PORTFOLIO OPTIMIZATION WITH JUMPS AND UNOBSERVABLE INTENSITY PROCESS

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1 PORTFOLIO OPTIMIZATION WITH JUMPS AND UNOBSERVABLE INTENSITY PROCESS Nicole Bäuerle Insiue for Mahemaical Sochasics, Universiy of Karlsruhe, Germany Ulrich Rieder Deparmen of Opimizaion and Operaions Research, Universiy of Ulm, Germany We consider a financial marke wih one bond and one sock. The dynamics of he sock price process allow jumps which occur according o a Markov-modulaed Poisson process. We assume ha here is an invesor who is only able o observe he sock price process and no he driving Markov chain. The invesor s aim is o maximize he expeced uiliy of erminal wealh. Using a classical resul from filer heory i is possible o reduce his problem wih parial observaion o one wih complee observaion. Wih he help of a generalized Hamilon- Jacobi-Bellman equaion where we replace he derivaive by Clarke s generalized gradien, we idenify an opimal porfolio sraegy. Finally, we discuss some special cases of his model and prove several properies of he opimal porfolio sraegy. In paricular we derive bounds and discuss he influence of uncerainy on he opimal porfolio sraegy. Key words: jump-diffusion process, filering, uiliy maximizaion, sochasic conrol, generalized HJB equaion, opimal porfolio sraegies, Bayesian conrol, sochasic comparison 1 Inroducion We consider an incomplee financial marke wih one bond and one sock. The sock price process allows for jumps of a random heigh where he jump ime poins are generaed by a Markov-modulaed Poisson process. There is an invesor who wans o maximize his uiliy from erminal wealh and who is only able o observe he sock price process. In paricular he is no informed abou he sae of he coninuous-ime Markov chain which drives he jump inensiy. Such a model is also called a Hidden Markov Model. For a general reamen of such models see e.g. Ellio e al. (1994). A model wih unobservable inensiy process is naural, since jumps in he sock price process are ofen generaed by various exernal evens whose impac on he sock marke Acknowledgemen: We are graeful o an anonymous referee for careful reading. Address correspondence o Nicole Bäuerle, Insiue for Mahemaical Sochasics, Universiy of Karlsruhe, D Karlsruhe, Germany, baeuerle@soch.uni-karlsruhe.de

2 canno compleely be analyzed. We are only able o draw some conclusions abou he jump inensiy from he observaion of he sock prices. Also i is more appropriae o allow a sochasically varying jump inensiy since a deerminisic jump inensiy seems only o be realisic for a shor period of ime. A coninuous-ime Markov chain can model he changing condiions which give rise o a changing jump behavior. This underlying Markov chain can be inerpreed as an environmen process which collecs facors which are relevan for he sock price dynamics like echnical progress, poliical siuaions, law or naural caasrophes. There is an exensive lieraure on porfolio opimizaion wih parial observaion as well as on porfolio opimizaion wih disconinuous sock price processes. In his paper we will rea hese wo aspecs in one model. Mos papers on problems wih parial observaion deal wih he case of an unobserved (sochasic) appreciaion rae process (µ ). Lakner (1995, 1998) for example reas he case where he appreciaion rae follows a linear Gaussian model. The mos recen papers by Honda (23), Sass and Haussmann (24), Haussmann and Sass (24) and Rieder and Bäuerle (25) consider a Hidden Markov Model for (µ ). We refer he reader o hese papers for a recen survey on financial models wih parial observaion. Of course i would be more realisic o assume ha boh he appreciaion rae and he jump inensiy depend on he hidden Markov chain bu his seems o be oo challenging a he momen. For a risk averse invesor i migh be more imporan o model an unobserved jump inensiy han an unobserved appreciaion rae, since poenial losses due o jumps can be much higher. In order o solve hese problems he usual echnique is o use he well-esablished filer heory o reduce he sochasic conrol problem wih parial observaion o one wih complee observaion. I is hen possible o solve his problem eiher wih sochasic conrol mehods or via he maringale approach (which is mosly done in case of a complee marke in he lieraure). On he oher hand here exis several papers on porfolio opimizaion problems wih disconinuous sock price processes, in paricular in he case where he price is modelled wih he help of a Lévy process. Empirical work has shown ha logreurns are in general no normally disribued and ha sock price models should conain a jump componen. In Framsad e al. (1999), he auhors deal wih he problem of opimal consumpion and porfolio selecion in a model where he sock price follows a geomeric Lévy process. They assume a power uiliy and solve he problem explicily by showing ha he value funcion is a classical soluion of he associaed Hamilon-Jacobi-Bellman equaion. Benh e al. (21) consider a similar quesion in he case ha he sock price is given by an exponenial Lévy process. They have o use he noion of a consrained viscosiy soluion o characerize heir soluion. As we will see laer, our price process canno be wrien as a funcional of a Lévy process. Imporan applicaions of opimizaion problems wih jumps are well-known in risk heory and insurance mahemaics (see e.g. Hipp and Plum (23), Schmidli (22)). Bu in hese papers he inensiy process is always observable. In his paper we combine he jump diffusion model wih an unknown jump inensiy. Our main conribuions are a non-sandard approach o solve he sochasic conrol problem by a generalized Hamilon-Jacobi-Bellman (HJB) equaion which migh be ineresing for oher porfolio problems as well and moreover, a sudy of he influence of uncerainy on he opimal porfolio sraegy. The ouline of he paper is as follows. In Secion 2 we give a precise mahemaical formulaion of our model and define he opimizaion problem. In Secion 3 we show how we can use filering heory o reduce he problem o one wih complee observaion. The reduced marke model we end up is no complee. Thus, he maringale approach canno be 2

3 applied direcly. In he case of a logarihmic uiliy funcion, i is shown in secion 4 ha he porfolio opimizaion problem can be solved (as usual) raher easily by a pahwise opimizaion. Secion 5 deals wih he power uiliy. Here we use he heory of sochasic conrol o solve he problem explicily. The value funcion is no a classical soluion of he corresponding Hamilon-Jacobi-Bellman equaion, however, we can characerize he value funcion as a soluion of a generalized HJB equaion where we use he Clarke generalized gradien. This is possible since he value funcion can be shown o be locally Lipschiz-coninuous and almos convex. This approach is non-sandard and has he advanage ha he opimal porfolio sraegy can be given raher explicily. Mos ineresing is he fac ha he expression for he opimal porfolio sraegy includes he value funcion bu no derivaive of i. Secion 5.1 conains he main resuls. In Secion 5.2 we deal wih some special cases and derive some imporan properies and comparison resuls for he opimal porfolio sraegy. In paricular we highligh he role of uncerainy in our model. I urns ou ha adding more uncerainy by jumps reduces he invesmen in he sock for all invesors wih power uiliy. Moreover, we look a he Bayesian model, i.e. when he jump inensiy is consan bu unobservable. We are able o derive bounds and compare he opimal porfolio sraegy in his case wih he sraegy we obain in a model where he consan jump inensiy is equal o he esimaed one. The comparison depends on wheher jumps go upwards or downwards and on he parameer of he power uiliy funcion. In his case furher uncerainy does no auomaically lead o a smaller invesmen in he sock. Some auxiliary resuls which are needed for he proof of our main heorems are given in Secion 6. Secion 7 finally conains he proofs of our main heorems in he case of a power uiliy. 2 The Model We consider a financial marke wih one bond and one risky asse. More precisely le (Ω, F, F = {F, T }, P ) be a filered probabiliy space. T > is a fixed ime horizon. The bond price process (B ) evolves according o db = rb d wih ineres rae r > and he sock price process evolves according o ) ds = S (µd + σdw + yn(d, dy), where σ >, µ IR are given consans and (W ) is a Brownian moion w.r.. F. N is he random couning measure of a Markov-modulaed compound Poisson process. Tha means N is consruced as follows: le us denoe by (Y ) a coninuous-ime Markov chain wih sae space {e 1,..., e d } where e k is he k-h uni vecor in IR d and (Y ) has he generaor Q = (q ij ). q ij is he inensiy of geing from sae e i in sae e j. Furher le us denoe by T =, T 1, T 2,... he jump ime poins of a Poisson process wih F-inensiy (λ ) := (λ Y ) where λ = (λ 1,..., λ d ) IR d +, i.e. as long as Y = e j, jumps arrive a rae λ j. Finally we assume ha (δ n ) is a sequence of independen and idenically disribued random variables bounded from above wih δ n > 1. The probabiliy disribuion of δ n is denoed by Q. Then we have N = n 1 ɛ (Tn,δ n) 3

4 where ɛ x is he one poin measure in x. δ n is he relaive jump heigh of he sock a ime T n. Noe ha he resricion δ n > 1 guaranees ha he sock price says posiive. The compound Poisson process which describes he jumps is hen obained by yn(ds, dy) = The economic inerpreaion of (Y ) is some kind of environmen process which collecs facors which are relevan for he sock price dynamics like e.g. echnical progress, poliical siuaions, laws or naural caasrophes. These facors change sochasically over ime. All processes are adaped w.r.. F and (W ) and N are independen as well as (δ n ) and (Y ). In wha follows we assume an invesor who is only able o observe he sock price process and who knows he disribuion of Y. This means ha he invesor is no informed abou he inensiy wih which he sock price process jumps. Of course i is more realisic o assume ha he appreciaion rae of he sock price also depends on he unobservable environmen process (Y ), however his case is much more challenging. The case of unobservable Markovmodulaed appreciaion rae has been invesigaed in Honda (23), Sass and Haussmann (24), Haussmann and Sass (24) and Rieder and Bäuerle (25) among ohers. Le F S = (F S ) be he filraion generaed by he sock price process (S ). Our aim is o solve he opimizaion problem our invesor faces, when he ries o find porfolio sraegies ha maximize he expeced uiliy from erminal wealh. We resric ourselves o self-financing porfolio sraegies and denoe by π [, 1] he fracion of he wealh invesed in he sock a ime. The resricion of he fracion o [, 1] ( no shor-sellings ) guaranees a posiive wealh process which is reasonable for logarihmic and power uiliy. Due o he jumps of he sock price a violaion of his resricions may lead o a negaive wealh wih posiive probabiliy. The process π = (π ) is called porfolio sraegy. An admissible porfolio sraegy has o be F S -predicable and akes values in [, 1]. Thus, we inroduce he se U[, T ] := {π = (π s ) s T π s [, 1] for all s [, T ], π is F S predicable.}. The wealh process under an admissible porfolio sraegy π U[, T ] is given by ) d X π = X ((r π + (µ r)π )d + σπ dw + π yn(d, dy). We assume ha X π = x is he given iniial wealh. Le U : IR + IR be an increasing, concave uiliy funcion. Then we define he value funcions for π U[, T ], [, T ], x > by [ Ṽ π (, x) := E,x U( X ] T π ) F S N n=1 δ n. Ṽ (, x) := sup Ṽ π (, x) π U[,T ] where he expecaion is aken w.r.. he probabiliy measure P,x wih X π = x. Noe ha Ṽ π (, x) and Ṽ (, x) are random variables, in paricular Ṽπ(, x) is F S -measurable. Moreover, Ṽ π (, x ) and Ṽ (, x ) depend on he disribuion of Y which is fixed. A porfolio sraegy π U[, T ] is opimal if Ṽ (, x ) = Ṽπ (, x ). 4

5 We have chosen he appreciaion rae and he volailiy o be consan. All he analysis which follows can be done in a similar way if hey are modelled by bounded, deerminisic (observable) processes. 3 The Reducion We can reduce he conrol problem o one wih complee observaion. This procedure is classical. The idea is o updae our belief abou he disribuion of he environmen sae Y coninuously and make i par of our sae space. Noe ha only he jump ime poins of he sock price conain relevan informaion for esimaing he environmen sae and hus he jump inensiy. This coninuous esimaion is done by he so-called Wonham filer. We proceed as in Brémaud (1981) p. 94 ff. Define p k () = P (Y = e k F S ), k = 1,..., d and p = (p 1 (),..., p d ()). p k () is he probabiliy ha he environmen process is in sae k a ime, given ha we have observed he sock price process unil ime. The process (p ) is called filer process. Recall form secion 2 ha he Markov-modulaed Poisson process is given by N(ds, dy) and couns he number of jumps in he sock price unil ime. This couning process has F-inensiy (λ ) = (λ Y ) which is equivalen o η := N(ds, dy) λ s ds being an F-maringale. The following saemens hold: Lemma 3.1: There exiss an F S maringale (ˆη ) such ha a) he filer processes p k () saisfy for dp k () = j ( λk q jk p j ()d + p k ( ) ˆλ ) dˆη ˆλ wih ˆλ := d k=1 λ k p k () = E[λ F S ]. b) λ d + dη = ˆλ d + dˆη. c) (W ) and (ˆη ) are independen. Par b) of Lemma 3.1 saes ha he Markov-modulaed Poisson process N(ds, dy) admis an F S -inensiy (ˆλ ) and can be compensaed in order o obain an F S -maringale ˆη := N(ds, dy) ˆλ s ds. 5

6 In wha follows we also need he compensaed random measure Noe ha ˆM(d, dy) = N(d, dy) ˆλ dq(dy). f(s, y)n(ds, dy) f(s, y)ˆλ s dsq(dy) is a maringale for arbirary f whenever he inegrals exis. The conrol model wih complee observaion is now characerized for π U[, T ] by he following d+1-dimensional sae process: ( ( ) dx π = X π r + π (µ r) + π ˆλ yq(dy) )d + σπ dw + π y ˆM(d, dy) X π = x dp k () = ( λk q jk p j ()d + p k ( ) ˆλ ) dˆη j ˆλ p k () = P (Y = k), k = 1,..., d where P (Y = k), k = 1,..., d, is he given disribuion of Y. A soluion of he sochasic differenial equaion for he wealh process is given by { X π = x exp (r + (µ r)π s 1 } 2 σ2 πs)ds 2 + σπ s dw s + ln(1 + π s y)n(ds, dy) { = x exp (r + (µ r)π s 1 2 σ2 πs 2 + ˆλ ) s ln(1 + π s y)q(dy) ds + σπ s dw s + ln(1 + π s y) ˆM(ds, dy) } By d we denoe he probabiliy simplex in IR d. The value funcions in he reduced model are for π U[, T ] and p d, [, T ], x > defined by V π (, x, p) := E,x,p [U(X π T )]. V (, x, p) := sup V π (, x, p) π U[,T ] where E,x,p is he condiional expecaion, given X π = x, p = p. The reduced model now solves our original problem. This is ofen aken for graned, however i has o be proved formally. The nex heorem saes ha he filer conains he necessary informaion in order o solve our original problem. Insead of he whole hisory F S i is sufficien o know p. More precisely, Ṽ (, x) depends on he hisory F S only hrough p. Theorem 3.2: For all π U[, T ] i holds ha V π (, x, p ) = Ṽπ(, x) and V (, x, p ) = Ṽ (, x) for all x >, [, T ]. Proof: From Lemma 3.1 and he sochasic differenial equaion for he wealh process i follows ha X T π = Xπ T a.s. for all π U[, T ] which obviously yields he saemen. In he reduced model, all processes are F S -adaped and admi an F S -inensiy respecively. 6

7 Therefore, we can solve his problem by sochasic conrol echniques. The following properies are easily derived. Lemma 3.3: a) For all π U[, T ], p d and x > we have d V π (, x, p) = p j V π (, x, e j ). j=1 b) The mapping p V (, x, p) is convex for all [, T ] and x >. Proof: Par a) is obained by condiioning. For b) le p, q d be wo iniial disribuions and α [, 1]. Then V (, x, αp + (1 α)q) = sup α V π (, x, e j )p j + (1 α) V π (, x, e j )q j π U[,T ] j j α sup V π (, x, e j )p j + (1 α) sup V π (, x, e j )q j π U[,T ] j π U[,T ] j = αv (, x, p) + (1 α)v (, x, q). 4 Logarihmic Uiliy In his secion we briefly summarize he resuls in he case of a logarihmic uiliy funcion U(x) = log(x). This is always he easies case. For π U[, T ] we obain from he explici soluion for X π V π (, x, p) = log(x) + h π (, p) where [ T h π (, p) = E,p r + (µ r)π s 1 2 σ2 πs 2 + ˆλ s log(1 + π s y)q(dy)ds ]. Noe ha h π does no depend on x. Obviously we obain he following resul: Lemma 4.1: a) For all [, T ], x >, p d we have V (, x, p) = log(x) + h(, p), where h(, p) = sup π U[,T ] h π (, p). 7

8 b) Suppose ha for all p d, u (p) maximizes u r + (µ r)u 1 2 σ2 u 2 + λ p log(1 + yu)q(dy) on [, 1] hen π = (π ) U[, T ] wih π = u (p ) is an opimal porfolio sraegy for he given porfolio problem. Noe ha π depends on F S only hrough p. I is easy o show ha in he case of complee observaion, i.e. when we know ha he sae of he Markov chain is for example e i, he opimal porfolio sraegy would be o inves a consan fracion u of he wealh in he sock, where u is he maximizer of u r + (µ r)u 1 2 σ2 u 2 + λ i log(1 + yu)q(dy) on [, 1]. Par b) of Lemma 4.1 shows ha he so-called cerainy equivalence principle holds, i.e. he unknown inensiy λ is replaced by he esimae ˆλ = E[λ F S ] in he opimal porfolio sraegy (cf. Kuwana (1991)). This means ha uncerainy abou he jump inensiy does no change he opimal porfolio sraegy in his case. The siuaion is compleely differen in he case of a power uiliy funcion as we will see in secion Power Uiliy In his secion we assume ha he uiliy funcion is given by U(x) = 1 γ xγ for γ < 1, γ. The value funcion under sraegy π U[, T ] is herefore where V π (, x, p) = 1 γ xγ g π (, p), [ { g π (, p) = E,p T exp γ(r + (µ r)π s 1 T 2 σ2 πs)ds 2 + γσπ s dw s T +γ ln(1 + π s y)n(ds, dy)} ]. Noe ha g π does no depend on x. If we define (1) hen i obviously holds ha g(, p) := sup g π (, p) π U[,T ] V (, x, p) = 1 γ xγ g(, p). 8

9 5.1 The Soluion of he Porfolio Opimizaion Problem In his secion we summarize he main resuls. We use a sochasic conrol approach o solve he problem. Unforunaely i is no clear wheher he value funcion is coninuously differeniable in p and and we hus are no able o obain a classical soluion for he associaed Hamilon- Jacobi-Bellman (HJB) equaion. The sandard way would hen be o show ha he value funcion is he unique viscosiy soluion of he HJB equaion. However his ype of soluion is quie weak and he uniqueness proof can be hard. In our seing i is possible o show ha he value funcion is locally Lipschiz-coninuous and hus almos everywhere differeniable. This is much more han coninuiy which is required for he viscosiy soluion. Therefore we decided o pursue a differen approach by considering a generalized HJB equaion where he classical derivaive is replaced by a generalized derivaive. A similar approach has been used by Davis (1993) for piecewise deerminisic models. We can show ha he value funcion is he unique soluion of he generalized HJB equaion and he maximizer yields an opimal porfolio sraegy. In his secion we only presen he resuls, proofs are posponed o secion 7. For he analysis, i is imporan o noe ha (p ) is a piecewise deerminisic process wih jumps appearing according o he F S -inensiy (ˆλ ). We denoe by φ k (, p ) = p k () + j q jk p j (s) p k (s)(λ k ˆλ s )ds, k = 1,..., d and φ(, p ) = (φ 1 (, p ),..., φ d (, p )) he evoluion of he filer beween jumps and by ( λ1 p 1 J(p) = λ p,..., λ ) dp d λ p he new sae of he filer direcly afer a jump from sae p. Moreover, we le S d be he inerior of he probabiliy simplex d. In order o obain a reasonable model we assume now ha all saes of he Markov chain Y communicae. Thus, he filer process p will for > always say in S d. Le us inroduce he following operaor L which acs on funcions v : [, T ] S d IR and u [, 1] Lv(, p, u) := v(, p)(r + (µ r)u (γ 1)σ2 u 2 ) ( ) + λ p v(, J(p)) (1 + yu) γ Q(dy) v(, p). γ In order o moivae he HJB equaion of his problem, we give some heurisic argumens. For his purpose, suppose ha he value funcion V is sufficienly differeniable. An applicaion of Io s Lemma gives: T T V (T, XT π, p T ) = V (, x, p) + V (s, Xs π, p s )ds + V x (s, Xs π, p s )dxs π d T + V pk (s, Xs π, p s )dp k (s) + 1 T V xx (s, Xs π, p s )σ 2( ) 2π X π 2 s k=1 2 s ds + [V (s, Xs π, p s ) V (s, Xs, π p s )] V x (s, Xs π, p s ) Xs π <s d V pk (s, Xs π, p s ) p k (s). k=1 9

10 I can be shown ha V (, X π, p ) is a maringale under he opimal porfolio sraegy and a supermaringale under any admissible sraegy. Thus, he drif erms in he preceding equaion have o be zero. Moreover, plugging in he form V (, x, p) = 1 γ xγ g(, p) yields as an opimaliy condiion: = 1 γ g (, p) + g(, p) (r + π(µ r) (γ 1)σ2 π 2) ( ) g(, J(p)) (1 + yπ) γ Q(dy) g(, p) + λ p γ + 1 d ( g pk (, p) γ k=1 j ) q jk p j p k (λ k λ p). However, since he value funcion (in paricular g defined in equaion (1)) is probably no differeniable w.r.. g k we replace he gradien by he Clarke generalized gradien. The resuling generalized Hamilon-Jacobi-Bellman equaion for our problem hen reads as follows = sup {Lg(, p, u)} + u [,1] { 1 sup θ g(,p) γ θ + 1 γ d ( )} θ k q jk p j p k (λ k λ p) wih boundary condiion g(t, p) = 1 for all p S d. The se g(, p) IR d+1 denoes he Clarke generalized gradien (see Clarke (1983)). This is a weaker noion for differeniabiliy which is defined as follows: le f : IR d IR be a locally Lipschiz coninuous funcion. For x, y IR d he upper generalized direcional derivaive of f a x in direcion y is defined by f (x; y) := lim sup z x,ε k=1 f(z + εy) f(z). ε The Clarke generalized gradien of f a x is now defined by he se f(x) := {θ IR n f (x; y) θy for all y IR d }. f(x) is a non-empy, convex, compac subse of IR d and if f is differeniable a x, hen f(x) := { f(x)}. Moreover, since f is locally Lipschiz coninuous, i is almos everywhere differeniable and we can find for every poin x IR d sequences of poins x n IR d such ha lim n x n = x and f is differeniable a x n. f(x) can hen be wrien as he closed convex hull of exising limis of sequences f(x n ), i.e. f(x) := co{lim sup f(x n ) n Our firs resul is a verificaion heorem: j lim x n = x}. n Theorem 5.1: Suppose here exiss a bounded funcion v : [, T ] S d IR + such ha for all p S d, v(, φ(, p)) is absoluely coninuous, v(t, p) = 1 and v saisfies he generalized HJB equaion. Furher assume ha u is a maximizer of he generalized HJB equaion, i.e. for all [, T ] and p S d, u (, p) maximizes u Lv(, p, u) on [, 1]. 1

11 Then V (, x, p) = 1 γ xγ v(, p) and he sraegy π = (π ) U[, T ] wih π := u (, p ) is an opimal feedback sraegy for he given porfolio problem. Noe ha π depends on F S only hrough p. The nex heorem saes he exisence of a soluion of he generalized HJB equaion. Theorem 5.2: The value funcion of our problem is given by V (, x, p) = 1 γ xγ g(, p) wih g defined by (1) above and g saisfies he generalized HJB equaion = sup {Lg(, p, u)} + u [,1] { 1 sup θ g(,p) γ θ + 1 γ d ( )} θ k q jk p j p k (λ k λ p) wih boundary condiion g(t, p) = 1 for all p S d. Moreover, π from Theorem 5.1 (wih v replaced by g) is an opimal porfolio sraegy. k=1 j 5.2 Special Cases and Properies of he Opimal Porfolio Sraegy In his secion we invesigae he opimal porfolio sraegy in some special cases in greaer deail and esablish some ineresing properies. In paricular we discuss he influence of uncerainy on he opimal porfolio sraegy. A) Jumps occur wih known and consan inensiy Suppose ha δ n δ ( 1, ) is deerminisic and ha he jumps in he sock price process occur wih known consan inensiy λ >, i.e. λ = λ 1 =... = λ d. This model is similar o he seup invesigaed in Øksendal and Sulem (24) and Framsad e al. (1999). In his case i is opimal o inves a consan fracion u δ (λ) (independen of ime) of he wealh in he sock. Specializing our HJB equaion (noe ha J(p) = p in his case), i is easy o see ha u δ (λ) is he maximum poin of he mapping u (µ r)u (γ 1)σ2 u 2 + λ γ (1 + δu)γ on [, 1]. In his case i can also be shown ha he value funcion is a classical soluion of he HJB equaion. In wha follows we wan o compare he opimal fracions which are invesed in he sock in differen models. In paricular we highligh he role of uncerainy. For his ask he following simple lemma is useful: Lemma 5.3: Le f, h : [, 1] IR be coninuous funcions and suppose ha h is increasing. If we denoe u f := argmax {f(u) u [, 1]} u f+h := argmax {f(u) + h(u) u [, 1]} 11

12 hen u f u f+h. Throughou he paper we use increasing and decreasing in he non-sric sense. A direc implicaion of he previous lemma is Lemma 5.4: If δ <, hen λ u δ (λ) is decreasing and if δ >, hen λ u δ (λ) is increasing. Of course his resul is no surprising. If we have downward jumps, we inves less in he sock if he jump inensiy increases. In order o invesigae he influence of furher uncerainy we have o add a jump maringale o he sock price o keep he expeced drif unchanged. Thus, suppose for a momen ha he sock price process evolves according o ds = S (ˆµd + σdw + δdη ), where η = N(ds, dy) λ. If we se ˆµ = µ+λδ we obain a sochasic differenial equaion for he sock price in he form given in secion 1. In he case wihou jumps (δ = ), we know ha he opimal fracion maximizes u (ˆµ r)u (γ 1)σ2 u 2 on [, 1]. In he case wih jumps (δ ), we know ha he opimal fracion maximizes u (ˆµ r)u (γ 1)σ2 u 2 + λ γ (1 + δu)γ λδu on [, 1]. Thus we obain he following comparison resul: Theorem 5.5: In he previous model we have u u δ(λ). Proof: In view of Lemma 5.3 i is sufficien o show ha h(u) := λ γ (1 + δu)γ λδu is decreasing for all δ > 1, δ. This can be done by showing ha h (u). Theorem 5.5 means ha he opimal fracion invesed in he sock in he model wih furher uncerainy coming from jumps is always less or equal o he opimal fracion in he model wihou jumps. Noe ha he expeced drif of he sock remains he same in boh scenarios. Since we have a risk averse invesor such a resul is no unexpeced. However also noe ha he saemen is rue for all γ < 1, γ. In B) we will observe a differen behavior. B) Jumps occur wih unknown and consan inensiy - he Bayesian case Suppose ha δ n δ ( 1, ) is deerminisic and ha he jumps in he sock price process occur wih unknown consan inensiy λ >. We assume ha λ can be one of he possible 12

13 values λ 1... λ d and ha he iniial probabiliy p S d for he values is given. Thus, we have a Bayesian conrol problem wih an unknown parameer. This is a special case of our model, if we formally se he inensiy marix of he Markov chain (Y ) o zero, i.e. Q = and he Markov chain says in he iniial sae. If we define p k () = P (Y = e k F S ) = P (λ = λ k F S ) and p = (p 1 (),..., p d ()), hen he following equaion holds ( λk p k () = p k () + p k (s ) ˆλ ) s dˆη s ˆλ s where (ˆη ) is defined as in Lemma 3.1. The opimal fracion π invesed in he sock depends on he ime and he esimae p, i.e π = u δ (, p ) and u δ maximizes u (µ r)u (γ 1)σ2 u 2 + λ p γ g(, J(p)) (1 + δu) γ on [, 1]. g(, p) I is possible o compare he opimal porfolio sraegy of his scenario wih he previous case A) of complee observaion. Theorem 5.6: The opimal fracion u δ (, p) invesed in he sock has he following properies: a) If δ < (downward jumps) i holds for all (, p) [, T ] S d ha u δ(λ d ) u δ(, p) u δ(λ 1 ). If δ > (upward jumps) he inequaliies are reversed. b) If δγ < i holds for all (, p) [, T ] S d ha If δγ > he inequaliy is reversed. u δ(λ p) u δ(, p). Proof: a) Suppose δ <. In view of Lemma 5.3 i suffices o show Recall ha λ p g(, J(p)) λ 1 g(, p) and λ p g(, J(p)) λ d g(, p). Now suppose π U[, T ] is fix. [, T ] and j. We obain d g(, p) = sup g π (, p) = sup p j g π (, e j ). π U[,T ] π U[,T ] j=1 Noe ha due o he definiion g π (, e j ) for all d d λ p g π (, J(p)) = p j λ j g π (, e j ) λ 1 p j g π (, e j ) = λ 1 g π (, p). j=1 j=1 Taking he supremum over all π U[, T ] hen yields he firs inequaliy. The case δ > and he second saemen obviously follows similarly. 13

14 b) Suppose δ < and < γ < 1. In view of Lemma 5.3 i suffices o show g(, J(p)) g(, p). Noe ha if p = e j, he couning process of jumps is simply a Poisson process wih inensiy λ j. Moreover, i is well-known ha if λ ˆλ >, hen a Poisson process wih inensiy λ pahwise sochasically dominaes a Poisson process wih inensiy ˆλ (see e.g. Sec in Müller and Soyan (22)). Thus, under an arbirary fixed π U[, T ] we have X π s ˆXπ where s is he usual sochasic order. Thus, he value funcion is decreasing in λ and we obain for all [, T ]. Thus, i follows ha g π (, e 1 )... g π (, e d ), d d d p j g π (, e j ) p j λ j p j λ j g π (, e j ) j=1 j=1 j=1 where his inequaliy is derived by applying he following general inequaliy (cf. Mironovic e al. (1993)): le α 1... α d and β 1... β d be real numbers and p 1,..., p d, d j=1 p j = 1. Then d d d p j α j p j β j p j α j β j. j=1 Taking he supremum over all π U[, T ] hen yields he saemen. j=1 j=1 Please noe ha in he case γ < we obain he inequaliy g π (, e 1 )... g π (, e d ), for all [, T ] since he value funcion is negaive. analogously. The case δ > can be shown Par a) of Theorem 5.6 means ha he opimal fracion which is invesed in he sock is bounded by he smalles and larges invesed fracion in he models wih known inensiy λ 1 and λ d. Par b) of his heorem is mos ineresing. For example in he case of downward jumps δ < and γ (, 1), he opimal fracion invesed in he sock in he model wih unknown jump inensiy in sae (, p) is larger han in he model wih known (average) inensiy λ p. Though our invesor is risk averse, his is a siuaion where more uncerainy leads o a higher invesmen in he risky sock. If γ < he siuaion is vice versa. An economic explanaion is ha he degree of risk aversion changes wih γ. From he Arrow-Pra absolue risk aversion coefficien which is U (x) U (x) = (1 γ) 1 x in he case of he power uiliy U(x) = 1 γ xγ, we see ha he risk aversion decreases wih γ for all wealh levels. If γ we obain he logarihmic uiliy case and we know from secion 4 ha here he opimal fracions invesed coincide, i.e. u δ (λ p) = u δ (, p). In paricular if γ (, 1) he invesor is less risk averse. A similar resul has been obained for a model wih unobservable appreciaion rae in Rieder and Bäuerle (25). 14

15 6 Auxiliary Resuls In his secion we summarize some resuls which are imporan for he proofs of our main heorems. Lemma 6.1 summarizes imporan properies of he funcion g defined in (1) which is par of he value funcion V. Lemma 6.1: Le g be defined by (1) in Secion 5. a) p g(, p) is convex for all [, T ]. b) g(, p) is decreasing (increasing) for all p S d if < γ < 1, (γ < ). c) g(, p) is bounded on [, T ] S d. d) g(, p) is locally Lipschiz-coninuous for all p S d. e) g(, φ(, p)) is locally Lipschiz-coninuous for all p S d. Proof: a) follows from Lemma 3.3 b). b) This is equivalen o showing ha V (, x, p) is decreasing. Bu his is clear since because of r > we ge a posiive reward over a small ime inerval by puing all he money in he sock. c) For γ < he saemen is obvious due o par b) and he fac ha g(, p) and g(t, p) = 1. For γ (, 1) i suffices o show ha g(, p) is bounded on S d. I is convenien o inroduce a new measure Q π by dq π = L π T dp, where π U[, T ] and Lπ is a soluion of he sochasic differenial equaion ( ) dl π = L π γσπ dw + ((1 + yπ ) γ 1) ˆM(d, dy) where ˆM(d, dy) := N(d, dy) ˆλ dq(dy) is he compensaed random measure defined before. The soluion is given by { T ( L π T = exp 1 2 γ2 σ 2 πs 2 ˆλ ) T s ((1 + yπ s ) γ 1)Q(dy) ds + γσπ s dw s T +γ I is easy o see ha for π U[, T ] [ g π (, p) = E,p Q π ln(1 + yπ s )N(ds, dy) { T ( exp γ r + (µ r)π s + 1 ) 2 (γ 1)σ2 πs 2 +ˆλ s ((1 + yπ s ) γ 1)Q(dy)ds} ]. Since π, δ n and ˆλ = λ p are bounded, i follows from his equaion ha g π (, p) is bounded on [, T ] S d and he bound is independen of π. }. 15

16 d) In his par we make he dependence of g on he ime horizon explici by wriing g π,t (, p). Firs noe he following: if π U[, T ] we define ˆπ by ˆπ s = π +s for s [, T ] which implies ha g π,t (, p) = gˆπ,t (, p). Now le 1 < 2 T. Then here exiss for every ε > a sraegy π U[, T 1 ] wih g( 1, p) g( 2, p) g π,t 1 (, p) g π,t 2 (, p) + ε π[ T { 1 K E,p Q exp T 2 γ ( r + (µ r)π s + 1 ) 2 (γ 1)σ2 πs 2 ] } +ˆλ s ((1 + yπ s ) γ 1)Q(dy)ds 1 + ε T 1 K 1 E,p Q π γ 2r + µ + σ 2 + λ ((1 + yπ s ) γ + 1)Q(dy)ds + ε T 2 K ε where λ = max k λ k. This implies he saemen if we le ε. Noe ha K 2 can be chosen independen of p and π. e) Le 1 < 2 T. Then g( 2, φ( 2, p)) g( 1, φ( 1, p)) = g( 2, φ( 2, p)) g( 2, φ( 1, p)) + g( 2, φ( 1, p)) g( 1, φ( 1, p)) g( 2, φ( 2, p)) g( 2, φ( 1, p)) + g( 2, φ( 1, p)) g( 1, φ( 1, p)). Since g is convex in p i is also locally Lipschiz-coninuous in p S d wih a module K 3 which can be chosen independen of (see e.g. Sec. 1 in Rockafellar (197)). Therefore we obain g( 2, φ( 2, p)) g( 1, φ( 1, p)) K 3 φ( 2, p) φ( 1, p) + K Bu obviously φ(, p) is also locally Lipschiz-coninuous wih a module independen of p which yields he resul. Recall ha = T < T 1 < T 2 <... are he jump ime poins of he Markov-modulaed Poisson process. Since g(, φ(, p)) is locally Lipschiz-coninuous, here exiss a funcion Dg(s, p s ) such ha g(t i, p Ti ) g(t i 1, p Ti 1 ) = Ti T i 1 Dg(s, p s )ds. Almos everywhere on he ime inerval [, T ], he derivaive of g(s, p s ) w.r.. s exiss and we can choose Dg(s, p s ) = g (s, p s ) + k g p k (s, p s ) φ k (s, p). Le us define he operaor Hv(, p, u) := Lv(, p, u) + 1 Dv(, p) γ 16

17 for all funcions v : [, T ] S d IR where he righ-hand side is well defined. Noe ha he HJB equaion can be wrien as = sup {Hv(, p, u)} = sup {Lv(, p, u)} + 1 Dv(, p) u [,1] u [,1] γ a hose poins (, p) where v is differeniable. Lemma 6.2: Suppose ha π U[, T ] is an arbirary sraegy. The value funcion V saisfies he following sochasic differenial equaion dv (, X π, p ) = (X π ) γ Hg(, p, π )d + dη π, where (η π ) is an F S -maringale wih zero expecaion. Proof: Le π U[, T ] be arbirary. Using Io s Lemma we can verify ha Z π := (X π ) γ saisfies he following sochasic differenial equaion dz π = γz π (r + (µ r)π (γ 1)σ2 π 2 )d +γz π σπ dw + Z π ((1 + yπ ) γ 1)N(d, dy). Moreover, since g(, φ(, p)) is absoluely coninuous, we can wrie g(, p ) = g(, p ) + Dg(s, p s )ds + g(s, p s ) g(s, p s ). <s Since V (, X π, p ) = 1 γ Zπ g(, p ), he produc rule implies Le us define V (, X π, p ) = V (, x, p) + + g(s, p s )Zs π σπ s dw s + η π,1 := 1 γ 1 γ + 1 γ <s <s g(s, p s )Z π s (r + (µ r)π s (γ 1)σ2 π 2 s)ds Zs π Z π s g(s, p s ) Z π s g(s, p s ). Z π s g(s, p s ) Z π s g(s, p s ) Zs πˆλ s (g(s, J(p s )) 1 γ Dg(s, p s)ds ) (1 + yπ s ) γ Q(dy) g(s, p s ) ds. According o Brémaud p.27, T8, (η π,1 ) is an F S maringale, since [ ] T E Zs π g(s, p s ) Zs g(s, π p s ) λ p s ds <. 17

18 Moreover, due o our boundedness condiions η π,2 := g(s, p s )Zs π σπ s dw s is also an F S -maringale. If we define now η π := η π,1 + η π,2 he saemen follows. Theorem 6.3: Bellman equaion Le τ [, T ] be an F S -sopping ime. Then V (, x, p) = sup E,x,p [V (τ, Xτ π, p τ )]. π U[,T ] The proof of he Bellman equaion follows he usual recipe and we skip i here. 7 Proofs for he Resuls in Secion 5.1 In his secion we provide he proofs of he Verificaion Theorem and he fac ha V is a soluion of he generalized HJB equaion. Proof of Theorem 5.1: Le π U[, T ] be an arbirary sraegy. Then we obain for Z π := (X π ) γ and G(, x, p) := 1 γ xγ v(, p) as in Lemma 6.2 T G(T, XT π, p T ) = G(, x, p) + Zs π Hv(s, p s, π s )ds + ηt π η π where (η π ) is an F S -maringale wih zero expecaion. A hose poins where v(s, p s ) is differeniable we have Hv(s, p s, π s ) since v saisfies he generalized HJB equaion. Moreover, s v(s, p s ) is almos everywhere differeniable which yields Thus, we obain T Z π s Hv(s, p s, π s )ds. G(T, X π T, p T ) G(, x, p) + η π T η π. Taking he condiional expecaion on boh sides (noe ha G(T, XT π, p T ) = U(XT π )) yields: E,x,p [U(X π T )] G(, x, p). Taking he supremum over all admissible sraegies gives V (, x, p) G(, x, p). Nex, noe ha he maximum poins of he HJB equaion rivially exis. If we use π we obain T Z π s Hv(s, p s, π s)ds = 18

19 and he resul follows. Proof of Theorem 5.2: Le τ be he ime of he firs jump of he sock price process (S ) afer ime. From he Bellman equaion (Theorem 6.3) we obain for every sraegy π U[, T ] and < T : V (, x, p) E,x,p [ V (τ, X π τ, p τ )]. From Lemma 6.2 we know ha for Z π = (X π ) γ τ V (τ, Xτ π, p τ ) = V (, x, p) + Zs π Hg(s, p s, π s )ds + ητ π ηπ. Insering his equaion in he preceding inequaliy yields [ ] τ E,x,p Zs π Hg(s, p s, π s )ds. Le π be now a fixed sraegy wih π s u [, 1] for s [, + ε), ε >. Thus, we ge [ ] 1 τ lim E,x,p Zs π Hg(s, p s, π s )ds = [ ] 1 = lim E,x,p Z π s Hg(s, p s, π s )ds < τ P ( < τ) + [ 1 τ ] + lim E,x,p Zs π Hg(s, p s, π s )ds τ P ( τ) Since P (τ ) 1 e λ( ) for, where λ = max k λ k, we obain a hose poins (, p) where g is differeniable Z π Hg(, p, u). From he definiion we see ha Z π > which yields Hg(, p, u). Le now (, p) be an arbirary poin, where g migh no be differeniable. We know ha g(, p) = co{lim sup n g( n, p n ), n } which means by definiion ha every θ g(, p) is a convex combinaion of θ m = lim sup n g( m n, p m n ) for sequences m n, along which g is differeniable. Since g is coninuous, we obain Lg(, p; u) + 1 γ θm + 1 d ( θ m k q jk p j p k (λ k γ ˆλ) ) k=1 j which yields he same inequaliy wih θ. Finally since u and θ are arbirary, we obain sup {Lg(, p; u)} + u [,1] { 1 sup θ g(,p) γ θ + 1 γ d ( θ k q jk p j p k (λ k ˆλ) )}. k=1 j On he oher hand, for ε > and < < T wih > small enough here exiss a sraegy π ε, U[, T ] wih [ ] V (, x, p) ε( ) E,x,p V (τ, Xτ πε,, p τ ). 19

20 Again wih Lemma 6.2 we obain ε( ) E,x,p [ τ Z πε, s Hg(s, p s, πs ε, )ds ]. Thus, we ge [ 1 τ ε E,x,p Zs πε, Hg(s, p s, π s ε, )ds [ 1 τ E,x,p Zs πε, sup Hg(s, p s, u)ds u [,1] ] ]. Denoe now by u () he maximum poin of u Lg(, φ(, p), u) on [, 1]. Since u () is coninuous we obain a hose poins (, p) where g is differeniable and since ε > is arbirary ε x γ sup Hg(, p, u) u [,1] sup Hg(, p, u). u [,1] The analysis if g is no differeniable a (, p) follows in he same way as before by using he convexiy of g(, p). Alogeher i follows ha V saisfies he generalized HJB equaion. References [1] Benh, F.E., K.H. Karlsen and K. Reikvam (21): Opimal porfolio selecion wih consumpion and nonlinear inegro-differenial equaions wih gradien consrain: a viscosiy soluion approach, Finance Soch. 5, [2] Brémaud, P. (1981): Poin processes and queues. Springer-Verlag, New York. [3] Clarke, F.H. (1983): Opimizaion and nonsmooh analysis. John Wiley & Sons, New York. [4] Davis, M. H. A. (1993): Markov models and opimizaion. Chapman & Hall, London. [5] Ellio, R.J., L. Aggoun and J. B. Moore (1994): Hidden Markov models: esimaion and conrol. Springer-Verlag, New York. [6] Framsad, N.C., B. Øksendal and A. Sulem (1999): Opimal consumpion and porfolio in a jump diffusion marke, in: Shiryaev, A. e al. (eds.) Workshop on mahemaical finance. Paris: INRIA, 9-2. [7] Haussmann U.G. and J. Sass (24): Opimal erminal wealh under parial informaion for HMM sock reurns, in Mahemaics of Finance (Conemp. Mah. 351), AMS, Providence, [8] Honda, T. (23): Opimal porfolio choice for unobservable and regime-swiching mean reurns, J. Econ. Dyn. Conr. 28,

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