Opimal Porfolio under Fracional Sochasic Environmen Ruimeng Hu Join work wih Jean-Pierre Fouque Deparmen of Saisics and Applied Probabiliy Universiy of California, Sana Barbara Mahemaical Finance Colloquium Universiy of Souhern California Jan 29, 2018 Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 1 / 27
Porfolio Opimizaion: Meron s Problem An invesor manages her porfolio by invesing on a riskless asse B and one risky asse S (single asse for simpliciy) { db = rb d ds = µs d + σs dw π amoun of wealh invesed in he risky asse a ime he wealh process associaed o he sraegy π X π dx π = π ds + Xπ π db (self-financing) S B = (rx π + π (µ r)) d + π σ dw Objecive: M(, x; λ) := sup π A(x,) E [U(X π T ) X π = x where A(x) conains all admissible π and U(x) is a uiliy funcion on R + Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 2 / 27
Porfolio Opimizaion: Meron s Problem An invesor manages her porfolio by invesing on a riskless asse B and one risky asse S (single asse for simpliciy) { db = rb d ds = µs d + σs dw π amoun of wealh invesed in he risky asse a ime he wealh process associaed o he sraegy π X π dx π = π ds + Xπ π db (self-financing) S B = (rx π + π (µ r)) d + π σ dw Objecive: M(, x; λ) := sup π A(x,) E [U(X π T ) X π = x where A(x) conains all admissible π and U(x) is a uiliy funcion on R + Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 2 / 27
Sochasic Volailiy In Meron s work, µ and σ are consan, complee marke Empirical sudies reveal ha σ exhibis random variaion Implied volailiy skew or smile Sochasic volailiy model: µ(y ), σ(y ) incomplee marke Rough Fracional Sochasic volailiy: Gaheral, Jaisson and Rosenbaum 14 Jaisson, Rosenbaum 16 Omar, Masaaki and Rosenbaum 16 We sudy he Meron problem under slowly varying / fas mean-revering fracional sochasic environmen: Nonlinear + Non-Markovian HJB PDE no available Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 3 / 27
Sochasic Volailiy In Meron s work, µ and σ are consan, complee marke Empirical sudies reveal ha σ exhibis random variaion Implied volailiy skew or smile Sochasic volailiy model: µ(y ), σ(y ) incomplee marke Rough Fracional Sochasic volailiy: Gaheral, Jaisson and Rosenbaum 14 Jaisson, Rosenbaum 16 Omar, Masaaki and Rosenbaum 16 We sudy he Meron problem under slowly varying / fas mean-revering fracional sochasic environmen: Nonlinear + Non-Markovian HJB PDE no available Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 3 / 27
Relaed Lieraure Opion Pricing + Markovian modeling: Fouque, Papanicolaou, Sircar and Solna 11 (CUP) Porfolio Opimizaion + Markovian modeling: Fouque, Sircar and Zariphopoulou 13 (MF) Fouque and H. 16 (SICON) Opion Pricing + Non-Markovian modeling: Garnier and Solna 15 (SIFIN), 16 (MF) Porfolio Opimizaion + Non-Markovian modeling: Fouque and H. (slow facor, under revision a MF) Fouque and H. (fas facor, under revision a SIFIN) Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 4 / 27
A General Non-Markovian Model Dynamics of he risky asse S { ds = S [µ(y ) d + σ(y ) dw, { (W Y : a general sochasic process, G := σ Y ) } -adaped, 0 u wih d W, W Y = ρ d. Dynamics of he wealh process X (assume r = 0 for simpliciy): dx π = π µ(y ) d + π σ(y ) dw Define he value process V by V := ess sup π A E [U(X π T ) F where U(x) is of power ype U(x) = x1 γ 1 γ, γ > 0. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 5 / 27
If Y is Markovian... For example, Y is a diffusion process dy = k(y ) d + h(y ) dw Y, V V (, x, y) characerized by a nonlinear HJB PDE Wih a disorion ransformaion 1 V (, x, y) = x1 γ 1 γ Ψ(, y)q, Ψ solves he linear PDE ( 1 Ψ + 2 h2 (y) yy + k(y) y + ρ 1 γ γ λ(y)h(y) y and has he probabilisic represenaion Ψ(, y) = [e Ẽ 1 γ T 2qγ λ 2 (Y s) ds Y = y. 1 Zariphopoulou 99 ) Ψ + 1 γ 2qγ λ2 (y)ψ = 0, Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 6 / 27
If Y is Markovian... For example, Y is a diffusion process dy = k(y ) d + h(y ) dw Y, V V (, x, y) characerized by a nonlinear HJB PDE Wih a disorion ransformaion 1 V (, x, y) = x1 γ 1 γ Ψ(, y)q, Ψ solves he linear PDE ( 1 Ψ + 2 h2 (y) yy + k(y) y + ρ 1 γ γ λ(y)h(y) y and has he probabilisic represenaion Ψ(, y) = [e Ẽ 1 γ T 2qγ λ 2 (Y s) ds Y = y. 1 Zariphopoulou 99 ) Ψ + 1 γ 2qγ λ2 (y)ψ = 0, Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 6 / 27
If Y is Markovian... For example, Y is a diffusion process dy = k(y ) d + h(y ) dw Y, V V (, x, y) characerized by a nonlinear HJB PDE Wih a disorion ransformaion 1 V (, x, y) = x1 γ 1 γ Ψ(, y)q, Ψ solves he linear PDE ( 1 Ψ + 2 h2 (y) yy + k(y) y + ρ 1 γ γ λ(y)h(y) y and has he probabilisic represenaion Ψ(, y) = [e Ẽ 1 γ T 2qγ λ 2 (Y s) ds Y = y. 1 Zariphopoulou 99 ) Ψ + 1 γ 2qγ λ2 (y)ψ = 0, Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 6 / 27
If Y is Markovian + Slowly Varying... For example, Y is a diffusion process dy = δk(y ) d + δh(y ) dw Y, V V (, x, y) characerized by a nonlinear HJB PDE Wih a disorion ransformaion 1 V (, x, y) = x1 γ 1 γ Ψ(, y)q, Ψ solves he linear PDE ( 1 Ψ + 2 δh2 (y) yy + δk(y) y + δρ 1 γ γ λ(y)h(y) y and has he expansion Ψ(, y) = ψ (0) (, y) + δψ (1) (, y) + δψ (2) (, y) +. 1 Zariphopoulou 99 ) Ψ+ 1 γ 2qγ λ2 (y)ψ = 0 Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 6 / 27
In General: Maringale Disorion Transformaion 2 The value process V is given by V = X1 γ [Ẽ (e 1 γ T ) 2qγ λ 2 (Y s) ds q µ(y) G, λ(y) = 1 γ σ(y) where under P, W Y := W Y + 0 a s ds is a sandard BM. The opimal sraegy π is [ π λ(y ) = γσ(y ) + ρqξ X γσ(y ) where ξ is given by he maringale represenaion dm = M ξ d W Y and M is M = [e Ẽ 1 γ T 2qγ 0 λ2 (Y s) ds G 2 Tehranchi 04: differen uiliy funcion, proof and assumpions Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 7 / 27
Remarks only works for one facor model assumpions: inegrabiliy condiions of ξ, X π and π γ = 1 case of log uiliy, can be reaed separaely degenerae case λ(y) = λ 0, M is a consan maringale, ξ = 0 V = X1 γ 1 γ e 1 γ 2γ λ2 0 (T ), π = λ 0 γσ(y ) X. uncorrelaed case ρ = 0, he problem is linear since q = 1 V = X1 γ [ 1 γ E e 1 γ T 2γ λ 2 (Y s) ds G, π = λ(y ) γσ(y ) X. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 8 / 27
Skech of Proof (Verificaion) V is a supermaringale for any admissible conrol π V is a rue maringale following π π is admissible Define α = π /X, hen wih he drif facor D (α ) dv = V D (α ) d + d Maringale D (α ) := α µ γ 2 α2 σ 2 λ2 2γ + q 1 γ a q(q 1) ξ + 2(1 γ) ξ2 + ρqα σξ. α and D (α ) = 0 wih he righ choice of a and q: ( ) 1 γ γ a = ρ λ(y ), q = γ γ + (1 γ)ρ 2. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 9 / 27
Skech of Proof (Verificaion) V is a supermaringale for any admissible conrol π V is a rue maringale following π π is admissible Define α = π /X, hen wih he drif facor D (α ) dv = V D (α ) d + d Maringale D (α ) := α µ γ 2 α2 σ 2 λ2 2γ + q 1 γ a q(q 1) ξ + 2(1 γ) ξ2 + ρqα σξ. α and D (α ) = 0 wih he righ choice of a and q: ( ) 1 γ γ a = ρ λ(y ), q = γ γ + (1 γ)ρ 2. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 9 / 27
Skech of Proof (Verificaion) V is a supermaringale for any admissible conrol π V is a rue maringale following π π is admissible Define α = π /X, hen wih he drif facor D (α ) dv = V D (α ) d + d Maringale D (α ) := α µ γ 2 α2 σ 2 λ2 2γ + q 1 γ a q(q 1) ξ + 2(1 γ) ξ2 + ρqα σξ. α and D (α ) = 0 wih he righ choice of a and q: ( ) 1 γ γ a = ρ λ(y ), q = γ γ + (1 γ)ρ 2. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 9 / 27
Muliple Asses Modeling Consider he following model of S 1, S 2,..., S n ds i = µ i (Y i )S i d + n j=1 σ ij (Y i )S i dw j, i = 1, 2,... n. Each S i is driven by a sochasic facor Y i, bu all facors Y i are adaped o he same single Brownian moion W Y wih he correlaion srucure: d W i, W j = 0, d W i, W Y = ρ d, i, j = 1, 2,..., n, nρ2 < 1. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 10 / 27
Maringale Disorion Transformaion wih Muliple Asses Then, he porfolio value V can be expressed as V = X1 γ [Ẽ (e 1 γ T ) 2qγ µ(y s) Σ(Y s) 1 µ(y s) ds q G, 1 γ he consan q is chosen o be: q = The opimal conrol π is given by [ π Σ(Y ) 1 µ(y ) = γ γ γ + (1 γ)ρ 2 n. + ρqξ σ 1 (Y ) 1 n X. γ Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 11 / 27
Fracional Processes A fracional Brownian moion W (H), H (0, 1) a coninuous Gaussian process zero [ mean E W (H) W s (H) = σ2 H 2 ( 2H + s 2H s 2H) H < 1/2: shor-range correlaion; H > 1/2: long-range correlaion A fracional Ornsein Uhlenbeck process solves dz H = az H d + dw (H) saionary soluion Z H = e a( s) dw s (H) = K( s) dw s Z Gaussian process wih zero mean and consan variance K is non-negaive, K L 2 Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 12 / 27
Fracional Processes A fracional Brownian moion W (H), H (0, 1) a coninuous Gaussian process zero [ mean E W (H) W s (H) = σ2 H 2 ( 2H + s 2H s 2H) H < 1/2: shor-range correlaion; H > 1/2: long-range correlaion A fracional Ornsein Uhlenbeck process solves dz H = az H d + dw (H) saionary soluion Z H = e a( s) dw s (H) = K( s) dw s Z Gaussian process wih zero mean and consan variance K is non-negaive, K L 2 Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 12 / 27
Fracional Processes A fracional Brownian moion W (H), H (0, 1) a coninuous Gaussian process zero [ mean E W (H) W s (H) = σ2 H 2 ( 2H + s 2H s 2H) H < 1/2: shor-range correlaion; H > 1/2: long-range correlaion A fracional Ornsein Uhlenbeck process solves dz H = az H d + dw (H) saionary soluion Z H = e a( s) dw s (H) = K( s) dw s Z Gaussian process wih zero mean and consan variance K is non-negaive, K L 2 Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 12 / 27
Meron Problem under Slowly Varying Fracional SV Consider a rescaled saionary fou process Z δ,h { [ ds = S µ(z δ,h ) d + σ(z δ,h ) dw, Z δ,h = Kδ ( s) dw Z s, K δ () = δk(δ), d W, W Z = ρ d. Our sudy gives, for all H (0, 1): The value process V δ := ess sup π A δ E [U(X π T ) F The corresponding opimal sraegy π Firs order approximaions o V δ and π A pracical sraegy o generae his approximaed value process Apply he maringale disorion ransformaion wih Y = Z δ,h V δ = X1 γ [Ẽ (e 1 γ T ) 2qγ λ 2 (Zs δ,h ) ds q G, 1 γ Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 13 / 27
Meron Problem under Slowly Varying Fracional SV Consider a rescaled saionary fou process Z δ,h { [ ds = S µ(z δ,h ) d + σ(z δ,h ) dw, Z δ,h = Kδ ( s) dw Z s, K δ () = δk(δ), d W, W Z = ρ d. Our sudy gives, for all H (0, 1): The value process V δ := ess sup π A δ E [U(X π T ) F The corresponding opimal sraegy π Firs order approximaions o V δ and π A pracical sraegy o generae his approximaed value process Apply he maringale disorion ransformaion wih Y = Z δ,h V δ = X1 γ [Ẽ (e 1 γ T ) 2qγ λ 2 (Zs δ,h ) ds q G, 1 γ Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 13 / 27
Meron Problem under Slowly Varying Fracional SV Consider a rescaled saionary fou process Z δ,h { [ ds = S µ(z δ,h ) d + σ(z δ,h ) dw, Z δ,h = Kδ ( s) dw Z s, K δ () = δk(δ), d W, W Z = ρ d. Our sudy gives, for all H (0, 1): The value process V δ := ess sup π A δ E [U(X π T ) F The corresponding opimal sraegy π Firs order approximaions o V δ and π A pracical sraegy o generae his approximaed value process Apply he maringale disorion ransformaion wih Y = Z δ,h V δ = X1 γ [Ẽ (e 1 γ T ) 2qγ λ 2 (Zs δ,h ) ds q G, 1 γ Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 13 / 27
Approximaion o he Value Process Theorem (Fouque-H. 17) For fixed [0, T ), he value process V δ akes he form V δ = X1 γ 1 γ e 1 γ 2γ λ2 (Z δ,h 0 )(T ) + X1 γ + δ H ρ X1 γ 1 γ e 1 γ 2γ λ2 (Z δ,h 0 )(T ) λ 2 (Z δ,h 0 )λ (Z δ,h 0 ) + O(δ 2H ), where φ δ is he random componen of order δ H γ [ T ( φ δ = E Zs δ,h e 1 γ 2γ λ2 (Z δ,h 0 )(T ) λ(z δ,h 0 )λ (Z δ,h 0 )φ δ ( ) 1 γ 2 (T ) H+ 3 2 ) Z δ,h 0 γ ds G. Γ(H + 5 2 ) Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 14 / 27
Approximaion o he Opimal Sraegy Recall ha [ π λ(z δ,h ) = γσ(z δ,h ) + ρqξ γσ(z δ,h X ) and ξ is from he maringale rep. of M = [e Ẽ 1 γ T 2qγ 0 λ2 (Zs δ,h ) ds G. Theorem (Fouque-H., 17) The opimal sraegy π is approximaed by [ π λ(z δ,h ) ρ(1 γ) (T ) H+1/2 = γσ(z δ,h + δh ) γ 2 σ(z δ,h ) Γ(H + 3 2 ) λ(z δ,h 0 )λ (Z δ,h 0 ) + O(δ 2H ) := π (0) + δ H π (1) + O(δ 2H ). X Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 15 / 27
Approximaion o he Opimal Sraegy Recall ha [ π λ(z δ,h ) = γσ(z δ,h ) + ρqξ γσ(z δ,h X ) and ξ is from he maringale rep. of M = [e Ẽ 1 γ T 2qγ 0 λ2 (Zs δ,h ) ds G. Theorem (Fouque-H., 17) The opimal sraegy π is approximaed by [ π λ(z δ,h ) ρ(1 γ) (T ) H+1/2 = γσ(z δ,h + δh ) γ 2 σ(z δ,h ) Γ(H + 3 2 ) λ(z δ,h 0 )λ (Z δ,h 0 ) + O(δ 2H ) := π (0) + δ H π (1) + O(δ 2H ). X Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 15 / 27
How Good is he Approximaion? Corollary In he case of power uiliy U(x) = x1 γ 1 γ, π(0) = λ(zδ,h X γσ(z δ,h generaes he ) approximaion of V δ up o order δ H (leading order + wo correcion erms of order δ H ), hus asympoically opimal in A δ. ) H = 1 2, Zδ,H becomes he Markovian OU process, boh approximaion coincides wih resuls in [Fouque Sircar Zariphopoulou 13. The corollary recovers [Fouque -H. 16. Skech of proofs: Apply Taylor ( expansion ) o λ(z) a he poin Z δ,h 0, and hen conrol he momens Z δ,h Z δ,h 0 δ H. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 16 / 27
How Good is he Approximaion? Corollary In he case of power uiliy U(x) = x1 γ 1 γ, π(0) = λ(zδ,h X γσ(z δ,h generaes he ) approximaion of V δ up o order δ H (leading order + wo correcion erms of order δ H ), hus asympoically opimal in A δ. ) H = 1 2, Zδ,H becomes he Markovian OU process, boh approximaion coincides wih resuls in [Fouque Sircar Zariphopoulou 13. The corollary recovers [Fouque -H. 16. Skech of proofs: Apply Taylor ( expansion ) o λ(z) a he poin Z δ,h 0, and hen conrol he momens Z δ,h Z δ,h 0 δ H. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 16 / 27
Meron Problem under Fas-Varying Fracional SV Consider a ɛ-scaled saionary fou process Y ɛ,h Y ɛ,h = ɛ H e a( s) ɛ dw s (H) = K ɛ ( s) dws Y, K ɛ () = 1 K( ɛ ɛ ) ogeher wih he risky asse [ ds = S µ(y ɛ,h ) d + σ(y ɛ,h ) dw, d W, W Y = ρ d. For power uiliies, we obain: The value process V ɛ and he corresponding opimal sraegy π Firs order approximaions o V ɛ and π A sraegy π (0) o generae his approximaed value process Using ergodic propery of Y ɛ,h, bu only valid for H ( 1 2, 1) Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 17 / 27
Meron Problem under Fas-Varying Fracional SV Consider a ɛ-scaled saionary fou process Y ɛ,h Y ɛ,h = ɛ H e a( s) ɛ dw s (H) = K ɛ ( s) dws Y, K ɛ () = 1 K( ɛ ɛ ) ogeher wih he risky asse [ ds = S µ(y ɛ,h ) d + σ(y ɛ,h ) dw, d W, W Y = ρ d. For power uiliies, we obain: The value process V ɛ and he corresponding opimal sraegy π Firs order approximaions o V ɛ and π A sraegy π (0) o generae his approximaed value process Using ergodic propery of Y ɛ,h, bu only valid for H ( 1 2, 1) Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 17 / 27
Meron Problem under Fas-Varying Fracional SV Consider a ɛ-scaled saionary fou process Y ɛ,h Y ɛ,h = ɛ H e a( s) ɛ dw s (H) = K ɛ ( s) dws Y, K ɛ () = 1 K( ɛ ɛ ) ogeher wih he risky asse [ ds = S µ(y ɛ,h ) d + σ(y ɛ,h ) dw, d W, W Y = ρ d. For power uiliies, we obain: The value process V ɛ and he corresponding opimal sraegy π Firs order approximaions o V ɛ and π A sraegy π (0) o generae his approximaed value process Using ergodic propery of Y ɛ,h, bu only valid for H ( 1 2, 1) Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 17 / 27
Approximaion o he Value Process V ɛ Theorem (Fouque-H. 17) For fixed [0, T ), he value process V ɛ akes he form V ɛ = X1 γ 1 γ e 1 γ 2γ λ2 (T ) + X1 γ + ɛ 1 H ρ X1 γ 1 γ e 1 γ 2γ λ2 (T ) λ + o(ɛ 1 H ), where φ ɛ is he random componen of order ɛ 1 H e 1 γ 2γ λ2 (T ) φ ɛ γ ( ) 1 γ 2 λλ (T ) H+ 1 2 γ aγ(h + 3 2 ) [ 1 T ( φ ɛ = E λ 2 (Ys ɛ,h ) λ 2) ds 2 G. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 18 / 27
Opimal Porfolio Theorem (Fouque-H., 17) The opimal sraegy π is approximaed by [ π λ(y ɛ,h ) ρ(1 γ) λλ (T ) H 1/2 = γσ(y ɛ,h + ɛ1 H ) γ 2 σ(y ɛ,h ) aγ(h + 1 2 ) Corollary + o(ɛ 1 H ) ɛ,h λ(y ) γσ(y ɛ,h In he case of power uiliy, π (0) = X generaes he ) approximaion of V ɛ up o order ɛ 1 H (leading order + wo correcion erms of order ɛ 1 H ), hus asympoically opimal in A ɛ. X Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 19 / 27
Ergodiciy of Y ɛ,h For H > 1/2, under appropriae assumpion of λ( ), 0 0 0 (λ 2 (Y ɛ,h s (λ(y ɛ,h s (λ(y ɛ,h s are small and of order ɛ 1 H. ) λ 2 ) ds, ) λ) ds, )λ (Ys ɛ,h ) λλ ) ds, Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 20 / 27
Comparison wih he Markovian Case The value funcion and he opimal sraegy are derived in [FSZ, 13: [ V ɛ (, X ) = X1 γ 1 γ e 1 γ 2γ λ2 (T ) 1 ( ) 1 γ 2 λθ ɛρ (T ) + O(ɛ) γ 2 [ π (, X, Y ɛ,h λ(y ɛ,h ) ) = γσ(y ɛ,h ) + ρ(1 γ) θ (Y ɛ,h ) ɛ γ 2 σ(y ɛ,h X + O(ɛ) ) 2 Formally le H 1 2 in our resuls: [ V ɛ = X1 γ 1 γ e 1 γ 2γ λ2 (T ) 1+ ( ) 1 γ 2 λ λλ ɛρ (T ) + o( ɛ) γ a [ π λ(y ɛ,h ) = γσ(y ɛ,h ) + ρ(1 γ) λλ ɛ γ 2 σ(y ɛ,h X + o( ɛ) ) a Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 21 / 27
Work in preparaion Rough Fas-varying fsv Surprisingly, Y ɛ,h order correcion: V ɛ Y ɛ,h = = X1 γ 1 γ e 1 γ 2γ λ2 (T ) Muliscale fsv ds = S [ Y ɛ,h = µ(y ɛ,h K ɛ ( s) dw Y s, H < 1 2 is no visible o he leading order nor in he firs, Z δ,h K ɛ ( s) dw Y, [ 1 + ( ) 1 γ 2 ɛρ D(T ) + o( ɛ) γ ) d + σ(y ɛ,h Z δ,h =, Z δ,h ) dw, K δ ( s) dw Z. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 22 / 27
Work in preparaion Rough Fas-varying fsv Surprisingly, Y ɛ,h order correcion: V ɛ Y ɛ,h = = X1 γ 1 γ e 1 γ 2γ λ2 (T ) Muliscale fsv ds = S [ Y ɛ,h = µ(y ɛ,h K ɛ ( s) dw Y s, H < 1 2 is no visible o he leading order nor in he firs, Z δ,h K ɛ ( s) dw Y, [ 1 + ( ) 1 γ 2 ɛρ D(T ) + o( ɛ) γ ) d + σ(y ɛ,h Z δ,h =, Z δ,h ) dw, K δ ( s) dw Z. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 22 / 27
Meron Problem under General Uiliy Maringale Disorion Transformaion is no available Sar wih a given sraegy π (0) A firs order approximaion o V π(0),δ obained by epsilon-maringale decomposiion 34 Opimaliy of π (0) in a smaller class of conrols of feedback form Denoe by v (0) (, x, z) he value funcion a he Sharpe-raio λ(z), we define π (0) by π (0) (, x, z) = λ(z) v x (0) (, x, z) σ(z) v xx (0) (, x, z) and he associaed value process V π(0),δ V π(0),δ 3 Fouque Papanicolaou Sircar 01 4 Garnier Solna 15 := E [ U(X π(0) T ) F. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 23 / 27
Meron Problem under General Uiliy Maringale Disorion Transformaion is no available Sar wih a given sraegy π (0) A firs order approximaion o V π(0),δ obained by epsilon-maringale decomposiion 34 Opimaliy of π (0) in a smaller class of conrols of feedback form Denoe by v (0) (, x, z) he value funcion a he Sharpe-raio λ(z), we define π (0) by π (0) (, x, z) = λ(z) v x (0) (, x, z) σ(z) v xx (0) (, x, z) and he associaed value process V π(0),δ V π(0),δ 3 Fouque Papanicolaou Sircar 01 4 Garnier Solna 15 := E [ U(X π(0) T ) F. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 23 / 27
Meron Problem under General Uiliy Maringale Disorion Transformaion is no available Sar wih a given sraegy π (0) A firs order approximaion o V π(0),δ obained by epsilon-maringale decomposiion 34 Opimaliy of π (0) in a smaller class of conrols of feedback form Denoe by v (0) (, x, z) he value funcion a he Sharpe-raio λ(z), we define π (0) by π (0) (, x, z) = λ(z) v x (0) (, x, z) σ(z) v xx (0) (, x, z) and he associaed value process V π(0),δ V π(0),δ 3 Fouque Papanicolaou Sircar 01 4 Garnier Solna 15 := E [ U(X π(0) T ) F. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 23 / 27
Meron Problem under General Uiliy Maringale Disorion Transformaion is no available Sar wih a given sraegy π (0) A firs order approximaion o V π(0),δ obained by epsilon-maringale decomposiion 34 Opimaliy of π (0) in a smaller class of conrols of feedback form Denoe by v (0) (, x, z) he value funcion a he Sharpe-raio λ(z), we define π (0) by π (0) (, x, z) = λ(z) v x (0) (, x, z) σ(z) v xx (0) (, x, z) and he associaed value process V π(0),δ V π(0),δ 3 Fouque Papanicolaou Sircar 01 4 Garnier Solna 15 := E [ U(X π(0) T ) F. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 23 / 27
Epsilon-Maringale Decomposiion Finding Q π(0),δ Then such ha Q π(0),δ T = V π(0),δ T = U(XT π(0) ), Q π(0),δ = M δ + R δ, where M δ is a maringale and R δ is of order δ 2H. V π(0),δ [ = E Q π(0),δ T F = Q π(0),δ R δ + E = M δ + E [RT δ F [R δ T F, and Q π(0),δ is he approximaion o V π(0),δ wih error of order O(δ 2H ) Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 24 / 27
Epsilon-Maringale Decomposiion Finding Q π(0),δ Then such ha Q π(0),δ T = V π(0),δ T = U(XT π(0) ), Q π(0),δ = M δ + R δ, where M δ is a maringale and R δ is of order δ 2H. V π(0),δ [ = E Q π(0),δ T F = Q π(0),δ R δ + E = M δ + E [RT δ F [R δ T F, and Q π(0),δ is he approximaion o V π(0),δ wih error of order O(δ 2H ) Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 24 / 27
Epsilon-Maringale Decomposiion Finding Q π(0),δ Then such ha Q π(0),δ T = V π(0),δ T = U(XT π(0) ), Q π(0),δ = M δ + R δ, where M δ is a maringale and R δ is of order δ 2H. V π(0),δ [ = E Q π(0),δ T F = Q π(0),δ R δ + E = M δ + E [RT δ F [R δ T F, and Q π(0),δ is he approximaion o V π(0),δ wih error of order O(δ 2H ) Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 24 / 27
Firs order approximaion o V π(0),δ Proposiion For fixed [0, T ), he F -measurable value process V π(0),δ is of he form V π(0),δ = Q π(0),δ where Q π(0),δ (x, z) is given by: (X π(0), Z δ,h 0 ) + O(δ 2H ), Q π(0),δ (x, z) =v (0) (, x, z) + λ(z)λ (z)d 1 v (0) (, x, z)φ δ + δ H ρλ 2 (z)λ (z)d1v 2 (0) (T )H+3/2 (, x, z) Γ(H + 5 2 ). For power uiliy, Q π(0),δ coincides wih he approximaion of V δ For he Markovian case H = 1 2, recovers he resuls in [Fouque-H. 16 Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 25 / 27
Firs order approximaion o V π(0),δ Proposiion For fixed [0, T ), he F -measurable value process V π(0),δ is of he form V π(0),δ = Q π(0),δ where Q π(0),δ (x, z) is given by: (X π(0), Z δ,h 0 ) + O(δ 2H ), Q π(0),δ (x, z) =v (0) (, x, z) + λ(z)λ (z)d 1 v (0) (, x, z)φ δ + δ H ρλ 2 (z)λ (z)d1v 2 (0) (T )H+3/2 (, x, z) Γ(H + 5 2 ). For power uiliy, Q π(0),δ coincides wih he approximaion of V δ For he Markovian case H = 1 2, recovers he resuls in [Fouque-H. 16 Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 25 / 27
Asympoically Opimaliy of π (0) Theorem (Fouque-H. 17) The rading sraegy π (0) (, x, z) = λ(z) σ(z) v x (0) (,x,z) v xx (0) (,x,z) is asympoically opimal in he following class: { Ã δ [ π 0, π 1, α := π = π 0 + δ α π } 1 : π A δ, α > 0, 0 < δ 1. Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 26 / 27
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Sochasic Volailiy wih Fas Facor Y S is modeled by: { ds = µ(y )S d + σ(y )S dw, dy = 1 ɛ b(y ) d + 1 ɛ a(y ) dw Y, wih correlaion dw W Y = ρ d. Theorem (Fouque-H., in prep.) Under appropriae assumpions, for fixed (, x, y) and any family of rading sraegies A 0 (, x, y) [ π 0, π 1, α, he following limi exiss and saisfies Ṽ ɛ (, x, y) V π(0),ɛ (, x, y) l := lim 0. ɛ 0 ɛ Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 27 / 27
Theorem (Fouque-H., in prep.) The residual funcion E(, x, y) := V π(0),ɛ (, x) v (0) (, x) ɛv (1) (, x) is of order ɛ, where in his case, v (0) solves v (0) 1 2 λ2 and v (1) = 1 2 (T )ρ 1BD 2 1 v(0) (, x). ( v (0) x v (0) xx ) 2 = 0, Ruimeng Hu (UCSB) Opimal Porfolio under Fracional Environmen Jan. 29, 2018 27 / 27