Mahemaical Problems in Engineering Volume 5, Aricle ID 788, 8 pages hp://dx.doi.org/.55/5/788 Research Aricle Nonzero-Sum Sochasic Differenial Porfolio Games under a Markovian Regime Swiching Model Chaoqun Ma, Hui Wu, and Xiang Lin Business School of Hunan Universiy, Changsha 48, China School of Finance, Zhejiang Gongshang Universiy, Hangzhou 8, China Correspondence should be addressed o Hui Wu; h wu8@6.com Received 7 Sepember 4; Acceped 5 December 4 Academic Edior: Chuangxia Huang Copyrigh 5 Chaoqun Ma e al. his is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We consider a nonzero-sum sochasic differenial porfolio game problem in a coninuous-ime Markov regime swiching environmen when he price dynamics of he risky asses are governed by a Markov-modulaed geomeric Brownian moion (GBM). he marke parameers, including he bank ineres rae and he appreciaion and volailiy raes of he risky asses, swich over ime according o a coninuous-ime Markov chain. We formulae he nonzero-sum sochasic differenial porfolio game problem as wo uiliy maximizaion problems of he sum process beween wo invesors erminal wealh. We derive a pair of regime swiching Hamilon-Jacobi-Bellman (HJB) equaions and wo sysems of coupled HJB equaions a differen regimes. We obain explici opimal porfolio sraegies and Feynman-Kac represenaions of he wo value funcions. Furhermore, we solve he sysem of coupled HJB equaions explicily in a special case where here are only wo saes in he Markov chain. Finally we provide comparaive saics and numerical simulaion analysis of opimal porfolio sraegies and invesigae he impac of regime swiching on opimal porfolio sraegies.. Inroducion he opimal porfolio selecion has been sudied exensively in modern finance. his research is of grea imporance from boh heoreical and pracical purposes. he pioneering work canberacedomarkowiz[], which provided quaniaive mehods o invesigae a single-period opimal porfolio selecion problem. hen Meron [, ] firs exended his single-period model o a coninuous-ime model. In he seminal work of Meron, he obained closed-form soluions o he opimal porfolio selecion problems. Meron s work has opened up an imporan field called coninuous-ime finance.wenoehaoneofhekeyassumpionsinmeron s opimal porfolio models is ha he model parameers are assumed o be consans and he price processes of risky asses are modeled by he classical geomeric Brownian moions. However, we know ha his key assumpion is no consisen wih he acual behavior of asse price dynamics. Empirical finance lieraure has found los of sylized facs in asse reurns, such as heavy ails in he asse reurns disribuions, ime-varying volailiy, long-erm memory, and regime swiching. hus i would be of pracical relevance and imporance o consider more realisic porfolio selecion models. Inhepashreedecadesorso,amonghoseesablished Meron s porfolio models, Markov regime swiching and differenial game models are wo main exensions. Regime swiching models are an efficien and convenien approach o capure he cyclical feaures of srucure changes in real macroeconomic fundamenals. Early works on regime swiching models can be raced o Quand [4]. Hamilon [5] pioneered he economeric applicaions of Markov regime swiching models. Since hen, here has been a growing ineres in applicaions of regime swiching models ino finance and economics. Guo e al. [6] builandsolvedareal opion model of invesmen decisions in which he growh rae and volailiy of decision variable such as growh rae and diffusion coefficien shif beween differen saes a
Mahemaical Problems in Engineering random imes. References [7 9] esablisheda dynamic capial srucure model and demonsraed how business-cycle variaions (regime swiching) in expeced growh raes, economic uncerainy, and risk premia influence firms financing and defaul policies. Some papers on opimal porfolio selecion under regime swiching models can refer o [ 6] and ohers. However, i seems ha one imporan issue ha may be overlooked by he exising lieraure on porfolio selecion is ineracive decision making problems. In his paper we develop a model of wo invesors o sudy ineracive decision making on porfolio selecion. he origin of he differenial game heory migh race o he 94s. In 965, Isaacs wroe he classical work Differenial Game. Ihaslaiddownhesolidmahemaicalandheoreical foundaions o he differenial game heory. Since hen, sochasic differenial game models have found a wide range of applicaions in finance. Some early works include [7 9] and some references herein. Some recen works include [ 4]. Browne [] formulaed various versions of a zerosum sochasic differenial game o invesigae dynamic opimal invesmen problems beween wo small invesors in coninuous ime. He provided he exisence condiions of Nash equilibrium and gave explici represenaions for he value funcions and opimal porfolio sraegies. Mannucci [5] sudied Nash equilibrium for wo-player, nonzero-sum sochasic differenial game. Ellio and Siu [6] exended he model o a coninuous-ime Markovian regime swiching seing and coninued o sudy he risk minimizaion porfolio selecion problem by using sochasic differenial game. Siu [7] considered opion pricing under regime swiching by a game heoreical approach. More recenly, Leong and Huang [8] developed a sochasic differenial game of capialism o sudy he role of uncerainy. Lin [9]sudiedanonzero-sum sochasic differenial porfolio game beween wo invesors. Ellio and Siu [] inroduced a model which covered economic risk, financial risk, insurance risk, and model risk o discuss an opimal invesmen problem of an insurance company using sochasic differenial game approach. Liu and Yiu [] considered sochasic differenial games wih VaR risk consrains beween wo insurance companies. hey provided explici Nash equilibria and derived closed-form soluions o value funcions. However, I seems ha he lieraure has no well sudied he opimal porfolio ineracive decision making problem under sochasic differenial game in a coninuous-ime Markovian regime swiching seing. o he bes of our knowledge, our model is mos relaed wih Browne [] andlin[9], since he wo papers have discussed sochasic differenial porfolio games. However, he disincion beween hem and our paper is also eviden. here are wo fundamenal differences beween he model considered by hem and he models considered here. Firsly, we incorporae our model wih a general coninuous-ime Markov regime swiching seing and ake ino accoun he Markov regime swiching risk and is impacs on he financial asse prices. Secondly, we consider a nonzero-sum sochasic differenial porfolio game which is differen from he several versions of zero-sum sochasic differenial game in Browne []. Furhermore, we use a sochasic opimal conrol approach for he curren nonzero-sum sochasic differenial porfolio game problem. his mehod is differen fromheapproachused inlin [9], namely, sochasic linear quadraic conrol. In his paper we rea he opimal porfolio selecion problem in a wide class of coninuous-ime Markovian regime swiching models. We consider he porfolio selecion beween wo small invesors; call hem A and B. (he invesors are called small as heir decision making behavior does no affec he marke prices of he underlying asses.) We consider a coninuous-ime financial marke wih hree primiive securiies, namely, a bank accoun and wo risky asses. he dynamic price processes of all he primiive securiies are assumed o be modulaed by a coninuousime Markovian chain. he raionale of using his regime swiching model is o incorporae he impac of regime shifs on asse prices aribued o srucure changes in marke or macroeconomy. he wo risky asses are correlaed wih each oher, only one of which is available o each invesor. Moreover, boh invesors are allowed o rade freely in he bank accoun. he invesors cooperae wih each oher by he choice of heir own porfolio sraegies when hey make decisions on invesmen. We formulae he sochasic differenial game as wo uiliy maximizaion problems. wo objecive funcions are considered here. One invesor is rying o maximize his payoff; simulaneously he oher invesor acs anagonisically o maximize he oher payoff. Each payoff is formulaed as expeced uiliy of he wealh sum process of he wo invesors. By using sochasic opimal conrol heory, we derive a pair of regime swiching HJB equaions for hevaluefuncions.moreover,weobainhefeynman-kac represenaions of value funcions. Closed-form expressions for opimal porfolio sraegies are also obained. Finally, we find ha Markov regime swiching in he model parameers has a significan effec on he opimal porfolio sraegies and value funcions. Aside from he inrinsic probabilisic and game heoreic ineres, such a model is applicable in many economic seings. As we know, diversificaion improves he abiliy of an invesor s risk-reurn rade-off. However, i can be difficul for a small invesor o hold enough socks which are well diversified. In addiion, mainaining a well-diversified porfolio can lead o high ransacion coss. If several invesors form a group, well-diversified porfolio and low ransacion coss can be realized. Differen invesors have differen aiudes owards risk, so he choice of he sock and he goal of invesmen for invesors are differen wih each oher. he res of he paper is organized as follows. he following secion presens he price and wealh dynamic processes in a coninuous-ime Markov regime swiching economy. In Secion, wefirsinroducewoopimalporfolioproblems wih differen objecive funcions. And hen we formulae a wo-invesor, nonzero-sum sochasic differenial porfolio game problem. In Secion 4, we derive a pair of regime swiching Hamilon-Jacobi-Bellman (HJB) equaions for he nonzero-sum differenial game problem and explici soluions for he opimal porfolio sraegies and value funcions of wo invesors are obained. In Secion 5 we discuss one special case for a wo-sae Markov regime swiching model. In Secion 6 weprovidehecomparaivesaicsandnumerical
Mahemaical Problems in Engineering simulaions analysis. Finally, we summarize he findings and ouline some poenial opics for fuure research.. Marke Model In his secion we will consider a coninuous-ime, Markov regime swiching financial marke model consising of a bank accoun and wo risky asses (e.g., socks or muual funds). hese asses are radable coninuously over a finie ime horizon [, ], where (, ). Denoe he ime horizon [, ] by. Same as [], he sandard assumpions of financial marke hold, such as no ransacion coss, infiniely divisible asse, and informaion symmeric. Ariple(Ω, F, P) is a probabiliy space where Ω is a se, F is a σ-field of subses of Ω, andp is a real world probabiliy measure on F. A subse N of Ω is negligible if here exiss B F such ha N B and P(B) =. he probabiliy space is complee if F conains he se of all negligible ses. o model uncerainies ha emerged in our model, we adop a complee probabiliy space wih filraion (Ω, F,{F }, P),whereF = F and {F } describes he flow of informaion available o invesors. We also assume he probabiliy space is rich enough o incorporae all sources of randomness arising from flucuaions of financial asse prices and srucural changes in macroeconomic condiions. We model he evoluion of he saes of he economy over ime by a coninuous-ime, finie sae, ime-homogeneous, observable Markov chain ξ := {ξ() } defined on (Ω, F, P) wih a finie sae space S := {s, s,...,s N } R N, where N. he saes of he Markov chain are inerpreed as proxies of differen observable macroeconomic indicaors, such as gross domesic produc (GDP), sovereign credi raings, and consumer price index (CPI). More precisely, we suppose ha he Markov chain is also righ-coninuous and irreducible. Wihou loss of generaliy, following he convenion of [], we idenify he sae space of he chain as a finie se of uni basis vecors E := {e, e,...,e N },wheree i R N and he jh componen of e i is he Kronecker dela, denoed by δ ij, for each i,j =,,...,N. Kronecker dela δ ij is a piecewise funcion of variables i and j where δ ij =if i=j;oherwise iiszero.hesee is called he canonical represenaion of he sae space of he Markov chain ξ and i provides a mahemaically convenien way o represen he sae space of he chain. Here ξ() = e i means ha macroeconomic indicaors are in sae i a ime. o specify he saisics properies or he probabiliy law of he Markov chain, we define saionary ransiion probabiliies P ij () = P(ξ() = e j ξ() = e i ),fori, j =,,...,Nand, iniial disribuion p i = P(ξ() = e i ), and he generaor Q:=[q ij ] i,j=,,...,n of he chain ξ under P as follows: P ij () { lim q ij = + { P lim ii () { + i =j, i=j, i,j=,,...,n. () he generaor Q is also called a rae marix or a Q-marix. Here for each i,j =,,...,N, q ij is he consan, insananeous inensiy of he ransiion of he chain ξ from sae e i o e j.noehaq ij,fori = j and N j= q ij =, so q ii.hereforeachi,j =,,...,N wih i = j,we assume ha q ij >.Soweobainhaq ii <.hen,wihhe canonical represenaion of he sae space of he chain, Ellio e al. [] provided he following semimaringale dynamic decomposiion for ξ: ξ () = ξ () + Q ξ (u) du + M (), () where denoes he ranspose of a marix or a vecor. Here {M() } is an R N -valued maringale wih respec o he filraion generaed by Markov chain ξ. hefilraion F ξ := {F ξ () } saisfies he usual condiions which are he righ-coninuous, P-compleed nauralfilraion. In wha follows, we will specify he price processes of he primiive securiies and describe how he sae of he economy represened by he chain ξ influences he price processes. Noe ha he sae space of he chain ξ is a se of uni basis vecors, so any funcion of Markov chain ξ() can be denoed by a scalar produc beween a vecor and ξ(). Suppose r() denoe he insananeous, coninuously compounded ineres rae of he bank accoun a ime for each. hen he chain deermines r() as r () := r (, ξ ()) = r, ξ (), () where, is he inner produc in R N and r := (r,r,...,r N ) R N wih r i >for each i =,,...,N. r i is inerpreed as he ineres rae of he bank accoun when he economy is in he ih sae. Here he inner produc is o decide which componen of he vecors of ineres rae r,drif rae μ, or volailiy rae σ is in force according o he sae of he economy described by he chain ξ() a a paricular ime. henhepriceprocessofhebankaccounb:={b() } evolves over ime according o db () =r() B () d,, B() =. (4) For each andeach k=,,supposeμ k (), σ k () denoe he appreciaion rae and he volailiy rae of he kh risky asseaime,respecively.hen,he chainξ also deermines he appreciaion rae and volailiy rae of he kh risky asse as μ k () := μ k (, ξ ()) = μ k, ξ (), (5) σ k () := σ k (, ξ ()) = σ k, ξ (), where μ k := (μ k,μ k,...,μn k ) R N, σ k := (σ k,σ k,...,σn k ) R N, μ i k >and σi k >,foreachi =,,...,N. μi k and σi k are he appreciaion rae and he volailiy rae of he kh risky asse a ime, respecively when he economy is in he ih saeahaime.furhermore,wesupposehaμ i k r i for i=,,...,nand ha μ i k s and σi k sarealldisinc.hecondiion μ i k r i is necessary o exclude he arbirage opporuniies in he marke.
4 Mahemaical Problems in Engineering WeconsiderwosandardBrownianmoionsW () and W () for on (Ω, F, P). o allow for complee generaliy, we allow he wo sandard Brownian moions o be correlaed, wih correlaion coefficien denoed by ρ; namely, E[W ()W ()] = ρ. Incaseherewouldonlybeonesource of randomness lef in he model, we also assume ha ρ <. AhesameimewealsoassumehewosandardBrownian moions are independen of he Markov chain ξ. For each k=,,les k := {S k () } denoehepriceprocess of he kh risky asse. hen, we assume ha he evoluion of S k over ime saisfies he following Markov regime swiching geomeric Brownian moion: ds k () =μ k () S k () d + σ k () S k () dw k (), S k () =s, where he marke price of risk for risky asse k is defined as θ k () = (μ k () r())/σ k (). We consider sochasic dynamic porfolio game in a coninuous-ime financial marke beween wo invesors; call hem A and B. Wihou loss of generaliy, we suppose ha hereisabankaccounhaisfreelyavailableobohinvesors andsimulaneously,hereareonlywocorrelaedriskyasses in he financial marke, only one of which is available o each invesor. Invesor A may be allowed o rade in he firs risky asse, S, and similarly, invesor B may be resriced o rade only in he second risky asse, S.heycooperaewih each oher on invesmen by he choice of heir individual dynamic porfolio rading sraegies in he risky asses and bank accoun. In he nex we describe he wealh dynamic processes of boh invesors. For each,lex(, π ) denoe he wealh process of invesor A a ime under a porfolio sraegy π := {π () } wih X() = x.invesorainvess an amoun of wealh π () in he risky asse S a ime.noe ha once π () is deermined, he remaining amoun invesed in he bank accoun is compleely specified as X(, π ) π (). Similarly, le Y(, π ) denoe he wealh process of invesor B a ime under a porfolio sraegy π := {π () } wih Y() = y.invesorbinvessanamounofwealhπ () in he risky asse S a ime. he remaining amoun Y(, π ) π () is in he bank accoun. Le Π be he space of all admissible porfolio sraegies π.heelemensinπ saisfy he following wo condiions: (i) F-progressively measurable and càdlàg (righ-coninuous wih lef limi) R-valued process (i.e., π is a nonanicipaive funcion) and (ii) E[ π ()d] <. he condiion (ii) is a echnical condiion. If π Π,wecallheporfoliosraegy π admissible. So Π is he se of all admissible porfolio sraegies of invesor A. Similarly, we can define he se of all admissible porfolio sraegies of invesor B and denoe i by Π. As in a sandard porfolio selecion problem, he porfolio sraegies(conrols)areassumedobepiecewiseconinuous. We also assume he porfolio sraegies of sochasic differenial game beween he wo invesors are feedback sraegies, more specifically, Markov conrol sraegies. Markov conrol is only dependen on he curren value of sae variables in he (6) sysem no upon he hisory. ha is, he value we choose a ime only depends on he sae of he sysem a his ime. Furhermore, he invesor can condiion his acion a each poin in ime on he basis of he sae of he sysem a ha poin in ime. In many cases, i suffices o consider Markov conrol. For more discussions on he sraegies employed in he differenial games, ineresed readers can refer o [4]. We place no oher resricions on porfolio sraegy π or π. For example, we allow π () < or (π () < ); his means he invesors are allowed o sell he risky asses shor. Whereas we allow π () > X(, π ) or (π () > X(, π )), his corresponds o a credi and i means he invesors have borrowed o purchase he risky asses. Here, we noe ha he invesor decides he wealh amoun allocaed o he risky asse according o he curren and pas marke prices informaion and observaions of marke or macroeconomic condiions. his is oally differen from some radiional opimal porfolio models, where he invesors only consider he price informaion in making heir opimal invesmen decisions. Under he self-financing assumpion, for each he dynamics of he wealh process X(, π ) associaed wih π of invesor A evolves over ime as he following Markov regime swiching sochasic differenial equaion: dx (, π ) =[r(, ξ ()) X(,π )+(μ (, ξ ()) r(, ξ ()))π ()]d +σ (, ξ ()) π () dw (). (7) Similarly, for each he dynamics of he wealh process Y(, π ) associaed wih π of invesor B is governed by he following Markov-modulaed sochasic differenial equaion: dy (, π ) =[r(, ξ ()) Y(,π )+(μ (, ξ ()) r(, ξ ()))π ()]d +σ (, ξ ()) π () dw (). (8) For each, denoe he sum of wealh processes by Z (π,π ) () = X(, π )+Y(,π ).SinceX(, π ) and Y(, π ) are diffusion processes conrolled by invesors A and B, respecively, hen Z (π,π ) () is a joinly conrolled diffusion process. Specifically, he evoluion of he sum process over ime is governed by he following Markov regime swiching sochasic differenial equaion: dz (π,π ) () =[r(, ξ ()) Z (π,π ) () +(μ (, ξ ()) r(, ξ ()))π () +(μ (, ξ ()) r(, ξ ()))π () ]d +σ (, ξ ()) π () dw () +σ (, ξ ()) π () dw (), (9) where Z (π,π ) () = x +y.
Mahemaical Problems in Engineering 5 For mahemaical convenience, we can rewrie (9) in a more compac form dz (π,π ) () =[r() Z (π,π ) () +(μ () r())π () +(μ () r())π () ]d +σ () π () dw () +σ () π () dw (). () In he nex secion we provide a uiliy-based sochasic differenial porfolio game wih respec o he process {Z (π,π ), } of (9) or (). We formulae he sochasic differenial porfolio game as a problem of maximizing he expeced uiliy of he sum of erminal wealh processes. More general resuls on zero-sum sochasic differenial porfolio games are discussed in, for example, Browne []. Moreover, for some resuls on nonzero-sum differenial games, ineresed readers can refer o Lin [9].. Nonzero-Sum Game Problem Formulaion In his secion, we consider nonzero-sum sochasic differenial porfolio game problem beween wo invesors. he differenial game is formulaed as a problem o maximizing expeced uiliy of he sum of erminal wealh processes of wo invesors, respecively, a some fixed ime. For each k =,, le U k : R + R denoe uiliy funcions of invesors A and B, respecively, which are boh sricly increasing, sricly concave, and coninuous differeniable (i.e., U k > and U k < ). More resuls abou risk preference can refer o [5 7]. Furhermore, we assume ha he uiliy funcions saisfy he following Inada condiions (echnical condiions): U k (+) = lim z +U k (z) =+, U k (+ ) = () lim z + U k (z) =. Inhecaseofwoinvesors,AandB,foreach and each i =,,...,N, a ypical differenial game is posed as follows. Given ξ() = e i and Z (π,π ) () = z,invesorachoose his own admissible porfolio sraegy π Π o maximize affecs he sae equaion. Furhermore, we assume ha he wo invesors choose heir porfolio sraegies simulaneously. Invesor A would like o choose an admissible sraegy π so as o maximize his payoff V (π,π ) (, z, e i ) for every possible choice of invesor B s porfolio sraegy, while invesor B is rying o choose an admissible sraegy π in order o maximize his payoff V (π,π ) (, z, e i ) for every possible choice of invesor A s porfolio sraegy. he game erminaes a a fixed duraion. hen he nonzero-sum sochasic differenial porfolio game can be formulaed as he following wo opimal porfolio selecion uiliy maximizaion problems of invesors A and B: V (, z, e i ):= sup V (, z, e i ):= sup V (π,π ) π Π V (π,π ) π Π (, z, e i ), (, z, e i ). (4) Here V (, z, e i ) and V (, z, e i ) are he value funcions of he opimal porfolio selecion problems associaed wih invesors A and B, respecively, over he ime horizon [, ]. hisisawo-player,nonzero-sum,sochasicdifferenial porfolio game beween wo invesors A and B. o solve he nonzero-sum sochasic differenial porfolio game, in he following, we firs give he definiion of Nash equilibrium for he differenial game beween wo invesors AandBdescribedabove. Definiion. For each ime, given ha he sae of macroeconomic is in he ih sae, le π be an admissible sraegy of invesor B. One defines he se of bes responses of invesor A o he admissible porfolio sraegy π as BR i (π ) := {π Π V (π,π ) (, z, e i )= sup π Π V (π,π ) (, z, e i )}. (5) V (π,π ) (, z, e i ) =E[U [Z (π,π ) ()] Z (π,π ) () =z,ξ () = e i ], () And similarly, one can define he se of he bes responses of invesor B o he sraegy π of invesor A as while invesor B choose his own admissible porfolio sraegy π Π o maximize BR i (π ) V (π,π ) (, z, e i ) =E[U [Z (π,π ) ()] Z (π,π ) () =z,ξ () = e i ], () := {π Π V (π,π ) (, z, e i )= sup π Π V (π,π ) (, z, e i )}. (6) wih boh uiliy maximizaion problems subjec o he sum process (9) or (). We assume ha each invesor is aware of he oher invesor s presence and how he oher s choice of his sraegy Apairofadmissibleporfoliosraegies(π,π )issaidobe a Nash equilibrium (i.e., saddle poin) for he nonzero-sum differenial game wih invesors A and B, sraegies spaces Π
6 Mahemaical Problems in Engineering and Π,andpayoffs() and () when he economy (Markov chain) is in sae e i if π BRi (π ), π BRi (π ). (7) Equivalenly, (π,π )issaidobeanashequilibriumif V + sup π Π {[r() z+(μ () r())π () +(μ () r())π ()] V z + [σ () π () +σ () (π ()) V (π,π ) (, z, e i )= sup V (π,π ) (, z, e i )= sup V (π,π ) π Π V (π π Π (, z, e i ),,π ) (, z, e i ), (8) +ρσ () σ () π () π () ] V z + V,Q ξ () }=, where π and π are referred o as invesor A s and invesor B s respecive equilibrium sraegies. If a Nash equilibrium exiss, hen he value funcions of he nonzero-sum differenial game can be obained. For more discussions on he implicaions of Nash equilibrium, ineresed readers can refer o [8]. 4. Regime Swiching HJB Equaion and he Opimal Condiions In his secion we will derive a pair of regime swiching HJB equaions and Feynman-Kac represenaions for he value funcions of he nonzero-sum differenial game formulaed inhelassecion.wewillalsoderiveaseofcoupled HJB equaions corresponding o he regime swiching HJB equaions. In he sequel, we consider he case of invesors wih risk averse exponenial uiliy funcions. Suppose ha uiliy funcions U of invesor A and U of invesor B are given by U (x) = γ e γ x, U (x) = γ e γ x, (9) where γ and γ are posiive consans, which represen he coefficiens of absolue risk aversion (CARA) of invesors. ha is, γ k = U k (x), k=,, () (x) U k wih U k (x) and U k (x) represening he firs and second derivaives of U k wih respec o x,foreachk=,. Le V i k := V k(, z, e i ) and V k := (V k,v k,...,vn k ),foreach k =, and each i =,,...,N.Wenowsolvehewo uiliy maximizaion problems via he dynamic programming principle in sochasic opimal conrol according o [9]. I can be shown ha value funcions V k of uiliy maximizaion ofhenonzero-sumdifferenialgamesaisfyhefollowing regime swiching HJB equaions: V + sup π Π {[r() z+(μ () r())π () +(μ () r())π ()] V z + [σ () (π ()) +σ () π () +ρσ () σ () π () π () ] V z + V,Q ξ () }=. () In wha follows of his secion, wih a sligh abuse of noaions, for each k =,, we sill le all he noaions r(, e i ), μ k (, e i ), σ k (, e i ),andπ k (, e i ) be denoed by r(), μ k (), σ k (), andπ k () for, unless oherwise saed. For each i =,,...,N,leV i k := V k(, z, e i ).Hence,he vecor V k of value funcions a differen regimes saisfies he following wo sysems of coupled regime swiching HJB equaions, respecively: V i + sup { { π Π [r () z+(μ () r())π () { +(μ () r())π ()] Vi z + [σ () π () +σ () (π ()) +ρσ () σ () π () π () ] V i z + j E q ij [V (, z, e j ) V (, z, e i )] } } } =, ()
Mahemaical Problems in Engineering 7 wih he erminal condiion V (, z, e i ) = U (z) = (/γ )e γz ; V i + sup { { π Π [r () z+(μ () r())π () { +(μ () r())π ()] Vi z + [σ () (π ()) +σ () π () π () = ρ (ρμ () r() V i / z σ () σ () V i/ z μ () r() σ () V i / z V i ), / z π () = ρ (ρμ () r() V i / z σ () σ () V i/ z μ () r() σ () V i / z V i ). / z (7) +ρσ () σ () π () π () ] V i z + q ij [V (, z, e j ) V (, z, e i )] } } =, j E } () wih he erminal condiion V (, z, e i ) = U (z) = (/γ )e γz. Inwhafollows,oabbreviaeheexpressionincurly brackes, for each k=,, we define he operaors Second, assume ha HJB equaions () or (5) and () or (6) have smooh soluions wih V i k,z >and Vi k,zz <for each k=,.hesubscripsonv i k,z and Vi k,zz denoe he firs and second parial differeniaion wih respec o variable z, respecively. Following he approach in [], hen we consider valuefuncionsareofhefollowingrialsoluions: V i =V (, z, e i )= e γ ze f(,ei ), (8) γ L (π,π ) V k (, z, e i ) =[r() z+(μ () r())π () +(μ () r())π ()] Vi k z + [σ () π () +σ () π () +ρσ () σ () π () π () ] V i k z + q ij [V k (, z, e j ) V k (, z, e i )], j E i=,,...,n. (4) hen he HJB equaions () and () can be simplified as follows: V i + sup L (π,π ) V (, z, e i )=, (5) π Π V i + sup L (π,π) V (, z, e i ) =. (6) π Π Firs, he firs-order condiions for maximizing he quaniy in he HJB equaions (5) and (6) give opimal porfolio sraegies where f(, e i ) C, o be deermined, is a suiable posiive funcion wih he boundary condiion f(, e i )=for all e i E. Consider V i =V (, z, e i )= e γ ze g(,ei ), (9) γ where g(, e i ) C, o be deermined, is a suiable posiive funcion wih he boundary condiion g(, e i )=for all e i E. Hence, from (7)we obain he associaed explici expressions of compeiive opimal porfolio sraegies π () = ρ (μ () r() γ σ () e ρ μ () r() γ σ () σ () e π () = ρ (μ () r() γ σ () e ρ μ () r() γ σ () σ () e ), ). ()
8 Mahemaical Problems in Engineering Las, subsiuing hese resuls ino HJB equaions ()or (5) and () or (6), we derive he following wo sysems of coupled linear ordinary differenial equaions: f (, e i ) g (, e i ) + γ ( ρ ) [ (μ () r()) γ σ () (μ () r()) γ γ σ () (μ () r()) γ σ () +ρ (μ () r())(μ () r()) γ σ ]f(,e () σ () i ) + q ij [f (, e j ) f(,e i )] =, f (, e i )=, j E + γ ( ρ ) [ (μ () r()) γ σ () (μ () r()) γ γ σ () (μ () r()) γ σ () +ρ (μ () r())(μ () r()) γ σ ]g(,e () σ () i ) + q ij [g (, e j ) g(,e i )] =, g (, e i )=. j E () () I is well known ha boh sysems of differenial equaions onlyhaveauniquesoluion,respecively.wewillgivean appropriae explanaion in Remark 8 of his secion. In summary, we obain he following heorem immediaely. heorem. If ρ =, hen he value funcions of invesors A and B a differen regimes of he uiliy maximizaion of nonzero-sum sochasic differenial porfolio game are given by V i =V (, z, e i )= e γ ze f(,ei ), γ V i =V (, z, e i )= e γ ze g(,ei ), γ () where f(, e i ) and g(, e i ) are soluions of coupled linear ordinary differenial equaions () and (). he opimal porfolio sraegies can be expressed as π () = ρ (μ () r() γ σ () e ρ μ () r() γ σ () σ () e π () = ρ (μ () r() γ σ () e ρ μ () r() γ σ () σ () e ), ). (4) Remark. From he expressions (4) of opimal porfolio sraegies, we can see ha he opimal porfolio sraegies are composed of wo erms and he firs erm is similar o Meron-ype soluion. Meanwhile, we find ha he porfolio sraegies also depend on he regime swiching of he macroeconomy. Boh of hem are direcly proporional o he expeced excess reurns on he corresponding risky asses in he marke when he correlaion ρ beween wo risky asses is negaive. However, when ρ >, he opimal sraegies are inversely proporional o he expeced excess reurn of he oher risky asse no available o he associaed invesors. Furhermore, even if he macroeconomy has N saes, only if he sae of economy is unchanged, we find ha he opimal porfolio sraegies are always consan. Namely, hey are independen of he wealh level of invesors. Remark 4. When he Markov chain has only one sae (i.e., S := {s }), hen he Markov regime swiching model considered here degeneraes ino a deerminisic case. And he marke parameers r, μ, and σ of he model become consans. hen he conclusions will no be influenced by Markov chain, so we can omi he index i in he value funcions and he opimal sraegies. Albei wih he difference in form, he linear ordinary differenial equaions which he funcions f and g in he value funcions saisfy coincide wih (4.) and (4.4) of lemma 4. of [9] respecively. Correspondingly he value funcions and opimal sraegies are consisen wih he resuls of heorem 4. of [9]. Corollary 5. If ρ =, he wo sandard Brownian moions (W () and W ())aremuual,independenof(ω, F, P).hen he value funcions of invesors A and B a differen regimes of he uiliy maximizaion of he nonzero-sum sochasic differenial porfolio game are denoed by V i =V (, z, e i )= e γ ze f(,ei ), γ V i =V (, z, e i )= e γ ze g(,ei ), γ and he opimal porfolio sraegies are given by π () = μ () r() γ σ () e, π () = μ () r() γ σ () e, (5) (6)
Mahemaical Problems in Engineering 9 where he funcions f and g are he soluions of he following linear ordinary differenial equaions: f (, e i ) g (, e i ) + γ [(μ () r()) γ σ () (μ () r()) γ σ () (μ () r()) γ γ σ () ]f(,e i ) + q ij [f (, e j ) f(,e i )] =, f (, e i )=, j E (7) + γ [(μ () r()) γ σ () (μ () r()) γ σ () (μ () r()) γ γ σ () ]g(,e i ) + q ij [g (, e j ) g(,e i )] =, g (, i) =. j E (8) Remark 6. If he wo risky asses are no correlaed wih each oher, from Corollary 5, wecanseehaheopimal invesmen sraegies are of he Meron soluion ypes a a paricular regime. Moreover, opimal invesmen sraegies are direcly proporional o he expeced reurn and inversely proporional o he variance of he associaed risky asses available o invesors a such a regime. In his special case, we know each invesor does no wan o consider he oher invesor s sraegy when he makes his opimal porfolio choice. I is equivalen o consider he classical uiliy maximizaion for individual s opimal porfolio choice under a regime swiching financial marke. We noe ha he funcions f(, e i ) and g(, e i ) in he expressions of he value funcions given in heorem saisfy wo sysems of coupled linear ordinary differenial equaions. Generally speaking, differenial equaions are harder o solve. In wha follows, we will derive anoher represenaion of f(, e i ) and g(, e i ) via he Feynman-Kac ype represenaions which no only are more convenien o invesigae he influence of he Markov-modulaion on he value funcions of he nonzero-sum sochasic differenial porfolio game problem bu also can beer inerpre and derive some properies of value funcions. o derive he Feynman-Kac represenaions of boh value funcions f(, e i ) and g(, e i ),firswemakehe following assumpions of noaion: f () := (f (, e ),f(,e ),...,f(,e N )), (9) g () := (g (, e ),g(,e ),...,g(,e N )), (4) a i := a (, e i ) γ = ( ρ ) [(μ () r()) γ σ () b i := b (, e i ) (μ () r()) γ γ σ () γ = ( ρ ) [(μ () r()) γ σ () (μ () r()) γ σ () +ρ (μ () r())(μ () r()) γ σ ], () σ () (4) (μ () r()) γ γ σ () (μ () r()) γ σ () +ρ (μ () r())(μ () r()) γ σ ]. () σ () (4) WenowgiveheFeynman-Kacrepresenaionsoff(, e i ) and g(, e i ) in he following heorem. heorem 7. Le A () := diag [a (, e ),a(,e ),...,a(,e N )] Q, B () := diag [b (, e ),b(,e ),...,b(,e N )] Q, (4) where diag [x] denoes he diagonal marix wih diagonal elemens given by he row vecor x. hen he Feynman-Kac represenaions of f(, e i ) and g(, e i ) are given by f(,e i )=E,i [exp ( a (u, ξ (u)) du)], (44) g(,e i )=E,i [exp ( b (u, ξ (u)) du)], (45) where a(, e i ) and b(, e i ) are denoed by (4) and (4) and E,i is he condiional expecaion given ha ξ() = e i under P. Proof. Firs, we will prove he Feynman-Kac represenaion of f(, e i ). From Lemma.5 of Appendix B in [], we know ha he process {M(s) s [, ]}, defined by M (s) := f (s, ξ (s)) f(, ξ ()) s (f (u, ξ (u)) + f (u),q ξ (u) ) du, (46) is an (F ξ, P)-maringale. Nex, we define a funcion R(s) by s R (s) := exp ( a (u, ξ (u)) du). (47)
Mahemaical Problems in Engineering hen by applying he produc rule of Io formula and he definiion of M(s),weobain s R (s) f (s, ξ (s)) f(, ξ ()) = R (u) dm (u). (48) Seing s=and aking expecaion, we obain secion given a general rae marix Q.Furhermore,we know ha he wo-sae Markov chain is rich enough o disinguish a bull marke and a bear marke. In his case, from heorem, we see ha he value funcions of invesors A andbforhewo-saeeconomyaredenoedbyhefollowing forms: f(,e i )=E,i [R ()] =E,i [exp ( a (u, ξ (u)) du)]. (49) We noe ha he proof of he Feynman-Kac represenaion of funcion g(, e i ) ishesameasheaboveprocess,soweomi i here. V i =V (, z, i) = γ e γ ze r( ) f (, i), i =,, V i =V (, z, i) = γ e γ ze r( ) g (, i), i =,, (5) From he expressions of f(, e i ) in (44) and g(, e i ) in (45), iisnodifficuloseef(, e i ) > and g(, e i ) >. Consequenly, for each k =,, V k (, z, e i ) given by () indeed saisfy V i k,z >and Vi k,zz <jusas described in he previous secion. Remark 8. By using he noaions of (A() or B()) and(f() or g()), he marix form of () and () canbedenoedby f () =A() f (), f () =, g () =B() g (), g () =, (5) where = (,,...,) R N. he differenial equaion sysems (5) indeed have unique coninuous soluions referred o in (page, heorem ) of [4]. For more similar discussions on he exisence of soluions of he differenial equaion sysem, ineresed readers can refer o Remark 4. of [4]. 5. One Special Case In order o invesigae how he Markov regime swiching influences opimal invesmen sraegies and value funcions and provide meaningful comparaive saics analysis, in his secion we presen one special case of he nonzero-sum sochasic differenial porfolio game problem esablished in Secions and 4. We will assume ha he ineres rae denoed by r(, ξ()) r, is no modulaed by Markov chain. Namely, he price of bank accoun is no affeced by he exernal condiions. his assumpion is relaively reasonable sincecomparedoriskyasses,hepriceofhebankaccoun is more sable. Furhermore, we assume S := {, }; hais, ξ := {ξ() } is a wo-sae Markov chain. We firs derive regime swiching HJB equaions for value funcions of his simplified model. We assume ha he rae marix Q of he Markov chain is given by Q=( q q ), (5) q q where q is a posiive real consan. Noe ha i is no necessary o consider he Markov chain wih only wo saes. I jus increases compuaion complexiy in he numerical analysis where f(, i) is he soluions of he following pair of linear ordinary differenial equaions: f (, ) f (, ) γ + ( ρ ) [ (μ r) γ [ (σ ) +q[f(, ) f(, )] =, γ + ( ρ ) [ (μ r) γ [ (σ ) q[f(, ) f(, )] =, (μ r) γ (σ ) +ρ (μ r)(μ r) γ σ σ (μ r) γ (σ ) +ρ (μ r)(μ r) γ σ σ (μ r) γ γ (σ ) ] f (, ) ] (μ r) γ γ (σ ) ] f (, ) ] (5) wih boundary condiions f(, ) = and f(, ) =, and g(, i) is he soluions of he following pair of linear ordinary differenial equaions: g (, ) γ + ( ρ ) [ (μ r) γ [ (σ ) +q[g(, ) g(, )] =, (μ r) γ (σ ) +ρ (μ r)(μ r) γ σ σ (μ r) γ γ (σ ) ] g (, ) ]
Mahemaical Problems in Engineering g (, ) γ + ( ρ ) [ (μ r) γ [ (σ ) q[g(, ) g(, )] =, (μ r) γ (σ ) +ρ (μ r)(μ r) γ σ σ (μ r) γ γ (σ ) ] g (, ) ] (54) wih boundary condiions g(, ) = and g(, ) =. For he sake of conciseness, we can rewrie (5)-(54) in he following simplified forms by using he noaions of a i in (4) and b i in (4): f (, ) f (, ) g (, ) g (, ) +(a q)f(, ) +qf(, ) =, f(, ) =, +(a q)f(, ) +qf(, ) =, f(, ) =, (55) +(b q)g(, ) +qg(, ) =, g(, ) =, +(b q)g(, ) +qg(, ) =, g(, ) = (56) and he opimal porfolio sraegies of invesors A and B for his case are, respecively, given by π (, ) = ρ ( μ r γ (σ e r( ) ρ μ r ) γ σ σ π (, ) = ρ ( μ r γ (σ e r( ) ρ μ r ) γ σ σ π (, ) = ρ ( μ r γ (σ e r( ) ρ μ r ) γ σ σ π (, ) = ρ ( μ r γ (σ e r( ) ρ μ r ) γ σ σ e r( ) ), e r( ) ), (57) e r( ) ), e r( ) ). (58) We know ha he above wo pairs of linear differenial equaions for funcions f and g (see (55) and (56)) canbe explicily solved. o solve hese wo differenial equaions, we firs give he following lemma. Lemma 9. Le r, and r,4 be he soluions of he following wo quadraic equaions, respecively: r +(a +a q) r + [(a q)(a q) q ]=, (59) r +(b +b q) r + [(b q)(b q) q ]=. (6) hen one can easily obain r, = (a +a q)± (a a ) +4q, r,4 = (b +b q)± (b b ) +4q, (6) where he noaions a i, b i,andq are given by (4), (4), and (5). Proof. he conclusion of his lemma follows direcly from he soluions of quadraic equaion. Meanwhile, he discriminans of he quadraic equaions are always bigger han zero, so he equaions have wo differen real roos. For mahemaical convenience, le us denoe C = (r +a )(r +a q), q(r r ) C = (r +a )(r +a q), q(r r ) C = (r +a ) (r r ), C 4 = (r +a ) (r r ), C 5 = (r 4 +b )(r +b q), q(r r 4 ) C 6 = (r +b )(r 4 +b q), q(r r 4 ) C 7 = (r 4 +b ) (r r 4 ), C 8 = (r +b ) (r r 4 ). (6) hen he soluions of f(, ), f(, ) and g(, ), g(, ) are explicily given by f (, ) =C e r ( ) +C e r ( ), f (, ) =C e r ( ) +C 4 e r ( ), g (, ) =C 5 e r ( ) +C 6 e r 4( ), g (, ) =C 7 e r ( ) +C 8 e r 4( ). (6) Remark. When he model parameers μ k and σ k of risky asses for each k =, in he wo regimes are idenical o each oher, respecively, his resul will lead o a =a := a and b =b := b. No maer wha he value of he parameer q in he rae marix of he Markov chain is, we find ha he opimal value funcions and opimal porfolio sraegies are robus wih respec o he change in he value of q.inhiscase, he above wo pairs of linear ordinary differenial equaions (5) or (55) and (54) or (56) wih he corresponding boundary condiions have unique soluions, respecively. In his case, we can easily obain he soluions of f and g.hais, f () =e a( ) g () =e b( ), (64)
Mahemaical Problems in Engineering where his resul is jus as described in Remark 4 in Secion 4. hiscaseisequivalenoheassumpionhamarkovchain has only one sae. 6 4 =6 6. Comparaive Saics and Numerical Simulaion o gain more insigh ino he economic significance of regime swiching on he opimal porfolio sraegies, in his secion we will consruc numerical analysis o invesigae how opimal invesmen sraegies change wih he parameers arising from our Markov-modulaed model, for example, he absolue risk aversion coefficiens of wo invesors and he correlaion coefficien beween he wo risky asses. We also make comparisons of he qualiaive behaviors of he opimal porfolio sraegies obained from our model (Model I) o hose arising from he model wihou regimes (Model II). o perform boh he comparaive saics analysis and he comparisonbeweenmodeliandmodeliisimulaneously, we implemen he procedure jus as [4]. For illusraion, we consider he special case described in Secion 5 where Markov chain is assumed o only have wo saes and ineres raeisfixed.herewealsosupposehahewosaesin Markov chain ξ represen economy (E) and economy (E), respecively. In order o make he comparison effecively, our numerical resuls are based on he following annualized baseline hypoheical values for he model parameers unless oherwise saed: he absolue risk aversion coefficiens γ =.4, γ =.,hecurrenime=6, he maure horizon =, he risk-free ineres rae r =.5, he drif raes of he wo risky asses μ =.8, μ =., andhevolailiy raes of he wo risky asses σ =., σ =.5. hese hypoheical parameer values are drawn from hose used in [9],becausewesimulaneouslywanomakecomparisons of he parameer sensiiviy analysis resuls obained from our model o he properies obained in [9]. For each k =,, when he drif raes μ k and he volailiy raes σ k of he risky asses in E are he same as heir corresponding parameer values in E, we say ha Model I and Model II are idenical o each oher. he numerical resuls in his case for he opimal porfolio sraegies obained from Model I are idenical o hose arising from Model II whaever he value of he parameer q inheraemarixofmarkov chain Q is. hus he resuls are robus wih respec o he change of q. Ineresed readers can refer o Remark for deails abou his descripion. In he nex, we implemen he numerical analysis for he opimal invesmen sraegies wih respec o ha paricular parameer. We will focus on how he opimal porfolio sraegies vary agains model parameers when he economy saemodeledbymarkovchainchanges,namely,howhe regime shifs in marke parameers μ k and σ k for each k=, affec porfolio sraegies. In he following wo subsecions, we analyze his issue along wo dimensions. Firsly, we invesigae he impac of regime shifs in drif raes μ k, k=,,on opimal porfolio sraegies agains he absolue risk aversion coefficiens and correlaion coefficien, respecively. Secondly, π π 4.5..5..5. μ ρ =.5 ρ =.5 Figure : Opimal sraegy π agains μ for differen ρ. 4 =6.5..5..5. μ ρ =.5 ρ =.5 Figure : Opimal sraegy π agains μ for differen ρ. we analyze he effecs of regime swiching in volailiies of financial asses on he porfolio sraegies agains he same parameers as hose wih respec o he analysis of he impac of regime shifs in drif raes. 6.. he Effec of Drif Raes. he main purpose of his subsecion is o see he effecs of he drif raes μ and μ of he wo risky asses in economy E on he opimal porfolio sraegies. We suppose here are hree cases for our model parameers, namely, μ >μ or μ >μ or he wo condiions μ >μ and μ >μ saisfying simulaneously. When only one of he hree condiions saisfies, hen economy E is said o be a good economy relaive o economy E. Or else E is a bad economy. In his case, Model I and Model II are said o be differen from each oher. However, for each k=,, when μ k =μ k (i.e.,eandecoincide),imeanshahewo saes are idenical. I is equivalen o say he model has no regime swiching and Model I and Model II are idenical o each oher. Figures and plo he curren opimal sraegies π agains he drif rae μ and π agains he drif rae μ for wo differen, opposie sign, paricular values of correlaion coefficien ρ, respecively. Boh figures show he fundamenal properies of opimal porfolio sraegies in a benchmark model in which here are no regimes in he financial markes.
Mahemaical Problems in Engineering =6 8 = 6, ρ =.5 5 7 6 π 5 5 5.8.6.4...4.6.8 ρ μ =.8 μ =. μ =. Figure : Opimal π in E agains ρ for differen μ k. π 5 4...4.5.6.7.8.9.. γ μ =.8 μ μ =. =. Figure 6: π in E agains γ for ρ =.5 and differen μ k. π π 9 =6 8 7 6 5 4.8.6.4...4.6.8 ρ μ =. μ μ =. =. Figure 4: Opimal π in E agains ρ for differen μ k. 4 = 6, ρ =.5 5...4.5.6.7.8.9.. γ μ =.8 μ μ =. =. Figure 5: π in E agains γ for ρ =.5 and differen μ k. From Figure, icanbeeasilyseenhaπ decreases as μ increases when ρ>, while i increases when ρ<.his propery of porfolio sraegy π similarly applies o π. Figures and 4 depic he plos of he opimal invesmen sraegies π and π agains he correlaion coefficien ρ for differen values of μ k,respecively.heploscomparehe wo cases when here is no regime (μ = μ =., μ = μ =.8) and when he sae changes o E. We can see ha he regime swiching in drif raes has a significan impac on he qualiaive behavior of opimal porfolio sraegies. According o (57) and (58), foreachk=,,wecaneasily know ha π k increases wih μ k.fromfigure, wecansee ha π increases along wih μ.whenρ<,isubsanially increases as μ. However, i significanly decreases as μ increases when ρ>. According o heorem 5. in [9], he opimal porfolio sraegy π decreases wih ρ when θ /γ < θ /γ.hereheparameersgiveninmodeliandmodelii saisfy his condiion. Likewise, from Figure 4, weseehahe opimal invesmen π subsaniallyincreasesalongwihμ andalsoincreasesasμ when ρ<.whileρ>, π slighly decreases as μ increases. Moreover, for he hree cases when μ =., μ =., andμ =., hereisacriicalpoina whichhereisareversalofhebehaviorofopimalinvesmen sraegy from decreasing o increasing agains differen values of ρ. Figures 5 and 6 depic he plos of opimal invesmen sraegy π agains he absolue risk aversion coefficien γ. Similaroheabove,inhesefigureswecomparehewocases when here is no regime (μ =μ =., μ =μ =.8)and when he sae changes o E. From Figures 5 and 6, forall he hree cases when μ =.8, μ =., andμ =., we see ha he opimal porfolio sraegy π decreases as risk aversion coefficien γ increases. Also, π increases along wih μ whenever he values of correlaion coefficien ρ ake. A he same ime, Figure 5 shows ha opimal sraegy π significanly decreases as μ increases when ρ>.however, wefindhahisresulisopposiewhenρ<in Figure 6.
4 Mahemaical Problems in Engineering π π = 6, ρ =.5 4...4.5.6.7.8.9.. γ μ =.8 μ μ =. =. Figure 7: π in E agains γ for ρ =.5 and differen μ k. 7 6 5 4 = 6, ρ =.5...4.5.6.7.8.9.. γ μ =.8 μ μ =. =. Figure 8: π in E agains γ for ρ =.5 and differen μ k. Figures 7 and 8 plo π agains he absolue risk aversion coefficien γ.fromfigures7 and 8, weseehaheopimal invesmen sraegy increases as μ increases and also subsanially decreases as μ increases when ρ >. While ρ<, i shows he opposie resul. he opimal invesmen significanly increases along wih μ.forallhehreecases when μ =., μ =., andμ =., π increases wih γ when ρ>.whenρ<,idecreaseswihγ. his resul is consisen wih heorem 5. in [9]. Figures 9 and and Figures and describe he effecs of μ k on he qualiaive behavior of he opimal porfolio sraegy π of invesor B agains he risk aversion coefficiens γ and γ for differen values of ρ. Since he explanaions for hese figures are similar o he above analysis of opimal porfolio sraegy of invesor A, we omi hem here. From he above figures, we can see ha regime swiching represened byheswichesofdrifraeμreally has a significan impac on he opimal invesmen sraegy. o consider regime swichingmodelswillhelpoincorporaehesrucurechangeofhe model and help invesors o adjus heir invesmen sraegies according o he changing condiions of he macroeconomy. 6.. he Effec of Volailiy Raes. In his subsecion we will focus on he effec of he volailiy raes σ and σ of he risky π π 4 = 6, ρ =.5.5.5.5.5...4.5.6.7.8.9.. γ μ =. μ μ =. =. Figure 9: π in E agains γ for ρ =.5 and differen μ k. 5 = 6, ρ =.5 4.5 4.5.5.5...4.5.6.7.8.9.. γ μ =. μ μ =. =. Figure : π in E agains γ for ρ =.5 and differen μ k. asses in E on he opimal porfolio sraegies. We suppose here are hree cases for our model parameers, namely, σ > σ or σ >σ or he wo condiions σ >σ and σ >σ saisfying simulaneously. When one of he hree condiions saisfies, hen economy E is said o be a bad economy relaive o E. Or else E is a good economy. In his case Model I and Model II are differen from each oher. For each k=,,whenσ k =σ k (i.e., E and E coincide), i means ha he model has no regime and we say Model I and Model II are idenical. From he following comparaive saic analysis, we can see ha he effecs of regime swiching on he opimal invesmen sraegies are significan. Figures and 4 describe he plos of opimal invesmen sraegies π and π agains he correlaion coefficien for differen values of σ k. he plos compare when here is no regime (σ = σ, σ = σ ) and when he sae changes o E. From Figure, we see ha he opimal invesmen π decreases when σ increases for ρ<.his resul is opposie for ρ >.FromFigure4, ishows ha he opimal invesmen π decreases when σ increases for ρ <. However, for ρ >, i is an increasing funcion of σ. his relaion for boh invesmen sraegies a a paricular regime is consisen wih heorem 5. in [9].
Mahemaical Problems in Engineering 5 π.5.5.5.5 = 6, ρ =.5.5...4.5.6.7.8.9.. γ μ =. μ μ =. =. Figure : π in E agains γ for ρ =.5 and differen μ k. π 5 4 =6.8.6.4...4.6.8 ρ σ =. σ =.5 σ =.7 Figure : π in E agains ρ for differen σ k. π 4.5 4.5.5.5 = 6, ρ =.5.5...4.5.6.7.8.9.. γ μ =. μ μ =. =. Figure : π in E agains γ for ρ =.5 and differen μ k. π.5.5.5.5 =6.8.6.4...4.6.8 ρ σ =.5 σ =.7 σ =.5 Figure 4: π in E agains ρ for differen σ k. And he monooniciy properies beween wo invesmen sraegies and correlaion coefficien for regime shifs in volailiies are he same as he analysis of drif raes in Figures and 4. Figures 5, 6, 7, and8 depic he plos of he opimal porfolio sraegy π agains he risk aversion coefficiens γ and γ for differen values of ρ when he regime swiching is represened by swiches of volailiies σ k.heresulsshowed in all hese figures reflec ha he regime swiching in volailiy has a significan impac on he opimal porfolio sraegy. he monooniciy properies beween invesmen and risk aversion coefficien γ are he same for differen values of ρ as wha are described in deail in he above subsecions for he effec of drif raes. From Figure 5, when ρ akes he posiive value and he volailiy σ increases from.o.5,wecanseehereisacriicalpoinawhichhereisa reversal of he opimal invesmen sraegy from less o more. However, in Figure 6, whenρ =.5 he opimal invesmen sraegy in no regime (σ =.) ismorehanhecasewhen here is regime. Based on he case for σ =.5, Figure5 shows ha he opimal invesmen sraegy becomes bigger when ρ =.5; however, his resul is opposie in Figure 6. In Figures 7 and 8, we can see he monooniciy properies of he opimal invesmen sraegy π agains he risk aversion γ are he exac opposie when ρ akes arbirary posiive or negaive value. And also compared o he no regime swiching case when σ =., he opimal invesmen sraegy π in he wo regime swiching cases (σ =.5 or σ =.5 and σ =.7)decreasessignificanly. Figures 9,,, and plo he opimal invesmen sraegy π agains he risk aversion coefficiens γ and γ for differen values of ρ when he regime swiching is represened by swiches of σ k.infigures9 and, wecan see he monooniciy relaions beween opimal invesmen sraegies π and γ are exacly opposie when ρ akes posiive ornegaivevalue,whileinfigures and, heopimal invesmen sraegy π decreases as γ increases for differen values of ρ. Since he analysis for hese figures is similar o he above analysis of opimal invesmen sraegy of invesor A, we omi i here.
6 Mahemaical Problems in Engineering π = 6, ρ =.5.8.6.4...4.6.8...4.5.6.7.8.9.. γ π = 6, ρ =.5.8.6.4..8.6.4....4.5.6.7.8.9.. γ σ =. σ =.5 σ =.7 Figure 5: π in E agains γ for ρ =.5 and differen σ k. σ =. σ =.5 σ =.7 Figure 8: π in E agains γ for ρ =.5 and differen σ k. π.5.5.5 = 6, ρ =.5...4.5.6.7.8.9.. γ σ =. σ =.5 σ =.7 Figure 6: π in E agains γ for ρ =.5 and differen σ k. π = 6, ρ =.5..9.8.7.6.5.4......4.5.6.7.8.9.. γ σ =.5 σ =.7 σ =.5 Figure 9: π in E agains γ for ρ =.5 and differen σ k. π.8.6.4.. = 6, ρ =.5.4...4.5.6.7.8.9.. γ σ =. σ =.5 σ =.7 π.8.6.4..8.6 = 6, ρ =.5.4...4.5.6.7.8.9.. γ σ =.5 σ =.7 σ =.5 Figure 7: π in E agains γ for ρ =.5 and differen σ k. Figure : π in E agains γ for ρ =.5 and differen σ k.
Mahemaical Problems in Engineering 7 π π..8.6.4. = 6, ρ =.5....4.5.6.7.8.9.. γ σ =.5 σ =.7 σ =.5 Figure : π in E agains γ for ρ =.5 and differen σ k..6.4..8.6.4 = 6, ρ =.5....4.5.6.7.8.9.. γ σ =.5 σ =.7 σ =.5 Figure : π in E agains γ for ρ =.5 and differen σ k. 7. Conclusion In his paper we deal wih a nonzero-sum sochasic differenial porfolio game problem beween wo invesors in a coninuous-ime Markov regime swiching model. We formulaed he sochasic differenial game as wo uiliy maximizaion problems. We derived a pair of regime swiching HJB equaions for his differenial game problem and hen obained wo sysems of coupled regime swiching HJB equaions a differen regimes. Explici soluions o he opimal invesmen sraegies of wo invesors were also obained. Furhermore, we derived he Feynman-Kac represenaions of he value funcions of he wo uiliy maximizaion problems. Numerical resuls for model parameers and he impac of he regime swiching were illusraed and discussed when hemarkovchainwasassumedoonlyhavewosaes.i showed ha he regime swiching in he model parameers had a significan impac on he opimal porfolio sraegies. here are some ineresing and poenial opics for fuure research. Firsly, he presen paper assumes ha he Markov chain can be observed. However, i is ineresing o consider a nonzero-sum sochasic differenial game under a hidden Markovian regime swiching economy. Secondly, we could formulae he nonzero-sum sochasic differenial game as a mean-variance porfolio selecion problem under a regime swiching economy. A recen relaed publicaion by Bensoussan e al. (4) [4] sudied noncooperaive nonzero sum games and he problem is formulaed as uiliy maximizaion of he difference beween he erminal wealh. Conflic of Ineress he auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmens he research for his paper was suppored by he Key Projec of Naional Naural Science Fund of China (Projec no. 748), Naional Naural Science Foundaion of China (Projec no. 775), Naional Naural Science Innovaion Research Group of China (Projec no. 7), and Hunan Provincial Innovaion Foundaion For Posgraduae (Projec no. CX4B4). References [] H. Markowiz, Porfolio selecion, he Journal of Finance,vol. 7, no., pp. 77 9, 95. [] R. C. Meron, Lifeime porfolio selecion under uncerainy: he coninuous-ime case, he Review of Economics and Saisics,vol.5,no.,pp.47 57,969. [] R. C. Meron, Opimum consumpion and porfolio rules in a coninuous-ime model, Journal of Economic heory, vol., no. 4, pp. 7 4, 97. [4] R. E. Quand, he esimaion of he parameers of a linear regression sysem obeying wo separae regimes, Journal of he American Saisical Associaion, vol.5,no.84,pp.87 88, 958. [5] J. D. Hamilon, A new approach o he economic analysis of nonsaionary ime series and he business cycle, Economerica, vol. 57, no., pp. 57 84, 989. [6] X. Guo, J. Miao, and E. Morellec, Irreversible invesmen wih regime shifs, Journal of Economic heory, vol., no., pp. 7 59, 5. [7] H. Chen, Macroeconomic condiions and he puzzles of credi spreads and capial srucure, Journal of Finance,vol.65,no.6, pp.7,. [8] F. Wen and X. Yang, Skewness of reurn disribuion and coefficien of risk premium, Journal of Sysems Science & Complexiy,vol.,no.,pp.6 7,9. [9]F.WenandZ.Liu, Acopula-basedcorrelaionmeasureand is applicaion in chinese sock marke, Inernaional Journal of Informaion echnology and Decision Making, vol.8,no.4,pp. 787 8, 9. [] N. Bauerle and U. Rieder, Porfolio opimizaion wih Markovmodulaed sock prices and ineres raes, IEEE ransacions on Auomaic Conrol,vol.49,no.,pp.44 447,4.
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