Optimal time-consistent investment in the dual risk model with diffusion

Transcription

1 Opimal im-conin invmn in h dual rik modl wih diffuion LIDONG ZHANG Tianjin Univriy of Scinc and Tchnology Collg of Scinc TEDA, Sr 3, Tianjin CHINA zhanglidong999@.com XIMIN RONG Tianjin Univriy School of Scinc Wijin Sr 9, Tianjin CHINA rongximin@ju.du.cn ZIPING DU Tianjin Univriy of Scinc and Tchnology Collg of Economic & Managmn Dagu Nanlu 3, Tianjin CHINA dux@u.du.cn Abrac: Th objciv of hi papr i o inviga opimal invmn ragy for a pharmacuical or prolum company undr man-varianc cririon. Th urplu of h company i modld a a dual rik modl. W aum ha h company can inv ino a rik-fr a and n riky a. Shor-lling and borrowing mony ar allowd. Sinc hi problm i im-inconin, w udy i wihin a gam horical framwork. Th ubgam prfc Nah quilibrium ragi namly im-conin ragi) ar drivd by olving h Exndd Hamilon-Jacobi-BllmanHJB) quaion in h claic diffuion dual modl and in h diffuion dual approximaion modl, paraly. Surpriingly whn a cofficin paramr ρ =, opimal im-conin invmn ragi and h valu funcion hav h am xprion in boh ca. Finally, W prn conomic implicaion and provid niiviy analyi for our rul by numrical xampl. Ky Word: Opimal im-conin invmn; Exndd Hamilon-Jacobi-Bllman quaion; Man-varianc cririon Inroducion Rcnly, qui a fw inring papr hav bn publihd on h dual rik modl. Th dual rik modl can b naural for compani ha hav occaional profi. For compani uch a pharmacuical or prolum compani, h jump can b inrprd a h n prn valu of fuur incom from an invnion or dicovry. Many cholar invigad h dual rik modl undr diffrn cririon. For xampl, Zhu and Yang [] udid h ruin probabiliy undr h dual modl, whil Avanzi al. [] [] conidrd h dividnd paymn problm in h dual modl. Dai al.[] and Yao al.[] invigad h dual modl undr h objciv of maximizing h xpcd prn valu of h dividnd minu capial injcion, whil h lar conidrd h ranacion co. Th objciv of hi papr i o inviga h opimal invmn problm in h dual modl undr man-varianc cririon. Mo of h liraur xploid an mbdding chniqu o dal wih dynamic man-varianc problm. Th mbdding chniqu wa propod by Li and Ng) and Zhou and Li). Sinc h irad xpcaion propry do no hold for h varianc opraor, h opimal - ragyprcommid ragy) drivd by h mbd- School of Managmn,Tianjin Univriy ding chniqu do no aify h Bllman Opimaliy Principl and i im-inconin. Thi prcommid ragy impli ha h ragy compud a will no ncarily coincid wih h ragy drivd a +. A a rul, a + h raional invor will implmn h ragy compud a. A raional invor will find a im-conin ragy which nur him o kp a conin aifacion. Anohr poibiliy i o ak h im-inconincy mor riouly and udy h problm wihin a gam horical framwork. Th baic ida i ha whn w dcid on a ragy a w hould xplicily ak ino accoun ha a fuur im w will hav a diffrn objciv funcional, which man our prfrnc chang in a mporally inconin way a im go by, and w can hu viw hi problm a a non-coopraiv gam. W hn look for a ubgam prfc Nah quilibrium poin for hi gam. Th gam horical approach o addr gnral im inconincy via Nah quilibrium poin ha a long hiory aring wih [7] whr a drminiic Ramay problm wa udid. Furhr work along hi lin in coninuou and dicr im wr providd in [], [] and [5]. Rcnly hr ha bn rnwd inr in h problm. In h inring papr [9] and [], h auhor conidrd opimal conumpion and invmn undr hyprbolic dicouning in drminiic E-ISSN: - 3 olum, 5

2 and ochaic modl from h abov gam horical poin of viw. Thy providd a prci dfiniion of h gam horical quilibrium concp in coninuou im. Björk and Murgoci [3][] udid im inconincy in dicr and coninuou im) and h auhor undrak a dp udy of h problm wihin a Winr drivn framwork. Zng and Li [9] udid h diffuion rik modl undr man-varianc cririon and obaind opimal im-conin invmn and rinuranc polici. Zng al.[] invigad h opimal im-conin invmn problm in h claic rik modl. In h following papr, Björk al.[5] invigad man-varianc porfolio opimizaion wih a dpndn rik avrion. Bid, hr ar om ohr invmn problm undr diffrn modl, Li al.[], Chang and Lu [], Chang al.[7] and rfrnc hrin. Howvr, all of h abov rfrnc do no conidr opimal invmn problm in h dual rik modl undr man-varianc cririon. Zhang al.[] udid opimal invmn ragy in a dual rik modl undr man-varianc cririon and hy aumd ha h financial mark conid of a rikfr a and a riky a which pric wa modld by a diffuion proc wihou jump. In our papr, w aum h pric of h riky a ar dcribd by a diffuion proc wih jump. Shorlling and borrowing mony ar allowd. Du o lack of Bllman Opimaliy Principl, w xploi h gam horical approach o dal wih hi problm and h im-conin invmn ragi ar invigad in h claic diffuion dual modl and h diffuion dual approximaion modl. Similarly a Björk and Murgoci [3], w giv a ri of lmnary dfiniion and clod xprion for opimal im-conin invmn and h corrponding valu funcion ar drivd by olving h xndd Hamilon-Jacobi-Bllman quaion. Thi papr i organizd a follow. In Scion, w dcrib h claic diffuion dual rik modl and h diffuion dual approximaion modl by conidring h invmn ragy, dfin h quilibrium - ragy and giv h vrificaion horm for h dual rik modl. In Scion 3, opimal im-conin invmn ragy and h valu funcion ar drivd by olving h xndd HJB quaion in h claic diffuion dual modl. In Scion w giv opimal im-conin invmn ragy and h valu funcion in h diffuion dual approximaion modl. Th final Scion provid conomic implicaion and niiviy analyi for our rul by numrical mhod. Problm formulaion In hi cion, w ar wih h claic diffuion dual modl and h urplu proc of h company i givn by N ) R) = x + Z j c + σ W ), R) = x, ) j= whr x i h iniial capial, c i h ra of xpn, {N )} i a Poion proc wih inniy λ and Z j i h iz of h jh poiiv incom or profi. Th incom ar i.i.d.wih h fir and cond momn µ z and σ z, rpcivly. W ) i a - andard Brownian moion which dno h uncrainy of profi. Th xpcd incra of h urplu pr uni im aifi h poiiv loading condiion: λ µ z c >. Th company i allowd o inv i urplu in a financial mark coniing of a rik-fr a and n riky a. Th oal amoun of mony invd in h ih riky a a im i dcribd a l i ). Dno h pric of h rik-fr a S by ds ) = )S )d, S ) =, ) whr i h iniial pric of h rik-fr a, ) rprn h rik-fr ra and i i a poiiv coninuou boundd funcion. Th pric proc S i ) of h ih riky a i =,,..., n) aifi h following ochaic diffrnial quaion ds i ) = S i ) r i )d + d σ ij )dw j ) j= ) N ) +d Y j, S i ) = i, j= 3) whr r i )> ) i h apprciaion ra, r i ) and σ ij ) ar poiiv coninuou boundd funcion. W ) = W ), W ),..., W d )) T i a d- dimnional andard Brownian moion. Y j i h jh jump ampliud of h riky a pric and Y j, j =,,... ar i.i.d. random variabl wih h fini firordr momn µ y and cond-ordr momn σy. Furhrmor, aum r i ) + λ µ y > ) which man i i br o inv mony ino riky a in h long rm and W ), W ), N ) j= Z j and N ) j= Y j ar indpndn and P Y i ) = which can mak h riky a pric rmain poiiv. Hr h uprcrip T dno h ranpo of a marix or a vcor and d n. L X l ) dno h ruling urplu proc afr incorporaing ragy l ino ). E-ISSN: - olum, 5

3 Th dynamic of X l ) can b prrvd a follow dx l ) = )X l ) + r T )l) c ) d +σ dw ) + l T )σ)dw ) N ) +d j= Z j + l T N ) ) d j= Y j, ) whr r) = r ) ), r ) ),...,r n ) )) T, σ) = σ ij ) n d and =,,..., ) T. Furhrmor, dno Σ) = σ)σ T ) and l I rprn h idniy marix. W alo aum ha Σ) + λ σ yi i rvribl for all [, T ]. Similarly a Schmidli [], h diffuion approximaion of ) can b dcribd a dr) = λ µ z c)d + λ σ zdw ) +σ dw ), 5) whr W ) i a andard Brownian moion. Aum ha W ), W ), W ) and N ) j= Y j ar indpndn xcp ha W ) i corrlad wih W ) wih corrlaion cofficin ρ, i.., EW )W )) = ρ. ) Whn h company inv i mony ino h financial mark which i dcribd abov, h ruling urplu proc Y l ) can b givn by dy l ) = )Y l ) + r T )l) + λ µ z c) d + σ dw ) + l T )σ)dw ) + λ σ zdw ) + l T ) d N ) j= Y j, 7) Dno C, Q) = { ϕ, x) ϕ,.) i onc coninuouly diffrniabl on [, T], and ϕ., x) i wic coninuouly diffrniabl on R } whr Q := [, T ] R, hn for ϕ, x) C, Q), h infiniimal opraor of h urplu proc X l ) i A l ϕ, x) = ϕ, x) + ϕ x, x) [ )x c +r T )l) ] + ϕ xx, x) σ + lt )Σ)l) ) +λ [Eϕ, x + Z) ϕ, x)] +λ [Eϕ, x + n i= l i)y ) ϕ, x)], and h infiniimal opraor of h urplu proc Y l ) i B l ϕ, x) = ϕ, x) + ϕ x, x) [ )x + λ µ z c + r T )l) ] + ϕ xx, x) σ + ρσ o λ σz +l T )Σ)l) + λ σz ) +λ [Eϕ, x + n i= l i)y ) ϕ, x)]. ) 9) Nx, w giv a ri of dfiniion on h claic diffuion dual proc X l ). Dfiniion. A ragy l ={l) = l ),l ),..., l n ))} i aid o b admiibl if ) l) i a F -adapd proc; ) l aifi h ingrabiliy condiion: E lt )Σ)l)d < almo urly, for all ; 3) SDE) ha a uniqu oluion corrponding o l. Dno h of all h admiibl ragi by U. W mainly conidr man-varianc cririon and h objciv of h company i find h maximizaion of h following funcion ] J, x; l) = E,x [X l T ) γ a,x[x l T )], ) whr γ i a pr-pcifid rik avrion cofficin, E,x [.] = E[. X l ) = x] and a,x [.] = ar[. X l ) = x]. Thi problm i a aic manvarianc problm. Du o hi cririon lacking h irad-xpcaion propry, i lad o a iminconin problm. I man ha h Bllman Opimaliy Principl do no hold. W formula an invmn problm in a gam horical framwork. Furhrmor, w ak hi problm a a noncoopra gam, wih on playr for ach im, whr playr can b rgardd a h fuur incarnaion a im. For any fixd, x), h objciv i o find up l U { [ J, x; l) = up l U E,x X l T ) ] γ ar [,x X l T ) ]}. ) Thi problm can b viwd a a dynamic manvarianc problm. Similarly a Björk and Murgoci [3], w provid h dfiniion of h quilibrium ragy and h vrificaion horm for problm). Dfiniion. Equilibrium Sragy) For any fixd chon iniial a, x) Q, Conidr an admiibl ragy l, x). Choo wo fixd ral numbr l > and ε > and dfin h following ragy: { l, l ε for, x) [, + ε] R, x) = l, x), for, x) [ + ε, T ] R. If for all l R + and, x) Q, w hav lim inf ϵ J, x, l ) J, x, l ε ) ε, ) hn l, x) i calld an quilibrium ragy, and h corrponding quilibrium valu funcion i dfind by, x) = J, x, l [ ) = E,x X l T ) ] γ ar [,x X l T ) ] 3). E-ISSN: - 5 olum, 5

4 By dfiniion, w know ha h quilibrium - ragy i im-conin. So h quilibrium ragy l i calld opimal im-conin ragy for problm ). To olv problm ), w u ochaic analyi chniqu dcribd in [3] or [9] o driv h xndd Hamilon-Jacobi-BllmanHJB) quaion and h vrificaion horm. Thorm 3. rificaion Thorm) If hr xi - wo ral funcion W, x), h, x) C, Q), which aify h following xndd HJB quaion whr up l U { A l W, x) A l γ ) h, x) } + γh, x)a l h, x) =, ) W T, x) = x, 5) A l h, x) =, ) ht, x) = x, 7) l = arg up { A l W, x) A l γ h, x) ) +γh, x)a l h, x) }. ) Thn, x) = W, x), E,x X l T )) = h, x) and l i opimal im-conin ragy. Thorm 3 can b provd by h am procdur ad in [3] or [9],whil h only diffrnc in h proof i ha diffuion proc and jump diffuion proc hav h diffrn infiniimal gnraor. Rmark. A ri of dfiniion on Y l ) can b omid, for w can do i in h am way a in claic diffuion dual proc X. In h following wo cion, opimal imconin ragi and h corrponding valu funcion can b xplicily drivd in h claic diffuion dual modl and h diffuion dual approximaion modl, rpcivly. funcion in h claic dual modl wih diffuion. Nx, w will conruc h oluion o problm ). Aum ha hr xi wo ral funcion W, x) and h, x) aifying h condiion ad in Thorm 3. By viru of h infiniimal opraor ), ) can b rwrin a up l U { W, x) + W x, x) )x c +r T )l) ) + W xx, x) γh x, x)) σ + lt )Σ)l) ) + λ E [W, x + Z) γ h, x + Z) h, x + Z) h, x))] +λ E [ W, x + l)y ) γ h, x + Y n i= l i )) h, x + n i= l i)y ) h, x))] λ + λ ) [ W, x) + γ g, x) ] } = ) bcom h, x) + h x, x) )x c + r T )l ) ) + h xx, x) σ + lt )Σ)l ) ) +λ E [h, x + Z) h, x)] +λ E [h, x + n i= l i )Y ) h, x)] = 9) ) whr l i drmind blow. Sinc h linar rucur of 9) and ), and h boundary condiion of W, x) and h, x) givn by 5) and 7) ar linar in x, i i naural o conjcur ha W, x) = M)x + N), MT ) =, NT ) =, h, x) = m)x + n), mt ) =, nt ) =. Th corrponding parial drivaiv ar W, x) = Ṁ)x + Ṅ), W x, x) = M), h, x) = ṁ)x + ṅ), h x, x) = m), W xx, x) =, h xx, x) =. ) ) 3 Opimal im-conin ragy and i quilibrium valu funcion in h claic dual modl wih diffuion Thi cion udi opimal im-conin invmn ragy and h opimal quilibrium valu Inring )-) ino 9), i yild { up Ṁ)x + Ṅ) + M )x c l U +r T )l) + λ µ z + λ µ y l T ) ) γ m ) [ σ + λ σz) + l T )Σ)l) +λ σ yl T )l) ] } =. 3) E-ISSN: - olum, 5

5 Nx, w conruc a funcion Ll) = Ṁ)x + Ṅ) + M )x c +r T )l) + λ µ z + λ µ y l T ) ) γ m ) [ σ + λ σ z) + l T )Σ)l) +λ σ yl T )l) ]. ) Diffrniaing Ll) wih rpc o l and ing h drivaiv o zro, w g M) r) + λ µ y ) γm ) Σ) + λ σ yi ) l) =. 5) I follow from 5) ha Σ) + λ l σyi ) r) + λ µ y ) ) = γ M)). ) m ) Inring ) ino 9) and ), w hav Ṁ) + r )M)) x + Ṅ) cm) whr +λ µ z M) γ m )σ + λ σ z) + M )ξ) γm ) =, ṁ) + )m)) x + ṅ) cm) +λ µ z m) + M)ξ) γm) =, ξ) = r) + λ µ y ) T Σ) + λ σ yi ) r) + λ µ y ). To nur h abov quaion hold, w rquir 7) ) Ṁ) + )M) =, MT ) =, Ṅ) + λ µ z c) M γ m )σ + λ σ z) + M )ξ) γm ) =, NT ) =, ṁ) + )m) =, mt ) =, ṅ) + λ µ z c) m) + M)ξ) γm) =, nt ) =. Solving h ym of ordinary quaion, w hav M) = )d, N) = λ µ z c) γ σ + λ σ z) + γ ξ)d, m) = )d, n) = λ µ z c) + γ ξ)d. u)du d u)du d u)du d 9) Subiuing 9) ino ), w hav Σ) + λ l σyi ) r) + λ µ y ) ) = γ. 3) )d According o h argumn lid abov, w can driv h xplici xprion for W, x) and h, x) and h rul can b ummarizd a h following horm. Thorm 5. In h claic dual modl, opimal imconin ragy l i givn by 3), and h quilibrium valu funcion i givn by and, x) = )d x + λ µ z c) u)du d γ σ + λ σ z) u)du d + γ ξ)d E,x X l T )) = )d x + λ µ z c) u)du d + γ ξ)d. 3) 3) By Thorm 5 and h dfiniion of h corrponding valu funcion givn by 3), w hav ar,x X l T )) = γ [h, x), x)] = σ + λ σz) + γ ξ)d. u)du d 33) Rmark. I i ay o ha h opimal ragy do no dpnd on h walh proc X l ) and h paramr of h urplu proc hav no impac on h opimal ragy; Th rik avrion cofficin and h cofficin of financial mark dcid h opimal ragy oghr. Rmark 7. Th fficin fronir for problm ) a iniial a, x) can b drivd from 3) and 33). E,x X l T )) = )d x + λ µ z c) { T [ ru)du d + ar,x X l T )) σ + λ σz) ] T ru)du d ξ)d }. 3) Thi fficin fronir i no a raigh lin bu a hyprbola in h man-andard dviaion plan. E-ISSN: - 7 olum, 5

6 Opimal im-conin ragy and i quilibrium valu funcion in h dual diffuion approximaion modl In hi cion, opimal im-conin invmn ragy and h opimal quilibrium valu funcion can b drivd in h dual diffuion approximaion modl by h am way a ha in Scion 3. Dno h of all admiibl ragi for Y ) by U. Nx, w will conruc h oluion o hi problm. Aum ha hr xi wo ral funcion Q, x) and g, x) aify h Exndd HJB quaion which hav h am xprion a in Thorm 3 xcp h diffrn infiniimal opraor. up l U { Q, x) + Q x, x) )x + λ µ z c + r T )l) ) + Q xx, x) γgx, x)) ) σ + ρσ o λ σz + λ σz + l T )Σ)l) +λ E [ Q, x + l)y ) γ g, x + n i= Y l i )) g, x + n i= l i)y ) g, x) )] λ [ Q, x) + γ g, x) ]} =, 35) QT, x) = x. 3) g, x) + g x, x) )x + λ µ z c + r T )l) ) + g xx, x) ) σ + ρσ o λ σz + λ σz + l T )Σ)l) +λ E [g, x + n i= l i)y ) g, x)] =, 37) gt, x) = x. 3) whr l i drmind by h incoming quaion ). Sinc 35)-3) ar linar in x, w can conjcur ha Q, x) and g, x) hav h following rucur Q, x) = D)x + F ), DT ) =, F T ) =, g, x) = d)x + f), dt ) =, ft ) =. 39) By a impl calculaion, h corrponding parial drivaiv ar givn by Q, x) = Ḋ)x + F ), Q x, x) = D), g, x) = d)x + f), g x, x) = d), Q xx, x) =, g xx, x) =. ) By inring39)-) ino 35), 35) can b rducd o h following quaion { up Ḋ)x + F ) + D )x c l U +r T )l) + λ µ z + λ µ y l T ) ) [ ) γ d ) σ + ρσ o λ σz + λ σz +l T )Σ)l) + λ σ yl T )l) ] } =. ) Diffrniaing h funcion in h lf brack of ) wih rpc o l and ing h drivaiv o zro, w g l) = D) Σ) + λ σyi ) r) + λ µ y ) γd. ) ) Inring ) ino 35) and 37), w hav whr Ḋ) + r )D) ) x + F ) cd) + λ µ z D) + D )ξ) γd ) γ d )σ + ρσ o λ σ z + λ σ z) =, ) d) + )d) x + f) cd) +λ µ z d) + D)ξ) γd) =, ξ) = r) + λ µ y ) T Σ) + λ σ yi ) r) + λ µ y ). 3) ) 5) In ordr o nur h abov quaion hold, w rquir h funcion D), F ), d), f) aify h following quaion Ḋ) + )D) =, DT ) =, F ) + λ µ z c) D + D )ξ) γ γd ) d ) σ + ρσ o λ σ z + λ σ z) =, F T ) =, d) + )d) =, dt ) =, f) + λ µ z c) f) + D)ξ) γd) =, ft ) =. ) E-ISSN: - olum, 5

7 Solving h ym of quaion, w hav D) = )d, F ) = λ µ z c) γ σ + ρσ o λ σ z + λ σ z) u)du d + γ d) = )d, f) = λ µ z c) + γ ξ)d. u)du d ξ)d, u)du d 7) Subiuing 7) ino ), w hav Σ) + λ σyi ) r) + λ µ y ) l) = γ. ) )d Summarizing h rul dicud abov, h xplici xprion for Q, x) and g, x) can b givn by h following horm. Thorm. For h diffuion dual approximaion modl, opimal im-conin ragy l i givn by Σ) + λ σyi ) r) + λ µ y ) l) = γ. 9) )d and h quilibrium valu funcion i givn by and, x) = Q, x) = + λ µ z c) )d x u)du d γ σ + ρσ o λ σ z + λ σ z) u)du d + γ E,x X l T )) = g, x) = + λ µ z c) u)du d + γ ξ)d )d x ξ)d. 5) 5) Rmark 9. From 5) and 5), h rlaionhip bwn h xpcaion and h varianc of h rminal walh can b obaind a blow: E,x X l T )) = )d x + λ µ z c) σ + ρσ o { T [ ru)du d + ar,x X l T )) λ σz + λ σz) ξ)d }. u)du d ] 5) 5) how ha h fficin fronir i alo a hyprbola in h man-andard dviaion plan. Rmark. I follow from 3) and 9) ha opimal im-conin invmn ragi hav h am xprion in h claic diffuion dual modl and h diffuion dual approximaion modl. Whn W ) i indpndn wih W ), in ohr word ρ =, h valu funcion and h am fficin fronir ar h am in boh ca. Thu, w can only prn conomic implicaion and provid niiviy analyi for our rul in h dual diffuion approximaion modl. Rmark. A pcial ca i conidrd in ordr o analyz h ffc of h paramr on h imconin ragy and h quilibrium valu funcion. Aum ha h walh can only inv ino a rikfr a and a riky a whr h dimnion d of W ) qual o and all h ohr paramr ar all conan. Th opimal invmn ragy, h corrponding valu funcion and h fficin fronir ar givn by h following quaion r + λ µ y l) = γ ). T ) σ + λ σy 53), x) = T ) x + λ µ z c) T ) ) γσ +ρσ o λ σz +λ σz ) + ξ γ T ) and whr T ) ) E,x X l T )) = T ) x + λ µ z c) T ) ) + { [ ar,x X l T )) σ +ρσo λ σz+λ σz) T ) )] ξt ) }, 5) 55) r = r, ξ = r + λ µ y ) σ + λ σy. 5) 5 Numrical analyi In h nx ubcion, w udy h ffc of paramr on opimal im-conin ragy and h corrponding valu funcion in h dual diffuion approximaion modl and provid om numrical xampl o illura h ffc. For convninc bu wihou lo of gnraliy, aum ha d =, E-ISSN: - 9 olum, 5

8 n = and all h ohr paramr involvd ar conan. For h following numrical illura, unl ohrwi ad, h baic paramr ar givn by =., r =.5, c =., x =, γ =., ρ =.3, λ =, µ z =, σ z =., λ =., µ y =.5, σ y =, σ =.3, σ =., T =. Du o all h paramr ar all conan, opimal invmn ragy, h corrponding valu funcion and h fficin fronir ar givn by 53)- 55). 5. Analyi of h im-conin ragy In hi ubcion, w will work on numrical analyi of im-conin ragy in h dual diffuion approximaion modl. From 53), w can conclud ha ) γ = l γ < which illura ha opimal im-conin invmn ragy i dcraing wih rpc o γ, namly, h mor h company dilik rik, h l amoun h company inv in h riky a, Figur a) ) = +r+λ µ y)t ) < which rval ha opimal im-conin invmn ragy γ T ) σ +λ σy) i dcraing wih rpc o, namly, h mallr h rik-fr ra i, h mor amoun h company inv in h riky a, Figur b). 3) r = > which rval γ T ) σ +λ σy) opimal im-conin invmn ragy i incraing wih rpc o r, namly, whn h apprciaion ra r incra, h company hould inv mor mony in h riky a, Figur c). ) σ = l σ +λ σ y < which ll ha opimal im-conin invmn ragy i dcraing wih rpc o σ, namly, whn h volailiy of h riky a incra, h company hould inv mor mony in h rik-fr a, Figur d). λ 5) µ y = > which illura ha opimal im-conin invmn ragy γ T ) σ +λ σy) i incraing wih rpc o µ y, namly, h biggr h xpcaion of ach jump ampliud of h riky pric, h mor amoun h company inv in h riky a, Figur ). ) = λ l < which illura σy σ +λ σy) ha opimal im-conin invmn ragy i dcraing wih rpc o σy, namly, h biggr h cond-ordr momn of ach jump ampliud of h riky pric, h l amoun h company inv in h riky a, Figur f). µ yσ 7) λ = rσy which illura γ T ) σ +λ σy) ha opimal im-conin invmn ragy i dcraing incraing) wih rpc o λ whn µy < σy r σ µ y σ y > r σ ). For xampl, whn µ y σ y < r σ, h biggr h inniy of h jump of h riky pric, h l amoun h company inv in h riky a, Figur g). 5. Analyi of h quilibrium valu funcion In hi ubcion, w will work on numrical analyi of h valu funcion in h dual diffuion approximaion modl. Figur how ha how h cofficin involvd impac on h valu funcion. For convninc bu wihou lo of gnraliy, aum ρ > xcp h analyi of ρ and σ >. By 5) and om impl calculaion, w can hav h following finding: ) γ = σ +ρσo λ σz+λ σz) T ) ) ξ T ) < which illura ha h valu funcion i dcraing wih rpc o h cofficin γ rik avrion γ, namly, h largr rik avrion h company ha, h mallr h opimal man-varianc uilii i, Figur a). ) ha a complx xprion wih rpc o. Thu w only analyz i impac on h valu funcion from numrical mhod. Whn all h paramr xcp ar fixd, our numrical xampl rval whn h rik-fr ra incra, h valu funcion incra, Figur b). 3) r = r+λ µ y γσ +λ σy )T ) > which illura ha h valu funcion i incraing wih rpc o r, namly, h biggr h apprciaion ra i, h biggr h opimal man-varianc uilii i, Figur c). ) = r+λ µ y ) T ) < which rval σ γσ +λ σy) ha h valu funcion i dcraing wih rpc o σ, namly, h biggr h volailiy of h mark riky a i, h mallr h opimal man-varianc uilii i, Figur d). γ+ ρ = λ σz σ ) 5) σ T ) ) < which rval ha h valu funcion i dcraing wih rpc o σ, namly, whn h volailiy of h riky a incra, h opimal man-varianc uilii dcra, Figur ). σ z λ +σ z ) ) λ = µ z T ) ) γ ρσo T ) ) which illura ha h valu funcion i dcraing incraing) wih rpc o λ whn µ z T ) ) < γ ρσo σz +σ λ z) T ) ) E-ISSN: - olum, 5

9 .5 γ.5 a) Th ffc of γ on opimal im-conin ragy µ y ) Th ffc of µ y on opimal im-conin ragy b) Th ffc of on opimal im-conin ragy.5.5. σ y.. f) Th ffc of σ y on opimal im-conin ragy r.... c) Th ffc of r on opimal im-conin ragy λ g) Th ffc of λ on opimal im-conin ragy σ.. Figur : Th ffc of paramr on opimal imconin ragy d) Th ffc of σ on opimal im-conin ragy E-ISSN: - olum, 5

10 µz ρσo σ z λ +σ z) T ) ) ). T ) ) > γ For xampl, whn h valu funcion i incraing wih rpc o λ, h biggr h inniy of h jump of h riky pric bcom, h biggr h opimal man-varianc uilii bcom, Figur f). γ ρσo λ σ z +λ ) 7) = σz T ) ) < which illura ha h valu funcion i dcraing wih rpc o σz, namly, h mallr h cond momn of h poiiv incom i, h biggr h opimal man-varianc uilii bcom, Figur g). ) µ z = λ T ) ) > which how ha h valu funcion i incraing wih rpc o µ z, namly, h biggr h xpcaion of h poiiv incom, h biggr h opimal man-varianc uilii bcom, Figur h). 9) λ = µ yσ rσ y+λ µ y σ y)r+λ µ y )T ) γσ +λ σ y) which illura ha h valu funcion i dcraing incraing) wih rpc o λ whn λ > rσ y µ yσ µ y σ y λ < rσ y µ yσ ). For xampl, whn λ > µ y σ y rσy µ yσ µ y, h biggr h inniy of h jump of h σy riky pric, h biggr h opimal man-varianc u- ilii bcom, Figur i). ) µ y = λ r+λ µ y)t ) > which illura h valu funcion i incraing wih rpc o γσ +λ σy) µ y, namly, h biggr h xpcaion of ach jump ampliud of h riky pric i, h biggr h opimal man-varianc uilii bcom, Figur j). ) σ y = λ r+λ µ y) T ) γσ +λ σ y) < which how ha h valu funcion i incraing wih rpc o σy, namly, h biggr h cond-ordr momn of ach jump ampliud of h riky pric, h mallr h opimal man-varianc uilii bcom, Figur k). λ σ z ) ρ = γσ T ) ) < which rval ha h valu funcion i dcraing wih rpc o ρ, namly, whn ρ incra, h opimal man-varianc uilii dcra, Figur l) γ.. a) Th ffc of γ on h quilibrium valu funcion b) Th ffc of on h quilibrium valu funcion.5..5 r..5 c) Th ffc of r on h quilibrium valu funcion Concluion In hi papr, w inviga h dual modl wih diffuion including h claic dual modl and h d- ual approximaion modl. W ar concrnd on opimal im-conin invmn ragy undr manvarianc cririon. W aum ha compani can in-... σ. d) Th ffc of σ on h quilibrium valu funcion E-ISSN: - olum, 5

11 5 5.5 σ.5 ) Th ffc of σ on h quilibrium valu funcion... λ. i) Th ffc of λ on h quilibrium valu funcion.5 λ.5 f) Th ffc of λ on h quilibrium valu funcion... µ y. j) Th ffc of µ y on h quilibrium valu funcion σ z. g) Th ffc of σ z on h quilibrium valu funcion 5.. σ y.. k) Th ffc of σ y on h quilibrium valu funcion µ z h) Th ffc of µ z on h quilibrium valu funcion... ρ. l) Th ffc of ρ on h quilibrium valu funcion Figur : Th ffc of paramr on h quilibrium valu funcion E-ISSN: - 3 olum, 5

12 v ino a financial mark which ha a rik-fr a and n riky a. Shor-lling and borrowing mony ar allowd. Surpriingly opimal im-conin invmn ragy and opimal valu funcion hav h am xprion in boh ca whn ρ =. Howvr, in pracic, h company can no b allowd o borrow a h rik-fr ra o inv in riky a. An inring furhr rarch opic i o inviga h prcommid ragy whr Shor-lling and borrowing mony ar no allowd. I i ncary o compar h prcommid ragy wih im-conin ragy. Anohr inring furhr rarch opic i o inviga h opimal ragy for h dual modl wih rgim wiching. Thi i a challng o driv h opimal ragy and opimal valu funcion. Acknowldgmn: Thi rarch i uppord by h Naional Naural Scinc Foundaion of ChinaGran No.335,337) and h rarch projc of h ocial cinc and humaniy on Y- oung Fund of h miniry of Educaion Gran No.YJC97) Rfrnc: [] B. Avanzi, H. Grbr, E. Shiu, Opimal dividnd in h dual modl, Inuranc: Mahmaic and Economic. ol., No., 7, pp. 3. [] B. Avanzi and H. Grbr, Opimal Dividnd in h Dual Modl wih Diffuion, Ain Bullin. ol.3, No.,, pp [3] T. Björk and A. Murgoci, A gnral hory of Markovian im inconin ochaic conrol problm, Sockholm School of Economic,, Working Papr. [] T. Björk and A. Murgoci, A hory of Markovian im-inconin ochaic conrol in dicr im, Financ and Sochaic. ol., No.3,, pp [5] T. Björk, A. Murgoci and X. zhou, Manvarianc porfolio opimizaion wih a dpndn rik avrion, Mahmaical Financ. ol., No.,, pp.. [] H. Chang and J. Lu, Uiliy porfolio opimizaion wih liabiliy and mulipl riky a undr h xndd CIR modl, WSEAS Tranacion on Sym & Conrol. ol.9,, pp. 55. [7] H. Chang, X. Rong and H. Zhao, Opimal Invmn and Conumpion Dciion Undr h Ho-L Inr Ra Modl, WSEAS Tranacion on Mahmaic. ol., 3, pp [] H. Dai, Z. Liu and N. Luan, Opimal dividnd ragi in a dual modl wih capial injcion, Mah Mh Opr R. ol.7, No.,, p- p [9] I. Ekland and A. Lazrak, Bing riou abou non-commimn: ubgam prfc quilibrium in coninuou im,, Univriy of Briih Columbia. [] I. Ekland and T. Pirvu, Invmn and conumpion wihou commimn, Mahmaic and Financial Economic. ol., No.,, p- p. 57. [] P. Krull and A. Smih, Conumpion and aving dciion wih quaigomric dicouning, Economrica. ol.7, No., 3, pp [] D. Li, X. Rong and H. Zhao, Opimal invmn problm wih ax, dividnd and ranacion co undr h conan laiciy of varianc CE) modl, WSEAS Tranacion on Mahmaic. ol., No.3, 3, pp [3] H. Markowiz, Porfolio lcion, Journal of Financ. ol.7, No., 95, pp [] B. Plg and E. Mnahm, On h xinc of a conin cour of acion whn a ar changing, Rviw of Economic Sudi. ol., 973, pp. 39. [5] R. Pollak, Conin planning, Rviw of Economic Sudi. ol.35, 9, pp.. [] H. Schmidli, Sochaic conrol in inuranc,, Springr. [7] R. Sroz, Myopia and inconincy in dynamic uiliy maximizaion, Rviw of Economic Sudi. ol.3, 955, pp. 5. [] D. Yao, H. Yang and R. Wang, Opimal dividnd and capial injcion problm in h dual modl wih proporional and fixd ranacion co, Europan Journal of Opraional Rarch. ol., No.3,, pp [9] Y. Zng and Z. Li, Opimal im-conin invmn and rinuranc polici for manvarianc inurr, Inuranc: Mahmaic and Economic. ol.9, No.,, pp [] Y, Zng, Z. Li and Y. Lai, Tim-conin invmn and rinuranc ragi for manvarianc inurr wih jump, Inuranc: Mahmaic and Economic. ol.5, No.3, 3, p- p [] L, Zhang, X. Rong, and Z, Du, Prcommid invmn ragy vru im-conin invmn ragy for a dual rik modl, Dicr Dynamic in Naur and Sociy, ol.,, pp. 3 E-ISSN: - olum, 5

13 [] J. Zhu and H. Yang, Ruin probabilii of a dual Markov-modulad rik modl, Communicaion in Saiic-Thory and Mhod. ol.37, No.,, pp E-ISSN: - 5 olum, 5