Optimal Consumption and Investment Portfolio in Jump markets. Optimal Consumption and Portfolio of Investment in a Financial Market with Jumps

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1 Opimal Consumpion and Invesmen Porfolio in Jump markes Opimal Consumpion and Porfolio of Invesmen in a Financial Marke wih Jumps Gan Jin Lingnan (Universiy) College, China Insiue of Economic ransformaion and Opening, Sun Ya-sen Universiy, Guangzhou 575 Cellphone: , lnsjg@mail.sysu.edu.cn ABSRAC We will address he problems ha in order o obain a maximum expeced uiliy from boh consumpion and erminal wealh, how should a small invesor, choose his securiies porfolio and his consumpion rae a every ime. We discuss hese in a financial marke wih jumps, wih Lagrangian mehods aking ino accoun o find ou he possible opimal soluions. hen we prove hey are really opimal. Finally, under he assumpions of deerminisic coefficiens on he model, we obain he opimal pair of porfolio/consumpion in an explici feedback form on he curren level of wealh. KEYWORDS: Opimal Consumpion and Porfolio, Lagrangian mehod, Dynamical Programming INRODUCION Lagrangian and dynamical programming mehods are very popular in financial sudies o solve opimal problems. Here we discuss he problems of maximizaion of uiliy from consumpion, maximizaion of uiliy from erminal wealh and maximizaion of uiliy from boh consumpion and erminal wealh in a financial marke wih jumps. And we use mahemaics very carefully. Alhough direc Lagrangian procedure can deliver almos all he opimal soluions ha he popular indirec-uiliies mehod can deliver, and seems ha i will do so more economically in he sense ha redundan laborious calculaions of any indirec uiliies can be avoided, he soluions i delivers are only possible soluions. We sill need o prove wheher he possible soluions are real opimal or no. And he proving procedure is very complicaing, whereas can no be avoided. All he mehods we use here can be exended o he problems of opimal invesmen, consumpion and life insurance. he erminology of financial marke wih jumps in his paper was defined in my paper las year, which was published in he proceeding of DSI 4. We should pay aenion o he iems of Price of a sock or a bond, wealh process, consumpion process, ec, Uiliy funcions Le U :(, ) be a sricly increasing, sricly concave and C funcion wih U() lim U( c ) ³-, c + c + U ( ) lim U ( c ) =. We allow he possibiliy ha c U () lim U ( c ) = and U () =-. A funcion wih hese properies will be called a uiliy funcion. he sric increase of Uc says ha he invesor prefers higher levels of consumpion and/or erminal wealh o lower levels. he sric concaviy of U () says ha he is also risk-averse, i.e., ha his marginal uiliy U ( c ) is decreasing, and ends o zero as c (a sauraion effec ).

2 Opimal Consumpion and Invesmen Porfolio in Jump markes ono Since U :[, ] [, U ()] is sricly decreasing, i has a sricly decreasing inverse ono I :[, U ()] [, ]. We exend I o be a coninuous funcion on he enirey of [, ] by seing Iy º for U () y. I is easily o verify ha U( I( y) )- yi( y) ³ U( c) - yc, c <, < y <, () by sric concaviy of U (). For some of our resuls in his paper, we shall need o impose he addiional condiions U is a C funcion and U is nondecreasing on (, ). () Lemma.. If f( x ) is convex and sricly decreasing (or respecively, increasing) on (, ), hen he inverse funcion g of f is convex (respecively, concave) and sricly decreasing (respecively, increasing) on ( f( ), f ()) (respecively, ( f(), f ( ))). Proof. We jus prove he convexiy. For any y < y < y f () (respecively, f() y < y < y ), here exis x < x < x (respecively, x < x < x ) such ha y = f( x), y = f( x), y = f( x ). Because f is convex, hen f( x )-f( x) f( x - ) f( x) f( x (respecively, )-f( x) f( x - ) f( x) ). x -x x-x x-x x -x For g is he inverse funcion of f, x = gy, x= gy, x = gy, hus we have y -y y - y y (respecively, -y y - y gy -gy gy -gy gy -gy gy -gy ). Muliply wo sides of he previous inequaliy by ( gy - gy )( gy - gy ) ( > ), we have ( y-y)( y -y) gy ( )-gy gy ( - ) gy gy ( (respecively, )-gy gy ( - ) gy ), y-y y -y y -y y-y so g is convex (respecively, concave). Under condiion (), U is convex, so he inverse funcion I of U is also convex by Lemma.. And I is coninuously differeniable on (, ) \ { U ()}. In he case U () <, we assume ha I ( U ()) =, hen because I ( y ) = for every y > U (), he ideniy du ( I( y) ) = U ( I( y) ) I ( y) = yi ( y) (3) dy becomes valid on (, ) ; in paricular, if we have he addiional condiion I ( U ()) = lim I ( y) (4) y U () hen I is coninuously differeniable on (, ). Besides, he ideniy (3) is also valid on (, ) if U () =. Maximizaion of uiliy from consumpion In his secion we shall ry o address he following quesion. In order o obain a maximum expeced uiliy from consumpion, how should a small invesor, endowed wih iniial wealh x >, choose his securiies porfolio p( ) and his consumpion rae C from among

3 Opimal Consumpion and Invesmen Porfolio in Jump markes admissible pairs ( p, C) Î A ( x) a every ime? Suppose ha we have a measurable, { F }-adaped and uniformly bounded discoun process g { g (), } and a uiliy funcion U. Le G - ( udu ) () e. We ry o maximize he expeced discouned uiliy from consumpion J( x; p, C) = E G( s) U( C( s) ) ds (5) over he subclass { } - A ( x) ( p, C) Î A ( x) : E G ( s) U ( C( s) ) ds < (6) J in (5) is well defined for every pair p C Î U () >-. We denoe by (, ) A ( x), and A( x) = A ( x)( ¹Æ) if V ( x) sup J ( x; p, C) ( p, C) ÎA ( x) he value funcion of his problem, and V( x ) =- if A ( x) =Æ for some x. Because uiliy comes only from consumpion, we may ignore he porfolio process p by considering only consumpion process C in D ( x). he following proposiion will show his. Proposiion.[Karazas, 989]. For every x ³, we have V ( x) = sup J ( x; p, C), and in paricular, ( p, C) ÎA ( x) CÎD( x) V() = U() E G( s) ds (8) - - Proof. Define D() Z() b() [ G () ] = z() [ G() ] for laer use. ake ( p, C) Î A ( x), and consider z E b ( s) C( s) ds x. If z >,define C ˆ ( x ) Cx, hen C Î z ˆ D ( x ) and here exiss a porfolio pˆ such ha ( pˆ, Cˆ ) Î A ( x). ˆ - - U ( C) U( C( )) since U is sricly increasing, hen U ( C() ) ³ U ˆ ( C()). hus Ĉ saisfies he condiion in (6) since C does. Consequenly, ( pˆ, C ˆ ) Î A ( x). If z =, hen C () =, a.e., a.s., and le ˆ( C ) x E b ( sds, ), which is a consan and belongs o D ( x). Again, here exiss a porfolio pˆ such ha ( pˆ, Cˆ ) Î A ( x), Ĉ saisfying (6), hen ( pˆ, C ˆ ) Î A ( x). In he above wo cases, hey boh have he propery J( x; p, C) J( x; pˆ, C ˆ ). According o [Karazas, 989], If x =, hen C () º, a.e.,a.s., and his leads o (8). Now we need only o solve he opimal consumpion problem V ( x) = sup E G( s) U ( C( s)) ds ( p, C) ÎA ( x) when he consrain is given by E z () C() d = x. Le us inroduce he Lagrangian (7)

4 Opimal Consumpion and Invesmen Porfolio in Jump markes L é é ( C, l) = E G( su ) ( Cs ) ds -le z( scsds ) -x ê ë ê ë (9) If C >, hen for any e = es( w), es( w) Î(, Cs ), we have G ( su ) ( Cs + e) - lz( s) ( Cs + e) -[ G( su ) ( Cs )-lz( scs ) ] e U( C( s) + e) -U( C( s)) G ( s) + lz( s) G ( su ) (min{ Cs, Cs + e} ) + lz( s) () e he las inequaliy derives from he following wo inequaliies implied from he concaviy of U : for e >, we have U( C( s)) - U( C( s) + e) ³ U ( C( s )) ( -e), i.e., U( C( s) + e) - U( C( s)) U ( C( s )) e; and for e = e ( w) s, <- es ( w) < Cs, we have U( C( s) + e) - U( C( s)) ³ U ( C( s ) + e) e, i.e., U( C( s)) - U( C( s) + e) U ( C( s ) + e) ( -e). If he righ hand side of () has finie l P -inegral (l here represens Lebesgue measure on [, ]), hen he dominaed convergence heorem ensures ha L ( C, l) is differeniable w.r.. C, and we can move he derivaive ino he sign of expecaion and inegral. For he same reason, L ( C, l) has he similar conclusion w.r.. l. Using Lagrangian mehods enable us o find ou he poin C, l where he formal derivaives of L w.r.. ( C,l ) are equal o zero. We can associae a unique porfolio p () () (up o equivalence) o C by heorem.8 or Proposiion.9 in [Jin, 4], such ha () () ( p, C ) is an admissible sraegy. I remains o check carefully ha his pair gives he maximum of J. From G U ( C ) = lz ( ) (obained by seing he formal derivaive of L w.r.. C o equal o zero), and E z ( s) C( s) ds = x, we ge ( ) C = I l D () E z( s) I l D ( s) ds = x () Le s inroduce he funcion X ( y) E z( s) I( yd ( s) ) ds = E b( s) I( yd ( s) ) ds,< y < (3) and assume ha X ( y) <, y Î (, ) (4) From now on, we should have he following addiional assumpion in force: (A ) For any e >, l Ä P{ Z() < e} >. Lemma... Under he condiion (), X inheri he convexiy of I on (, );. Under condiion (4) and assumpion (A ), he funcion X ( y ) defined in (3) is coninuous and sricly decreasing on (, ) wih X () lim X ( y ) =, y +

5 Opimal Consumpion and Invesmen Porfolio in Jump markes X ( ) lim X ( y ) =. y Proof. Omied. - herefore, X has an inverse Y X, and here is a unique l = Y ( x ) which saisfies () for any given x >. hen he corresponding consumpion process in (6.7) becomes ( C ) = I( Y ( x) D ), (5) which belongs o D ( x ), and according o heorem.8 and Proposiion.9 in [Jin, 4],, here is a unique (up o equivalence) porfolio process p () () such ha ( p,c ) Î A ( x), () and he corresponding wealh process X ( ) is nonnegaive on [, ) and vanishes a =, given by ( ) é ( ) X b = E C ( s) b( s) ds ê F ë ( ) ( ) = x - b s C s ds + b s p ( s) s s dw ( s) + If U =, hen ( ) ( b( s) p ) ( s) r( s) dq s ( X ) ( ) is posiive on [, ). Indeed, a his ime, I ( ) = I ( ) > on [, ). For x Î (, ), < Y ( x ) <, and, and ( g- ) D = u r u du Z e <, a.s. on [, ) (his is derived from < <, a.s. () since EZ ( ) = EZ ( ) =, ), hen C = I( Y ( x) D ) >, a.s. on [, ). () é ( hus from he firs equaliy of (6), we have ) = X E b > b ê C s s ds F, ë a.s., Î [, ]. heorem.3[karazas, 989].. Assume ha (4) holds. hen for any x > and wih C () Î D ( x ) given by (5), he pair ( p,c ) consruced above belongs o A ( x ) and is opimal for he problem of (7), i.e., ( V( x) = E G( s) U( C ) ( s) ) ds (7) Proof. Omied. In order o guaranee finieness of he value funcion V in (7) and o obain a useful represenaion for i, we impose he condiion E G( s) U( I( yd ( s) )) ds <, < y < (8) heorem.4. Under he condiions (4) and (8), he value funcion V () saisfies V( x) = G( Y ( x) ), < x < (9) where he funcion G :(, ) given by (6) G ( y) E G U ( I ( yd )) d, < y < ()

6 Opimal Consumpion and Invesmen Porfolio in Jump markes Furhermore, if U also saisfies () and I () is coninuous on (, ), hen X and G are boh coninuously differeniable, as well as wih G ( y) = yx ( y), < y <, () and be valid, and hen X ( y) = E G D I ( yd ) d () V ( x) = Y ( x) and V ( x) = Y ( x), < x < (3) follow from (9); in paricular, G is sricly decreasing, V is sricly increasing, sricly concave and V Î C (, ). Proof. Omied. Proposiion.5. If U >-, hen (8) implies (4). Proof. From he concaviy of U, we have ha C U ( C) U( C) - U U( C) + U for any C Î [, ), whence yd I( yd ) U( I( yd )) + U by aking C = I ( yd( )) in he previous inequaliy. And from his inequaliy i follows ha yx ( y) = ye G D I ( yd ) d U ( ) E G d + E G U ( I ( yd )) d <, < y <. Maximizaion of uiliy from erminal wealh We now ake up he problem of he maximizaion of he expeced discouned uiliy from erminal wealh J ( x; p, C) E[ G U ( X )] (4) over he subclass - A( x) {( p, C) Î A ( x) : Eé G U ( X ) ë û < } (5) and i is easy o verify ha A( x ) º A ( x ) if U >-. We denoe he value funcion of he problem V ( x) sup J ( x; p, C) (6) ( p, C) ÎA ( x) his is in fac he complemenary problem o ha of 6. U is he uiliy funcion. Again, he case x = is rivial. Indeed, for every ( p, C) Î A( x) º A, we have X( )= a.s. from (.3) in [Jin, 4], and hen V = U EG( ); his can be achieved by p º and C º. So we only ake x > from now on. In his seing, because uiliy comes now only from erminal wealh, he agen obviously ries o maximize he uiliy from his erminal wealh, wihin he consrains imposed by he level of his iniial capial and quanified by he budge consrain, i.e., E [ X b ] = E[ X z ] x (7)

7 Opimal Consumpion and Invesmen Porfolio in Jump markes which mandaes ha he expeced erminal wealh, discouned or deflaed down o = (discouned or deflaed down respecively in wo differen probabiliy spaces), should no exceed he iniial capial. Because uiliy comes now only from erminal wealh, i is quie reasonable ha he laer should be increased wihin he consrains mandaed by he level of he iniial endowmen as quanified by (7), by considering porfolio processes p in he class P ( x ) of Definiion.7 in [Jin, 4]. Proposiion 3.. For every x >, we have V ( x) = sup J ( x; p,). ( p,) ÎA ( x) p ÎP( x) Proof. Omied. o solve he opimal erminal wealh problem V ( x) = sup E[ G U ( X )] ( p, C) ÎA ( x) under he consrain E [ X ( ) b ( ) ] = E[ X ( ) z ( ) ] = x, le us inroduce he Lagrangian L ( X, l) = E[ G U ( X )]-le[ X z -x]. Analogue o he previous secion, le he formal parial derivaives of L wih respec o ( X( ),l ) be equal o zero, we obain G U ( X ) = z( ) l, and hen ( X ) = I D l (8) where l is decided by ( l ) EéI z êë D = x (9) I is he inverse of U (In order o make Lagrangian valid, we should have some supplemenary condiions, cf. 6). Le us now subsaniae he heurisics of he preceding elemenary Lagrangian muliplier consideraions. We sar by inroducing he funcion X ( y) E[ z I ( yd )] = E [ b I ( yd )] < y < (3), and assume ha for any y Î (, ) X ( y ) < (3) And by he similar procedure discussed previously, we can prove ha he soluion of Lagrangian is he opimal, and i will make a maximum uiliy from erminal wealh. Maximizaion of uiliy from boh consumpion and erminal wealh We now consider an invesor who derives uiliy boh from living well (i.e., from consumpion) and from becoming rich (i.e., from erminal wealh). His endowmen is an iniial posiive wealh x bu all he ime he has o share his wealh according o a sock porfolio p( ) and a consumpion rae C( ). His aim is o maximize he uiliy of his consumpion and erminal wealh. His expeced oal uiliy is hen

8 Opimal Consumpion and Invesmen Porfolio in Jump markes J( x; p, C) J ( x; p, C) + J ( x; p, C) = E G U ( C ) d + E[ G U ( X )] (3) and he ries o maximize J( x; p, C ) over A ( x ) A ( x) Ç A ( x) :, V( x) sup J( x; p, C) ( p, C) ÎA ( x), (33) Unlike he wo problems sudied already, his one calls for balancing compeing objecives. Single-minded deerminaion o maximize J ( x; p, C ) mandaes no consumpion a all. On he oher hand, single-minded maximizaion of J ( x; p, C ) will leave he invesor broke a he end (cf. heorem.3 and Proposiion.9 in [Jin, 4]). We will draw a proper compromise beween hese wo compeing objecives. A ime = he invesor simply divides he endowmen x ino wo pars x ³ and x ³ wih x+ x = x; for he amoun x (respecively, x ), he invesor will face, form hen on, an opimizaion problem wih uiliy coming only from consumpion (respecively, only from erminal wealh). I will be shown jus how x, x should be deermined, in order for he resuling procedure o be opimal. hroughou his secion i will be assumed ha U, U are uiliy funcions, wih (), (4), (8) and (3) hold, and Assumpion (A ), (A ) also in force. Proposiion 4.. For any x > and an arbirary porfolio/consumpion pair ( p,c) Î A, ( x), le x E b C d (34) hen here exiss a pair ( p,c) Î ( x) A, such ha Jx ( ; p, C) Jx ( ; p, C ) = V( x) + V( x-x) (35) In paricular, V( x) V ( x) max [ V ( x ) + V ( x )] = max [ G ( y ) + G ( y )] * x, xî [, ) y, yî (, ] x+ x= x X( y) + X( y) = x y Y ( x ), y Y ( x ) (36) Proof. Omied. Corollary 4.. In he problem (33), he consrain condiion can be an equaliy, i.e., é æ ö E z X + z( s) C( s) ds = E b X + b( s) C( s) ds = x ê ú ç è ë û (37) ø Proof. Omied. herefore, from Proposiion 4., he quesion is o find x, x for which he maximum in (36) is achieved, because hen he oal expeced uiliy corresponding o he pair ( p,c ) in Proposiion 4. will be exacly equal o V* ( x ); his will in urn imply V( x) = V ( x) (38) from (36), and hus ( p,c ) will be shown o be opimal for he problem of (33). In order o find dv [ ( x) + V( x) ] ou he above menioned x, x, le =, hen we have dx V ( x )- V ( x ) =, i.e., Y ( x ) - Y ( x ) =, where x = x-x. And because *

9 Opimal Consumpion and Invesmen Porfolio in Jump markes d [ V( x) + V( x) ] = V( x) + V ( x) = Y( x) + Y ( x) < dx (i.e., V ( x ) + V ( x ) is sricly concave on (, )), he maximum over x, x Î (, ), x + x = x indicaed in (36) is obained by x, x > which saisfy Y( x) = Y ( x). And he consrained maximizaion in he las expression of (36) is achieved by y = Y ( x ) = Y ( x ) = y y (39) where his common value is deermined uniquely by X ( y ) + X ( y) = x (4) (Uniqueness can be proved by he sric decrease of X, X ). In oher words, we find hose values of x, x for which he marginal expeced uiliies V ( x), V ( x ) from he wo individual opimizaion problems are idenical. In order o solve he problem (33) under he consrain of (37), we inroduce he Lagrangian é é L( XC,, l) = E G +G - l z + z - ê () U( C ()) d U X E ë ê () Cd () X x. ë Imiae o he previous secions, le he formal derivaives of L wih respec o ( X, C,l ) be equal o zero, we obain X * = I D l *, C * = I D l * (4) ( ) where l * saisfies é * * E z() I( D l ) d + z I( D l ) = x ê (4) ë Compare wih (4), l * in (4) is exacly he y in (4). he consequence of Lagrangian is idenical wih he preceding discussions (in order o make he Lagrangian feasible, we need some supplemenal condiions, please refer o he corresponding par of previous discussion). Opimizaion problem in he case wih deerminisic coefficiens he heory developed previously provides a precise characerizaion of he value funcion for he opimiion problem, as well as explici formulas for he opimal processes of * * consumpion rae C and erminal wealh X ( ). Bu for he opimal porfolio process p *, he maringale mehodology ha we have employed so far is able o ascerain only is exisence; in general, here is no useful characerizaion ha could lead o is compuaion. In his secion supplemenary assumpions of deerminisic coefficiens on he model enable us o obain he opimal pair of porfolio/consumpion in an explici feedback form on he curren level of wealh. Specifically, we shall assume hroughou his secion ha g, r, b, s, r, d, l are all deerminisic, and assume ha U >-, U >-. And we will consider he consumpion/invesmen problem wih iniial ime ³. Furhermore, we shall assume ha all sochasic processes in his secion are Fs -adaped (for he definiion of F s ). We wrie he analogues of he wealh equaion as

10 Opimal Consumpion and Invesmen Porfolio in Jump markes dx( s) = ( r( s) X( s) - C( s) ) ds + p ( s)[ b( s) + d( s) -r( s) ] ds + p ( s) s( s) dw( s) + p ( s) r( s) dq( s), X () = x>, s, and he value funcion é V(, x) = sup E G ( s) U( C( s) ) ds +G U( X ), ( p, C) Î (, x) ê A ë é V(, x) = sup E G ( s) U( C( s) ) ds, ( p, C) Î (, x) ê A ë V(, x) = sup EéG U( X ) ë û, ( p, C) ÎA(, x) (43) (44) where A ( x, ) consiss of hose admissible porfolio/consumpion process pairs for which he corresponding wealh process X ( s ) decided by (43) remains nonnegaive, a.s., and s g( sdu ) G ( s) = e - for s (45) Now we ake he dynamic programming principle ino accoun. For any a Î [, ], we have s - ( udu ) ( udu ) e g - g U( C( s) ) ds + e U( X ) s s a - ( udu ) ( udu ) ( udu ) e g - U( C( s) ) ds e g - U( C( s) ) ds e g ( ) U X a s a s a - ( udu ) ( udu ) g - g g( udu ) g( udu ) ( ) é - - a ( ) a êa = + + = e U C s ds + e e U C s ds + e U ( X ) ú ë û. Using dynamical programming principle, we have ( p, C) ÎA(, x) a { Vx (, ) = sup E G ( su ) ( Cs ) ds é a a ü +G ( a) sup E G ( s) U( C( s)) ds + G U( X) ï ý ( p, C) Î ( a, X( a)) ê A ë a ú û ï ïþ é a = sup E G ( s) U( C( s)) ds +G ( a) V( a, X( a)), ( p, C) Î (, x) ê A ë where G ( a) V( a, X( a)) is he discouned opimal value expeced a ime a when we have used he conrol pair ( p,c ) and V(, x ) is he value funcion. hus for each pair ( p, C) Î A (, x), (46) shows ha é a V(, x) ³ E G ( s) U( C( s) ) ds +G ( a) V( a, X( a) ) ê (47) ë In his secion, we should have he following assumpions in force. (A 3 ) Vsy (, ) defined in (44) is coninuously differeniable wih respec o s. hen by he previous discussion, Vsy (, ) is a funcion of wice-coninuously differeniable in y and once-coninuously differeniable in s. Le us inroduce some noaion for laer use. (46)

11 Opimal Consumpion and Invesmen Porfolio in Jump markes Vuy (, ) Ku (, p, Cy, ) U( C) + é yru C p ( bu d( u) ru ) y ë û Vuy (, ) + p s( u) y é Vuy (, ) + l( u) Vuy (, + p r( u)) -Vuy (, )-p r( u), êë y d + u, p Î, C ³, y ³, u VsXs (, ()) LM( u) G ( s) p ( s) s( s) dw( s) X u + G ( s) é V( s, Xs p ( s) r( s)) VsXs (, ) ë û dqs. I is obvious ha LM( u ) is a local maringale. Apply Iô s formula o he process G ( sv ) ( s, X( s)) on [, + d], < d < -, where Xs is he soluion of (43), we arrive o heorem 5.. Suppose ha LM( u), u is a maringale and (H ) holds, hen for any (, x) Î [, ] +, V saisfies he following HJB (Hamilon-Jacobi-Bellman) equaion Vx (, ) - g() V(, x) + sup K (, p(), C (), x) =, <, ( p, C) ÎA (, x) (49) Vx (, ) = U( x), and he opimal pair ( p *, C * ) can be found ou by solving sup K (, p, C, x) ( p, C) ÎA(, x) (48) in (49). Proof. Omied. Corollary 5.. Wih all he condiions in heorem 5., and he supplemenary hypohesis ha no jump occurs in he complee financial model, i.e., s() = s() is a d d marix, r () º, and s s is srongly nondegenerae for any Î [, ], hen he opimal pair ( p *, C * ) is given by * C () = I( Y(, x)), * - Y(, x) (5) p () =- [ s() s ()] [ b() + d() -r()]. Y (, x) Proof. Omied. CONCLUSION We ge he formulae for opimal consumpion and porfolio of invesmen in a financial marke wih jumps. And under some condiions, we can ge he closed-form soluions for hese problems alhough i is impossible in general. And we should emphasize ha all he discussions were carried ou under he condiion ha all socks prices can be expressed by linear sochasic equaions. However, he discussions can be exended o he siuaion when socks prices are expressed by nonlinear sochasic equaions. I will do his exension laer. REFERENCES Chow, G.C., 997, Dynamic economics: opimizaion by he Lagrange mehod, Oxford Univ. x

12 Opimal Consumpion and Invesmen Porfolio in Jump markes Press. Jeanblanc-Picquè, M. & M. Ponier, 99, Opimal porfolio for a small invesor in a marke model wih disconinuous prices, Appl. Mah. Opim., Jin, G., 4, porfolio managemen deermined by iniial endowmen or erminal wealh in a consumer finance marke wih jumps, Proceedings of DSI 4, Karazas, I., 989, Opimizaion problems in he heory of coninuous rading, SIAM J. Conrol Opim. 7(6), -59. Karazas, I., Lehoczky, J.P., Sehi, S.P. & Shreve, S.E., 986, Explici soluion of a general consumpion /invesmen problem, Mah. Oper. Res., Karazas, I., Lehoczky, J.P. & Shreve, S.E., 987, Opimal porfolio and consumpion decisions for a small invesor on a finie horizon, SIAM J. Conrol Opim. 5, Karazas, I. & Shreve, S.E., 988, Brownian moion and sochasic calculus, Springer-Verlag, New York. Meron, R.C., 994, Coninuous-ime finance, Blackwell. Peng, S., 994, Backward sochasic differenial equaions, Lecure noes on sochasic calculus and applicaions o mahemaical finance, CIMPA school, Beijing.