Copyrigh by Tinrn Geng 207
The Disserion Commiee for Tinrn Geng cerifies h his is he pproved version of he following disserion: Essys on forwrd porfolio heory nd finncil ime series modeling Commiee: Thlei Zriphopoulou, Supervisor Rfel Mendoz-Arrig Mihi Sîrbu Sephen Wlker Gordn Žiković
Essys on forwrd porfolio heory nd finncil ime series modeling by Tinrn Geng, B.S., M.A. DISSERTATION Presened o he Fculy of he Grdue School of The Universiy of Texs Ausin in Pril Fulfillmen of he Requiremens for he Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN My 207
Dediced o my moher.
Acknowledgmens Firs, I wish o express my sincere griude o my supervisor, Professor Thlei Zriphopoulou, for her guidnce nd suppor during my grdue sudies. Her words of wisdom nd srong work ehic hve been consn source of inspirion nd moivion for me. I would lso like o cknowledge he members of my disserion commiee: Professors Rfel Mendoz-Arrig, Mihi Sîrbu, Sephen Wlker, nd Gordn Žiković, for heir ime nd dvice. I m greful for he suppor nd ssisnce I hve received over he yers from he fculy nd sff he Deprmen of Mhemics, in priculr, Elis Bss, Sndr Cle, Ev Hernndez, nd Professor Dn Knopf. Finlly, I hnk ll my friends nd collegues for heir friendship. I m forever indebed o my moher, Huiln Wng, for her uncondiionl love nd suppor hrough he yers. This disserion is dediced o her. v
Essys on forwrd porfolio heory nd finncil ime series modeling Publicion No. Tinrn Geng, Ph.D. The Universiy of Texs Ausin, 207 Supervisor: Thlei Zriphopoulou This disserion conins four self-conined essys h explore he pplicion of sochsic nd sisicl modeling echniques o he problem of opiml porfolio choice nd finncil ime series nlysis. The firs essy presens urnpike-ype resuls for he risk olernce funcion in n incomplee Iô-diffusion mrke seing under ime-monoone forwrd performnce crieri. We show h, conrry o he clssicl cse, he emporl nd spil limis do no coincide. Rher, we esblish h hey depend direcly on he lef- nd righ-end of he suppor of n underlying mesure, used o consruc he forwrd performnce crierion. We provide exmples wih discree nd coninuous mesures, nd discuss he sympoic behvior of he risk olernce for ech cse. The second essy exmines he long erm behvior of he opiml welh nd opiml porfolio weighs processes in n Iô-diffusion mrke under he vi
ime-monoone forwrd performnce crieri. We show h he underlying mesure µ ssocied wih he forwrd performnce crierion defines he risk profile of he invesor, nd in urn deermines he opiml porfolio sregy nd opiml welh in he long run. The hird essy considers wo fund mngers who rde under relive performnce concerns, depending on ech oher s sregies, in n Iô-diffusion mrke, We nlyze boh he pssive nd he compeiive cses, nd under boh sse specilizion nd diversificion. To llow for dynmic model revision nd flexible invesmen horizons, we inroduce he concep of relive forwrd performnce for he pssive cse, nd he noion of forwrd Nsh equilibrium for he compeiive one. For homoheic forwrd crieri, we provide explici soluions for ll cses. In he fourh essy, we ssess he dynmics of relized bes, relive o he dynmics in he underlying mrke vrince nd covrinces wih he mrke, using 5-minue high-frequency sse prices of he DJIA componen socks from Jnury, 200 o December 3, 204. We find h, unlike he relized vrinces nd covrinces which flucue widely nd re highly persisence, he relized be series, on he oher hnd, disply much less persisence. We hen consruc simple uoregressive plus noise DLM ime series model for he relized be, where he mesuremen error follows norml disribuion cenered zero wih sympoiclly vlid vrince given in [7]. This pproch helps us obin smples from filered nd smoohed rue underlying be series nd forecs fuure bes. vii
Tble of Conens Acknowledgmens Absrc Lis of Tbles Lis of Figures v vi xi xii Chper. Temporl nd spil urnpike-ype resuls under forwrd ime-monoone performnce crieri. Inroducion.............................2 The model nd he invesmen performnce crierion...... 9.2. A moiving exmple................... 4.3 Spil sympoic resuls.................... 8.4 Temporl (urnpike) sympoic resuls............. 22.5 Spil nd emporl limis for he relive prudence funcion. 35.6 Exmples.............................. 36.6. Finie sum of Dirc funcions............... 36.6.. Temporl sympoic expnsion of h ( ) (x 0, ) for lrge....................... 37.6..2 Spil sympoic expnsion of h ( ) (x, 0 ) for lrge x....................... 37.6..3 Spil nd emporl sympoics of r(x, )... 39.6.2 Lebesgue mesure..................... 40.6.2. Temporl sympoic expnsion of h ( ) (x 0, ) for lrge....................... 4.6.2.2 Spil sympoic expnsion of h ( ) (x, 0 ) for lrge x....................... 44.6.2.3 Spil sympoics of r(x, 0 ) for lrge x.... 48.7 Exensions............................. 50 viii
Chper 2. On he opiml welh nd porfolio weighs processes under ime-monoone forwrd performnce crieri in n Iô-diffusion mrke 52 2. Inroducion............................ 52 2.2 The model nd he invesmen performnce crierion...... 54 2.3 Long erm behvior of opiml welh process......... 59 2.4 Long erm behvior of opiml porfolio weighs process.... 65 Chper 3. Pssive nd compeiive invesmen sregies under relive forwrd performnce crieri 72 3. Inroducion............................ 72 3.2 Asse specilizion nd forwrd relive performnce crieri. 77 3.2. Forwrd relive performnce crieri........... 8 3.2.. The CRRA cse.................. 86 3.2.2 Forwrd Nsh equilibrium................. 89 3.2.2. The CRRA cse.................. 9 3.3 Diversificion (no sse specilizion) nd relive forwrd performnce crieri......................... 93 3.3. Pssive forwrd performnce crieri........... 94 3.3.. The CRRA cse.................. 97 3.3.2 Forwrd Nsh equilibrium................. 98 3.3.2. The CRRA cse.................. 99 3.4 Exensions............................. 00 Chper 4. Modeling relized be ime series using high-frequency inr-dy sse prices 02 4. Inroducion............................ 02 4.2 Theoreicl frmework...................... 03 4.3 Empiricl nlysis......................... 07 4.4 DLM frmework.......................... 0 4.4. Sep : prior specificions nd sufficien sisics... 2 4.4.2 Sep 2: FFBS........................ 3 4.4.3 Empiricl nlysis..................... 5 4.5 Conclusion............................. 5 4.6 Tbles nd figures......................... 6 ix
Appendix 33 Bibliogrphy 42 x
Lis of Tbles 4. The Dow Jones Thiry...................... 7 4.2 The Dynmics of Monhly Relized Mrke Vrince, Covrinces, nd Bes......................... 8 xi
Lis of Figures 4. Time Series Plo of Monhly Relized Mrke Vrince.... 9 4.2 Voliliy Index (VIX) of S&P 500............... 20 4.3 Time Series Plos of Monhly Relized Covrinces...... 2 4.4 Time Series Plos of Monhly Relized Bes.......... 22 4.5 Smple Auocorrelions of Monhly Relized Mrke Vrince, Medin Smple Auocorrelions of Monhly Relized Covrinces, nd Medin Smple Auocorrelions of Monhly Relized Bes............................. 23 4.6 Smple Auocorrelions of Monhly Relized Covrinces.. 24 4.7 Smple Auocorrelions of Monhly Relized Bes...... 25 4.8 95% Confidence Inervls for Monhly Be, Five-minue Smpling26 4.9 95% Confidence Inervls for Monhly Be of BA/IBM/DD, Five-minue Smpling...................... 27 4.0 95% Confidence Inervls for Qurerly Be of BA/IBM/DD, Dily Smpling.......................... 28 4. Smples from Poserior Disribuion of, b, nd σ 2 for Monhly Relized Be of IBM....................... 29 4.2 Time Series Plo of Medin Smoohed Smples nd Acul Relizion for Monhly Relized Be for IBM.......... 30 4.3 Time Series Plo of Medin Smoohed Smples nd 95% Confidence Bnds for Monhly Relized Be for IBM........ 3 4.4 00 Forecsing Phs for he Nex 2 Monhs of Monhly Relized Be for IBM nd heir 95% Confidence Inervl.... 32 xii
Chper Temporl nd spil urnpike-ype resuls under forwrd ime-monoone performnce crieri. Inroducion Turnpike resuls in mximl expeced uiliy models yield he behvior of opiml porfolio funcions when he invesmen horizon is long, under sympoic ssumpions on he invesor s risk preferences. The essence of he urnpike resul (sed, for simpliciy, for single log-norml sock wih coefficiens µ nd σ) is he following: ssume h he invesmen horizon is [0, T ] nd h he invesor s uiliy U T behves like power funcion for lrge welh levels, i.e., U T (x) γ xγ, x lrge. (.) Then, if his horizon is very long, he ssocied opiml porfolio funcion π (x, ; T ) is close o he one corresponding o his power uiliy, i.e., for ech x > 0, [0, T ], π (x, ; T ) x µ, T lrge. (.2) σ 2 γ
In oher words, he sympoic spil behvior of he erminl dum dices he long-erm emporl behvior of he porfolio funcion for every welh level. We recll h he funcion π (x, ; T ) is he one he deermines he opiml welh process in feedbck form, in h he opiml welh process X, [0, T ], is genered by he invesmen sregy π = π (X, ; T ). Turnpike resuls cn be found in [20] where coninuous-ime model ws firs considered, nd he urnpike properies were esblished using coningen clim mehods. Their resuls were ler exended in [35] using n uonomous equion h he funcion π (x, ; T ) sisfies nd rgumens from viscosiy soluions. Duliy mehods were used in [22] for complee mrkes nd he incomplee mrke cse ws sudied in [33]. More recenly, he uhors of [] esblished he re of convergence in log-norml model, showing h here exis posiive consn c nd funcion D (x), such h, for ll x > 0, π (x, ; T ) µ σ 2 γ x D (x) e c(t ). A closer look hese urnpike resuls yields h we re essenilly working in single invesmen horizon seing, [0, T ], which is ken o be very long. As resul, however, one needs o commi o mrke model for his long horizon, bu his choice cnno be modified ler on, if ime-consisency is desired. Furhermore, one pre-commis iniil ime o uiliy funcion for very fr in he fuure, T. We lso remrk h no mer how big T is, 2
he opiml invesmen problem is no defined beyond his poin, becuse he uiliy funcion is given for T only. Herein, we ke n lernive poin of view. Insed of commiing o single long horizon [0, T ], we define n invesmen problem for ll imes [0, ). Moreover, insed of choosing n iniil ime he uiliy U T for he remoe horizon T, we choose he uiliy his iniil ime. We lso depr from he log-norml seing nd work wih generl Io-diffusion mulisecuriy incomplee mrke model. We mesure he performnce of invesmen sregies vi he so-clled forwrd invesmen performnce crierion. This crierion ws inroduced by Musiel nd one of he uhors in [55] nd offers flexibiliy for performnce mesuremen nd risk mngemen under model dpion nd mbiguiy, lernive mrke views, rolling horizons, nd ohers. We recll is definiion nd refer he reder o, mong ohers, [57], [59], for n overview of he forwrd pproch. Herein, we focus on he clss of ime-monoone forwrd performnce crieri, sudied in [58] nd briefly reviewed in he nex secion. They re given by ime-decresing nd dped o he mrke informion process, U (x, ), (x, ) R + [0, ), of he form U (x, ) = u (x, A ), where u (x, ) is deerminisic funcion (see (.4)) nd A wih he process λ being he mrke price of risk. = 0 λ s 2 ds, Noe h U (x, ) is 3
compilion of deerminisic invesor-specific inpu, u (x, ), nd sochsic mrke-specific inpu, A. The opiml invesmen process π urns ou o be, for 0, π = σ + λ r (X, A ) wih r (x, ) := u x (x, ) u xx (x, ), (.3) where σ + is he pseudo-inverse of he voliliy mrix, nd X, 0, he opiml welh genered by his invesmen sregy π (cf. (.2)). The funcion r (x, ) is he (locl) risk olernce nd will be he min objec of sudy herein. Conrry o he clssicl cse, in which erminl dum is pre-ssigned for T nd he soluion is hen consruced for [0, T ), in he forwrd seing, he crierion is defined for ll imes, sring wih n iniil (nd no erminl) dum u 0 (x) = U (x, 0). In nlogy o he clssicl urnpike seing, we re hus moived o sudy he following quesion: if he iniil condiion u 0 (x) is such h u 0 (x) γ xγ, x lrge, (.4) does his imply h, for ech x > 0, r(x, ) x γ, lrge? There re fundmenl differences beween he clssicl nd he forwrd seings, for one is no mere vriion of he oher by ime reversl. Rher, he clssicl problem is well-posed while he forwrd is n inverse problem. 4
Nurlly, vrious properies used for he clssicl urnpike resuls fil, wih he mos imporn being he lck of comprison principle for vrious PDEs (cf. (.4) nd (.22)) hnd. The firs sriking difference beween he wo seings is he disinc nure of he emporl nd spil limis. Indeed, in he rdiionl urnpike resuls in [35] nd [], he emporl limi in (.2) coincides wih he spil one, in h for fixed ime T 0 nd welh level x 0, π (x, ; T 0 ) lim x x = lim T π (x 0, ; T ) x 0. However, his is no he cse in he forwrd seing. Indeed, he emporl nd spil limis of he funcion r(x,) x do no coincide. This cn be seen, for insnce, in he moivionl exmple in secion 2.. limis The im herein hen becomes he sudy of he spil nd emporl r(x, 0 ) lim x x nd r(x 0, ) lim, (.5) x for fixed 0 > 0, x 0 > 0, respecively, under pproprie condiions for he sympoic behvior of he iniil dum u 0 (x), for lrge x. Pivol role for deermining hese limis is plyed by n underlying posiive finie Borel mesure, µ, which is he defining elemen for he consrucion of he forwrd performnce process. Indeed, i ws shown in [58] h he bove funcion u is uniquely (up o n ddiive consn) reled o hrmonic funcion h : R [0, ) R +, nd, furhermore, he ler is 5
uniquely chrcerized by n inegrl rnsform, specificlly, u x (h (z, ), ) = e x+ 2 wih h (z, ) = for 0 b. b e zy 2 y2 µ (dy), (.6) An immedie consequence of his generl soluion is h he iniil dum is direcly reled o his mesure µ, in h (u 0) ( ) needs o be of he inegrl form (u 0) ( ) (x) = b x y µ (dy). As resul, i is nurl o expec h he sympoic properies of u 0 (x), which ener in he urnpike ssumpions, re lso direcly linked o he form nd properies of µ. Furhermore, his mesure lso ppers in he specificion of he risk olernce funcion. Indeed, we deduce from (.3) nd (.6) h r (x, ) cn be represened s wih boh h x nd h ( ) depending on µ. r (x, ) = h x ( h ( ) (x, ), ), (.7) The min resuls herein re h, if he suppor of he mesure is finie, b <, hen he spil limi coincides wih he righ-end poin of he suppor while he emporl limi wih he lef-end one, nmely, r(x, 0 ) lim x x = b nd lim r(x 0, ) x =. (.8) The firs sep in obining he bove limis is o undersnd he connecion beween he sympoic behvior of he iniil (mrginl) dum nd he 6
finieness of he mesure s suppor. We sudy he following wo cses, which correspond o he spil nd emporl limis, respecively. he mrginl, We firs show h he sympoic ssumpion (.4), sed in erms of u 0 (x) x γ, (.9) if nd only if he righ end of he mesure s suppor sisfies boh b = γ nd µ ({b}) =. In oher words, condiion (.9) implies h he mesure mus hve finie suppor wih is righ boundry equl o γ nd, furhermore, wih mss his poin. Conversely, for he mesure o hve hese properies, condiion (.9) mus hold. We hen esblish he firs limi in (.8) using represenion (.6), he equion (.4) sisfied by u (x, ), nd vrious convexiy properies of h nd is derivives. We sress h he requiremen h µ ({b}) 0 cnno be relxed. Indeed, we show in Exmple 6.2, where he mesure is he Lebesgue one, h he spil urnpike propery fils. For he second cse, we rele he finieness of he mesure s suppor wih relxed version of (.9). We show h if here exiss γ <, γ 0, such h for ll γ (γ, ) nd γ < γ, u 0 (x) lim x x γ u 0 (x) = 0 nd lim x x γ =, (.0) hen he righ boundry of he mesure s suppor mus sisfy b = γ, nd vice-vers. This regulr vriion ssumpion is weker hn (.9), needed for he spil limi nd, nurlly, yields weker resul. Indeed, while he 7
suppor hs o be finie wih righ boundry equl o, i does no need o γ hve mss γ. We in urn esblish he second limi in (.8), which is he genuine nlogue of he clssicl urnpike resul. Obining his limi is considerbly more chllenging hn in he clssicl cse due o he ill-posed nure of he problem. Indeed, he mehodology used in [35] is inpplicble becuse of lck of comprison resuls for he ergodic version of he equion sisfied by r (x, ). The pproch of [] does no pply eiher becuse of he lck of connecion beween he soluions of he ill-posed he equion nd Feynmn-Kc ype sochsic represenion of is soluion. Therefore, one needs o work direcly wih he funcion r (x, ), which, from (.7) nd (.6), is given in he implici form r (x, ) = b ye yh( ) (x,) 2 y2 µ (dy), where however he spil inverse h ( ) is involved. The key sep in obining he emporl limi is o show h h ( ) (x, ) lim = 2, where is he lef end poin of he mesure s suppor. Then he emporl convergence in (.8) nd he re of convergence is shown using he implici represenion r (x, ) x = b ( ) (y ) e y h ( ) (x,) 2 y µ (dy). In ddiion o he generl spil nd emporl convergence resuls, we presen wo represenive exmples. In he firs, he mesure is finie sum 8
of Dirc funcions while, in he second, i is ken o be he Lebesgue mesure. We clcule he limis of (.8), nd lso provide sympoic expnsions for he risk olernce funcion. The pper is srucured s follows. In secion 2, we presen he mrke model, he invesmen performnce crierion nd moiving exmple demonsring h he emporl nd spil limis do no in generl coincide. In secions 3 nd 4, we nlyze respecively he spil nd emporl sympoic behvior of he relive risk olernce, while in secion 5 we nlyze he sympoic properies of he relive prudence funcion. In secion 6 we presen he wo represenive exmples, nd conclude in secion 7 wih fuure reserch direcions..2 The model nd he invesmen performnce crierion The mrke environmen consiss of one riskless nd k risky securiies. The prices of he risky securiies re modelled s Iô processes, nmely, he price S i of he i h risky sse follows ds i = S i ( ) µ i d + Σ d j=σ ji dw j, wih S0 i > 0, for i =,..., k. The process W = ( ) W,..., W d, 0, is sndrd Brownin moion, defined on filered probbiliy spce (Ω, F, P). The coefficiens µ i nd σ i = ( ) σ i,..., σ di, i =,..., k, 0, re F dped processes nd vlues in R nd R d, respecively. We denoe by σ he voliliy 9
mrix, i.e. he d k rndom mrix ( ) σ ji, whose i h column represens he voliliy σ i of he i h sse. We my, hen, lernively, wrie he bove equion s ds i = S i ( µ i d + σ i dw ). The riskless sse, he svings ccoun, hs price process B sisfying db = r B d wih B 0 =, nd for nonnegive F dped ineres re process r. Also, we denoe by µ he k-dimensionl vecor wih coordines µ i nd by he k-dim vecor wih every componen equl o one. The processes µ, σ nd r sisfy he pproprie inegrbiliy condiions. We ssume h µ r Lin ( σ T ), where Lin ( σ T ) denoes he liner spce genered by he columns of σ T. Therefore, he equion σ T z = µ r hs soluion, known s he mrke price of risk, λ = ( σ T ) + (µ r ). (.) I is ssumed h here exiss deerminisic consn c > 0, such h λ c nd h lim 0 λ s 2 ds =. Sring = 0 wih n iniil endowmen x 0, he invesor invess ny ime > 0 in he risky nd riskless sses. The presen vlue of he mouns invesed re denoed by he processes π 0 nd π, i i =,..., k, respecively, which re ken o be self-finncing. The presen vlue of her invesmen is hen given by he discouned welh process X π = π, i > 0, which solves dx π = σ π (λ d + dw ) (.2) 0
wih he (column) vecor π = (π i ; i =,..., k). I is ken o sisfy he nonnegiviy consrin X π 0, > 0. The se of dmissible policies is given by A = {π : self-finncing, π F, E P 0 } σ s π s 2 ds <, X π 0, > 0. The performnce of dmissible invesmen sregies is evlued vi he soclled forwrd invesmen performnce crieri, inroduced in [55] (see, lso [56], [57] nd [59]). We review heir definiion nex. [0, ). We inroduce he domin noion D + = R + [0, ) nd D = R Definiion.2.. An F -dped process U(x, ) is forwrd invesmen performnce if for (x, ) D, i) he mpping x U(x, ) is sricly incresing nd sricly concve; ii) for ech π A, E P (U(X π, )) + <, nd for s, U (X π, ) E P (U(X π s, s) F ), iii) here exiss π A such h for s, U ( X π, ) ( = E P U(X π s, s) ) F. Herein we focus on he clss of ime-monoone forwrd performnce processes. For he reder s convenience, we rewrie some of he resuls we
sed in he inroducion. Time-monoone forwrd processes were exensively sudied in [58], nd re given by U(x, ) = u(x, A ), (.3) where u : D + R + is sricly incresing nd sricly concve in x, sisfying u 2 x u =. (.4) 2 u xx The mrke inpu processes A nd M, 0, re defined s M = 0 λ s dw s nd A = 0 λ s 2 ds = M. (.5) The opiml porfolio process π is given by π = σ + λ r(x, A ), where he (locl) risk olernce funcion r (x, ) : D + R + is defined s r (x, ) := u x (x, ) u xx (x, ). (.6) Cenrl role in he consrucion of he performnce crierion, he opiml policies nd heir welh plys hrmonic funcion h : D R +, defined vi he rnsformion u x (h(z, ), ) = e z+ 2. (.7) I solves, s i follows from (.4) nd (.7), he ill-posed he equion h + 2 h zz = 0. (.8) Moreover, i is posiive nd sricly incresing in z. I ws shown in [58], h such soluions re uniquely represened by h(z, ) = b e yz 2 y2 ν(dy) + C, y 2
where = 0 + or > 0, b nd C generic consn. The mesure ν is defined on B + (R), he se of posiive Borel mesures, wih he ddiionl properies h, for z R, b eyz ν(dy) < nd b ν(dy) < y. To simplify he presenion nd wihou loss of generliy, we choose C := b ν(dy) nd, lso, inroduce he normlized mesure µ (dy) = ν(dy). y y Then, he funcion h hs, for (z, ) D, he represenion h(z, ) = b wih b yeyz µ(dy) <, = 0 +, > 0, b. e yz 2 y2 µ(dy), (.9) We esily deduce h for ech 0 0, he funcion h (., 0 ) is bsoluely monoonic, since i h (z, 0 ) / z i > 0, i =, 2... Such funcions sisfy, for ech 0 0, i =, 2,..., he inequliy i+ h (z, 0 ) i h (z, 0 ) z i+ z i ( ) i 2 h (z, 0 ) > 0. (.20) From (.7), (.6) nd (.9), we obin h he risk olernce funcion is represened s r(x, ) = h z ( h ( ) (x, ), ) = b z i ye yh( ) (x,) 2 y2 µ(dy). (.2) Furhermore, he firs equliy ogeher wih (.8) yields h i sisfies he (ill-posed) non-liner equion wih r(x, 0) = b yeyh( ) (x,0) µ(dy). r + 2 r2 r xx = 0, (.22) 3
We lso hve h r x (x, ) = h ( zz h ( ) (x, ), ) r (x, ) = r (x, ) b y 2 e yh( ) (x,) 2 y2 µ(dy) > 0. (.23) Furhermore, r xx (x, ) = where we used (.20). ( hzzz (z, ) h r 3 z (z, ) h zz (z, ) 2) (x, ) z=h > 0, (.24) ( ) (x,) We noe h we will frequenly differenie under he inegrl sign in (.9), which is permied s explined in [58]. I cn be lso seen direcly since, fer differeniion, one cn show h he relevn inegrnds re joinly coninuous in heir respecive rgumens nd hus uniformly loclly inegrble. This llows us o differenie under he inegrl sign (see, for exmple, Theorem 24.5 in [3] nd he remrks following i). As sed in he inroducion, he im herein is o invesige he spil nd emporl limis in (.5), wih r (x, ) s in (2.3) when he mesure hs finie suppor. We firs provide n exmple which shows h, conrry o he clssicl cse, hese wo limis do no in generl coincide..2. A moiving exmple Le he underlying mesure µ be Dirc funcion, γ <. From γ (.9) nd (.7) we hve h, for 0, h(x, ) = e γ x 2( γ ) 2 nd u x (x, ) = x γ e γ 2( γ). 4
Therefore, he locl risk olernce funcion is given by r(x, ) = γ x nd hus he spil nd emporl limis coincide, r(x, 0 ) lim x x for fixed 0, x 0 respecively. = γ nd r(x 0, ) lim = x 0 γ, Nex, le he mesure µ be he sum of wo Dirc funcions poins = θ nd b = γ µ = δ θ such h b = 2, wih 0 < θ < nd γ <, i.e., + δ γ wih Then, (.9) nd (.7) yield h h(x, 0) = e θ x + e γ x, γ = 2 θ. (.25) u x (x, 0) = 2 θ ( + 4x ) θ nd u ( ) x (x, 0) = x θ + x γ. In urn, u x (x, 0) lim x x γ Moreover, expression (.9) gives, for > 0, h(x, ) = e (.26) ( ) 2 2( γ) 2(γ ) + 4x = lim =. (.27) x x γ θ x 2 ( θ) 2 + e 2 θ x 2 2 ( θ) 2, nd, hus, h ( ) (x, ) = e ( θ) 2 + 4x e ( + ( θ) ln θ 2 θ) 2 In urn, rnsformion (.7) yields ( γ u x (x, ) = 2 θ e ( 2 θ ) e ( θ) 2 + 4x e ( θ) ) 2.. (.28) 5
Differeniing he bove o obin u xx (x, ) (or using (.9), (.28) nd (2.3)), we deduce h he risk olernce funcion is given by r(x, ) = x 4x + e ( θ) 2 γ e. (.29) ( θ )2 + 4x + e ( θ )2 Therefore, for ech 0 0, while, for ech x 0 > 0, r(x, 0 ) lim x x = 2 θ = γ. (.30) r(x 0, ) lim = x 0 θ. (.3) Therefore, he spil nd emporl limis do no coincide. Nex, we mke he following wo imporn observions. Firsly, noe h (.25) yields h he suppor of he mesure is supp (µ) = { } θ,. γ Therefore, he spil limi coincides wih he righ-end of he suppor while he emporl limi wih he lef-end one. Secondly, for ech x 0 > 0 he emporl limi of he rio h( ) (x 0,) equl o hlf of he lef-end poin, since (.28) yields is h ( ) (x 0, ) lim ( = lim θ + θ ( ( ln e ( θ )2 + 4x 2 e ( θ )2 ))) = 2 ( θ). 6
In secion 4 we will show h hese wo properies re lwys vlid. In priculr, we will see h i is he limi of he bove rio h plys he key role in esblishing he emporl urnpike limi for generl mesures. To juxpose he bove resuls wih he ones in he rdiionl expeced erminl uiliy seing, we compue he nlogous quniies nd ssocied limis for he cses nlyzed in [35] nd [] for log-norml mrkes. Wihou loss of generliy, we consider mrke wih riskless bond of zero ineres re nd single log-norml sock wih men re of reurn µ nd voliliy σ. To his end, we fix n rbirry horizon T > 0 nd, in nlogy o (.26), we ke he erminl inverse mrginl uiliy, I (x) = (U ) ( ) (x), o be I (x) = x θ + x γ, for x > 0 nd θ, γ s in (.25). This corresponds o erminl mrginl uiliy ( +4x ) θ U (x) = nd, hus, in nlogy o (.27), 2 U (x) lim =. x x γ We now consider he vlue funcion, sy u (x, ; T ) of he ssocied Meron problem, for [0, T ]. Leing τ = T be he ime o he end of he invesmen horizon, we deduce, using well known resuls, h he funcion ũ (x, τ) u (x, T ; T ), sisfies, for (x, τ) R + [0, T ), he Hmilon- Jcobi-Bellmn equion ũ τ + 2 λ2 ũ2 x ũ xx = 0. 7
The inverse spil mrginl vlue funcion ṽ : R + [0, T ) R + hen solves ṽ τ = 2 λ2 x 2 ṽ xx + λ 2 xṽ x, wih ṽ(x, 0) = I (x). We esily deduce h ṽ(x, τ) = e ατ x α + e βτ x 2α, wih α = 2 λ2 θ nd β = λ 2 +θ. Noe h β > 2α. ( θ) 2 ( θ) 2 Tking he spil inverse of ṽ(x, τ) yields ũ x (x, τ) = ( e ατ + e 2ατ + 4xe βτ 2x ) θ. Therefore, he ssocied risk olernce funcion is given by r(x, τ) = θ 2x + + 4xe + 8x 2 ( (β 2α)τ e (2α β)τ + 2. e (2α β)τ + 4x) In urn, for ech τ 0 > 0 nd x 0 > 0, we obin, respecively, he spil nd he emporl limis, r(x, τ 0 ) lim x x = θ nd r(x 0, τ) lim = τ x 0 θ..3 Spil sympoic resuls We exmine he spil sympoic behvior of he risk olernce funcion s x, for ech 0 0, under sympoic ssumpions for lrge welh levels of he invesor s iniil risk preferences. In ccordnce wih similr 8
ssumpions in [35] nd [], we impose his sympoic ssumpion on he mrginl u 0 (x) insed of he funcion iself. Assumpion : The iniil dum u 0 sisfies, for some γ <, lim x u 0(x) =. (.32) xγ The nex resul yields necessry nd sufficien condiions on b, he righ end of he suppor of he mesure, for he bove ssumpion o hold. Lemm.3.. Assumpion (.32) holds if nd only if he ssocied mesure µ sisfies b = γ ({ }) nd µ =. (.33) γ Proof. From (.32), (.7) nd he fc h h(x, 0) is sricly incresing nd of full rnge, we hve = lim x u x (x, 0) x γ = lim z u x (h(z, 0), 0) (h(z, 0)) γ Therefore, represenion (.9) gives lim z b = lim z ( ) γ h(z, 0). (.34) e γ z e z(y γ ) µ(dy) =. (.35) If = b, hen (.33) follows direcly. If < b, hen, i mus be h γ, oherwise, we ge conrdicion. In urn, for ε > 0, b b ( e z(y γ ) µ(dy) e z(y γ ) µ(dy) e εz µ [ γ +ε γ ) + ε, b]. (.36) 9
Sending ε 0 nd using (.35) yield h µ((, b]) = 0, nd hus, supp(µ) γ (, ]. Moreover, we hve from (.35) h γ ( γ ) = lim e z(y γ ) µ(dy) + µ({ }) = µ({ z γ γ }), nd we conclude. The res of he proof follows esily. We nex se he min spil sympoic resul. Proposiion.3.. Suppose h he iniil dum u 0 sisfies he sympoic propery (.32). Then, for ech 0 0, he relive risk olernce converges o he righ-end of he suppor of he mesure µ, r(x, 0 ) lim x x = γ. (.37) Proof. Le 0 0. From represenion (.36) we hve h h (z, 0 ) = ( γ ) e zy 2 0y 2 µ(dy) + e γ z 2( γ ) 2 0, nd, in urn, he domined convergence heorem implies lim z h(z, 0 ) e γ z 2( γ ) 2 0 =. (.38) Therefore, from (.7), ogeher wih he sric monooniciy nd full rnge of h(z, 0 ), we deduce h since lim x lim x u x (x, 0 ) x γ e γ 2( γ) 0 u x (x, 0 ) x γ e = lim z γ 2( γ) 0 =, (.39) e z+ 02 h γ (z, 0 )e γ 2( γ) 0 20
Nex, we clim h = lim z lim x ( h(z, 0 ) e γ z 2( γ ) 2 0 u xx (x, 0 ) x γ 2 e γ 2( γ) 0 ) γ =. = γ. (.40) To prove his, i suffices o show h for ny 0 0, u x (x, 0 ) is convex since he bove would hen follow from he rgumens in Lemm 3. (ii) in [35]. To his end, differeniing (.7) yields u xxx (h(z, 0 ), 0 ) (h z (z, 0 )) 2 + u xx (h(z, 0 ), 0 )h zz (z, 0 ) = e z+ 0 2. (.4) The sric convexiy of h nd he sric concviy of u hen give u xxx (h(z, 0 ), 0 ) > 0, (.42) nd using he sric monooniciy nd full rnge of h we conclude. Combining (.39) nd (.40) yields = lim x ( r(x, 0 ) lim x x u x(x, 0 ) x γ e γ 2( γ) 0 ( = lim u ) x(x, 0 ) x xu xx (x, 0 ) ( ) ) uxx (x, 0 ) x γ 2 e γ 2( γ) 0 = γ. We sress h ssumpion (.32), or equivlenly (.33), cnno be wekened. Indeed, s we will see in exmple 6.2, where we ke he mesure o be he Lebesgue on [, b], nd hus here is no mss b, he spil urnpike propery does no hold. 2
Corollry.3.. Suppose h he iniil dum u 0 sisfies he sympoic propery (.32). Then, for ech 0 0, lim r x (x, 0 ) = x γ. (.43) Proof. From (.24) we hve h, for ech 0 0, lim x r x (x, 0 ) exiss, nd we esily conclude..4 Temporl (urnpike) sympoic resuls We invesige he emporl sympoic behvior of he relive risk olernce s, for ech x 0 > 0, under sympoic ssumpion of he iniil mrginl uiliy for lrge welh levels. This is he genuine urnpike nlogue of similr resuls in clssicl expeced uiliy models nd he min finding herein. I shows h he relive risk olernce will converge o he lef-end of he suppor of he underlying mesure µ. As in he spil cse, we firs rele he properies of he mesure o he sympoic behvior of he iniil (mrginl) dum. Assumpion 2: There exiss γ < such h for ll γ (γ, ), u 0 (x) lim x x γ = 0, (.44) while, for ll γ < γ, u 0 (x) lim x x γ =. (.45) 22
As we show nex, he bove ssumpion is direcly reled o condiion inroduced in [36] nd [22], for discree nd coninuous-ime cse, respecively. Lemm.4.. Assumpion 2 is equivlen o he funcion u 0 (x) vrying regulrly infiniy wih exponen γ, i.e. for ll k > 0, u lim 0(kx) x u 0(x) = kγ. (.46) Proof. We firs show h condiion (.46) implies (.44) nd (.45). We rgue by conrdicion. Suppose h (.44) does no hold. Then, here exiss γ (γ, ) nd ε > 0 such h for x lrge enough, u 0 (x) x γ > ε. On he oher hnd, u condiion (.46) implies h, for ll k > 0 nd x lrge enough, 0 (kx) u < 0 (x)kγ ε. Thus, for lrge enough x, 0 < u 0(kx) (kx) γ = Since γ γ < 0, lim k esblish (.45). u 0(kx) u 0(x)k γ u 0(x) x γ kγ γ < ( + ε) u 0(x) x γ kγ γ. u 0 (kx) (kx) γ = 0, nd we conclude. Working similrly, we Nex, we show he reverse direcion. Assume h (.45) nd (.44) hold. Then, for ll δ, k > 0 nd x lrge enough, u 0(kx) (kx) γ+δ < nd x γ δ u 0(x) <. Muliplying hese wo equions nd rerrnging gives, for ll δ > 0, u 0(kx) u 0(x) < (kx)γ+δ x γ δ = k γ+δ x 2δ. 23
Similrly, i follows from inerchnging kx nd x in he bove wo inequliies h u 0(kx) u 0(x) > (kx)γ δ x γ+δ = k γ δ x 2δ, nd condiion (.46) follows by sending firs δ 0 nd hen x. Assumpion 2 is weker hn Assumpion, nd implies, s we show nex, h he mesure µ hs suppor wih righ-end poin, bu wihou γ necessrily hving mss herein. Lemm.4.2. Assumpion 2 holds if nd only if he mesure µ hs finie suppor wih is righ boundry γ, nmely, inf {y > 0 : µ ((y, )) = 0} = γ. (.47) Proof. We show h Assumpion 2 implies propery (.47). For ech γ (γ, ), we deduce from (.44) h nd, hus, 0 = lim x u x (x, 0) x γ = lim z u x (h(z, 0), 0) (h(z, 0)) γ lim z b = lim z ( h (z, 0) e z γ ) γ e z ( y γ )µ (dy) = 0. (.48) Nex, observe h if b, hen i will conrdic he bove limi, nd hus we need o hve b <. Assume now h here exiss γ (γ, ) wih b = γ. Then, for ech γ (γ, γ ) we hve γ < γ enough, nd he bove gives, for ε smll, 24
lim z ( ( γ +ε) Therefore, i mus be h µ e z(y γ ) µ (dy) + b γ +ε e z(y γ ) µ (dy) ) = 0. ( [ γ + ε, b] ) = 0. Sending ε 0, gives µ (( ]), b = γ 0, which is conrdicion. Thus, we mus hve b. Similrly, using (.45) γ we obin h b, nd, hus, b =. γ γ To show he reverse direcion, we firs observe h propery (.47) nd he domined convergence heorem yield h, for ny ε > 0, lim z h(z, 0)e ( γ +ε)z = lim γ z e z(y ( γ +ε)) µ(dy) = 0. Then, seing γ such h γ = γ + ε, we deduce (.44) for ll γ (γ, ). The res of he proof follows esily nd i is hus omied. We hve so fr esblished h under Assumpion 2 he ssocied mesure µ hs finie righ boundry (bu no necessrily mss) γ, nd vice-vers. by, where We now urn our enion o he lef boundry of he suppor, denoed := inf{y 0 : µ ((0, y]) > 0}. (.49) In he upcoming proofs we will frequenly use he ideniy x 0 = γ e yh( ) (x 0,) 2 y2 µ(dy), (.50) 25
for x 0 > 0, which follows direcly from (.9) for b = γ. Lemm.4.3. Le h ( ) : D + R be he spil inverse of h, nd s in (.49). Then, for ech x 0 > 0, lim h ( ) (x 0, ) exiss nd, moreover, for 0, 2 h( ) (x 0, ) Proof. Le x 0 > 0 nd observe h (.8) yields (x 0, ) = ( h xx h ( ) (x 0, ) ) 2 h x (h ( ) (x 0, ), ) = 2 h ( ) nd hus inequliy (.5) holds, for ll 0. 2 ( γ). (.5) γ γ y 2 e yh( ) (x 0,) 2 y2 µ(dy) ye yh( ) (x 0,) 2 y2 µ(dy) To show h lim h ( ) (x 0, ) exiss, i suffices o show h h ( ) (x 0, ) is decresing in ime. Indeed, direc clculions yield h ( ) (x 0, ) = γ ( yh ( ) (x 0, ) ) 2 2 y2 e yh( ) (x 0,) 2 y2 µ(dy) γ ye yh( ) (x 0,) 2 y2 µ(dy) < 0. (.52) Alernively, differeniing h ( h ( ) (x 0, ), ) = x 0 wice yields, seing z = h ( ) (x 0, ), ( ) 2 h ( ) (x 0, )h x (z, )+ h ( ) (x 0, ) hxx (z, )+2h ( ) (x 0, ) h x (z, )+h (z, ) = 0. We hve h boh h x, h xx > 0, s i follows direcly from (.9) nd differeniion. Furhermore, he bove qudric in h ( ) (x, ) remins posiive, 26
which would hen yield h h ( ) (x 0, ) < 0. Indeed, h 2 x (z, ) h xx (z, ) h (z, ) = h 2 xxx (z, ) h xx (z, ) h xxxx (z, ) < 0, s i follows from (.20). We re now redy o presen one of he min findings herein, which yields he limi s of he rio h( ) (x 0, ). We show h i converges o hlf of he lower-end of he mesure s suppor. Some reled weker resuls cn be found in [63]. Proposiion.4.. Le h ( ) : D + R be he spil inverse of he funcion h (cf. (.9)) nd le, b be he lef nd righ end of he suppor, respecively, wih = 0 + or > 0, nd b <. Then, for ech x 0 > 0, he following sserions hold. i) I holds h ii) Le h ( ) (x 0, ) lim (x 0, ) := h( ) (x 0, ) = 2. (.53) 2. (.54) If > 0, hen ( ) (x 0, ) ln µ [, ] γ, if (x 0, ) < 0, (.55) x 0 nd x 0 µ ([, + (x 0, )]) e 2 (x0,), if (x 0, ) > 0. (.56) 27
If = 0 +, hen (x 0, ) > 0, nd, moreover, for ech θ (0, ), x 0 µ ([ (x 0, ), ( + θ) (x 0, )]) e 2 ( θ2 ) 2 (x 0,). (.57) Proof. i). Le x 0 > 0 fixed. Recll h h ( ) (x 0, ) > 0 (cf. (.5)) nd, hus, lim h ( ) (x 0, ) exiss. Moreover, rewriing (.50) s x 0 = γ ( ) e y h ( ) (x 0,) 2 y µ(dy), (.58) we see h lim h ( ) (x 0, ) =, oherwise, sending we ge conrdicion. In urn, from Lemm 7 nd L Hospil s rule, we deduce h nd hus A(x 0 ) := lim h ( ) (x 0, ) 2 A(x 0) = lim h ( ) (x 0, ), (.59) 2( γ). (.60) Nex, we clim h A (x 0 ) < 2( γ). Le > 0. If = γ, hen = b nd h( ) (x 0, ) = ln x γ nd he resul follows direcly. 0 +, 2 γ Le 0 < < γ. Assume h here exiss x 0 such h A (x 0 ) = 2( γ). Then, for ε > 0, here exiss 0 (x 0, ε) such h, for 0, ε h( ) (x 0, ) 2( γ) ε. In urn, for δ > 0 smll enough, he bove nd (.50) yield x 0 ( γ 2ε δ) e y( 2( γ) ε y) γ 2 µ(dy) + e y( γ 2ε δ 28 2( γ) ε 2 y) µ(dy),
which yields conrdicion s, becuse he firs inegrl would converge o. Nex, ssume h here exiss x 0 > 0 such h 2 < A(x 0) < Then, for ε, δ > 0 smll enough we hve 2( γ). (.6) < 2(A(x 0 ) ε) δ < 2(A(x 0 ) ε) < γ. (.62) From (.50), we hen deduce h, for 0 (x 0, ε), x 0 γ e (y(a(x 0) ε) 2 y2) µ(dy). If µ ({}) 0, hen x 0 e 2 (2(A(x 0) ε) ) µ ({}), nd sending yields conrdicion. If µ ({}) = 0, hen x 0 γ e (y(a(x 0) ε) 2 y2) µ(dy) 2(A(x0 ) ε) δ Consider he qudric B (y) = y(a(x 0 ) ε) 2 y2. We hve e (y(a(x 0) ε) 2 y2) µ(dy). B (y ) = B (y 2 ) = 0, for y = 0 nd y 2 = 2 (A(x 0 ) ε), (.63) B (y) > 0, for 0 < y < 2 (A(x 0 ) ε), nd B (y) chieves mximum y = A(x 0 ) ε. h Nex, we look is minimum, y = min y 2(A(x) ε) δ (y), nd clim y = 2(A(x 0 ) ε) δ. (.64) Indeed, if 0 < y, hen choosing δ <, direc clculions yield () > (y ). If y <, hen (.62) yields < y < y 2, nd, hus, he minimum lso occurs y. 29
Clerly, becuse y < y < y 2, we hve B (y ) = 2 δ (2(A(x 0) ε) δ) > 0. Therefore, for 0 (x 0, ε), x 0 2(A(x0 ) ε) δ e B(y ) µ(dy). (.65) As, he righ hnd side of (.65) converges o, unless i holds h µ ([, 2(A(x 0 ) ε) δ]) = 0. Sending δ 0 nd ε 0, we hen hve µ([, 2A(x 0 )]) = 0, which, however, conrdics (.6). Therefore, i mus be h h, for ll x > 0, A(x 0 ), nd we esily conclude. 2 If = 0 +, similr rgumens yield h for every θ (0, A (x 0 )], we hve h µ([θ, 2A(x 0 )]) = 0. Sending θ 0 yields µ (0, 2A (x 0 )] = 0, which conrdics (.6). ii). Le > 0. If (x 0, ) < 0, from (.50) we hve x 0 = e (x γ 0,) nd (.55) follows. γ e y( (x 0,)+ 2 ( y)) µ(dy) e 2 y( y) µ (dy) e (x 0,) µ ([ ]),, γ If (x 0, ) > 0, hen (.53) yields h, for ε smll enough nd 0 (x 0, ε), 0 < h( ) (x 0,) 2 < ε. Choosing ε such h ε < 2( γ) 2 yields 0 < h( ) (x 0,) <, nd using h <, gives 2 2( γ) 2 γ h( ) 2 + (x 0, ) 30 γ.
From (.28) we hen deduce h x 0 2 + h( )(x 0,) ( ) e y h ( ) (x 0,) y 2 µ (dy). ( ) The qudric H (y) := y h ( ) (x 0,) y 2 in he bove inegrnd becomes zero y = 0 nd y 3 = 2 h( ) (x 0,) > nd, herefore, is minimum occurs one of he end poins or + h( ) (x 0,). Noe h < + h( ) (x0,) < y 2 2 3. If i occurs, hen H () = (x 0, ), while if i occurs ( ) ( ) + 2 h ( ) (x 0,), hen H + h( ) (x0,) = + h( ) (x0,) (x 2 2 2 0, ) > (x 2 0, ). Combining he bove gives x 0 2 + h ( ) (x0,) e 2 (x 0,) µ (dy) = µ ([, + (x 0, )]) e 2 (x 0,). Finlly, le = 0 +. Then, (x 0, ) = h( ) (x 0,). Recll h lim h ( ) (x 0, ) =, nd hus h( ) (x 0,) > 0, for lrge. ( h For ε ( ) (x 0,), 2 h( ) (x0 ),) we hen hve x 0 ε h ( ) (x 0,) ( ) e y h ( ) (x 0 ( ),) y ε 2 µ (dy) e ε h ( ) (x 0,) ε 2 µ (dy). h ( ) (x 0,) Seing ε = ( + θ) h( ) (x 0,), (.57) follows. We re now redy o prove one of he min resuls herein. Theorem.4.. Le be he lef end of he suppor of he mesure µ. Then, for ech x 0 > 0, r (x 0, ) lim =. (.66) x 0 3
Furhermore, here exiss funcion G (x 0, ) given by γ G (x 0, ) := y y( (y )e 2 ) µ(dy), (x 0, ) < 0 2 (x 0, ) x 0 + 2 (x γ 0,)+ y +2 (x 0,)(y )ey( 2 ) µ(dy), (x 0, ) > 0, sisfying wih lim G (x 0, ) = 0 nd, for lrge enough, 0 r(x 0, ) x 0 G (x 0, ). (.67) Proof. We presen wo lernive convergence proofs. The firs yields (.66) while he second gives he re of convergence G (x 0, ). To his end, differeniing (.7) gives ( ) u x (x 0, ) = 2 h( ) (x 0, ) u x (x 0, ). (.68) Moreover, (.4) nd (.6) imply h u (x 0, ) = 2 u x(x 0, )r(x 0, ) nd, in urn, u x (x 0, ) = 2 u xx (x 0, ) r (x 0, ) 2 u x (x 0, ) r x (x 0, ). (.69) Combining he bove we deduce nd from Proposiion 8 nd (.59) On he oher hnd, lim c 0 + 2 r x(x 0, ) = h ( ) (x 0, ), (.70) lim r x (x 0, ) = lim 2h ( ) (x 0, ) =. (.7) x0 c r x (ρ, )dρ = r(x 0, ) lim r(c, ). c 0 + 32
Using he fc h, for ll 0, lim x 0 + r(x, ) = 0 (see [58]), we ge h, for x 0 > 0, r(x 0, ) = x0 r x (ρ, )dρ. (.72) Finlly, we deduce from (.70) nd (.52) h r x (x 0, ) < 0, nd hus, for x 0 > 0, we hve for y (0, x 0 ], r x (y, ) r x (x 0, 0). However, for ll x 0 > 0, r x (x 0, 0) <. This follows direcly from (2.3),(.9) nd he full rnge of h (x, 0), since γ r x (h (z, 0), 0) = h zz (z, 0) h z (z, 0) = y 2 e yz 2 2y µ (dy) γ ye yz 2 2y µ (dy) γ. Using he domined convergence heorem nd pssing o he limi s in (.70), we deduce (.66). Nex, we give he second convergence proof, which lso yields he re of convergence. Firs noe h 0 r(x 0, ) x 0. (.73) This follows direcly from (2.3), (.9) nd (.50), for r (x 0, ) = γ ye (y h( ) (x 0,) 2 y2) µ(dy) γ Furhermore, from (2.3), (.9), (.50) nd (.54), we hve r(x 0, ) x 0 = γ e (y h( ) (x 0,) 2 y2) µ(dy). (y )e y( 2 (x 0,)+ y 2 ) µ(dy). (.74) If (x 0, ) < 0 (which occurs only if > 0, s shown in he previous proof), hen he bove yields r(x 0, ) x 0 γ 33 y y( (y )e 2 ) µ(dy),
nd (.67) follows direcly wih G () := γ y y( (y )e 2 ) µ(dy). Le (x 0, ) > 0 nd > 0 or = 0 +. If =, hen he resul γ follows rivilly. For < γ, observe h for lrge enough, 0 < + 2 (x 0, ) < γ, nd hus represenion (.74) gives r (x 0, ) x 0 = (+2 (x0,)) (y )e y( 2 (x 0,)+ y 2 ) µ(dy) + γ +2 (x 0,) (y )e y( 2 (x 0,)+ y 2 ) µ(dy). Le C (x 0, ) := (+2 (x 0,)) 2 (x y( 0,)+ y (y )e 2 ) µ(dy), nd observe h (+2 (x0,)) C (x 0, ) 2 (x 0, ) e y( 2 (x 0,)+ y 2 ) µ(dy) 2 (x 0, ) x 0, where we used (.50). Thus Le lso C 2 (x 0, ) := γ (y ) e y( 2 (x 0,)+ y 2 ), y +2 (x 0,) lim C (x 0, ) = 0. (.75) 2 (x 0,)+ y (y )ey( 2 ) µ(dy) nd F (y,, x 0 ) := ]. Then, F ( + 2 (x 0, ),, x 0 ) = [ + 2 (x 0, ), γ 2 (x 0, ), nd hus lim F ( + 2 (x 0, ),, x 0 ) = 0. Furhermore, for ech ( ] y + 2 (x 0, ),, we lso hve lim γ F (y,, x 0 ) = 0. In urn, he domined convergence heorem gives lim C 2 (x 0, ) = 0. (.76) Seing G (x 0, ) := C (x 0, )+C 2 (x 0, ), nd using (3.4) nd (.76), we obin (.67). 34
.5 Spil nd emporl limis for he relive prudence funcion We now rever our enion o he relive prudence funcion p (x, ) defined, for (x, ) D +, s wih u solving (.4). p (x, ) = xu xxx (x, ) u xx (x, ), (.77) Proposiion.5.. For (x, ) D +, we hve h p (x, ) > 0. Moreover, he following spil nd emporl limis hold. i) If Assumpion holds, hen, for ech 0 0, lim p(x, 0) = 2 γ. (.78) x ii) If Assumpion 2 holds, hen, for ech x 0 > 0, +, if > 0 lim p(x 0, ) =, if = 0 +. (.79) Proof. Using (.77) nd (.6), we deduce h, for ech 0 0, p (x, 0 ) = x r (x, 0 ) ( + r x (x, 0 )), nd he fc h p (x, 0 ) > 0 nd (.78) follow direcly from (.23) nd (.37), respecively. From (.77) nd equion (.4) we lso obin h, for ech x 0 > 0, u x (x 0, ) u x (x 0, ) = 2 Using h lim h ( ) (x 0, ) = 2 r (x 0, ) p (x 0, ) = x 0 2 h( ) (x 0, ). (.80) we esily conclude. 35
.6 Exmples We presen wo represenive exmples in which he mesure is, respecively, sum of Dirc funcions nd he Lebesgue mesure. The firs exmple generlizes he resuls of he exmple in subsecion 2., while he second demonsres h he spil urnpike propery fils if here is no mss he righ end of he mesure s suppor..6. Finie sum of Dirc funcions We ssume h µ = N n= δ yn, wih 0 < y < < y N = γ. Then, h(z, 0) = N n= eynz nd, hus, lim z h (z, 0) e zy N =. In urn, (.34) yields u x (x, 0) lim =, x x γ which verifies he resuls of Lemm 2. We lso hve, for (z, ) D, h(z, ) = N n= (cf. (.9)), nd, herefore, for x > 0, Furhermore, x = N n= ( exp y n z ) 2 y2 n. ( ( h ( ) (x, ) exp y n )) 2 y n. (.8) h ( ) (x, ) 2 y y ln x. (.82) 36
.6.. Temporl sympoic expnsion of h ( ) (x 0, ) for lrge We clim h, for ech x 0 > 0, s, h ( ) (x 0, ) = 2 y + y ln x 0 + o(). (.83) Indeed, using he limi (.53), we hve ( h ( ) (x 0, ) lim ) { 2 y < 0, < n N n = 0, n =. Therefore, s, ll he erms in (.8) vnish excep for he firs one, nd hus, ( x 0 = lim exp y h ( ) (x 0, ) ) 2 y2. (.84) Tking logrihm nd rerrnging erms yields (.83)..6..2 Spil sympoic expnsion of h ( ) (x, 0 ) for lrge x We clim h, for ech 0 0, h ( ) (x, 0 ) = ( γ) ln x + To obin his, we firs esblish h h ( ) (x, 0 ) lim x ln x Indeed, fix 0 0, le δ (0, ) nd ssume h γ 2 ( γ) 0 + o(). (.85) = ( γ). (.86) lim inf x h ( ) (x, 0 ) ln x < + δ. γ 37
Then, using (.8) nd h h ( ) (x, 0 ) > 0, for lrge x, we hve x = x n= nd using h γ N ( ( ) h ( ) (x, 0 ) exp y n ln x 2 ) ln x y2n 0 n= N ( ( )) h ( ) (x, 0 ) exp y n ln x ln x δ( γ) = +δ +δ( γ) γ Since δ is rbirry, we deduce h Nx h ( ) (x, 0 ) γ ln x, < 0, we ge conrdicion s x. h ( ) (x, 0 ) lim inf x ln x ( ) Similrly, ssume h for δ 0,, γ lim sup x h ( ) (x, 0 ) ln x ( γ). (.87) > δ. γ Then, (.82) gives ( > x exp xh( ) γ ln (x, 0 ) ln x 2 ( ) ) 2 0 γ nd using h γ = x h ( ) (x, 0 ) γ ln x e 2( γ ) 2 0 = γ δ δ( γ) δ( γ) Since δ is rbirry, we deduce h > 0, we ge conrdicion s x. lim sup x h ( ) (x, 0 ) ln x ( γ), (.88) nd we esily conclude. 38
Nex, we rewrie (.8) s = = N n= N n= ( exp y n h ( ) (x, 0 ) ) 2 y2 n 0 ln x ( ( h ( ) (x, 0 ) exp y n ln x ) ) ln x y n 2 y2 n 0. Noe h from he limi in (.86) we hve h ( h ( ) (x, 0 ) lim ) = x ln x y n { < 0, n < N = 0, n = N. (.89) Therefore, s x, he firs N erms in (.89) vnish, nd we deduce h lim exp x ( γ h( ) (x, 0 ) ln x 2 ( ) ) 2 0 =. γ We hen obin (.85) by king he logrihm nd rerrnging he erms..6..3 Spil nd emporl sympoics of r(x, ) From represenion (2.3), we hve for he risk olernce funcion r(x, ) = N n= Le x 0 > 0. Then, (.82) gives r(x 0, ) N n= = y x 0 + ( y n exp y n h ( ) (x, ) ) 2 y2 n. (.90) ( y n exp y n ( 2 y + ln x 0 ) ) y 2 y2 n N n=2 ( ) yn y n exp 2 y n(y y n ) x y 0. 39
Therefore, he emporl sympoic expnsion of r(x 0, ) s is given by ( ) r(x 0, ) = y x 0 + O e 2 y 2(y y 2 ). (.9) Nex, le 0 0. Then, lim r(x, 0) = lim x x nd, hus, s x, r(x, 0 ) = n= N n= ( y n exp y n (( γ) ln x + 2 ( γ) 0) ) 2 y2 n 0, N ( ) y n exp 2 y n 0 ( γ y n) x ( γ)yn + o(). (.92) Therefore, for ech x 0 > 0 nd 0 0, we hve he emporl sympoic expnsion (.9) yields r(x 0, ) r (x, 0 ) lim = y nd lim x 0 x x = y N = γ, nd hese limis re consisen wih he findings in Proposiion 3 nd Theorem 9, respecively..6.2 Lebesgue mesure We consider cse of mesure wih coninuous suppor bu wihou mss is righ boundry. We derive he ssocied limis nd lso show h he spil urnpike propery fils. Lebesgue mesure on [ ],, > 0 γ 40
Consider he funcions ϕ(z) := e z2 2 nd Φ(z) := z ϕ(y)dy, for z R. Then, represenions (.9) nd (.50) yield, respecively, nd x = h(z, ) = γ γ e yz 2 y2 dy = ez2 /2 γ z/ z/ ϕ(y)dy, (.93) ( ) e y h ( ) (x,) 2 y dy = e h( ) (x,) 2 h ( ) (x,) γ 2 ϕ(y)dy. (.94) h( ) (x,).6.2. Temporl sympoic expnsion of h ( ) (x 0, ) for lrge We clim h for x 0 > 0, s, h ( ) (x 0, ) = 2 + ( ln + ln x 0 + ln ) + o(). (.95) 2 To show his, we firs esblish h x 0 = lim Using (.94) nd h, for z < 0, we hve, for lrge enough, e (h( )(x0,) 2 ) 2. (.96) Φ(z) ϕ(z) z, (.97) x 0 ( ) ( h ( ) (x 0, ) 2 exp Φ ) + h( ) (x 0, ) 2 exp h( ) (x 0,) ( h ( ) (x 0, ) 2 2 ) ( ϕ ) + h( ) (x 0, ) 4
In urn, Nex, we show h = e(h( ) (x 0,) 2 ) h ( ) (x 0, ). x 0 lim inf e (h( ) (x 0,) 2 ) h ( ) (x 0, ). (.98) x 0 lim sup e (h( ) (x 0,) 2 ) h ( ) (x 0, ), which wih (.98) will yield (.96). To his end, we use h for ny b > > 0, he inequliy Φ(b) Φ() (ϕ() ϕ(b)) b holds. Le < k <. From (.94) nd he bove, we hve, for lrge ( γ) enough, h x 0 e h( ) (x 0,) 2 2 ( ( Φ k ) ( h( ) (x 0, ) Φ )) h( ) (x 0, ) = k h( ) (x 0,) ( ( ϕ ) h( ) (x 0, ) ϕ e h ( ) (x0,) 2 2 ( k )) h( ) (x 0, ) ) (e (h( ) (x 0,) k h ( ) 2 ) e k(h( ) (x 0,) 2 k). (x 0, ) From Proposiion 8 nd since k >, we hve e k(h( )(x0,) 2 k) lim k h ( ) (x 0, ) = lim ( e k2 ( h( ) (x 0,) k 2 ) k h( ) (x 0,) ) = 0. 42
Therefore, x 0 lim sup lim sup ) (e (h( ) (x 0,) k h ( ) 2 ) e k(h( ) (x 0,) 2 k) (x 0, ) e k(h( ) (x 0,) 2 k) k h ( ) (x 0, ) lim e k(h( ) (x 0,) 2 k) k h ( ) (x 0, ) nd sending k we conclude. = lim sup e (h( ) (x 0,) 2 ) k h ( ) (x 0, ), Nex, we uilize he Lmber-W funcion W (x), defined s he inverse funcion of F (x) = xe x, o derive he explici sympoic expnsion of h ( ) (x 0, ) s. Reclling he noion (x 0, ) = h ( ) (x 0, ) 2, we deduce from (.96) h here exiss ε() wih lim ε() = 0, such h Rewriing i yields e (x 0,) (x 2 0, ) = x 0( + ε()). ( ) 2 (x 0, ) e ( 2 (x0,)) = x 0 ( + ε()) e 2 2, Using h he lef hnd side is of he form F (( 2 (x 0, ))), we obin ( ( ) 2 (x 0, )) = W x 0 ( + ε()) e 2 2, nd, in urn, (x 0, ) = 2 ( ) W x 0 ( + ε()) e 2 2. I is esblished in [8] h he sympoic expnsion of W (x), for lrge x, is given by W (x) = ln x ln(ln x) + o(). 43
Therefore, (x 0, ) = 2 ( ) ln x 0 ( + ε()) e 2 2 + ( ) ln ln x 0 ( + ε()) e 2 2 + o() = ( ln x 0 + ln( + ε()) + ln ( 2 2 + ln Using h s, ln( + ε()) = o() nd h sserion (.95) follows. x 0 ( + ε()) ( ) ( ) ln 2 2 + ln = ln x 0 ( + ε()) 2 2 + o(), )) + o()..6.2.2 Spil sympoic expnsion of h ( ) (x, 0 ) for lrge x Le 0 0. We show h, s x, h ( ) (x, 0 ) = We firs esblish h ( ) 2( γ) 0 + ( γ) ln x + ln ln x ln + o(). (.99) γ h ( ) (x, 0 ) lim x ln x Indeed, le f(z, ) := z e γ z 2( γ ) 2. Then, = lim z = lim z ( γ ( = ( γ). (.00) h(z, 0 ) lim z f(z, 0 ) = lim γ ze z(y γ ) 2 (y2 ( γ ) 2 ) 0 dy z (z y 0 )e z(y γ ) 2 (y2 ( γ ) 2 γ ) 0 dy + y 0 e z(y γ ) 2 (y2 ( γ ) 2 ) 0 dy ) e ( γ )z 2 (2 ( γ ) 2 ) 0 + γ y 0 e z(y γ ) 2 (y2 ( γ ) 2 ) 0 dy =, ) 44
where we used h < γ heorem. Therefore, for ech 0 0, nd, for he hird erm, he monoone convergence h(x, 0 ) lim x f (x, 0 ) =. (.0) We now use resul on he inverses of sympoic funcions (see [23]) o prove he limi in (.00) by verifying he necessry ssumpions for his resul o hold. To his end, consider he funcion g(z) := ( γ) ln z, nd noice h g(f(z, 0 )) = ( γ) ln z + z 2 ( γ) 0 z, s z. Thus, lim z z g(f(z, 0 )) =. Since, on he oher hnd, lim z f(z, 0 ) =, we deduce h f ( ) (x, 0 ) g(x), s x. Moreover, g(x) is sricly incresing nd he rio gx(x, 0) g(x, 0 = O( ), for sufficienly lrge x. I hen ) x ln x x follows from he foremenioned resul h g(x) h ( ) (x, 0 ), s x, nd (.00) follows. Nex, we clim h, for ech 0 0, e lim x γ (h( ) (x, 0 ) 2 γ 0) x ln x = γ. (.02) Indeed, for 0 = 0, we hve from (.94) h x = γ e yh( ) (x,0) dy = ) (e h ( ) γ h( ) (x,0) e h( ) (x,0), (.03) (x, 0) nd (.00) yields h lim x e γ h( ) (x,0) x ln x 45
= lim x e γ h( ) (x,0) e γ h( ) (x,0) e h( ) (x,0) For 0 > 0, we deduce from (.94) h h ( ) (x, 0) ln x = γ. x = ( ) ( e h ( ) (x,0 ) 2 2 0 (Φ 0 h( ) (x, 0 ) Φ )) 0 h( ) (x, 0 ). 0 γ 0 0 (.04) Then, using (.97), we hve, for lrge x, x 0 exp ( h ( ) (x, 0 ) 2 ) ( ) Φ 0 h( ) (x, 0 ) γ 0 2 0 ( ) x e h ( ) (x,0 ) 2 2 0 h ( ) (x, 0 0 ) ϕ 0 h( ) (x, 0 ) 0 0 γ 0 γ nd, in urn, lim inf x = lim inf x (e = e γ (h( ) (x, 0 ) 2 γ 0) x(h ( ) (x, 0 ) γ 0), γ (h( ) (x, 0 ) 2 γ 0) xh ( ) (x, 0 ) e γ (h( ) (x, 0 ) 2 γ 0) xh ( ) (x, 0 ) = lim inf x Similrly, we use h, for < b < 0, lim x e γ (h( ) (x, 0 ) 2 γ 0) xh ( ) (x, 0 ) ) h ( ) (x, 0 ) h ( ) (x, 0 ) γ 0 h ( ) (x, 0 ) h ( ) (x, 0 ) γ 0. (.05) Φ(b) Φ() ϕ() ϕ(b), (.06) 46
nd deduce from (.04) h, for lrge x, x e h ( ) (x,0 ) 2 2 0 0 0 h( ) (x, 0 ) 0 ( ϕ( 0 h( ) (x, 0 ) ) ϕ( 0 = e γ (h( ) (x, 0 ) 2 γ 0) For he second erm, we hve Therefore, lim x x(h ( ) (x, 0 ) 0 ) e h( ) (x, 0 ) x(h ( ) (x, 0 ) 0 ) e = lim x exp lim sup x = lim sup x γ 2 0 = lim x ( ( h ( ) (x, 0 ) ln x ln x (e e = lim sup x γ (h( ) (x, 0 ) 2 γ 0) x(h ( ) (x, 0 ) 0 ) e γ (h( ) (x, 0 ) 2 γ 0) x(h ( ) (x, 0 ) 0 ) lim x = lim sup x γ (h( ) (x, 0 ) 2 γ 0) xh ( ) (x, 0 ) = lim sup x ) 0 h( ) (x, 0 ) ) 0 e(h( )(x,0) 2 0) x(h ( ) (x, 0 ) 0 ). e h( )(x,0) ln x 2 0 e h ( ) (x, 0 ) 0 )) = 0. h ( ) (x, 0 ) 0 e γ (h( ) (x, 0 ) 2 γ 0) x(h ( ) (x, 0 ) 0 ) lim x e γ (h( ) (x, 0 ) 2 γ 0) xh ( ) (x, 0 ) From (.05) nd (.07), we hen obin ) e(h( )(x,0) 2 0) x(h ( ) (x, 0 ) 0 ) e (h( ) (x, 0 ) 2 0) x(h ( ) (x, 0 ) 0 ) xh ( ) (x, 0 ) x(h ( ) (x, 0 ) 0 ). (.07) lim sup x e γ (h( ) (x, 0 ) 2 γ 0) xh ( ) (x, 0 ) =, 47