Portfolio Optimization with Nondominated Priors and Unbounded Parameters

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1 Porfolio Opimizaion wih Nondominaed Priors and Unbounded Parameers Kerem Uğurlu Saurday 14 h July, 218 Deparmen of Applied Mahemaics, Universiy of Washingon, Seale, WA keremu@uw.edu Absrac We consider classical Meron problem of erminal wealh maximizaion in finie horizon. We assume ha he drif of he sock is following Ornsein-Uhlenbeck process and he volailiy of is following GARCH(1) process. In paricular, boh mean and volailiy are unbounded. We assume ha here is Knighian uncerainy on he parameers of boh mean and volailiy. We ake ha he invesor has logarihmic uiliy funcion, and solve he corresponding uiliy maximizaion problem explicily. o he bes of our knowledge, his is he firs work on uiliy maximizaion wih unbounded mean and volailiy in Knighian uncerainy under nondominaed priors. Mahemaics Subjec Classificaion: 91B28;93E2 Keywords: Meron Problem; Knighian uncerainy; Robus opimizaion 1 Inroducion Saring wih he pioneering works of 23, 1, 8, 3, 4], he underlying risky asses are modelled as Markovian diffusions, where here exiss a fixed underlying reference probabiliy measure P has rerieved from hisorical daa of he price movemens. However, is mosly agreed has impossible o precisely idenify P. Hence, as a resul, model ambiguiy, also called Knighian uncerainy, in uiliy maximizaion is ineviably aken ino consideraion. Namely, he invesor is diffiden abou he odds, and akes a robus approach o he uiliy maximizaion problem, where she minimizes over he priors, corresponding o differen scenarios, and hen maximizes over he invesmen sraegies. 1

2 he lieraure on robus uiliy maximizaion in mahemaical finance, (see e.g. 3, 7, 8, 1, 29, 14, 32]), mosly assumes ha he se priors is dominaed by a reference measure P. Hence, i presumes a seing where volailiy of risky asses are perfecly known, bu drifs are uncerain. Namely, hese approaches assume he equivalence of priors. In paricular, hey assume he equivalence of probabiliy measures P wih a dominaing reference prior P. In his direcion o menion some of he relaed works, 15] proposed o weaken he srong independence axiom (also called sure hing principle used previously by 16] and 17]) o jusify (subjecive) expeced uiliy. Laer, 18] inroduced coheren risk measures in he spiri of he consrucion as in 15]. his heory of coheren risk measures has been generalized in several direcions aferwards, (see e.g. 11, 12, 19, 22, 24, 2, 33, 34] among ohers). By conras, we are sudying he case, where he se of priors, denoed by P(Ω), are nondominaed. Hence, here exiss no dominaing reference prior P. Some of he relaed works in his direcion are as follows: 6] sudied he case, where uncerainy in he volailiy is due o an unobservable facor; 9] sudied a similar seing in discree ime and has shown he exisence of opimal porfolios; 26, 7] works in a jump-diffusion conex, wih ambiguiy on drif, volailiy and jump inensiy. 27] esablishes a minimax resul and he exisence of a wors-case measure in a seup where prices have coninuous pahs and he uiliy funcion is bounded. 5] works in a diffusion conex, where uncerainy is modelled by allowing drif and volailiy o vary in wo consan order inervals. he opimizaion using power uiliy of he from U(x) = x γ for < γ < 1 is hen performed via a robus conrol (G- Brownian moion) echnique, which requires he uncerain volailiy marix is resriced o be of diagonal ype. We refer he reader o 21] for a deailed exposure on G-Brownian moion and is applicaions. In a more general seing 13] works in a coninuous ime seing, where he sock prices are allowed o be general disconinuous semi-maringales and sraegies are required o be compac. Conrary o he curren lieraure, we assume ha he drif of he sock process is modelled by Ornsein-Uhlenbeck (OU)-process, and volailiy of he sock process is modelled by GARCH(1)-process. In paricular, boh mean and volailiy of he sock ake unbounded values in R. We assume ha here is Knighian uncerainy on he parameers of boh mean and volailiy. We furher ake he invesor has logarihmic preference bu being diffiden abou he underlying dynamic parameers wans o reconsider he opimal parameers a prespecified small ime inervals. o he bes of our knowledge, his is he firs work on uiliy maximizaion wih unbounded mean and volailiy in Knighian uncerainy under nondominaed priors. he res of he paper is as follows. In Secion 2, we describe he model dynamics of he 2

3 3 problem and sae our general main problem. In Secion 3, we solve he invesor s problem using logarihmic uiliy. Finally, in Secion 4, we discuss our resuls and conclude he paper. 2 Model Dynamics and Invesor s Value Funcion 2.1 Framework We le Ω = C, ] be he se of all coninuous pahs (ω ) saring a ω = aking values in R over he finie ime horizon, ] equipped wih uniform norm ω = sup ω., ] We furher ake he corresponding induced meric on Ω in he usual way d(f, g) := f g, and ake he opology generaed by he open ses in his meric as he opology of uniform convergence on, ]. We furher define W as he coordinae funcional, i.e. for ω Ω and, ], we le W (ω) := ω. We denoe he corresponding Borel sigma algebra on Ω by F = σ{w s (ω); s }, and he sandard Wiener measure by P on Ω. Namely, P is he unique measure on Ω, which saisfies he following properies: P (ω Ω : W (ω) = ) = 1, For any f Cb (R), C b (R) being infiniely many imes coninuously differeniable bounded funcions, he sochasic process is an (F, P )-maringale, (, ω) f(w (ω)) 1 2 f(w s (ω))ds We ake he Wiener measure P as our reference measure and P(Ω) as he Polish space equipped wih weak opology of probabiliy measures on Ω. We furher denoe by P(Ω F ) for he Polish space of regular condiional probabiliy measures (r.c.p.m. s) on Ω (see 31], hm 5.1.9). 2.2 Model Dynamics We consider a marke consising of one risky asse, S, and one riskless asse R. We assume ha he semimaringale S represening he price of he risky asse saisfies he following

4 4 dynamics S = s, s > ds = S (µ d + σ S dw S ), P a.s.. (2.1) Here W S and W σ are independen (Ω, F ) Brownian moions, and boh he drif erm µ and he volailiy σ S are sochasic processes saisfying he following dynamics dµ = θ µ (η µ µ )d + σ µ dw µ, (2.2) d(σ S ) 2 = θ σ (η σ (σ S ) 2 )d + ξ(σ S ) 2 dw σ. (2.3) Here, θ µ, η µ, θ σ, η σ are piecewise consan on, +1 ) for i =,..., N, wih = and N = and sricly posiive, whereas σ µ, ξ > are posiive consans. Namely, we assume ha he drif erm, µ, is an Ornsein-Uhlenbeck (OU) process, whereas he volailiy (σ S ) 2 is a GARCH(1) process. Is easy o see ha he explici soluions of µ and (σ S ) 2 are (σ S ) 2 = (σ S ) 2 θs σ ηs σ exp{ µ = µ exp{ θ µ s ds} + s (θ σ r ξ2 )dr + ξ(w W s )}ds exp{ s θ µ r dr}σ µ dw s We furher assume ha he riskless asse s dynamics saisfies dr = rrd wih r >. 2.3 Model Uncerainy We denoe by γ (θ µ, θ σ, η µ, η σ ) as he quadruple of piecewise consan posiive funcions and le γ Γ be a collecion of hem. We assume ha Γ is uniformly bounded, i.e. γ() M for some consan M for all γ Γ and. Remark 2.1. he quadraic variaion erms of µ and σ can be deduced wih cerainy from he pah ω, bu he drif parameers of hese processes can only be esimaed from hisorical daa. Hence, is naural o inroduce uncerainy only in he corresponding drif erms of µ and σ, whereas he quadraic variaion erms ξ, σ µ are fixed known consans. We nex give he following a priori esimaes ha are o be used in he res of he paper. Lemma 2.1. Le (σ S ) 2 and µ be as defined in (2.2) and (2.3), respecively. hen, we have max γ Γ (σs ) 2n d] < max γ Γ (µ ) n d] <

5 5 for any n N. In paricular, for any, ]. max γ Γ (σs s ) 2n ds] < P -a.s. max γ Γ (µ s) n ds] < P -a.s. Proof. For ease of noaion, denoing (σ S ) 2 = ν and using consan C inerchangeably below, we have ν θs σ ηs σ exp ( (θr σ ξ2 )dr ) + ξ W W s ds Since θ σ, η σ and ξ are uniformly bounded, we have ( ) max ν n C exp C( s) + C W W s ds γ Hence, for any n 1, via Jensen s inequaliy we have ( ) max ν n C exp C(1 + W W s ) ds. γ s hus, and max ν n ] C γ Eexp(C(1 + W W s ))]ds < E max ν n ] <. γ For µ and n 1, we have he following inequaliies modulo P -a.s. µ C + C ( µ 2n C + C e s θµ r dw s e s θµ r dw s ) 2n max µ 2n ] C + C e 2n s θµ r dr ds γ Γ C(1 + ) <

6 6 Since x n x 2n + 1 for x > and any n 1, we have Hence, we conclude he he proof. max γ Γ µ n ] < (max γ Γ µ n )d] <. 2.4 Alernaive Models A each fixed ime, ] wih = < 1 <... < N = we consider he class of alernaive models denoed by Q i, ]. hese are he se of r.c.p.m. s on Ω ha are induced by he process S( ) as in Equaion (2.1). Namely, Q i, ] = { P P(Ω F i ) P is he r.c.p.m. induced by (2.4) S saisfying ds u = S u (µ u du + σ S u dw S u )du, P a.s. for u, ] given {S r (ω), r } wih (θ µ u, θ σ u, η σ u, η µ u) u Γ }. We noe here ha by heorem 11.2 in 25], here exiss a srong soluion o he Equaion (2.1) on (Ω, F, P ). Namely, denoing C, ] = Ω, here exiss an F measurable mapping G : Ω Ω such ha X( ) G(x i, W ( )) solves Equaion (2.1) on (Ω, F, P ), as in Definiion 1.9 in 25]. We furher noe ha here is a one-o-one correspondence beween he se of r.c.p.m. s Q i, ] and he compac se Γ. Namely, γ = (θ µ, θ σ, η σ η µ ) Γ uniquely defines S on, ] P -a.s. We denoe he r.c.p.m. induced by S for a fixed γ Γ and A F as P γ (A) wih P γ (A) =. P (ω Ω : {S u (ω), u } A; ) given {S r (ω) : r }. We furher ake he convex hull of Q i, ] and denoe i by Q c, ]. Namely, Q c, ] = { P P(Ω F i ) P (A) = m α i P i (A) }, (2.5) i=1 for all A F wih P i Q i, ], α i 1 wih m i=1 α i = 1 for i = 1,..., m.

7 7 2.5 Financial Scenario We consider he problem of invesing in a risky asse S and riskless asse R = e r, where r > is he fixed ineres rae. We assume ha risky asse s dynamics as in (2.1) is ds = µ S d + σ S dw S, P -a.s. For a given iniial endowmen x >, he invesor rades in a self financing way. Namely, denoing ˆπ as an F adaped sochasic process, which sands for he oal amoun of money invesed in he risky asse S a ime,, we have dx ˆπ dx ˆπ = ˆπ S 1 ds + (X ˆπ ˆπ )rd = ˆπ (µ d + σ S dw S ) + (X ˆπ ˆπ )rd We furher represen he amoun of money invesed in he risky asse as a fracion of curren wealh via ˆπ = X π π for, where π sands for he corresponding fracion a ime. Hence, for X = x, he dynamics of wealh in his seing are given by dx π = X π π (µ d + σ S dw S ) + rx π (1 π )d X π = x exp 2.6 Invesor s Problem {π s µ s + r(1 π s ) 1 2 π2 s(σ S s ) 2 }ds + π s σ S s dw S s. he invesor is risk-averse and maximizes he minimal expeced logarihmic uiliy over her se of r.c.p.m s represening alernaive models unil ime horizon, as follows. She is diffiden abou he underlying dynamics of he sock S and wans o reevaluae her opimizaion problem a prespecifed fixed ime epochs = < 1 <... < N =. A each ime for i =,..., N 1, we wrie he opimizaion problem of he invesor as V (, x) = sup inf Q Q i, ] { E Q log(x π ) ]}, (2.6) where E Q ] sands for he condiional expecaion wih respec o Q Q i, ] as defined in (2.4). For ime no equal o s, he invesor sicks o her opimal γ Γ, assumes he model is correc unil +1 and solves he classical expeced logarihmic uiliy problem based on V (, x) = sup E P log(x π ) X π = x ] (2.7) Here, he process S saisfies ds = S (µ d) + σ S dw S,

8 8 where dµ = θ µ (η µ µ )d + σ µ dw µ, d(σ S ) 2 = θ σ (η σ (σ S ) 2 )d + ξ(σ S ) 2 dw σ. wih < < +1, where γ = (θ µ, η µ, θ σ i, η σ i ). Hence, he invesor reevaluaes her model by prespecified ime inervals o be robus agains he model risk. Moreover, Π ad sands for he se of admissible cash-values for he invesor a ime episode, ], which is defined as follows. Definiion 2.1. Le π = (π s ) { s } denoe he B(, ]) F progressively measurable process represening he cash-value allocaed in he risky asse for, ]. We call (π s ) {s } admissible and denoe i by π Π ad, if x π s 4 ds F ] <, P -a.s. and X π > for all s, P -a.s. We say in his case ha π L 4 (, ]) and noe ha Π ad is nonempy, since π. We furher say ha π n π, if x π s π n s 4 F ]ds, P -a.s., as n. We say in his case ha π n converges o π in L 4 (, ]). o aack problem (2.6), we firs solve (2.7). he opimizaion problem reads as By Io s formula, we ge ha sup log(x π )] sup log(x π )]. = log(x ) + sup 1 ] 2 π2 s(σs S ) 2 ds (π s (µ s r) + r) By concaviy on π, we conclude ha checking firs order condiion inside he expecaion on π is sufficien and ge ha (µ s r) σ 2 sπ s =.

9 9 Hence, we have π s = µ s r σ 2 s for all s, P a.s., and he opimal value funcion reads as log(x π )] = log(x ) + E P r + (µ s r) 2 ds ]. 2σs 2 Going back o he robus opimizaion problem a each ime for i =,..., N, we have { sup inf E P log(x π ) ]}. Q Q i, ] o proceed, we firs sae our minimax resul. heorem 2.1. Le Q c, ] be as in Equaion (2.5), π Π ad be as defined in Definiion 2.1 and X π have he dynamics as in Equaion (2.1). hen, we have { } { } sup inf log(x π )] = inf sup log(x π )] Q Q c, ] Q Q c, ] heorem 2.1 is an applicaion of Sion s minimax heorem, which we recall here for convenience. heorem ] Le X be a compac convex subse of a linear opological space and Y a convex subse of a linear opological space. Le f be a real-valued funcion on X Y such ha f(x, ) is upper semi coninuous and quasi-concave on Y for each x X. f(, y) is lower semi coninuous and quasi-convex on X for each y Y. hen min sup x X y Y f(x, y) = sup y Y min f(x, y). x X We define firs a suiable opology o work wih he mapping P E P log(x π )] for P Q c, ]. Definiion 2.2. A family of condiional probabiliy measures on Ω, denoed by S i, ], is called relaively compac, if for every sequence {P n } in S i, ], here exiss a subsequence

10 1 {P m } of {P n } and a probabiliy measure on Ω (no necessarily in S i, ]) such ha {P m } converges weakly o P. has E Pm i g] E P g] for every g : Ω R wih g C b (Ω), where C b (Ω) is he space of coninuous bounded funcions on Ω. We say in his case ha P m converges weakly o P, and denoe i as P m P. Lemma 2.2. Q c, ] Definiion 2.2. as in Equaion (2.5) is compac wih respec o opology defined in Proof. Le {P n } be a sequence in Q c, ]. We will show ha every sequence in {P n} has a convergen subsequence in Q c, ]. By aking a subsequence if necessary, one can assume ha P n = m αj npn, j where αn j α j wih α j, 1] and Pn j Q i, ] for j = 1,..., m. Recall ha Pn j for j = 1,..., m are r.c.p.m s ha are induced by he processes of he form wih W S,j ds j = µj Sj d + σs S j dw j P a.s. for s, (2.8) given {S j r : r s} is an (F, P ) Wiener process as in Equaion (2.1). Denoing ϕ S j ( ) as he char- acerisic funcion of he condiional disribuion induced by he process Equaion (2.8), we have P n P in he sense of Definiion 2.2, if and only if he characerisic funcions ϕ S (z) = e izsj ] converges poinwise o some characerisic funcion of some F measurable random variable S for any z R, where S induces he probabiliy measure P. Bu by dynamics of (σ 2, µ ) as in Equaion (2.2) and (2.3), respecively, and since any sequence in Γ has a convergen subsequence in Γ, his is only rue if (θ n, µ θn, σ ηn, σ η n) µ (θ µ, θ σ, η σ, η µ ). Hence, we have ha he sequence of r.c.p.m. s P n converges o some r.c.p.m. P on Ω, which is induced by some F measurable random variable S, whose characerisic funcion is of he form m α ϕ j S j (z). Hence, we conclude he resul. We coninue wih he following lemma. Lemma 2.3. Le Q Q c, ] wih Q = m α j Q j,

11 11 wih α j, m α j = 1 and Q j Q i, ] for j = 1,..., m, where Q i, ] is as in Equaion (2.4) and le g : Ω R be an F measurable mapping wih E Q g(ω) ] <. hen, we have for any m N E Q g ] = m α j E Q j g ] wih m α j = 1 and α j. In paricular, he mapping Q E Q log(x π ) ] is quasi-convex i.e. E αq 1+(1 α)q 2 log(x π ) ] max{e Q 1 log(x π ) ], E Q 2 log(x π ) ]} Proof. Denoing he null ses of Q 1,..., Q m as N Q1,..., N Qm respecively, we noe ha N Q = m N Qj. Wihou loss of generaliy, by aking g(ω), Q-a.s. we ge via an approximaion of Lebesgue inegraion ha n c j I Aj (ω) g(ω), Q a.s. for some real consans c j, for i = j,..., n, wih n A j = Ω and A j A k = for j k. Here, I Aj ( ) is he indicaor funcion for he se A j F i. Hence, we conclude via an approximaion argumen E Q g] = m α j E Q j g] Since max γ Γ EP log(x π ) ] <, we have Q E Q log(x π )] is convex, in fac linear. Hence, we conclude he proof. We coninue wih he following lemma. Lemma 2.4. Le π and π n be in Π ad wih π n π in L 4 (Ω;, ]) as n, π n (s) = π(s) for s for all n N, and le P Q c, ] be fixed. hen, he mapping is coninuous P -a.s. Namely, as π n π in L 4 (, ]) P a.s. π E P log(x π )] E P log(x πn )] EP log(x π )]

12 12 Proof. By Lemma 2.3, we have ha for any P Q c, ] E P log(x π )] = m α j E P j log(x π )]. Hence, showing he coninuiy of π E P j log(x π )], for P j Q i, ] and hence for fixed γ Γ is sufficien o show he coninuiy of P Q c, ]. Since π n π in L 4 (, ]), we noe ha and using Lemma 2.1 we have sup n log(x π,γ )] EP log(x πn,γ )] = C, (π s π n s )(µ s r) C as n. hus, we have (π n s ) 4 ] <, P a.s. ( π 2 s max (µ s r) 2 ds] 1/2 γ Γ σs 2 (π s π n s )(π s + π n s ) 1 σ 2 s ds] (π s + π n s ) 4 ds] 1/2 max γ Γ (πn s ) 2 σ 2 s ) ds ] (π s π n s ) 2 ds] 1/2 1 (σ S s ) 8 ds]1/2 + C log(x πn,γ )] log(x π,γ )], (π π n s ) 2 ds] as π n π for n P a.s. Hence, we conclude he proof. Lemma 2.5. Le π be in Π ad. Le Q Q c, ] be fixed. hen, he mapping is concave, in paricular quasi-concave in π. π E Q log(x π )]

13 13 Proof. By Lemma 2.3, is enough o show he saemen for Q Q i, ]. Hence, for fixed γ Γ by Io lemma, we have log(x π,γ )] = log(x ) + x i π s (µ s r) 1 ] 2 π2 s(σs S ) 2 ds, from which is easy o see ha π log(x π )] is concave, hence quasi-concave in π. Hence, we conclude he resul. Nex, we coninue wih he following lemma. Lemma 2.6. Le π Π ad be fixed and Q, Q n Q c, ]. hen, he mapping is lower semi-coninuous P a.s.. Namely, lim inf n Q n E Qn log(x π )] { E Q n log(x π )] } E Q log(x π )], as Q n Q in he sense of Definiion 2.2 as n, P -a.s. Proof. We have Q n = m αnq j j n where α j n 1 and Q j n Q i, ] for j = 1,..., m. By Lemma 2.3, his means ha E Qn log(x π )] = m αne j P log(x π,µj n,σ S j n )], where (µ j n, σ j n) for j = 1,..., m defines uniquely P j Q, ]. Hence, we have where E Pn log(x π )] E Pn log(x π ) ] max γ Γ log(xπ,γ ) ] <, log(x π,γ ) = log(x ) + ( ( πs (µ s r) + r 1 2 πs 2 σs 2 ds ) + Nex, we runcae our uiliy funcion log(x) as k if log(x) k V k (x) = log(x) if log(x) k k if log(x) k π 2 s(σ S s ) 2 dw S s ).

14 14 for x > and for k >. We noe ha log(x π,γ )] + ɛ(k) EP V k (X π,γ )], uniformly for all γ Γ for some ɛ(k) depending on k only wih ɛ(k) as k. Indeed, we have log X π,γ I { log X π,γ >k} ] max γ Γ EP for k large enough. Furhermore, since by Lemma 2.1 log X π,γ I { log X π,γ >k} ] < ɛ, is inegrable wih we have lim inf n max γ Γ log(xπ,µ,σ,r ) ] < P -a.s. V k (x) log(x) for any k (2.9) V k (X π )] max γ Γ log(xπ ) ] < P a.s., ( m ) EPn log(x π )] = lim inf αne j P n i log(x π,µn j,σn j )] ( m ) αne j P V k (X π,µn j,σn j )] ɛ(k) lim inf n = E P V k (X π )] ɛ(k), P a.s. where he las equaliy is due o convergence P n P. Finally, by leing k wih ɛ(k) and via Lebesgue dominaed convergence heorem by Equaion (2.9), we conclude he resul. Proof of heorem 2.1. We have ha Π ad is a convex se by Definiion 2.1. Similarly, Q c, ] as in Equaion (2.5) is a convex opological subse of P(Ω F i ). Furher, by Lemma 2.3 we have ha E P log(x π )] = m α j log(x π,µ j,σ j )]. Hence, E P log(x π)] is real valued for any fixed π Π ad and any P Q c, ]. Furher, by Lemma 2.2, 2.4 and 2.6, we have ha he condiions for heorem 2.2 are saisfied. Hence, we conclude he resul via Sion s minimax heorem.

15 15 3 he Opimal Soluion of he Invesmen Problem We have he following series of equaions inf Q Q i, ] { sup E Q log(x π )] } = inf Q Q c, ] = sup Q Q c, ] sup sup E Q log(x π )] inf inf Q Q i, ] inf Q Q i, ] E Q log(x π )] E Q log(x π )] sup E Q log(x π )] where he firs equaliy is by Lemma 2.3, he second equaliy is by heorem 2.1, whereas he firsnequaliy is by Q c, ] Q, ] and second inequaliy is by classical minimax inequaliy. hus, we have V (, x, ν, σ) = inf Q Q i, ] { sup E Q log(x π )] } V (, µ, ν) = log(x i ) + inf γ Γ {EQ r (µ s r) 2 ] ds. For ease of noaion supressing index s for µ s and (σs S ) 2 and denoing ν = (σs S ) 2 he corresponding Hamilon-Jacobi-Bellman(HJB) equaion reads as ( = min r + 1 (µ r) 2 + V + V µ (θ µ (η µ µ)) + V ν (θ σ (η σ ν)) γ Γ 2 ν V µµ(σ µ ) ) 2 V ννξ 2 ν 2 For µ < r, hence V µ < 1. If µ η µ min, ηµ max], choose η µ = η µ max and θ µ = θ µ max. 2. If µ < η µ min, choose ηµ = η µ max and θ µ = θ µ max. 3. If µ > η µ max, choose η µ = η µ min and θµ = θ µ max. For µ > r, hence V µ > 1. If µ η µ min, ηµ max], choose η µ = η µ min and θµ = θ µ max. 2. If µ < η µ min, choose ηµ = η µ min and θµ = θ µ min. 3. If µ > η µ max, choose η µ = η µ min and θµ = θ µ max. σ 2 s

16 16 We have V ν < for any ν >. 1. If ν η σ min, η σ max], choose η σ = η σ max and θ σ = θ σ max. 2. If ν > η σ max, choose η σ = η σ min and θ σ = θ σ min. 3. If ν < η σ min, choose η σ = η σ min and θ σ = θ σ max. Hence, we have ha a each ime inerval, +1 ), here exiss a unique parameer se γ = (θ µ, η µ, θ σ, η σ ) Γ saisfying = r + 1 (µ r) 2 + V µ (θ µ (η µ µ)) + V ν (θ σ (η σ ν)) (3.1) 2 ν V µµ(σ µ ) V ννξ 2 ν 2 V (, µ, ν) = log(x ) (3.11) he soluion o (3.1) and (3.11) is well-known by Feynman-Kac V (, µ, ν, x) = log(x i ) + E M r + 1 ] (µ s r) 2 ds 2 ν s µ = µ, ν i = ν subjec o dµ = θ µ (η µ µ )d + σ µ dw M dν = θ ν (η σ ν )d + ξ(σ S ) 2 dw M, a =, where W M is a 2 dimensional Brownian moion under probabiliy measure M, and he iniial condiions are µ i = µ, ν i = ν and E M ] sands for he expecaion aken wih respec o M. Hence, he invesor solves a each ime using one of he end poins of he corresponding inervals in Γ. We noe ha due o coninuiy of he sae variables µ and ν, he opimal parameer se γ says he same for some ime inerval, and hen as he regions for he sae variables change, here is a swich from one corner o he oher. Hence, (γ ) being a piecewise consan funcion on, ] belongs o Γ by Lemma 2.1. Similarly, he opimal parameer π reads as µ r, which is an elemen of Π σ S ad for all. Hence, he opimal conrols (π, γ ) are as specifed in he admissible policy ses (Γ, Π ad ) jusifying he consisency of he opimal conrol problem (2.6). 4 Conclusion We have sudied a logarihmic uiliy maximizaion problem, where here is uncerainy on drif µ and volailiy σ. We assume ha he drif and volailiy erms are OU and GARCH(1)

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