OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL WITH DIFFUSION ABSTRACT KEYWORDS

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1 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL WITH DIFFUSION BY BENJAMIN AVANZI, JONATHAN SHEN, BERNARD WONG ABSTRACT Th ual mol wih iffusion is aroria for comanis wih coninuous xnss ha ar offs by sochasic an irrgular gains. Examls inclu rsarch-bas or commission-bas comanis. In his conx, Avanzi an Grbr (28) show how o rmin h xc rsn valu of ivins, if a barrir sragy is follow. In his ar, w furhr inclu caial injcions an allow for (roorional) ransacion coss boh on ivins an caial injcions. W rmin h oimal ivin an (unconsrain) caial injcion sragy (among all ossibl sragis) whn jums ar hyrxonnial. This sragy hans o b ihr a ivin barrir sragy wihou caial injcions, or anohr ivin barrir sragy wih forc injcions whn h surlus is null o rvn ruin. Th lar is also shown o b h oimal ivin an caial injcion sragy, if ruin is no allow o occur. Boh h choic o injc caial or no an h lvl of h oimal barrir n on h aramrs of h mol. In all cass, w rmin h oimal ivin barrir an show is xisnc an uniqunss. W also rovi clos form rrsnaions of h valu funcions whn h oimal sragy is ali. Rsuls ar illusra. KEYWORDS Dual mol, iffusion, ivins, caial injcions, HJB quaion Th sabiliy roblm 1. INTRODUCTION Wha cisions shoul a comany ma in orr o nsur sabl oraions? Criria ha ar us in h acuarial liraur o arss his sabiliy roblm (s, for insanc, Bühlmann, 197) inclu h robabiliy of ruin (s Asmussn an Albrchr, 21, for an xclln broa rfrnc) an h Asin Bullin 41(2), oi: /AST by Asin Bullin. All righs rsrv.

2 612 B. AVANZI, J. SHEN AND B. WONG xc rsn valu of ivins (as inrouc by Fini, 1957). Mor rcnly, som auhors inrouc caial injcions an roos o maximis h xc rsn valu of h iffrnc bwn ivins an caial injcions. Th xc rsn valu of ivins as an alrnaiv o h robabiliy of ruin was firs roos by Fini (1957). If a comany mas cisions so ha h robabiliy of ruin is minimis, hn i is imlici ha i shoul l is surlus grow o h infiniy. As his bhaviour is arguably unralisic, Fini (1957), in his mol, allow som surlus o b isribu. Ths laags ar lily o bnfi h comany s ownrs, hnc xlaining hir qualificaion of ivins. Usually, h way hs ar isribu (h ivin sragy ) is rmin such ha h xc rsn valu of ivins is maximis; s Albrchr an Thonhausr (29) an Avanzi (29) for rviws of h rla liraur. Th im valu of mony rovis an incniv o isribu ivins arlir an mor ofn. Whn hs ar maximis, ruin is usually crain. In som cass, i may b rofiabl (or rquir) o rscu h comany by injcing som caial. Irrsciv of ruin, injcing caial may hav a osiiv n rsn valu. This ia gos bac o Borch (1974, Char 2) an Porus (1977), an rcn rfrncs on caial injcions inclu Avram al. (27) for scrally ngaiv rocsss, Løa an Zrvos (28) an H an Liang (28) in h Brownian ris mol, Yao al. (21) in h ual mol, Dai al. (21) in h ual mol wih iffusion. In h cas of h Cramér- Lunbrg mol wihou iffusion, Kulno an Schmili (28) rovi a roof of h oimaliy of a barrir sragy unr gnral jum isribuions whn caial injcions ar forc (ha is, whn ruin is no allow o occur). I is worhwhil noing ha h broar issu is rlvan o ohr fils as wll, such as corora financ. In hir xclln rviw of h liraur on ivin ayou olicy, Alln an Michaly (23, Char 7) sa: W bliv ha [ ] how ayou olicy inracs wih caial-srucur cisions (such as b an quiy issuanc) ar imoran qusions an a romising fil for furhr rsarch. In his ar, w ar inrs in rmining h join oimal ivin an caial injcion sragy in h ual mol wih iffusion as scrib in h nx scion Th ual mol wih iffusion W consir h ual mol wih iffusion. In his mol, h comany surlus a im is scrib as U ( ) = x- c+ S( ) +s W( ), $, (1.1) whr U( ) = x $ is h iniial surlus, c > is h xns ra r uni of im an whr {S()} is a comoun Poisson rocss wih innsiy l. Th

3 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 613 rocss {W()} is a sanar Brownian moion which is innn of {S()}, wih volailiy of s r uni of im. Such a mol is aroria for comanis wih sochasic gains an rminisic xnss, such as rsarch-bas comanis ha vlo invnions or ans. Such comanis ma iscovris a ranom ims, an can crysallis h gain by slling h associa inllcual rory o a buyr, or rquiring an licnc fs from firms using h chnology (s, for insanc, Sharma an Clar, 28). Ohr xamls inclu commission-bas firms such as ral sa agns. Th Brownian moion rm rflcs aiional uncrainy in h firm s xnss an gains. Th ual ris mol was firs nam so by Mazza an Rullièr (24) bcaus of is ualiy o h Cramér-Lunbrg mol. Wihou iffusion, Avanzi al. (27) an Chung an Dric (28) rovi rsuls whn a ivin barrir sragy is ali, whras Ng (29) consirs hrshol sragis. Mol (1.1) is ual o h Cramér-Lunbrg mol wih iffusion as inrouc by Dufrsn an Grbr (1991). In his framwor, rsuls abou ivins wih a barrir sragy ar riv in Avanzi an Grbr (28). W will assum ha h isribuion P of h jums in {S()} is a mixur of xonnials, namly: P( y) n -bi y = y ( ) = / w i b i, for y >, (1.2) y wih n i = 1 / wi = 1, wi > for all i, an < b1 < b2 < f < b n < 3. (1.3) i = 1 Mixurs of xonnials can b us o aroxima crain long-ail isribuions such as h Paro an Wibull. In h cas of comlly monoon robabiliy isribuion funcions, algorihms ar raily availabl (s, for insanc, Flmann an Whi, 1998). Th broar class of combinaions of xonnials (for which w i > is no mor rquir) is also usful o aroxima robabiliy isribuions (s, for insanc, Dufrsn, 27). Alhough h oimaliy rsuls of his ar o no xn o combinaions, h clos form soluions for h valu funcions ar sill vali unr mil assumions (s also Rmar 2.1). Furhrmor, no ha (1.2) can b inrr in h following way. If a rsarch an vlomn firm has n iffrn armns, ach wih gains isribuion bing xonnial wih aramr b i, xnss w i c, an iniial invsmn w i x (i = 1,, n), hn (1.1) rrsns is global surlus (bcaus of h roris of comoun Poisson rocsss); s also Rmar Formulaion of h gnral oimal conrol roblm In his ar, w consir wo ys of conrols: ivin aymns (surlus ouflows) an quiy issuanc (surlus inflows). W assum ha a coml

4 614 B. AVANZI, J. SHEN AND B. WONG filr robabiliy sac (W, F, {F } $, P) is givn, such ha {U()} is aa. Th conroll surlus rocss is X( ) = U( ) - D( ) + E ( ), $. (1.4) Hr, {D ()} rrsns h aggrga ivins isribu u unil im, accoring o sragy. A ivin sragy is sai o b amissibl if {D ()} is a non-crasing, {F }-aa rocss wih D ( ) =. W assum ha {D ()} has càlàg saml ahs. In aiion, w rsric h ossibl conrol rocsss so ha a firm canno ay ou an amoun of ivins ha is largr han h currn surlus. Tha is, DD () X ( -) for all, (1.5) whr D D ( ) = D ( ) -D ( -) (1.6) rrsns h siz of h ivin ai a im. On h ohr han, {E ()} rrsns h aggrga caial injc u unil im. W assum ha {E ()} has càlàg saml ahs. A caial injcion sragy is amissibl if {E ()} is a non-crasing, {F }-aa rocss wih E ( ) =. An amissibl join conrol sragy is hn no by = (D, E ), an h s of amissibl conrol sragis is no by P so ha! P. Our objciv is o rmin h oimal conrol sragy ha maximiss h xc rsn valu of ivins lss caial injcions unil ruin, which w fin o b (1.7) x - lim su / / - cj D - " 3 Jx ( ; ) : = E < ( s) E ( s) mf, whr is h im of ruin, a / b nos h minimum of a an b, an whr E x is h coniional xcaion givn h iniial surlus x. W assum ha ivins ar ai ou of h surlus o h sam grou of invsors ha injc caial ino h surlus, an h forc of inrs > rflcs h im rfrnc of hos invsors. Proorional coss on ivin ransacions ar an ino accoun hrough h valu of j, wih < j 1 rrsning h n roorion of laags from h surlus rciv by invsors afr ransacion coss hav bn ai. Proorional ransacion coss on caial injcions ar an ino accoun hrough h valu of, wih 1 < 3 rrsning h oal coss of injcing a singl ollar of caial, whr hs ar fin o b h amoun of caial injc, lus any ransacion coss rquir o injc his caial. Givn iniial caial x $, w fin h valu of h oimal sragy o b V( x ; ) : = su J( x ; ). (1.8)! P

5 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 615 I follows from rsuls in h iscr-im sing of Miyasawa (1962) an Tauchi (1962) ha h barrir sragy shoul b h oimal ivin sragy in h ual mol, alhough i has y o b formally rovn. In h cas whr h ual mol is rurb by a iffusion rm, Bayraar an Egami (28, wihou caial injcions) an Dai al. (21) rov ha h barrir sragy is oimal if h gains isribuion is xonnial an has a fini righ noin, rscivly. No ha som ars forc caial injcions whn h surlus is null o rvn ruin. Such a comulsion may b jusifi by sricly ngaiv surlus a ruin (bcaus of ownwars jums) or by rgulaion (in h cas of insuranc comanis). Ths rasons ar lss rlvan in h ual mol, which givs us grouns for allowing any caial injcion sragy as abov Srucur of h ar In orr o solv h gnral oimal conrol roblm as scrib abov, w n o consir wo sub-roblms firs. Scion 2 rsrics h roblm o ivins only an shows ha a barrir sragy is oimal, whhr h rif of (1.1) is osiiv or no. Furhrmor, a clos form rrsnaion of h valu funcion is vlo, which i no aar in Avanzi an Grbr (28). In Scion 3, caial injcions ar forc whn h surlus his o rvn ruin. Again, i is shown ha a ivin barrir sragy is oimal irrsciv of h rif of (1.1), an a clos form rrsnaion for h valu funcion is givn. Th oimal join sragy as wll as a clos form for (1.8) ar vlo in Scion 4. Th soluion of h roblm is a combinaion of h wo subroblms abov. Whras h barrir sragy is always oimal for ivins, h cision whhr caial shoul b injc or no an h lvl of h oimal barrir n on h aramrs of h mol. This gnral soluion is illusra in Scion OPTIMALITY OF THE BARRIER WITHOUT CAPITAL INJECTIONS W firs xamin h oimal ivin roblm wihou quiy issuanc, such ha E () / for all. This is a scial cas of (1.4), whr X ( ) = U( ) - D ( ),. (2.1) $ An amissibl conrol sragy is hn no by = (D, E ), such ha! P. Th im of ruin for such a sragy is fin as : = inf{ : X ( -) }, (2.2) = bcaus of iffusion an bcaus h surlus rocss is scrally osiiv.

6 616 B. AVANZI, J. SHEN AND B. WONG Our objciv is o rmin h oimal conrol sragy ha maximiss h xc rsn valu of ivins unil ruin, which w fin o b x - - Jx ( ; ) : = E 9j D ( ) C. (2.3) Hr h ur limi of h ingral is o rflc h fac ha in gnral, X()! X( ) u o h ossibiliy of a jum in h comoun Poisson rocss. Givn iniial caial x >, w consir h xc rsn valu of ivins unr h oimal sragy, no by V( x = su J( x! P ; ): ; ) (2.4) whr h s of amissibl sragis is P := { = (D, E )! P}. W will inify h form of h valu funcion V(x; ) an h oimal sragy Hamilon-Jacobi-Bllman (HJB) quaion Suos ha for a givn lvl of iniial surlus x $, h valu funcion unr is no by G(x). Accoring o h Hamilon-Jacobi-Bllman (HJB) quaion for his roblm, if h valu funcion G is wic coninuously iffrniabl hn w xc i o saisfy max" ( A - ) Gx ( ), j - G (x), = wih G() =, (2.5) whr h oraor A is h infinisimal gnraor A f( x) = s f (x) - cf (x) - l f( x) + l f( x+ y) P( y). (2.6) 2 Th HJB (2.5) can b obain from h following hurisic argumn. Consir h small im inrval (, ). Suos ha on his im inrval, w follow an arbirary sragy whrby surlus is rlas a a ra l $ o covr ivin isribuion lus ransacion coss, an hrafr, an oimal sragy is ali. By coniioning on h numbr of jums ha occur, h siz of h jum if i os occur, an h valu of W(), w s ha h xc rsn valu of ivins unil ruin unr his sragy is (by Taylor xansions) lj + ( 1 -) $ ( 1 - l) E7G ( x- ( c+ l) + sw( )) A 3 + l E7G ( x + y -( c + l) + sw( )) AP( y) + o( ) = Gx ( ) + l6j -G ( (2.7) s G ( x) -cg (x) -( l+ ) G( x) + l G ( x+ y) P( y)} + o( ). 2 (2.8)

7 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 617 Sinc G(x) is h oimal valu, is valu mus b grar han or qual o h valu of quaion (2.8). Thus, i follows ha h xrssion in bracs mus hav maximal valu of zro, suggsing max l6 j - G ( x )@ + ( A - ) G(x) - =. (2.9) l $ No ha if G (x) < j w can ma h firs ar of (2.9) unboun by ling l n o infiniy, so w mus rsric h firs rivaiv o G (x) $ j. (2.1) Convrsly, whn G (x) $ j, h firs ar of (2.9) is lss han or qual o zro for any l $. Now sinc (2.9) hols whn l =, w mus hav ( A - ) Gx ( ). (2.11) Sinc w allow h iniial surlus x $ o b arbirary, (2.1) an (2.11) mus hol for any x $. Thus, w can rwri (2.9) by sliing i ino wo ars, as givn in h HJB quaion (2.5). Th bounary coniion G() = hols bcaus if h iniial surlus is zro, hn by finiion h firm is immialy ruin Consrucion of a cania soluion W conjcur ha h barrir sragy is oimal. L x 9 b - - j b ( C (2.12) - Gx ( ) = E D ) no h xc rsn valu of h ivins isribu unil ruin using a barrir sragy wih lvl b, givn an iniial surlus of x. I follows from h rsuls in Avanzi a n Grbr (28) ha G(x) saisfis h ingro-iffrnial quaion (IDE) s G ( x) - cg ( x) -( l + ) G( x) + l G ( x+ y) P( y) =, x b, 2 (2.13) laing o G( xb ; ) x! [, b ]; an Gx ( ) : = (2.14) j( x - b ) + G( b ; b ) x! ( b, ); 3 whr w fin ( ; b = C b = r x $ Gx ) : / ( ), for x, (2.15)

8 618 B. AVANZI, J. SHEN AND B. WONG an whr h r s ar h roos of h characrisic quaion f( bi z) = s z -cz-( l + ) + l/. 2 b - z = (2.16) n w i i = 1 I is asy o show ha h r s saisfy h following inrwaving roo coniion: r < < r < b < < r < b < r. (2.17) 1 1 f n n n+ 1 Th oimal barrir b an h associa n + 2 cofficins C (b ) ar h soluion of h following n + 3 quaions: ( = G( ; b ) = / C b ) =, (2.18) ( b ; ) C( b = rb G - b = / r ) = j, (2.19) G 2 ( b ; b r C ( b rb - ) = / ) =, an (2.2) = b ir r b ( i- r / C b ) = j, for i = 12,, f, n. (2.21) b = i Coniions (2.18) an (2.2) ar quivaln o Coniions (3.6) an (5.3) of Avanzi a n Grbr (28), rscivly, an Coniions (2.19) an (2.21) ar analogous o Coniions (3.7) an (3.5) of Avanzi an Grbr (28), rscivly, wih h incororaion of h ransacion coss j. Th lar ar riv using a similar aroach. No ha Coniions (2.18), (2.19) an (2.21) hol for any lvl of barrir b, whras (2.2) is h coniion for h oimal barrir b only. Rmar 2.1. Whn h cofficins w i in (1.2) ar allow o b ngaiv, ha is, whn jums ar isribu accoring o a combinaion of xonnials, Coniion (2.17) crucial for oimaliy os no ncssarily hol any mor. Nvrhlss, clos form xrssions for h valu funcions, hroughou h ar, hol as long as all r ar ral an isinc Exlici form of h valu funcion In his scion, w focus on h oimal sragy b an firs solv quaions (2.19)-(2.21) o g a clos form rrsnaion for h C (b ) s. W hn show ha (2.18) las o a uniqu oimal barrir b >, an ha his on xiss if an only if h rif of h rocss {U()}, n wi m : = E 7 U( + 1) - U() A = l / - c, $, (2.22) b i = 1 i

9 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 619 is sricly osiiv. If m, h oimal barrir is null; his is iscuss in Scion Drmining h C (b ) cofficins W sar by fining h raional funcion Q: Q( z) 2 rb r C = ( b ) : /. (2.23) z - r = Th objciv in his scion is o fin an quivaln rrsnaion of Q, an o us h fac ha 2 - Q z ( ) lim ( z r ) ( ) = r C b for = 1,2, f, n (2.24) z " r rb o rmin h C (b ) cofficins. W obsrv ha Q saisfis h following roris: (P1) By facorising h nominaor of (2.23), w s ha Q is a raional funcion wih h nominaor bing a olynomial of gr n + 2. Th numraor is a olynomial of gr n sinc h cofficin of z is zro u o (2.2). (P2) Is ols ar r, r 1, r 2,, r n, r ; (P3) Q() = j u o (2.19); (P4) Q(b i ) = for i = 1, 2,, n, by facorising h iffrnc bwn (2.21) an (2.19). Th four oins (P1)-(P4) uniquly rmin Q. (P1) an (P2) giv us h form of h nominaor, an hs can b combin wih (P3) an (P4) o rmin h form of h numraor. Hnc, w can wri Q( z) =-j % r % j j = i = 1 % j = n z - bi b ( z - r ) j i. (2.25) Alying (2.24) w fin r ( j - b j C b ) r n r- bi =- r r - r for =, 1, f, n+ 1. % % (2.26) j b j = i = 1 i j!

10 62 B. AVANZI, J. SHEN AND B. WONG Bcaus of (2.17), for all b $, C b ), C b ), an (2.27) ( < lim ( =-3 b " 3 C ( b ) >, lim C ( b ) =, = 1,2, f,. (2.28) b " 3 As a rsul, (2.15) wih h oimal barrir b, can now b xlicily wrin as -rb G( x ; b ) =-j r n j r- bi rx : / r % r - r %, x $, (2.29) = j = j! j b i = 1 i whr b is rmin by coniion (2.18), which can now b rwrin as -j 1 -rb + r b / r. (2.3) n = n j r - i % r - r % = j j b = i = 1 i j! Subsiuing (2.14), (2.19) an (2.2) ino h IDE (2.13) wih x = b yils G( ; b ) jm = (2.31) b which is h rsn valu of a ruiy of jm using forc of inrs. Rmar 2.2. From (2.29) w can s ha h inclusion of roorional ransacion coss on h ivins hrough j simly scals h siz of h valu funcion. A hurisic argumn for his rory is as follows: suos ha hr ar no ransacion coss an h oimal barrir is b. Thn inrouc roorional ransacion coss on ivins. Th inroucion of h coss os no affc h surlus rocss, sinc whnvr ivins ar ai ou, h sam amoun is rmov from h surlus, bu h invsors simly rciv lss ivins. Thus, i is sill oimal o us h sam barrir b. Howvr, sinc only j of ach ollar is isribu as ivins, h valu funcion is scal by j. In ligh of his rmar, w no ha quaion (2.31) is an ua vrsion of h analogous formulas from Grbr (1972), Avanzi al. (27) an Avanzi an Grbr (28), who foun ha in h absnc of ransacion coss on ivins, G( b ; b m ) = in h Brownian ris mol, ual mol an ual mol wih iffusion, rscivly. No ha i can b shown ha m,, h r an h b i saisfy h following lgan rlaionshi,

11 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 621 m = = r 1 - b 1. n / / (2.32) i = 1 i which os no sm o hav any aricular inrraion. I is rmarabl ha h wighs w i o no aar on h righ-han si. Rmar 2.3. Th aroach of fining a raional funcion an fining an quivaln rrsnaion o rmin h form of h C s was us in Scion 6 of Dufrsn an Grbr (1991) an Scion 4 of Albrchr al. (21) o solv roblms on ruin robabiliis an h iscoun naly funcion rscivly. Rmar 2.4. I shoul b no ha h C (b ) riv hr is a gnral form which alis o ohr roblms in h ual mol wih iffusion, rovi ha h gains isribuion is a mixur of xonnials, G (b ; b ) = j an G (b ; b ) =. This fac will b us in Scion 3 (wih caial injcions), which uss a iffrn bounary coniion Exisnc an uniqunss of b L us firs fin ): = ( ), = x( b / C b b $, (2.33) such ha (2.18) is quivaln o x ( b ) =. (2.34) Th roblm is now o show ha x(b ) has a uniqu roo. W firs no ha = - = x ( b ) / r C ( b ) <, (2.35) bcaus r an C (b ) hav h sam sign for all ; s (2.17), (2.27) an (2.28). Hnc, x is a crasing funcion in b. Sinc () (b ; jm x = G b ) = (2.36) an lim x ( b ) =-3 (2.37) b " 3 bcaus of (2.27), (2.28) an h coninuiy of x, i follows ha (2.3) has a uniqu osiiv soluion ha xiss if an only if m >. This also shows ha h oimal barrir b is innn of h iniial surlus x.

12 622 B. AVANZI, J. SHEN AND B. WONG 2.4. Vrificaion of all h coniions of h HJB quaion By consrucion, our cania soluion saisfis G (x) = j for x! [ b, 3), (A )G(x) = for x! [, b ] an h bounary coniion G() =. Furhrmor, ( ) G( x) 1 2 A - = s G ( x) -cg (x)-( l+ ) G( x) + l 2 = -cj-( l + ) 6j( x - b ) + G( b ; b )@ 3 + l 6j( x + y - b ) + G( b ; b )@ P( y) =-j( x -b ) <, x > b. 3 G ( x + y) P( y) (2.38) whr w hav us (2.31) o go from h scon o h las lin. Hnc, i only rmains o show ha G (x) $ j, x b. (2.39) Bcaus G (b ; b ) = an 3 rx (x = ( > = G ; b ) / r C b ), x b, (2.4) G (x) is ngaiv an G (x) crasing whn x b. I follows hn from (2.19) ha (2.39) hols Vrificaion lmma Lmma 2.1. If non-ngaiv funcion G! C 1 (R + ) is also wic coninuously iffrniabl xc a counably many oins an saisfis 1. (A ) G (x), x $, 2. G (x), x $, 3. G (x) $ j, x $, hn Gx ( ) $ V( x; ), x$. (2.41) Morovr, if hr xiss a oin b! R + such ha G! C 1 (R + ) + C 2 (R + \ {b }) wih 4. (A ) G (x) =, G (x) $ j for x! [, b ], 5. (A ) G (x) <, G (x) = j(x b ) + G(b ) for x! ( b, 3),

13 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 623 in which h ingro-iffrnial oraor A is fin by (2.6), hn G( x) = V(x; ), x! R+, an, (2.42) b ( = 1{ X ( -) > b } X ( - - D ) ( X ( ) b ) + L ), $, (2.43) is oimal, whr L ( ) L ( s), $, (2.44) b b X = 1{ X ( s) = b } X is h local im of h rocss X a h barrir b, rrsning ivins u o oscillaions of h Brownian Moion whn h surlus is a h barrir, an ( X ( ) ) (2.45) - - b 1 { X ( -) > b } rrsns h ivin isribu a im if h surlus rocss jums abov h barrir. A roof is iscuss in Anix A Th cas m In h rvious scions, w foun ha hr is a uniqu osiiv barrir b ha maximiss h valu funcion G(x) if an only if m >. W now consir h cas whn m an will show ha b = if an only if m. This mans ha if h businss is no rofiabl, h oimal sragy is o rmov any surlus ha is availabl as a final ivin an so h businss. This is no ncssarily rivial whn j < Cas 1: b = & m Suos ha b =. This mans ha h valu funcion G(x) is maximis whn h barrir is a zro, an i is oimal o immialy rlas h nir surlus as ivins. In his cas, i follows ha G(x) = jx. (2.46) Howvr, w now from h HJB quaion (2.5) ha any oimal sragy shoul saisfy (A ) G(x) for all x $. Uon subsiuion wih (2.46), his coniion rucs o m. Thus, w s ha if h oimal barrir is b =, hn h rif m shoul saisfy m. Going bacwars, i follows ha if m, hn G(x) = jx saisfis h HJB quaion (2.5).

14 624 B. AVANZI, J. SHEN AND B. WONG Cas 2: m & b = Consir an alrnaiv sragy, say, whrby h surlus x is immialy ai as a ivin, so ha ruin occurs immialy. Th valu unr his sragy is J(x; ) = jx. Howvr, his sragy mus hav valu lss han h oimal sragy, so i follows ha J(x; ) = jx V(x; ). Morovr, w show ha h funcion G(x) = jx saisfis h HJB quaion in h cas whn m. Thus, i follows from Lmma 2.1 ha G(x) = jx $ V(x; ). Bas on hs wo argumns, i follows ha V(x; ) = jx, an so, ha h oimal barrir is b =. 3. DIVIDEND MAXIMISATION WITH FORCED CAPITAL INJECTIONS TO PREVENT RUIN In his scion, as a sing son in solving h gnral oimal conrol roblm, w firs assum ha h s of amissibl conrol sragis is rmin such ha h surlus X is nvr ruin. This can b achiv by injcing xra caial in orr o h surlus abov zro. Th surlus rocss bcoms X ( ) = U( ) - D ( ) + E ( ),, (3.1) $ whr h s of amissibl sragis is P = : = { ( D, E )! P such ha X ( -) $ for all $ }. (3.2) In his mol, ruin os no occur. Th objciv funcion for his roblm is Jx ( ; ): = E ; limsuaj D ( s) - E ( s) E. x " 3 (3.3) Givn iniial surlus x >, w consir h xc rsn valu of ivins isribu lss h oal coss of quiy issuanc unr h oimal sragy, no by V( x ; ) : = su J( x ). (3.4) ;! P W will inify h form of h valu funcion V(x; ) an h oimal sragy HJB quaion Suos ha for a givn lvl of iniial surlus x $, h valu funcion unr h oimal join ivin an caial injcion sragy is no by H(x).

15 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 625 Accoring o h Hamilon-Jacobi-Bllman (HJB) quaion for his roblm, if h valu funcion H is wic coninuously iffrniabl hn w xc i o saisfy max" ( A -) H( x), j - H ( x), H ( x) -, = wih H () =. (3.5) Using h sam chniqus as scrib in Scion 2.1 an allowing for caial injcion m, h analogous rsul o (2.8) is H(x ) + $ l7j - H ( x) A+ m7h ( x )- A+ ( A - ) H( x). + o(). (3.6) Sinc H(x) is h oimal valu, i follows ha h xrssion in bracs mus hav maximal valu of zro, suggsing max $ l7j - H ( x) A+ m7h ( x) - A+ ( A - )H( x). =. (3.7) l$, m$ W rsric hn h firs rivaiv of h valu funcion such ha j H (x), (3.8) ohrwis w can ma h firs or scon ar of (3.7) unboun, by ling l or m n o infiniy rscivly. Now sinc (3.7) hols for l = m =, w rquir ( A - ) H (x). (3.9) Sinc w allow h iniial surlus x $ o b arbirary, quaions (3.8) an (3.9) mus hol for any x $, an w can rwri (3.7) by sliing i ino hr ars, laing o (3.5). Th bounary coniion can b xlain by h following hurisic argumn. Consir wo saml ahs of h surlus rocss: on saring a som small >, an anohr saring a zro. If h lar ah movs own o, an h formr ah movs aralll o his ah, w mus hav H() = H ( ) -. (3.1) Subracing H() from boh sis, iviing by an ling n o zro shows ha H () =. This is also suor by h following iscussion. Consir h rrsnaion of h xc rsn valu of ivins lss caial injcions unr h arbirary sragy givn in (3.6). Th oimal valu H(x) is obain whn h valu of h xrssion in bracs is maximis. W now consir h valu of m ha will maximis his xrssion. Sinc (A ) H(x) an l [j H (x)] ar innn of m, w wish o consir max $ m7h (x) - A.. (3.11) m $

16 626 B. AVANZI, J. SHEN AND B. WONG I is clar ha h valu of m ha maximiss his xrssion will n on h valu of H (x). Howvr, bcaus our objciv funcion (3.3) is nalis by caial injcions, w will minimis m whnvr ossibl. Toghr wih j H (x) i follows ha a any im >, h aroria valu of m is rmin by H (X ()) in h following way (wih sligh abus of noaion): < hn m = ; If H ( X ()) (3.12) = hn m! [, 3]. Sinc w wish o minimis m whnvr ossibl (bcaus of ransacion coss), hn ially w woul li o s m = a all ims. Howvr, in h roblm formulaion oulin a h sar of Scion 3, w ar rquir o injc caial o rvn ruin. Wih his bing h cas, h only im whn i is ossibly oimal o injc caial is whn H (X ()) =, an his shoul only han whn h surlus is null. Inuiivly, his is bcaus iscouning will unncssarily nalis caial injcions ha ar ma bfor hy ar absoluly ncssary, an hs can b absoluly ncssary only whn h surlus is null (o avoi imminn ruin) Consrucion of a cania soluion W conjcur ha h oimal ivin sragy is a barrir sragy b. Furhr mor, u o h fac ha our objciv funcion (3.3) is nalis by caial injcions, an hs caial injcions ar iscoun for im, w conjcur ha h oimal caial injcion sragy is o issu h minimum amoun of caial, an o lay h injcion of caial for as long as ossibl. W will hn consir a sragy ha only injcs caial whn h surlus rocss {X ()} his h lvl of zro. W consruc our cania soluion o saisfy j H (x) = abov h barrir, an (A ) H(x) = blow h barrir, which yils H( x ) : = Hx ( ; b) x! [, b] ; an (3.13) j( x - b ) + Hb ( ; b) x! ( b, 3), whr w fin = = rx H( x; b ) : / C ( b ), x $. (3.14) Hr, h r s rmain h soluions of (2.16). Th C (b ) s an h oimal barrir b hav o saisfy h following coniions: = H (; b) = / r C ( b) = (3.15)

17 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 627 rb H ( b -; b ) = / r C ( b ) = j (3.16) = 2 H ( b -; b ) = / r C ( b) = (3.17) = bi ( rb bi - r = rb r / C b ) = j, for i = 1,2, f, n. (3.18) Coniion (3.15) is h bounary coniion of h HJB quaion. Coniions (3.16)-(3.18) ar obain by analogous rasoning o Scion 2.3. As (3.16)-(3.18) ar inical o (2.19)-(2.21), wih b subsiu for b, i follows ha j ( -r b i C b ) r n r - b =-j r % r - r % (3.19) j = j! j b i = 1 i for =, 1,, an all b >. Th oimal barrir b is hn rmin by (3.15), as xlain in h following scion. W hav hn b -r b n j i rx -j / r % r r, x, - % $ j bi = j = i = 1 (3.2) j! H( x; ) : = r r - b whr b is rmin by (3.15), which can b rwrin as r b - / j. (3.21) = - r n b j r - i % r - r % = j j b = i = 1 i j! Rmar 3.1. No ha in his roblm H(; b ) is no longr zro bcaus quiy is issu o rvn ruin. Givn h iniial surlus of zro, if h rsn valu of h oal coss of injcing fuur caial ouwighs h rsn valu of h ivins isribu in h fuur hn H(; b ) will b ngaiv. Sinc b is fin o b h uniqu osiiv soluion o h quaion G(; b ) =, an G( ; b ) an H( ; b ) hav h sam form, i follows ha H(; b ) = if an only if b = b Th oimal ivin barrir b Now ha w hav rmin h form of h C (b ), w show ha hr is a uniqu valu of b ha solvs (3.15) in conjuncion wih (3.19). Using h funcion x as fin in (2.33), w fin h rla funcion = = ( = x ( b ) : -x ( b ) / r C b ). (3.22) 1

18 628 B. AVANZI, J. SHEN AND B. WONG W wan o show ha hr is a uniqu soluion o (3.15), which is quivaln o x. (3.23) 1 ( b ) = W firs no ha x () = H ( b -; b ) = (3.24) 1 bcaus of (3.17), an ha 1 ( 3 = ( = x b ) / r C b ) >, (3.25) bcaus r an C ( ) hav h sam sign for all ; s (2.17), (2.27) an (2.28). Hnc, x 1 is an (incrasingly) incrasing funcion in b. Sinc x1 ( ) = H ( b-; b) = j (3.26) an sinc lim x = 3 (3.27) b " 3 1 ( b) i follows from (3.8) ha hr xiss a uniqu non-ngaiv soluion o (3.15) ha is innn of h iniial surlus x. Furhrmor, his hols for any m (osiiv, null or ngaiv). Rmar 3.2. No ha b = if an only if j = = 1, an ha in his cas h valu funcion is H(x) = x + m/. Tha is, if hr ar no roorional ransacion coss on ivin isribuions or caial injcions, hn h oimal sragy is o ay ou all of h surlus as a ivin, an o offs all fuur surlus cash flows by ivins or caial injcions (wih rsn valu m/). As hs ar no nalis, hr is no bnfi in holing any surlus. Rmar 3.3. Equaion (3.21) shows ha h oimal barrir b is now nn on j, which is no h cas whn only ivins ar consir; s Rmar 2.2. Howvr, as h r s ar innn of j an, only h raio of o j mars Vrificaion of all h coniions of h HJB quaion By consrucion, our cania soluion saisfis H (x) = j for x! [ b, 3) an (A ) H(x) = for x! [, b ]. Hnc, i only rmains o show ha ( A - ) H( x), x > b, H (x) $ j, x < b, an H ( x), x $. (3.28) (3.29) (3.3)

19 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 629 Th roof of (3.28) is similar o h on vlo in Scion 2.4. Consiring (3.13) wih Coniions (3.15) an (3.16) imlis ha H (x) gos from o j as x gos from o b, an hn says qual o j for x $ b. Sinc $ j, in orr o show ha (3.29) an (3.3) hol, i suffics o show ha H (x) crass monoonically ovr x < b. This follows from H (b ) = bcaus of Coniion (3.17) an from h obsrvaion ha 3 rx H (x) = / r C ( b) >, x < b. = Rmar 3.4. Th obsrvaion ha H (x) > for x b allows us o uc h concaviy of h valu funcion. An alrnaiv roof of h concaviy for gnral jum isribuions is also rovi in Anix B. Unforunaly, his roof os no hol whn ruin is allow, hnc h n o xlicily rmin h sign of G (x) in Scions 2 an Vrificaion lmma W us h following vrificaion lmma o rov ha in h cas whn ruin is no allow, h oimal join ivin an caial injcion sragy is o isribu ivins accoring o a barrir sragy, an o injc caial only whn h surlus rachs h lvl of zro. This vrificaion lmma xns h lmma from Scion 2.5 by inroucing caial injcions. Lmma 3.1. If funcion H! C 1 (R + ) is also wic coninuously iffrniabl xc a counably many oins an saisfis 1. (A ) H(x), x $, 2. H (x), x $, 3. j H (x), x $, hn H(x) $ V( x; ), x $. (3.31) Morovr, if hr xiss a oin b! R + such ha H! C 1 (R + ) + C 2 (R + \ {b }) wih 4. (A ) H(x) =, H (x) $ j for x! [, b ], 5. (A ) H(x) <, H(x) = j(x b ) + H(b ) for x! (b, 3), in which h ingro-iffrnial oraor A is fin by (2.6), an 6. H () =, hn H(x) V( x; ) x! R, (3.32) = +

20 63 B. AVANZI, J. SHEN AND B. WONG an h join sragy D () = ( X ( -) - b ) 1 + L (), $, (3.33) { X ( - ) > b } b X an $ E () = LX (),, (3.34) is oimal, whr b b L () = 1 L ( s), $, (3.35) X { X ( s) b} X = is h local im of h rocss X a h barrir b, rrsning ivins u o oscillaions of h Brownian Moion whn h surlus is a h barrir, ( X ( -) ) 1{ ( - } (3.36) - b X ) > b rrsns h ivin ai a im if h surlus rocss jums abov h barrir, an X { X (s) } X = L L () = 1 ( s),, (3.37) $ rrsns caial injc whn h surlus is a h lvl of zro. A roof is iscuss in Anix A. 4. THE OPTIMAL JOINT DIVIDEND AND CAPITAL INJECTION STRATEGY In his scion w consir h gnral oimal conrol roblm as fin in Scion 1.3. Sinc hr ar now no rsricions on caial injcions, h surlus may bcom ngaiv. Th im of ruin for a givn conrol sragy is hn fin as : = inf{ : X ( -) <. (4.1) } No h sric inqualiy, which is rquir bcaus of h caial injcions. In fac, i is ossibl ha = 3. W consir h valu funcion (1.8). Sinc V(x; ) is h oimal sragy from h unrsric s of amissibl sragis P, i follows ha w mus hav V( x; ) $ max { V( x; ), V( x; )}, (4.2) whr V(x; ) an V(x; ) ar fin as in quaions (2.4) an (3.4). In his scion, w rmin V(x; ) an h oimal sragy.

21 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL HJB quaion an vrificaion lmma W firs us h following vrificaion lmma o rov h oimaliy of any concav soluion of h HJB quaion max" ( A -) V(x ), j -V ( x), V ( x) -, = wih max{ -V( ), V ( ) - } =. (4.3) Th bounary coniions ar xlain as follows. If V () > hn caial is injc u o a lvl a such ha V (a) =. This os no ma sns bcaus if caial is injc, ruin os no han an hn i is uslss o h surlus a a highr lvl han. W rsric hn V (). Howvr, if V () = hn caial is injc whn h surlus is null o rvn ruin. This can only ma sns if V() $. Ohrwis, h xc rsn valu of caial injcions woul b highr han ha of h ivins, an h comany woul hn nvr choos o injc caial, which las o a conraicion. Lmma 4.1. If non-ngaiv funcion V! C 1 (R + ) is also wic coninuously iffrniabl xc a counably many oins an saisfis 1. (A ) V(x), x $, 2. V (x), x $, 3. j V (x), x $, hn V(x) $ V( x; ), x $. (4.4) A roof is iscuss in Anix A Characrisaion of h oimal sragy In his scion w characris h oimal sragy o maximis J(x; ) an show how i ns on h rif m an h rlaionshi bwn h barrirs b an b rmin in h rvious scions. Thorm 4.2. L {X ()},, an V(x; ) b as fin in Scion 1, an l m b as in (2.22). Furhrmor,, b, an b ar h oimal sragis an associa oimal ivin barrirs as vlo in Scions 2 an 3, rscivly. Th oimal join ivin an caial injcion sragy is hn characris as follows: = if m, (4.5) = if m > an b > b, (4.6) = if m > an b < b, an (4.7) = or if m > an b = b. (4.8)

22 632 B. AVANZI, J. SHEN AND B. WONG In h nx four scions, w rovi a roof of Thorm 4.2 by showing (4.5)- (4.8) squnially Proof of (4.5) From Scion 2.6 w s ha V(x; ) = jx, an V(x; ) $ V(x; ) bcaus of quaion (4.2). If w can show ha V(x; ) V(x; ), hn w hav rov ha h oimal sragy is o us a barrir of zro. In orr o o his, w n o vrify ha V(x; ) saisfis h coniions of h HJB quaion (4.3). W hav rviously shown ha max$ ( A - ) V(x; ), j-v (x ; ). =, (4.9) so i rmains o show ha max{ V (x; )- } = wih max{ -V(; ), V (; ) - } =. (4.1) W hav which also mans ha V (x; ) = j, x $, (4.11) V (; )- = j -. (4.12) In aiion, - V (; ) = - j $ =, (4.13) which comls h roof Proof of (4.6) From Lmma 2.1 w s ha G(x) = V(x; ), an V(x; ) $ G(x) bcaus of quaion (4.2). If w can show ha V(x; ) G(x) for b b, hn i follows ha h oimal join ivin an caial injcion sragy is o us a barrir of b o isribu ivins, an o issu no caial. In orr o o his, w n o vrify ha G(x) saisfis h coniions of h HJB quaion (4.3). By consrucion, G() = ; s (2.18). I rmains hus o show ha G (x), x $. (4.14) In Scion 2.4, w show ha G (x) < for x! [, b ), an sinc G is linar on [b, 3), i follows ha G(x) is concav. Hnc, (4.14) hols if an only if G (), which follows from

23 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 633 ( r ( ( 1 b C b G 1 b rc( b = = / / (4.15) x ) = ) = () x ) = ) = H () = bcaus x 1 (z) is an incrasing funcion; s Scion 3.3. Du o his rsul an Lmma 2.1, G(x) saisfis h coniions of Lmma 4.1. Hnc G(x) $ V(x; ) so ha V(x; ) = V(x; ), which comls h roof Proof of (4.7) From Lmma 3.1 w s ha H(x) = V(x; ) an V(x; ) $ H(x) u o quaion (4.2), so i is sufficin o show ha V(x; ) H(x) if b b. As abov, w wish o vrify ha H(x) saisfis h bounary coniions in HJB quaion (4.3), an roc in a similar way. Du o Lmma 3.1, all coniions of h HJB quaion (4.3) hav bn confirm xc for H() $. This follows from H $ x ( = = / / (4.16) x( b ) = C ( b ) = () ( b ) = C b ) = G() = bcaus x(z) is a crasing funcion; s Scion Du o his rsul an Lmma 3.1, H(x) saisfis h coniions of Lmma 4.1. Hnc, H(x) $ V(x; ) so ha V(x; ) = V(x; ), which comls h roof Proof of (4.8) Bcaus of quaion (4.2), V(x; ) $ max{g(x), H(x)}. Furhrmor, i follows from h roofs of (4.6) an (4.7) ha G(x ) $ V( ; ) b b, an (4.17) x, Bu H(x) $ V x ; ) b b. (4.18) (, b = b, H( x ) = G( x ) ; (4.19) s Rmar 3.1. Hnc, V(x; ) = V(x; ) = V(x; ), which comls h roof. No ha his mans ha whn h surlus his, managmn will b iniffrn bwn injcing caial o rscu h businss an soing h businss. Rmar 4.1. Thr ar wo alrnaiv rrsnaions o h coniions on b an b in Thorm 4.2. From (4.15) an (4.16) i follows ha b < b, G (; b ) > = H (; b ), H(; b ) > = G(; b ), (4.2)

24 634 B. AVANZI, J. SHEN AND B. WONG an vic vrsa. This is inrr using similar argumns o h ons vlo o xlain h coniions in (4.3). If (4.2) hols, hn caial injcions ar rofiabl for low lvls of surlus (bcaus G () > ), which rsuls in H() > an =. Convrsly, if G () < hn caial will nvr b injc an H() <, so ha =. Rmar 4.2. Injcing caial can b consir as a ral oion (s, for insanc, Dixi an Pinyc, 1994). This oion has an aggrga osiiv valu qual o H(x) G(x) whn (4.2) is saisfi. Rmar 4.3. Grbr an Shiu (26) consir h mrgr of wo comanis whn hir surlus is a ur iffusion. Thr, mrgr is consir as rofiabl whn W( x + x ; b m ) > W( x ; b 1) + W( x ; b 2), (4.21) whr W(x; b) is h xc rsn valu of ivins unil ruin whn a barrir sragy b is ali, whr x i an b i ar h iniial surlus an oimal barrir of comany i (i = 1, 2), rscivly, an whr b m is h oimal barrir of h mrg surluss. This wor givs ris o wo rmars. Firsly, his aroach can asily b xn o h ual mol wih iffusion as h sum of wo (innn) comoun Poisson rocsss wih mixur of xonnial jums is comoun Poisson wih mixur of xonnial jums again, as nnc can sill b mol bwn h wo iffusion comonns. Numrical calculaions inica ha caial injcions ar mor lily o b oimal for lowr lvls of nnc. Sconly, mrgr can b sn as a cha way of injcing caial, as h aggrgaion of h surluss is no nalis by (roorional) ransacion coss. Howvr, h lvl of h barrir b m is lily o chang, rsuling in an inrmina n rofi. On h ohr han, if on of h comanis is comaraivly small hn is imac on h oimal barrir will b ngligibl, an h mrgr will b mor rofiabl as h surlus is lowr (sinc W (x; b) > 1 an crasing for x < b). No also ha in racic, a mrgr woul arac ransacion coss, bu inclusion of hs is rivial as hy only n o b subrac from x 1 + x 2 on h lf-han si of (4.21). 5. NUMERICAL ILLUSTRATIONS 5.1. Th choic bwn an L c =.2, =.8, l = 1, s = 5 an such ha m =.8 >. ( ) y y = c 3 m +.5^ y h

25 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 635 FIGURE 1: Valu funcions whn j =.9 an = 1.1 on h lf an whn j =.8 an = 1.1 on h righ. W firs consir j =.9 an = 1.1. In his cas, b = < b = 9.145, so i is oimal o injc caial an V(x; b ) > V(x; b ) for all x. If w incras h ransacion coss so j =.8 an = 1.1, hn i is no longr oimal o injc caial, sinc b = < b = In his cas, V(x; b ) > V(x; b ) for all x. Ths wo cass ar shown in Figur 1. No ha his also illusras Rmar Th ffc of h rif In his xaml, w consir h sam aramrs as in Scion 5.1, using j =.9 an = 1.1, bu vary h rif of h rocss by changing c in orr o suy is imac on h oimal sragy. Figur 2 shows wo cass, whn s =.5 an s = 5, rscivly. FIGURE 2: Oimal ivin barrirs accoring o, an whn h rif changs, for s =.5 an s = 5. Th imac of h rif on b is monoon for all lvls of volailiy. As h rif crass (c incrass), his barrir incrass slowly o ry o avoi caial injcions. In conras, m has a mix imac on h barrir b. Thr, wo conflicing forcs ar a wor. On on han, a lowr rif incrass ris which calls for a

26 636 B. AVANZI, J. SHEN AND B. WONG highr barrir. On h ohr han, whn h rif gs closr o, i is br o isribu a grar roorion of h surlus ha is availabl as a ivin bcaus of ba roscs. In h limi m = (c = 1), b =. In h cas s = 5, h scon forc ominas. Th oimal ivin barrir accoring o, min{b, b }, is shown in gry. W obsrv ha injcing caial is in gnral br whn h rif is high. As ris incrass, h oimal sragy swichs from o for highr lvls of rif Th ffc of h forc of inrs W now consir h ffcs of a chang in h forc of inrs. Incrasing h forc of inrs crass h valu of ivins, bu also crass h cos of injcing caial. W lo h lvls of h barrirs for h mixur from Scion 5.1 wih aramrs = 1.1, j =.9, l = 1 an c =.5. W loo a h cass whn h Brownian moion volailiy is s =.5 an s = 5 as h forc of inrs varis from o.2. FIGURE 3: Snsiiviy of h Oimal Barrirs o changs in h Forc of Inrs, for s =.5 an s = 5. Th wo grahs show ha h rlaionshi bwn b an b (as a funcion of ) ns on h volailiy of h surlus. If h volailiy is low, hn b sms o b always lowr han b as changs. Howvr, if h volailiy is high, hn as incrass, h cras valu of ivins is no sufficin o jusify furhr invsmns, aricularly sinc h high volailiy mans ha mor caial will n o b injc, an h rsn valu of h caial injcions will far ouwigh h rsn valu of h ivins isribu. ACKNOWLEDGMENTS Th auhors acnowlg financial suor of an Ausralian Acuarial Rsarch Gran from h Insiu of Acuaris of Ausralia. Jonahan Shn is inb o Mr Ewin Blacar for roviing him wih h EJ Blacar Honours Scholarshi. Th auhors ar graful o anonymous rfrs for hlful commns.

27 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 637 A. Proofs of Lmmas 2.1, 3.1 an 4.1 APPENDIX This anix ails h roofs of Lmma 4.1 an h scon scion of Lmma 3.1. Similar aroachs o h ons an hr can b us o rov h firs scions of Lmmas 2.1 an 3.1, an h scon scion of 2.1, rscivly. W will firs rov Lmma 4.1. Th firs scions of Lmma 2.1 an Lmma 3.1 can b rov by maing h following moificaions. For Lmma 2.1, s E as h my s, an rlac V wih G. For Lmma 3.1, rlac V wih H an rlac / wih. Proof of Lmma 4.1. For a givn sragy! P, w fin h following ss: D = { s : D( s-)! D( s) an S( s- ) = S( s)}; (A.1) E = { s : E ( s-)! E ( s) an S( s- ) = S( s)}; (A.2) ^DE h D, E. (A.3) = Tha is, D is h s conaining h jum ims of h rocss {D ()} u o ivin isribuions ha o no occur a h sam im as h jums in h comoun Poisson rocss, E is h s conaining h jum ims of h rocss {E ()} u o caial injcions ha o no occur a h sam im as h jums in h comoun Poisson rocss, an (DE) is h s conaining h ims whn h ivin an/or caial injcion rocsss jum, bu h comoun Poisson rocss os no jum. Also, l Z (c) no h coninuous ar of arbirary rocss Z, fin as: () c Z ( ) : = Z( ) - / [ Z(s) - Z( s-)]. (A.4) s By h Iô formula for jum-iffusion rocsss, w hav -( / -) V( X ( / -)) / - / / 2 - V ( X( s)) s - s (x ( s - s = V )- V( X )) s V ( X ( s)) X ( s) + DX( s)! / s 2 + [ V( X ( s-) + DX ( s)) -V( X ( s-))]. s / - () c (A.5) Th summaion rm in (A.5) rrsns changs u o jums in h comoun Poisson rocss {S()}, h aggrga ivin rocss {D ()} an h

28 638 B. AVANZI, J. SHEN AND B. WONG caial injcion rocss {E ()}. Using a similar aroach as in Bayraar an Egami (28, wih ivins only), w sli hs jums ino hr cagoris: 1. Jums in ihr or boh h ivin rocss an h caial injcion rocss, ha o no occur a h sam im as a jum in h comoun Poisson rocss; 2. All jums u o h comoun Poisson rocss; an 3. Th xra jums u o jums in ihr or boh h ivin rocss an caial injcion rocss ha occur a h sam im as a jum in h comoun Poisson rocss. Thus, w can wri h summaion as DX() s! / s / - [ V( X ( s-) + DX ( s)) -V( X ( s-))] / = [ VX ( ( s)) -VX ( ( s-))] s! ( DE) / - / [ V( X ( s-) + y) -V( X ( s-))] N( s, y) / [ V( X ( s)) - V( X ( s-) + y)] N( s, y). (A.6) Noing ha X (c), h coninuous ar of X, saisfis () c () c () c X () =- c + sw( ) - D () + E ( ), (A.7) an xrssing h firs ingral in (A.6) wih h comnsa jum masur, w can wri (A.5) as: -( / -) V( X ( / -)) / - / - A ( s ( ( = Vx ( ) + ( - ) VX ( s)) s+ V( X s)) Ws) () c / - - V ( X ( s)) D ( s) + V ( X ( s)) E ( s) + [ V( X ( s)) -V( X ( s-))] s! ( DE) / - / [ V( X ( s-) + y) -V( X ( s-))]( N( s, y) -n( s, y)) / [ V( X ( s)) - V( X ( s-) + y)] N( s, y). / () c (A.8)

29 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 639 Rwriing D (c) (s) an E (c) (s) using h comosiion as in (A.4) yils -( / -) V( X ( / -)) / - / - = Vx ( ) + ( A - ) VX ( ( s)) s+ s V ( X( s)) W( s) / - / - - j D ( s) + [ j -V ( X ( s))] D ( s) / - / - + E ( s) + [ V ( X ( s)) -] E ( s) + [ V( X ( s)) -V( X ( s-)) -[ X ( s) -X ( s-)] V ( X ( s-))] s! ( DE) / - / [ V( X ( s-) + y) -V( X ( s-))]( N( s, y) -n( s, y)) / [ V( X ( s)) - V( X ( s-) + y)] N( s, y) / / [ X ( s) -X ( s-) - y] V ( X ( s-) + y) N( s, y). (A.9) W no ha V is a concav funcion u o oin 2 of h vrificaion lmma, an in conjuncion wih oin 3, w hav j V (X ()) so ha h sochasic ingral wih rsc o h Brownian moion in (A.9) is a uniformly ingrabl maringal, bcaus > an V (x) is boun. In aiion, (A )V(x), (j V (X (s)) an (V (X (s) ) u o oins 1 an 3 of h vrificaion lmma, an combining V (x) from oin 2 of h vrificaion lmma wih h Man Valu Thorm, w hav V( y) -V( x) -( y -x) V ( x) for y $ x; an Vx ( )-V( y) -( x -y) V ( y) for y$ x. Afr alying all of hs rsuls o (A.9), aing xcaions an rarranging, i follows ha - ( -) / - x / x V( x) $ E 8 V( X ( / -)) B + E ; j D ( s) E x - E ; / - E ( s). E (A.1) Using h fac ha V() $ an coniioning on h valu of, w can hn wri

30 64 B. AVANZI, J. SHEN AND B. WONG liminf " 3 -( / -) = liminf " 3 + liminf " 3 - ( -) VX ( ( / -)) -( / -) -( / -) = V( X ( -)) 1 + liminf V( X ( )) 1 - ( -) VX ( ( / -)) 1 $ V() 1 $. VX ( ( / -)) 1 { -< 3} { -= 3} - { -< 3} " 3 { -= 3} { -< 3} Finally, aing limis as " 3 in (A.1), w fin x / - / - V( x) $ E < limsucj D( s) - E( s) mf = J( x; ). " 3 (A.11) Sinc h sragy is arbirary, i follows ha V( x) $ V( x; ). (A.12) W now roc by roving h scon scion of Lmma 3.1. A similar aroach o h on an hr can b us o rov h scon scion of Lmma 2.1 by sing E as h my s, rlacing H wih G, rlacing wih, rlacing b wih b an rlacing wih ( / ) in h ur limis of h ingrals. Proof. W now ha (A ) H(X (s)) / bcaus X (s) b. Afr aing xcaions on boh sis of (A.9), rlacing V wih H, rlacing / wih an using oins 4 an 5 of Lmma 3.1, w can wri E x 9 - H( X ()) C x = Hx ( )-E : H ( X ( s)) D ( s) D x + E : x / H ( X ( s)) E ( s) D + E < 8H( X ( s)) -H( X ( s-)) -( X ( s) -X ( s-)) H ( X ( s-)) BF s! ( DE) 3 x + E : [ H( X ( s)) - H( X ( s-) + y)] N( s, y) D 3 x -E : [ X ( s) -X ( s-) - y] H ( X ( s-) + y) N( s, y) D. (A.13) Th las hr rms ar zro u o h finiion of h roos join ivin an quiy sragy. By consrucion hr ar no jums in h quiy rocss. Hnc, h only jums in h rocss {X ()} occur u o jums in

31 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 641 h ivin an/or comoun Poisson rocss. Th summaion rm in (A.13) is summing ovr all oins in im whn hr is a jum in h ivin rocss, bu no jum in h comoun Poisson rocss. This can only ossibly occur a im zro, if h iniial surlus x is largr han h barrir b, so w hav X ( ) = x, X () = b, H ( X ( ) ) = j, HX ( ( ) ) = j( x- b) + Hb ( ) an HX ( ()) = Hb ( ), from which i follows ha h summaion rm is zro. Th las wo ingral rms in (A.13) aly o h oins in im whn hr is a jum in boh h ivin rocss an h comoun Poisson rocss. A hs oins, h surlus riss abov h barrir, an h valu funcion is linar, so w hav X ( s) = b, HX ( ( s)) = Hb ( ), H ( X ( s-) + y) = j, an HX ( ( s-) + y) = j( X ( s-) + y- b ) + Hb ( ), so h sum of h las wo ingral rms is zro. I follows ha (A.13) simlifis o x - x E 9 H( X ()) C = H( x) -E 9 H ( X ( s)) D ( s) C x + E 9 H ( X ( s)) E ( s) C. (A.14) Du o h finiion of h roos ivin sragy w can wri x x { X ( ) $ b} E 9 H ( X ( s)) D ( s) C = E 9 H ( X ( s)) 1 D ( s) C = E x 9j D ( s). C (A.15) Similarly, using oin 6 of h vrificaion lmma, x x { X ( ) = } E 9 H ( X ( s)) E ( s) C = E 9 H ( X ( s)) 1 E ( s) C = E x 9 E ( s). C (A.16) Subsiuing hs ino (A.14), rarranging an ling " 3, i follows ha j - - x H(x) = E ; limsub D ( s) - E ( s) le. " 3 (A.17)

32 642 B. AVANZI, J. SHEN AND B. WONG B. Proof of Concaviy of Valu Funcion Th following roof is an aaaion o h ual mol wih iffusion of h roof of concaviy rovi in Kulno an Schmili (28, in h Cramér- Lunbrg mol). This roof hols for any jum isribuion, albi only whn ruin is guaran no o occur. Consir wo surlus rocsss Y() an Z() of y (1.1) wih inical aramrs bu for hir iniial surluss y $ an z $, rscivly. Furhrmor, consir h amissibl sragis, y = (D, y, E, y ),, z = (D, z, E, z )! P an l a y, a z! (, 1) wih a y + a z = 1. Dfin D ( ) : = ayd ( ) + azd ( ), an (B.1) w, y, z, E ( ) : = a E ( ) + a E ( ). (B.2) w, y y, z z, No ha D, w () = D (,a y ) y+ a z bu ha in gnral E ( ) E ( ). z! w, W,ayy+ azz hav a Y( ) + a Z( ) - D ( ) + E ( ) = y z w, w, ay `Y( ) - D ( ) + E ( ) j + az `Z( ) - D ( ) + E ( ) j $. y, y, z, z, $ $ (B.3) Thus h sragy w = (D, w, E, w ) is amissibl. In aiion, w mus hav E ( ) aye ( ) az ( ), ayy a + E + z y,, ohrwis w rach h conraicion ha z z, h valu of h sragy ( D, E ) w, is infrior o h valu of h sragy,ayy+ azz ( D, E ). Thn w, w, V( y+ a z; ) a y $ E; limsubj $ E; limsubj D ( s) - E ( s) le `a D ( s) + a D ( s) j- `a E ( s) + a E ( s) jle = E; limsuba `jd ( s) - E ( s) j + a `jd ( s) -E ( s) jle = a J( y; ) + a J( z; ). y z " 3 " 3 " 3 w,, ayy+ azz y y, z z, y y, z z, y y, y, z z, z, y, z z, (B.4) L P, x no h s of amissibl sragis for h iniial caial x such ha ruin is guaran no o occur. Taing h surmum ovr all amissibl sragis w fin V( a y + a z; ) $ a su J( y; ) + a su J(; z ) y z y, y z, z y,! Py, z,! Pz, = a V( y; ) + ). y y, a (; z zv z, (B.5) (B.6)

33 OPTIMAL DIVIDENDS AND CAPITAL INJECTIONS IN THE DUAL MODEL 643 Rmar B.1. Whn ruin is allow o occur (such as in Scions 2 an 4), h ur bouns in (B.4) bcom funcions of,, w,, y an, z an h las inqualiy canno b guaran any mor. REFERENCES ALBRECHER, H., GERBER, H.U. an YANG, H. (21) A irc aroach o h iscoun naly funcion. Norh Amrican Acuarial Journal, 14(4), ALBRECHER, H. an THONHAUSER, S. (29) Oimaliy rsuls for ivin roblms in insuranc. RACSAM Rvisa la Ral Acamia Cincias; Sri A, Mahmáicas, 1(2), ALLEN, F. an MICHAELY, R. (23) Payou Policy, volum 1A of Hanboo of h Economics of Financ, char 7, Elsvir. ASMUSSEN, S. an ALBRECHER, H. (21) Ruin Probabiliis, volum 14 of Avanc Sris on Saisical Scinc an Ali Probabiliy. Worl Scinic Singaor, 2 iion. AVANZI, B. (29) Sragis for ivin isribuion: A rviw. Norh Amrican Acuarial Journal, 13(2), AVANZI, B. an GERBER, H.U. (28) Oimal ivins in h ual mol wih iffusion. Asin Bullin, 38(2), AVANZI, B., GERBER, H.U. an SHIU, E.S.W. (27) Oimal ivins in h ual mol. Insuranc: Mahmaics an Economics, 41(1), AVRAM, F., PALMOWSKI, Z. an PISTORIUS, M.R. (27) On h oimal ivin roblm for a scrally ngaiv Lévy rocss. Annals of Ali Probabiliy, 17(1), BAYRAKTAR, E. an EGAMI, M. (28) Oimizing vnur caial invsmns in a jum iffusion mol. Mahmaical Mhos of Oraions Rsarch, 67(1), BORCH, K. (1974) Th Mahmaical Thory of Insuranc. Lxingon Boos, D.C. Hah an Comany, Lxingon (Massachuss), Torono, Lonon. BÜHLMANN, H. (197) Mahmaical Mhos in Ris Thory. Grunlhrn r mahmaischn Wissnschafn. Sringr-Vrlag, Brlin, Hilbrg, Nw Yor. CHEUNG, E.C.K. an DREKIC, S. (28) Divin momns in h ual mol: Exac an aroxima aroachs. Asin Bullin, 38(2), DAI, H., LIU, Z. an LUAN, N. (21) Oimal ivin sragis in a ual mol wih caial injcions. Mahmaical Mhos of Oraions Rsarch, 72(1), DE FINETTI, B. (1957) Su un imosazion alrnaiva lla oria colliva l rischio. Transacions of h XVh Inrnaional Congrss of Acuaris, 2, DIXIT, A.K. an PINDYCK, R.S. (1994) Invsmn Unr Uncrainy. Princon Univrsiy Prss. DUFRESNE, D. (27) Fiing combinaions of xonnials o robabiliy isribuions. Ali Sochasic Mols in Businss an Inusry, 23(1), 23)-48. DUFRESNE, F. an GERBER, H.U. (1991) Ris hory for h comoun Poisson rocss ha is rurb by iffusion. Insuranc: Mahmaics an Economics, 1(1), FELDMANN, A. an WHITT, W. (1998) Fiing mixurs of xonnials o long-ail isribuions o analyz nwor rformanc mols. Prformanc Evaluaion, 31, GERBER, H.U. (1972) Gams of conomic survival wih iscr- an coninuous-incom rocsss. Oraions Rsarch, 2(1), GERBER, H.U. an SHIU, E.S.W. (26) On h mrgr of wo comanis. Norh Amrican Acuarial Journal, 1(3), 667. HE, L. an LIANG, Z. (28) Oimal financing an ivin conrol of h insuranc comany wih roorional rinsuranc olicy. Insuranc: Mahmaics an Economics, 42(3), KULENKO, N. an SCHMIDLI, H. (28) Oimal ivin sragis in a Cramér-Lunbrg mol wih caial injcions. Insuranc: Mahmaics an Economics, 43(2), LØKKA, A. an ZERVOS, M. (28) Oimal ivin an issuanc of quiy olicis in h rsnc of roorional coss. Insuranc: Mahmaics an Economics, 42(3), MAZZA, C. an RULLIÈRE, D. (24) A lin bwn wav govrn ranom moions an ruin rocsss. Insuranc: Mahmaics an Economics, 35(2), MIYASAWA, K. (1962) An conomic survival gam. Journal of h Oraions Rsarch Sociy of Jaan, 4(3),