Uiliy maximizaion in incomplee markes Marcel Ladkau 27.1.29 Conens 1 Inroducion and general seings 2 1.1 Marke model....................................... 2 1.2 Trading sraegy..................................... 2 2 Exponenial uiliy 2 2.1 Uiliy funcion...................................... 2 2.2 Admissible sraegies................................... 3 2.3 Problem formulaion................................... 3 2.4 Approach o a soluion.................................. 4 2.5 Calculaing a soluion.................................. 4 2.6 Usefull ools........................................ 5 3 Appendix 6 4 References 7 1
1 Inroducion and general seings 1.1 Marke model we consider a small invesor rading in an incomplee marke in a finie ime inerval [, T ] probabiliy space (Ω, F, P) wih a m-dimensional Brownian moion (W ) [,T ] denoes he Euclidian norm in R m le C be a closed subse of R m, a R m dis C (a) = min a b b C Π C (a) = {b C : a b = dis C (a)} is he se of elemens of C a which he minimum is obained financial marke consiss of one bond wih ineres rae zero and d m socks for d < m we face an incomplee marke he price process of sock i evolves according o he equaion ds i S i = b i d + σ i dw i = 1,..., d b i (σ i ) is a R(R m )-valued predicable uniformly bounded sochasic process θ = σ T (σ σ T ) 1 b is assumed o be uniformly bounded 1.2 Trading sraegy d-dim. F-predicable process π = (π ) T defined, ha means π σ 2 d < P-a.s. is called rading sraegy if π ds S process π i, 1 i d describes he amoun of money invesed in sock i a ime is well he wealh process X π of a rading sraegy π wih iniial capial x saisfies he equaion X π = x + d i=1 π i,u S i,u ds i,u = x + π u σ u (dw u + θ u du), [, T ] rading sraegies are self financing, an invesor uses his iniial capial x o inves in socks or he bond 2 Exponenial uiliy 2.1 Uiliy funcion uiliy funcions are sricly monoon increasing sric concave for α > he exponenial uiliy funcion is given by U(x) = exp( αx) 2
2.2 Admissible sraegies we assume ha he invesor wans o maximize his final wealh X π T F F is a liabiliy a ime T we assume ha our sraegies ake heir values in a closed se C R d Definiion 1 (Admissible sraegies wih consrains) Le C be a closed se in R d. The se of admissible rading sraegies à consiss of all d- dimensional predicable processes π = (π ) T which saisfy E[ π σ 2 d] < and π C λ P-a.s., as well as is a uniformly inegrable family. {exp( αx π τ ) : τ sopping ime wih values in [, T ]} he square inegrabiliy condiion guaranees ha here is no arbirage ogeher wih he boundedness of θ we ge E[sup T (X π ) 2 ] < Appendix (A1) (X, π σ ) is he unique soluion of X = X T (π s σ s )dw s for [, T ], ω F define he se C (ω) R m by C (ω) = Cσ (ω) (π s σ s )θ s ds he enries of he marix-valued process σ are uniformly bounded, herefore (, ω) C (ω) is a closed se p := π σ, [, T ] min{ a : a C (ω)} k 1, k 1, for λ P-a.e. (, ω) The se of admissible sraegies à is equivalen o a se A of Rm -valued predicable processes p, wih p A if E[ p 2 d] <, p (ω) C (ω) P-a.s. and is a uniformly inegrable family. 2.3 Problem formulaion {exp( αx p τ ) : τ sopping ime wih values in [, T ]} p A is called a sraegy and X p denoes he corresponding wealh process we ge he following maximizaion problem ds V (x) := sup E[ exp( α(x + π F ))] π à S x is he iniial wealh, V is called value funcion 3
2.4 Approach o a soluion consruc a family of sochasic processes R p wih R p T = exp( α(xp T F )) p A R p = R is consan p A R p is a supermaringale p A and here exiss a p A such ha R p is a maringale R p and is iniial value R depend on x ake a sraegy p A and p A and compare he expeced uiliies E[ exp( α(x p T F ))] R (x) = E[ exp( α(x p T o consruc his family R p we se R p := exp( α(x p Y )), where (Y, Z) is a soluion of he BSDE Y = F 2.5 Calculaing a soluion Z s dw s F ))] = V (x) [, T ], p A f(s, Z s )ds, [, T ] we choose a funcion f for which R p is a supermaringale p A and here exiss a p A such ha R p is a maringale we wrie R as he produc of a (local) maringale M p and a decreasing process D p D p is consan for some p M p = exp( α(x Y )) exp( D p = exp( in order o ge a decreasing process f has o saisfy α(p s Z s )dw s 1 2 α 2 (p s Z s ) 2 ds) ( αp s θ s + αf(s, Z s ) + 1 2 α2 (p s Z s ) 2 ) ds) }{{} :=v(s,p s,z s) v(, p, Z ) p A and v(, p, Z ) = for some p A for [, T ] se f(, z) = α 2 dis2 (z + 1 1 αθ, C) + zθ + 2α θ 2 we ge v(, p, z) for p Π C(ω)(Z + 1 α θ ) we obain v(, p, z) = Theorem 1 The value funcion of our opimizaion problem is given by sup E[ exp( α(x + p A p (dw + θ d) F ))] V (x) = exp( α(x Y )) where Y is defined by he unique soluion (Y, Z) H (R) H 2 (R m ) Y = F Z s dw s f(s, Z s )ds wih f(, z) = α 2 dis2 (z + 1 1 θ, C) + zθ + α 2α θ 2. There exiss an opimal rading sraegy p A wih p Π C(ω)(Z + 1 α θ ), [, T ], P a.s. 4
2.6 Usefull ools Lemma 1 see [HIM] Le (a ) [,T ], (σ ) [,T ] be R m respecively R d m valued predicable process, C R d a closed se and C = Cσ, [, T ]. a) The process d = (dis(a, Cσ )) [,T ] is predicable. b) There exiss a predicable process a wih Lemma 2 [HIM] a Π C (a ) [, T ] Le (Y, Z) H (R) H 2 (R m ) be a soluion of he BSDE Y = F and le p be given by Lemma 1 for a = Z + 1 αθ. Then he processes Z s dw s f(s, Z s )ds are P BMO 2 -maringales Proof of Theorem 1 Z s dw s, p sdw s Exisence of a soluion of he BSDE (H1) in [Kob] saes: sufficien condiion for he exisence of a soluion: consans c, c 1 such ha f(, z) c + c 1 z 2 consider he se of admissible sraegies min{ a : a C (ω)} k 1 for λ P-a.e.(, ω) for z R m, [, T ] f is predicable (Lemma1) and we ge dis 2 (z + 1 α θ, C ) 2 z 2 + 2( 1 α θ + k 1 ) 2 (H1) follows from he boundedness of θ Appendix (A4) we have a leas one soluion (Y, Z) H (R) H 2 (R m ) Uniqueness of he soluion we suppose wo soluions (Y 1, Z 1 ), (Y 2, Z 2 ) H (R) H 2 (R m ) and consider he difference Y 1 Y 2 = (Z 1 s Z 2 s )dw s (f(s, Z 1 s ) f(s, Z 2 s ))ds we rewrie for [, T ], z 1, z 2 R m f(, z 1 ) f(, z 2 ) = α 2 [dis2 (z 1 + 1 α θ, C ) dis 2 (z 2 + 1 α θ, C )] +(z 1 z 2 )θ using Lipschiz propery of he disance funcion for a closed se (see Appendix (A5))we obain f(, z 1 ) f(, z 2 ) c 1 z 1 z 2 + c 2 ( z 1 + z 2 )( z 1 z 2 ) c 3 (1 + z 1 + z 2 )( z 1 z 2 ) 5
se β() = { f(,z 1 ) f(,z2 ) Z 1 Z2 if Z 1 Z 2 if Z 1 Z 2 = we obain β() c 3 (1 + Z 1 + Z 2 ), [, T ] from he boundedness of Y 1 and Y 2 he P BMO 2 -propery of Zi sds, i = 1, 2, follows (Lemma 2) herefore β(s)dw s is a P BMO 2 maringale Y 1 Y 2 = (Z 1 s Z 2 s )[dw s β(s)ds] his is a maringale under he equivalen probabiliy measure Q wih densiy E( β()dw ) wih respec o P Appendix (A2) Y 1 T = F = Y 2 T Y 1 = Y 2 and Z 1 = Z 2 p A D p (ω) = 1 λ P-a.s. (p s Z s )dw s is a P BMO 2 maringale E( (p s Z s )dw s ) is a maringale, Appendix (A6) R p is uniformly inegrable Y is a bounded process {exp( αx p τ ) : τ sopping ime wih values in [, T ]} is a uniformly inegrable family Supermaringale propery of R p for p A M p = M E( α (p s Z s )dw s ) is a local maringale D p is a decreasing process, so R p τ n = M τn D p τ n is a supermaringale for any se A F s we have E[R p τ n 1 A ] E[R p s τ n 1 A ] {R p τ n } and {R p s τ n } are uniformly inegrable and Y is bounded n E[R p 1 A ] E[R p s1 A ] 3 Appendix Appendix (A1) see [EP Q] Theorem 2.1 Consider he BSDE dy = f(, Y, Z )d Z dw, Y T = ξ. For a pair (f, ξ) of sandard parameers here exiss a unique pair (Y, Z) H 2 T (Rd ) H 2 T (Rn d ) which solves he BSDE. Wih Y = X, Z = π σ, f(, Z ) = Z θ we have ha (f, ξ) are sandard parameers. ξ = X T L 2 T cause E [ ] sup T X < and is FT -measurable f(, z 1 ) f(, z 2 ) = θ (z 1 z 2 ) θ z 1 z 2 K z 1 z 2 π σ 2 d < (admissible sraegies) Appendix (A2) see [Kaz] le G be a compelee, righ-coninious filraion, P a probabiliy measure and M a coninious local (P; G)-maringale saisfying M = M is in he normed linear space BMO 2 if M BMO2 := sup E[ M T M τ G τ ] 1/2 < τ G-sopping ime 6
M BMO2 := sup E[ M T M τ G τ ] 1/2 < τ G-sopping ime M = ξ sdw s is a BMO 2 -maringale M BMO2 := sup E[ ξ s 2 ds G τ ] 1/2 < τ G-sopping ime τ his condiion is saisfied for bounded inegrands due o finie ime horizon Appendix (A3) [Kob] Le α, β, b R and c be a coninuous increasing funcion. We say he coefficien f saisfies (H1) wih α, β, b, c if for all (, v, z) R + R R d wih f(, v, z) = a (, v, z) + f (, v, z), β a (, v, z) α a.s. f (, v, z) b + c( v ) z 2 a.s. } (H1) Appendix (A4) [Kob] Le f saisfy (H1) wih α, β, b R, c : R + R + coninuous increasing, ξ L (Ω) and T bounded from above. Then he BSDE has a leas one soluion (Y, Z) H T (R) H2 T (Rm ) [Kob] Appendix (A5) dis(z 1, A) dis(z 2, A) = min a A z 1 a min a A z 2 a = z 1 a z 2 a = (z 1 z 2 ) (z 2 a ) z 2 a z 1 z 2 Appendix (A6) Novikov condiion [KT ] For a coninous local maringale M he sochasic exponenial U = exp(m 1 2 M ) is uniformly inegrable if E[exp( 1 2 M )] < In our case we have M = α(p s Z s )dw s. Therefore 1 2 M = α2 (p s Z s ) 2 ds Wih p and Z squareinegrable and T < we ge E[exp(M 1 2 M )] < and U uniformly inegrable. 4 References [EP Q] [HIM] [Kaz] [Kob] [KT ] El Karoui, N.; Peng, S.; Quenez, M.C. Backward sochasic differenial equaions in finance. Mah. Finance 7 (1997), 1-71. Hu, Y., Imkeller, P., Müller, M. Uiliy maximizaion in incomplee markes Kazamaki, N. Coninuous Exponenial Maringales and BMO. Lecure Noes in Mahemaics 1579, Springer, Berlin, 1994. Kobylanski, M. Backward sochasic differenial equaions and parial differenial equaions wih quadraic growh. Ann. Probab. 28 (2),558-62. Kazamaki, N.; Sekiguchi, T. UNIFORM INTEGRABILITY OF CONTINUOUS EXPONENTIAL MARTINGALES Tôhoku Mah. Journ. 35(1983), 289-31. [P P ] Pardoux, E.; Peng, S.G. Adaped soluion of a backward sochasic differenial equaion. Sysems Conrol Le. 14 (199), 55-61. 7