Total Duration of Negative Surplus for a Diffusion Surplus Process with Stochastic Return on Investments

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1 Applid Mahmais hp://dxdoiog/1436/am Plishd Onlin Novm 1 (hp://wwwsirpog/jonal/am) Toal Daion of Ngaiv Spls fo a Diffsion Spls Poss wih Sohasi Rn on Invsmns Honglong Yo Chann Yin Shool of Mahmaial Sins Qf Nomal Univsiy Qf China mail: yin@mailqfndn yohonglong815@163om Rivd Ags 31 1; visd Spm 3 1; apd Oo 7 1 ABSTRACT In his pap w onsid a Bownian moion is modl wih sohasi n on invsmns Using h song Maov popy xploiing h iaion ida w div h Lapla-Siljs Tansfom (LST) of h oal daion of ngaiv spls In addiion wo xampls a also psn Kywods: Ngaiv Spls; Rin Poailiy; Lapla-Siljs Tansfom 1 Inodion Assm ha h insan sinss is dsid y h is poss U B 1 (11) H is h iniial apial; is h fixd a of pmim inom; B 1 x is a sad Bownian moion; is a onsan psning h diffsion volailiy Sppos ha h ins is allowd o invs in an ass o invsmn pofolio Following Palsn Gjssing [1] w modl h sohasi n as a Bownian moion wih posiiv dif Spifially h n on h invsmn gnaing poss is R B (1) wh a posiiv onsans In (1) is a fixd ins a; B is anoh sad Bownian moion indpndn of B1 x sing fo h nainy assoiad wih h n on invsmns a im L h is poss X dno h spls of h ins a im nd his invsmns assmpion Ths X assoiad wih (11) (1) is hn h solion of h following lina sohasi ingal qaion: d X U X s R s (13) By Palsn [] h solion of (13) is givn y wh 1 d X R R s U s (14) R B No ha X is a homognos song Maov poss s g Palsn Gjssing [1] Th is poss (14) an win as d db1 s X sds X X s B s Bas h qadai vaiaional posss of X sdbs db1s X s db s a h sam wh B is a sad Bownian moion y Ida Waana [3 p 185] hy hav h sam disiion Ths in disiion w hav d X sd s X X s B s (15) Th a many paps onning opaion ims fo diffn is modls Fo xampl fo h lassial spls poss wih posiiv safy loading gdio dos Ris [4] divd h momn gnaing fnion of h oal daion of h ngaiv spls y maingal mhods whih was xndd in Zhang W [5] o h lassial spls poss pd y diffsion Chi Yin [6] divd xplii fomla fo h dol Lapla-Siljs Tansfom (LST) of h opaion im in h xponnial as fo h ompond Poisson modl Copyigh 1 SiRs

2 H L YOU C C YIN 1675 wih a onsan ins H al [7] givd h LST of h oal daion of ngaiv spls fo h lassial is modl wih di ins Mo nly Wang H [8] onsidd h Bownian moion is modl wih ins divd h LST of oal daion of ngaiv spls In his pap w onsid a Bownian moion is modl wih sohasi n on invsmns W will s h iaion ida o oain h LST of oal daion of ngaiv spls Th maind of h pap is oganizd as follows In Sion w giv som pinay sls In Sion 3 y xploiing h iaion ida ogh wih h sls oaind in Sion w oain h LST of h oal daion of ngaiv spls In h las sion w psn wo xampls Pinay Rsls Givn a wh T inf X a a dfin if h s is mpy T a T inf X if h s is mpy T T inf X if h s is mpy T PT X P T Lmma 1 Th is poss (15) has h song Maov popy: fo any fini sopping im T h gla ondiional xpaion of X T givn F T is X T X T ha is X T F X T X T as T wh F T is h infomaion ao h poss p o im T h qaliy holds almos sly Lmma Fo h following odinay diffnial qaion x f x x f x f x (1) has wo indpndn solions wh 1 K x 1 x x K d () 1 x x K d (3) x xp aan Poof Fom xampl of Palsn Gjssing [1] w g h sl Lmma 3 Fo a J a J T T dfin hn a J a I X Ja J a I X J a aa a a a a a wh a givn y () (3) Poof Th sl an fond in Chap 16 of Biman [9] a hn Lmma 4 Fo any S P T Ta S wh aion z s x ds s a S a S (4) S x d z is a solion of h q- x f x Poof By Dynin s fomla S x f x TTa S X T Ta LS X s ds wh L is h gnao of diffsion (15) I fo llows ha s X LS X s S X s X s S X s Copyigh 1 SiRs

3 1676 H L YOU C C YIN Thfo S X T Ta S Sin T Ta is fini i as vals T a wih poailiy P Ta < T T wih h ompnay poailiy Ling w an ass y dominad onvgn ha S S X T T a X S xping h xpaion on h lf w hav S a I S I S < T T T < T a a This ogh wih P Ta < T P T < Ta 1 givs h sl (4) Lmma 5 Fo h in poailiy fo h is modl (15) is givn y P T < (5) Th poailiy ha h spls poss hi h lvl is givn y wh P T < z s ds s h z X (6) Poof By Lmma 4 on an div (5) (6) 3 Toal Daion of Ngaiv Spls In his sion w will div h main sl of his pa p W assm ha h is poss (15) dos no aain h iial lvl Fo onvnin w assm ha h iniial spls is posiiv L h oal daion of ngaiv spls I < X < = d Fo < < dfin wo sqns of sopping ims of h poss (15): 1 inf > X ( 1 if h s is mpy) 1 inf 1 X X s 1 s ( 1 if h s is mpy) in gnal fo 3 sivly dfin inf > 1 X ( if h s is mpy) inf X X s s ( if h s is mpy) L 13 Givn fo som 1 fom h song Maov popy of h spls poss w oain ha h piods i i 13 a mally indpndn hav a ommon disiion L N dno h nm of S T N i1 i hom w hav By h monoon onvgn T Fis w giv h xpssion fo T in i (31) h following Thom 3 1 Thom 31 Fo h LST of T is givn y T (3) wh a givn y Lmmas 3 5 Poof Fom Lmma 1 w an g T I T T T I T T (33) (34) Fom song Maov popy of h spls poss w g Copyigh 1 SiRs

4 H L YOU C C YIN 1677 i T i1 P N IN i i1 I X This ogh wih (33) (34) givs (3) Thom 3 Fo h LST of oal daion of ngaiv spls is givn y 1 1 h z d z (35) wh hz a givn y Lmmas 3 5 Poof I follows fom (31) (3) ha 1 T Fom Lmma 5 i follows ha 1 1 hzdz 1 hzdz hzdz 1 hzdz hzdz h z By Ł Hospial s l w g dz (36) (37) h h h zdz 1 hzdz This ogh wih (36) (37) givs (35) 4 xampls In his sion w onsid wo xampls xampl 41 Ling in (15) w g h is modl d d (41) X X s s B s Fom Cai al [1] w now ha h wo indpndn solions of h diffnial qaion a x f x x f x f x x 3 x (4) xxp M 1 Copyigh 1 SiRs

5 1678 H L YOU C C YIN x 1 1 x x U xp (43) wh M U a alld h onfln hypgomi fnions of h fis sond ind spivly Mo dail on onfln hypgomi fnions an fond in Aamowiz Sgn [11] By Lmmas 3 5 w g T T I T < T (44) wh d h z z (45) (46) z z h z Aoding o Thoms 31 3 w g T wh (47) (48) hzdz a givn y (4) (45) a nd (4 6) Rma 41 Th sls (47) (48) oinid wih h main sls in Wang H [7] xampl 4 Ling in (15) w g h is modl X B (49) I is asy o oain ha h wo indpndn solions of h odinay diffnial qaion a f x f x f x x x x x By Lmmas 3 5 w g T I T < h z z Aoding o Thoms 31 3 w hav T =1 5 Conlsion In his pap w hav sdid h diffsion modl inopoaing sohasi n on invsmns W find h LST of h oal daion of ngaiv spls of his poss Howv if h is modl (11) is xndd o a ompond Poisson spls poss pd y a diffsion i is diffil o ma o W lav his polm fo fh sah RFRNCS [1] J Palsn H K Gjssing Opimal Choi of Dividnd Bais fo a Ris Poss wih Sohasi Rn on Invsmns Insan: Mahmais onomis Vol No pp 15-3 doi:1116/s (97)11-5 [] J Palsn Ris Thoy in a Sohasi onomi nvionmn Sohasi Posss Thi Appliaions Vol 46 No 1993 pp doi:1116/ (93)91- [3] N Ida S Waana Sohasi Diffnial qaions Diffsion Posss Noh-Holl Plishing Company Amsdam 1981 Copyigh 1 SiRs

6 H L YOU C C YIN 1679 [4] A D gdio dos Ris How Long Is h Spls low Zo? Insan: Mahmais onomis Vol 1 No pp 3-38 doi:1116/ (93) [5] C S Zhang R W Toal Daion of Ngaiv Spls fo h Compond Poisson Pos s Tha Is Pd y Diffsion Insan: Mahmais onomis Vol 39 No 3 pp [6] S N Chi C C Yin On Opaion Tims fo a Ris Poss wih Rsv-Dpndn Pmim Sohasi Modls Vol 18 No 1 p p doi:1181/stm [7] J M H R W H Y Zhang Toal Daion of Ngaiv Spls fo h Ris Modl wih Di Ins Saisis Poailiy Ls Vol 79 No 1 9 pp doi:1116/jspl95 [8] W Wang J M H Toal Daion of Ngaiv Spls fo a Bownian Moion Ris Modl wih Ins Aa Mahmaia Sinia 1 (Smid) [9] L Biman Poailiy Addison-Wsly Rading 1968 [1] J Cai H U G H L Yang Opimal Dividnds in an Onsin-Uhln Typ Modl wih Cdi Di Ins Noh Amian Aaial Jonal Vol 1 No 6 pp [11] M Aamowiz I A Sgn Hoo of Mahmaial Fnions: Wih Fomlas Gaphs Mahmaial Tals Unid Sas Dpamn of Comm US Govnmn Pining Offi Washingon DC 197 Copyigh 1 SiRs