bicrieria problem of expeced value minimum variance, ha is: 5
6 ( x σ x ) Min z( ), () (.4) x D In his sudy we have chosen o subsiue he variance of he sochasic objecive for is sandard deviaion given ha, in his way, he wo objecives of he problem (.) are expressed in he same unis of measuremen. In any case, since he square roo funcion is sricly increasing, he efficien ses of boh problems coincide. Finally, we define he conceps of opimal soluion minimum risk of aspiraion level u and he opimal soluion wih probabiliy β. Boh soluions are obained by applying he minimum risk and Kaaoka crieria, respecively, referred o in he lieraure as crieria of maximum probabiliy or saisfying crieria, due o he fac ha, as we shall see in he following definiion, in boh cases he crieria o be used provide, in one way or anoher, good soluions in erms of probabiliy. Definiion 4: Minimum risk soluion of aspiraion level u Poin x D is he minimum risk soluion of aspiraion level u for he problem (.) if i is he opimal soluion o he problem: Max x D P ( ~ z ( x, ~ c ) u) (.5) Tha is, o obain a minimum risk opimal soluion o he problem (.) we apply wha in he lieraure is referred o as he minimum risk crierion. This consiss of fixing a level for he sochasic objecive funcion u R, o which we apply he erm aspiraion level, and of maximising he probabiliy ha he objecive will be less han or equal o ha level: P( z ~ ( x, ~ c ) u). In his way, he fixed level u can be inerpreed as being he uppermos level ha he Decision Maker (DM) is capable of admiing for he sochasic objecive ha we wish o minimise. 6
7 The firs sudies in which his crierion, also ermed model P, is proposed as a means of solving he problem (.), are hose carried ou by Charnes and Cooper (963) and Bereanu (964). Definiion 5: Kaaoka soluion wih probabiliy β Poin x D is he Kaaoka soluion wih probabiliy β o he problem (.) if here exiss u R such ha (x, u) is he opimal soluion o he problem: s. Min u x, u P( ~ z( x, ~ c) u) = β (.6) x D Thus, he obaining of opimal soluions wih probabiliy β is derived from he applicaion of he βfracile or Kaaoka crierion o he problem (.), inroduced by Kaaoka (963). I consiss of fixing a probabiliy β, for he objecive and deermining he lowes level u, which canno exceed he objecive funcion wih ha probabiliy. The objecive funcion of he sochasic problem goes on o become par of he feasible se as a probabilisic or chance consrain and he deerminisic equivalen problem is a problem wih n+ decision variables: he n decision variables of he sochasic programming problem carried in vecor x and variable u. Once hese five soluion conceps for sochasic programming problems have been defined, each one associaed wih a crierion which is considered adequae for he resoluion of he iniial problem, a comparaive analysis of he abovemenioned conceps enables us o commen on he following differences which emerge beween hem:. In order o apply he maximum probabiliy crieria (minimum risk and Kaaoka) i is necessary o fix previously a parameer: level u, if we apply he minimum risk crierion, or probabiliy β, if we apply he Kaaoka crierion, 7
8 while in he firs hree crieria his is unnecessary. Moreover, he minimum risk and Kaaoka opimal soluions will depend, generally, on he values assigned o hose parameers.. Whils he firs hree soluion conceps (expeced value, minimum variance and expeced value sandard deviaion efficiency) can be obained provided he expeced value and variance of he objecive funcion are known, he wo crieria of maximum probabiliy depend on he disribuion funcion of he sochasic objecive funcion and, herefore, in order o obain hem i is essenial o know he laer. 3. The las hree soluion conceps (expeced value sandard deviaion efficiency and he wo relaed o maximum probabiliy) give rise, in general, o a se of soluions: he se of expeced value sandard deviaion efficien soluions for he firs of hese and a soluion for each level, u, or probabiliy β o be fixed for he las wo conceps which are no generally comparable. From wha has been saed above i can be concluded ha he diversiy of soluion conceps leads o he need o choose one or he oher o solve he sochasic programming problems and, in his sense, we can sae ha he resoluion of hese problems always implicily conains a decision process. Obviously, he choice of a soluion will have o be made bearing in mind he characerisics of he problem o be solved and he preferences of he DM. However, as we shall see below, if specific hypoheses are shown o hold rue, he soluions which we have jus defined are relaed, alhough a he beginning one may hink he opposie, due o he fac ha hese are soluions o deerminisic equivalen problems which draw on saisical characerisics differen from he sochasic objecive. 8
9 3. Relaions beween he Soluion Crieria for Sochasic Programming Problems In his secion we deal wih he analysis of he exisence of relaions beween he soluions o he problem (.) previously defined, which are obained by applying o he problem he crieria discussed in he previous secion: expeced value, minimum variance, expeced value sandard deviaion efficiency, minimum risk and Kaaoka. Specifically, relaions are esablished beween he minimum risk opimal soluions and he Kaaoka ones and beween hese and he expeced value sandard deviaion efficiency soluions for a sochasic programming problem. Prior o ha, we shall consider some of he resuls of Muliple Objecive Programming which will be applied hroughou his secion. In addiion, as an immediae consequence of hese resuls we shall esablish relaions beween he opimal soluions expeced value and minimum variance and he expeced value sandard deviaion efficien soluions. Le us consider he following muliple objecive programming problem: Min x D ( f ( x),..., f ( x) ) q (3.) where f is a vecorial funcion, f: H R n R q, and he problem resuling from he applicaion of he weighing mehod o i gives us: Min µ f ( x) µ f ( x) x D q q (3.) wih vecor of weighs m, generaliy, we assume o be normalised. µ k 0, for all k {,,, q}, which, wihou losing The following heorem relaes o he efficien soluions o he muliple objecive problem (3.) and he opimal soluions o he associaed weighed problem, (3.). 9
10 Theorem.( Sawaragi, Nakayama and Tanino (985). Le us assume ha he funcions f,..., f q are convex and ha D is a convex se. Thus: a) If x* is a properly efficien soluion for he muliple objecive problem (3.), here exiss a weigh vecor m wih sricly posiive componens such ha x* is he opimal soluion for he weighed problem (3.). b) If x* is he opimal soluion o he weighed problem (3.), for a vecor of weighs wih sricly posiive componens, x* is a properly efficien soluion for he muliple objecive problem (3.). c) If x* is he only soluion o he problem (3.), wih m 0, hen x* is an efficien soluion o he problem (3.). If i is no he only soluion, he soluions obained are weakly efficien for (3.). This heorem of muliple objecive programming allows us o relae he expeced value and minimum variance soluions o he problem (.) o he expeced value sandard deviaion efficien se. Therefore, le us consider he problems (.), (.) and (.3). The weighed problem associaed o he problem (.3), for weighs µ and  µ, µ [0, ], is: Min µ z( x) + ( µ ) σ( x) x D From Theorem we can sae ha:. If he expeced value opimal soluion o he problem (.) is unique hen i is an expeced value sandard deviaion efficien soluion. Where i is no unique i can only be assured ha he expeced value opimal soluions are expeced value sandard deviaion weakly efficien soluions, bu we canno sae ha hey are expeced value sandard deviaion efficien soluions.. If he variance of he sochasic objecive is a sricly convex funcion, he minimum variance problem has a unique soluion (since we assume ha se D 0
11 is closed, bounded, convex and no empy), and so is opimal soluion is an expeced value sandard deviaion efficien soluion o he sochasic programming problem. Should he minimum variance problem have more han one opimal soluion, hese soluions are expeced value sandard deviaion weakly efficien soluions and he only hing we can be cerain (as in poin above) is he weak efficiency of hese soluions. We shall now analyse he relaions beween he minimum risk and Kaaoka problems. Following his, we shall reurn o deal wih he concep expeced value sandard deviaion efficien soluion and we shall relae i o he Kaaoka and minimum risk soluions o he sochasic programming problem. 3.. Relaions beween he minimum risk problem and he Kaaoka problem Given he sochasic programming problem (.) le us consider he deerminisic equivalen problem (.5) corresponding o he crierion of minimum risk of aspiraion level u, and he problem (.6) corresponding o he applicaion of he Kaaoka crierion for a probabiliy β (0, ). The following heorems esablish a relaion beween he opimal soluions o hese wo problems. Theorem Le us assume ha he disribuion funcion of he random variable ~ z ( x, ~ c ) is sricly increasing. Then x* is he minimum risk soluion of aspiraion level u* if and only if ( x *, u *) is he Kaaoka soluion wih probabiliy β*, wih u* and β* so ha: P( ~ z ( x *, ~ c ) u*) = β *. Proof: We demonsrae he heorem by process of reducio ad absurdum. ) If x* is he soluion o he problem (.5) i is rue ha: P ( ~ z ( x, c~ ) u *) P( ~ z ( x*, ~ c ) u *) = β*, x D
12 Le us assume ha ( x *, u *) is no he soluion o he problem (.6). In his case here exiss a vecor ( x, u ), feasible in (.6) proving ha u < u*, ha is: ~ x ~ β and u < u*. x D, here exis u R such ha P( z(, c) u) = * However as he disribuion funcion of he random variable ~ z ( x, ~ c ) is sricly increasing, i is also rue ha: P ( ~ z ( x, ~ c ) u* ) = P( ~ z ( x, ~ c ) u) + P( u < ~ z ( x, ~ c ) u *) = β * + θ ~ ~ ) > where θ = P( u < z( x, c u *) 0 ~ ~ ) β x D for which P( z ( x, c u *) > *, from which we deduce ha here exiss a vecor, which conradics he hypohesis. ) By hypohesis ( x *, u *) is he opimal Kaaoka soluion of level β *, herefore i proves ha: ~ ~ ) β a) I is feasible in (.6): x* D and P( z ( x*, c u *) = * b) If ( x, ) ~ x ~ ) β, hen u* u. u proves ha x D and P( z (, c u) = * Le us assume ha x* is no he opimal soluion o he problem (.5). Then, here exiss a vecor x D for which: P ( ~ z ( x, ~ c ) u *) > β * As he disribuion funcion of he random variable ~ z ( x, ~ c ) is sricly increasing, i holds rue ha here exiss a u < u* for which: which conradics he hypohesis b). P ( ~ z ( x, ~ c ) u) = β * From Theorem we can affirm ha if he disribuion funcion of he sochasic objecive is a sricly increasing funcion here exiss a reciprociy beween he minimum risk and he Kaaoka soluions, such ha, if we have he minimum risk soluion (Kaaoka soluion) we can affirm ha i is he Kaaoka soluion (minimum risk soluion), ha is:
13 . For each fixed aspiraion level u, he minimum risk soluion is also he Kaaoka soluion wih a probabiliy β equal o he maximum probabiliy obained in he minimum risk problem. For each fixed β he Kaaoka soluion is also he minimum risk soluion, if, for his laer, we esablish an aspiraion level u equal o he opimal value of he Kaaoka problem. 3.. Relaions beween he opimal soluions of Kaaoka and he expeced value sandard deviaion efficien soluions We shall now analyse he exisence of some relaion beween hose soluions o he sochasic programming problem (.) which correspond o he applicaion of he Kaaoka crierion and he se of expeced value sandard deviaion efficien soluions o he same problem. Up ill now, he relaions ha we have esablished beween he soluion crieria for sochasic programming problems have been of a general naure, ha is, applicable o any problem of sochasic programming. However, in his secion he relaions o esablish are exclusively applicable o sochasic programming problems of specific characerisics, given ha for he Kaaoka crierion o be applied i is necessary o know he disribuion funcion of he sochasic objecive, somehing which is, on he whole, complicaed, even when he probabiliy disribuion of vecor ~ c is known. This fac leads us o he heme of our sudy, assuming he objecive funcion of he problem o be of he linear ype, so ha from his poin onwards, he problem we are dealing wih is: Min D x cx % (3.3) Furhermore, we esablish a hypohesis regarding he probabiliy disribuion of vecor ~ c, and, specifically, we consider wo cases which correspond o he hypohesis ha vecor ~ c is normally disribued (normal case) 3
14 and o he consideraion ha vecor c ~ depends linearly on a single random variable (simple randomizaion case). In boh cases i is possible o deermine he probabiliy disribuion funcion of he sochasic objecive raise and solve he problem (3.3) by applying he Kaaoka crierion. ~ c x and, herefore, o To carry ou his analysis we begin by raising he problem of expeced value sandard deviaion efficiency of he problem (3.3) and he problems corresponding o he applicaion of he Kaaoka crierion in he cases menioned, and, following his, we analyse he relaions beween hem. Le c be he expeced value vecor of c ~ and V be is variance and covariance marix, which we assume is posiive definie. Then he expression of he expeced value of ~ c x is c x and is variance is x Vx, and, herefore, he se of expeced value sandard deviaion efficien soluions for he problem (3.3) is ha of he problem: / ( cx xvx ) Min, ( ) (3.4) x D As we assume ha he feasible se D is convex, by hypohesis from he / origin, and he funcion ( x Vx) is convex (see SancuMinasian (984) pages 9495), he bicrieria problem expeced value sandard deviaion is convex, and we can sae, on he basis of Theorem, ha he se of expeced value sandard deviaion properly efficien soluions o he problem (3.4), p E Eσ, can be generaed from he resoluion of he weighed problem: Min µ c x + ( µ )( x x D Vx) / (3.5) where µ (0, ). Subsequenly we go on o obain he deerminisic equivalen problems for (3.3) corresponding o he applicaion of he Kaaoka crierion, in he cases previously cied. a) Normal linear case 4
15 Le us suppose ha 0 D and c ~ is a random vecor mulinormal wih expeced value c and posiive definie marix of variances and covariances V. In his case he random variable ~ c x is a normal variable wih expeced value c x and variance x Vx, which implies ha: (( ~ ) / ( ) / c x c x ( x Vx) u c x ( x Vx) ) = Φ( u c x) ( x ) / ) ~ P( c x u) = P Vx where Φ is he sandardised normal disribuion funcion. This implies ha he feasible se of he problem (.6) is, according o he esablished hypoheses: { ( x, u) D R / u = c x + Φ ( β )( x Vx ) / } and he Kaaoka problem, for a probabiliy β, is: s.a u = c Min u x, u x + Φ x D ( β)( x Vx) / equivalen o: If α Min c x + Φ x D ( β )( x Vx) / Φ ( β), we can express he preceding problem as follows: = Min x D c x + α( x Vx) / (3.6) b) Simple linear randomizaion case Le us assume ha vecor c ~ depends linearly on a single random variable: ~ c c ~ = + c, where ~ is a coninuous random variable of expeced value, sandard deviaion υ > 0, υ <, and disribuion funcion F, sricly increasing. Moreover, le us suppose ha c x > 0 for all x D. Wih hese hypoheses he expeced value of ~ c is c = c + c and is marix of variances and covariances is V = υ c c. Proceeding from he previous hypoheses i follows ha: P( c ~ x u) = P( ~ c x u c x) = P ( ~ ( u c x) c x) = F ( u c x) c x) 5
16 so ha he Kaaoka problem is equivalen o: Min x D c x + F ( β ) c x As in he normal case, making α = ( ( β ) ) υ F we can express he preceding problem by means of he problem (3.6). Therefore, in he wo cases analysed, he problem corresponding o he Kaaoka crierion is problem (3.6), where α = Φ ( β ) in he normal case and ( ( β ) υ α = ) in he simple randomizaion case. F In order o esablish he relaions beween he opimal soluion of he Kaaoka problem in hese wo cases and he se of expeced value sandard deviaion properly efficien soluions we have only o relae he opimal soluions of he problems (3.5) and (3.6). Proposiion If α >0 and µ (0, ), wih µ = ( + α), hen problems (3.5) and (3.6) are equivalen. Proof: Since µ (0, ) he soluion o problem (3.5) coincides wih ha of: Min x D c x + ( µ )( x Vx) / µ and simply doing α = ( µ ) µ > 0 or, wha is equivalen, µ = ( + α) we obain he equivalence beween problems (3.5) and (3.6). Thus, from he Proposiion, we esablish an equivalence beween problem (3.6) and he weighed problem (3.5). If we denoe x*(α) he opimal soluion o problem (3.6), we can be sure ha: U α> 0 x *( α) = Le us see wha inerpreaion his resul has in he cases analysed. In boh cases he parameer α muliplies he sandard deviaion of he sochasic objecive E p Eσ 6
17 and depends on he fixed probabiliy for he applicaion of he Kaaoka crierion. Specifically:. In he normal case α = Φ ( β ), and consequenly his parameer is posiive if and only if he fixed probabiliy, β, is greaer han In he simple randomizaion case α = ( F ( β ) ) σ and herefore α > 0 if he probabiliy o be fixed in he Kaaoka problem β > F ( ), ha is greaer han he probabiliy of he random variable ~ being less han or equal o he expeced value. Consequenly, in boh cases, he fac ha α is sricly posiive can be inerpreed as fixing a high probabiliy. From he resuls obained we can sae ha: If he probabiliy fixed in he Kaaoka problem is high (β > 0.5 in he normal case and β > F ( ) in he simple randomizaion case), we can affirm ha he opimal soluion o his problem is a expeced value sandard deviaion properly efficien soluion of he sochasic problem (.). Given a expeced value sandard deviaion properly efficien soluion o he problem (.), in he cases analysed here exiss a high probabiliy, β, such ha his soluion is a Kaaoka opimum. Obviously, based on hese resuls one has o ask he quesion of wha happens when, in eiher of hese wo cases under sudy, he fixed probabiliy is such ha α 0. Noe ha for α = 0 he problem ha is obained in boh cases is he expeced value problem, previously menioned in he sudy, and i corresponds o he fixing of a probabiliy β = 0.5 in he normal case and β = F () in he simple randomizaion case. 7
18 Wha ype of soluion is obained when he probabiliy fixed is low? When his is so he opimal soluion o he Kaaoka problem does no have o be a expeced value sandard deviaion properly efficien soluion o he sochasic problem. In his sense i mus be noed ha he expeced value sandard deviaion efficiency crierion is appropriae when he individual is risk averse. So, in he cases analysed, he hypohesis of risk aversion is ranslaed ino he fixing of a high probabiliy. In addiion o his, if we analyse he objecive funcion of he Kaaoka problem, we noe ha when he fixed probabiliy is low, he sandard deviaion of he sochasic objecive is muliplied by a negaive facor. This ells us ha, in some way, a leas in he cases analysed, he minimum variance crierion may no be appropriae if he DM wans a risky soluion o he sochasic problem. On he oher hand, i mus be emphasised ha, as for hese problems, he disribuion funcion of he sochasic objecive is sricly increasing, his deermines as rue he reciprociy analysed in he previous secion amongs he opimal Kaaoka and minimum risk soluions, so ha i can be saed ha ha he minimum risk opimal soluions mainain wih he se of expeced value sandard deviaion properly efficien soluions, he same relaions ha we have obained beween hese laer and he opimal Kaaoka soluions. Finally, we esablish a furher connecion beween he problems (3.6) and (3.5) in Proposiion. Specifically we relae he weighs assigned o he expeced value and he sandard deviaion in he weighed problem (3.5) wih he value of parameer α in problem (3.6), whose opimal soluion coincides wih ha of (3.5). Given ha he weigh assigned, µ, in problem (3.5) can be considered as a measuremen of he relaive imporance which is given o he expeced value of he objecive funcion as opposed o he sandard deviaion in he obaining of expeced value sandard deviaion efficien soluions, we can also pose he quesion of 8
19 wheher here exiss any relaion beween he value of he parameer µ and he value of he parameer α which will weigh he sandard deviaion in problem (3.6). Proposiion Le x ( µ ), x( µ ' ), x( α) and x( α' ) be opimal soluions o he problems (3.5) for µ, (3.5) for µ ', (3.6) for α and (3.6) for α ', respecively wih µ, µ ' (0, ), α ( µ ) µ and α ' = ( µ ') µ ' =. Thus, µ < µ ' if and only if α ' < α. Proof: ) Since µ <µ, i is eviden ha:  µ ' < µ and µ ' < µ, consequenly: ) Obvious from ( ). ( µ ') µ ' < ( µ ) µ ' < ( µ ) µ α α ' = = Given ha α = Φ ( β ) in he normal case and α = ( F ( β ) ) σ in he simple randomizaion case, he resul is ha a weigh, µ, higher for he expeced value in he weighed problem (3.5), corresponds o a lower probabiliy in he Kaaoka problem and, herefore, more risky in erms of probabiliy. To illusrae he resuls obained in his secion, we shall now propose and solve he following example. 4. Example Le us consider he following problem of sochasic programming: 9
20 Min x cx % = cx % + cx % s.. 3x + x x 3x 8 x + x 5 x, x 0 where ~ c c ~ = + c, wih c = ( 4, 4), c = (, ) (4.) and ~ a random exponenial variable of parameer λ=, so ha is disribuion funcion is F ( η) = exp( η ) and is expeced value and sandard deviaion are = 0.5 y υ = 0.5, respecively, and consequenly he expeced value and he sandard deviaion of he random variable ~ c x are c x =.5x + and 4 x σ ( x) = 0.5x + 6 x, respecively. Le us proceed o solve he above problem using he soluion crieria seen in he sudy and illusrae he resuls obained. The expeced value and minimum variance problems associaed wih he sochasic problem have a unique soluion. These soluions are x E =(0, ) and x V =(/3, 0), respecively, as indicaed in Figure. Figure 0
21 The se of expeced value sandard deviaion efficien soluions, E Eσ, can be obained by solving he bi crieria problem: Min x s. ( 4.5x + x,0.5x + 6x ) 3x x + x 3x x + x x, x (4.) and herefore he payoff marix for his problem is shown in Table. Crierion Soluion Expeced Value Sandard Deviaion Expeced value x E =(0, ) 6 Variance x V =(/3, 0) 3 /3 Table : PayOff Marix The se of expeced value sandard deviaion efficien soluions is: { (, ) /3,, 0 } E = E x x R x + x = σ x x This se is he segmen of he feasible se which joins he verices x E and x V in Figure. Noe ha hese wo soluions (which are unique opima for he expeced value and minimum variance problems respecively) are expeced value sandard deviaion efficien soluions. In addiion, given ha (4.) is a linear problem, i is held as rue ha he ses of efficien and expeced value sandard deviaion properly efficien soluions coincide, ha is: Eσ p E Eσ E =. Le us now consider he soluion of he problem using Kaaoka crierion. For his, he feasible poins of he problem mus verify c x = x + x > 0. I is easy o prove ha in his problem his condiion holds rue. The soluions o he deerminisic equivalen problem which correspond o he applicaion of he
Niche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationForecasting and Information Sharing in Supply Chains Under QuasiARMA Demand
Forecasing and Informaion Sharing in Supply Chains Under QuasiARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationIssues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d
These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION
QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationEfficient Risk Sharing with Limited Commitment and Hidden Storage
Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing
More informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More informationBAYESIAN CONFIDENCE INTERVALS FOR THE NUMBER AND THE SIZE OF LOSSES IN THE OPTIMAL BONUS MALUS SYSTEM
QUANTITATIVE METHODS IN ECONOMICS Vol. XIV, No., 203, pp. 93 04 BAYESIAN CONFIDENCE INTERVALS FOR THE NUMBER AND THE SIZE OF LOSSES IN THE OPTIMAL BONUS MALUS SYSTEM Marcin Dudziński, Konrad Furmańczyk,
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationadaptive control; stochastic systems; certainty equivalence principle; longterm
COMMUICATIOS I IFORMATIO AD SYSTEMS c 2006 Inernaional Press Vol. 6, o. 4, pp. 299320, 2006 003 ADAPTIVE COTROL OF LIEAR TIME IVARIAT SYSTEMS: THE BET O THE BEST PRICIPLE S. BITTATI AD M. C. CAMPI Absrac.
More informationCan Individual Investors Use Technical Trading Rules to Beat the Asian Markets?
Can Individual Invesors Use Technical Trading Rules o Bea he Asian Markes? INTRODUCTION In radiional ess of he weakform of he Efficien Markes Hypohesis, price reurn differences are found o be insufficien
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationTail Distortion Risk and Its Asymptotic Analysis
Tail Disorion Risk and Is Asympoic Analysis Li Zhu Haijun Li May 2 Revision: March 22 Absrac A disorion risk measure used in finance and insurance is defined as he expeced value of poenial loss under a
More informationOptimal Monetary Policy When LumpSum Taxes Are Unavailable: A Reconsideration of the Outcomes Under Commitment and Discretion*
Opimal Moneary Policy When LumpSum Taxes Are Unavailable: A Reconsideraion of he Oucomes Under Commimen and Discreion* Marin Ellison Dep of Economics Universiy of Warwick Covenry CV4 7AL UK m.ellison@warwick.ac.uk
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationA OneSector Neoclassical Growth Model with Endogenous Retirement. By Kiminori Matsuyama. Final Manuscript. Abstract
A OneSecor Neoclassical Growh Model wih Endogenous Reiremen By Kiminori Masuyama Final Manuscrip Absrac This paper exends Diamond s OG model by allowing he agens o make he reiremen decision. Earning a
More informationImproper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].
Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,
More informationTHE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS
VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely
More informationOn the Role of the Growth Optimal Portfolio in Finance
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 14418010 www.qfrc.us.edu.au
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationSEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, Email: toronj333@yahoo.
SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.
More informationResearch Article Optimal Geometric Mean Returns of Stocks and Their Options
Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationJumpDiffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
umpdiffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More informationInventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds
OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp. 1079 1097 issn 0030364X eissn 15265463 06 5406 1079 informs doi 10.1287/opre.1060.0338 2006 INFORMS Invenory Planning wih Forecas Updaes:
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationThe effect of demand distributions on the performance of inventory policies
DOI 10.2195/LJ_Ref_Kuhn_en_200907 The effec of demand disribuions on he performance of invenory policies SONJA KUHNT & WIEBKE SIEBEN FAKULTÄT STATISTIK TECHNISCHE UNIVERSITÄT DORTMUND 44221 DORTMUND Invenory
More informationAn Optimal Strategy of Natural Hedging for. a General Portfolio of Insurance Companies
An Opimal Sraegy of Naural Hedging for a General Porfolio of Insurance Companies HongChih Huang 1 ChouWen Wang 2 DeChuan Hong 3 ABSTRACT Wih he improvemen of medical and hygienic echniques, life insurers
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationOptimal Consumption and Insurance: A ContinuousTime Markov Chain Approach
Opimal Consumpion and Insurance: A ConinuousTime Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationTime Series Analysis Using SAS R Part I The Augmented DickeyFuller (ADF) Test
ABSTRACT Time Series Analysis Using SAS R Par I The Augmened DickeyFuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationStatistical Analysis with Little s Law. Supplementary Material: More on the Call Center Data. by SongHee Kim and Ward Whitt
Saisical Analysis wih Lile s Law Supplemenary Maerial: More on he Call Cener Daa by SongHee Kim and Ward Whi Deparmen of Indusrial Engineering and Operaions Research Columbia Universiy, New York, NY 1799
More informationUsing RCtime to Measure Resistance
Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a builin ADC (Analog o Digial Converer)
More informationAnalysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer
Recen Advances in Business Managemen and Markeing Analysis of Pricing and Efficiency Conrol Sraegy beween Inerne Reailer and Convenional Reailer HYUG RAE CHO 1, SUG MOO BAE and JOG HU PARK 3 Deparmen of
More informationOptimal Life Insurance Purchase and Consumption/Investment under Uncertain Lifetime
Opimal Life Insurance Purchase and Consumpion/Invesmen under Uncerain Lifeime Sanley R. Pliska a,, a Dep. of Finance, Universiy of Illinois a Chicago, Chicago, IL 667, USA Jinchun Ye b b Dep. of Mahemaics,
More informationOptimal Life Insurance Purchase, Consumption and Investment
Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationThe Application of Multi Shifts and Break Windows in Employees Scheduling
The Applicaion of Muli Shifs and Brea Windows in Employees Scheduling Evy Herowai Indusrial Engineering Deparmen, Universiy of Surabaya, Indonesia Absrac. One mehod for increasing company s performance
More informationPrice elasticity of demand for crude oil: estimates for 23 countries
Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh
More informationHouse Price Index (HPI)
House Price Index (HPI) The price index of second hand houses in Colombia (HPI), regisers annually and quarerly he evoluion of prices of his ype of dwelling. The calculaion is based on he repeaed sales
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationUSE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationAnalogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar
Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 073807 Ifeachor
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationRelationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith**
Relaionships beween Sock Prices and Accouning Informaion: A Review of he Residual Income and Ohlson Models Sco Pirie* and Malcolm Smih** * Inernaional Graduae School of Managemen, Universiy of Souh Ausralia
More informationDensity Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).
FW 662 Densiydependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Longerm
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More information