On Hyperbolic Growth of Long-term Bayesian Optimal Power-utility - - PowerPoint PPT Presentation

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On Hyperbolic Growth of Long-term Bayesian Optimal Power-utility

Hideaki Miyata† and Jun Sekine‡

†Kyoto University and ‡Osaka University

Workshop on Stochastic Processes and Their Applications, March 8–9, 2012, NCTS, Hsinchu, Taiwan

Key Finding

§1. Introduction ⊲ Key Finding ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Bayesian CRRA-utility maximization: U(T,γ)(x) := sup

π Eu(γ)(Xx,π T ),

u(γ)(x) := x1−γ 1 − γ , γ > 1.

■ 1-riskless asset (r: risk-free interest rate) and n-risky assets. ■ λ : market price of risk, hidden and unobservable r.v. ■ available information is generated by risky assets prices.

Key Finding

§1. Introduction ⊲ Key Finding ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Bayesian CRRA-utility maximization: U(T,γ)(x) := sup

π Eu(γ)(Xx,π T ),

u(γ)(x) := x1−γ 1 − γ , γ > 1.

■ 1-riskless asset (r: risk-free interest rate) and n-risky assets. ■ λ : market price of risk, hidden and unobservable r.v. ■ available information is generated by risky assets prices.

Are interested in ∂T log U (T,γ)(x) as T ≫ 1, U (T,γ)(x) = u(γ)(x) exp T ∂t log U (t,γ)(x)dt

• .

Hyperbolic Growth

§1. Introduction ⊲ Key Finding ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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∂T log U (T,γ)(x) = (1 − γ)r − n 2 1 T + ǫ(T) as T → ∞.

■ ǫ(T): “smaller” than 1/T, (depends on the law of λ).

Hyperbolic Growth

§1. Introduction ⊲ Key Finding ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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∂T log U (T,γ)(x) = (1 − γ)r − n 2 1 T + ǫ(T) as T → ∞.

■ ǫ(T): “smaller” than 1/T, (depends on the law of λ).

Sharp contrast to “non-Bayesian” case with exponential growth ∂T log U (T,γ)(x) = (1 − γ)

• r + 1

2γ |λ|2

• .

Lifetime Portfolio Selection with Hyperbolic Discount

§1. Introduction ⊲ Key Finding ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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U (γ)(x) := sup

(π,c)

E ∞ exp

• −

t ρ(u)du

• u(γ)(ct)dt,

ρ(t) :=δ + β 1 + αt,

■ agent’s wealth (Xx,π,c

t

)t≥0.

■ Merton (1969, 1971): β = 0. ■ Zervos (2008), Bjork and Murgoci (2010), Eckland, Mbodji and

Pirvu (2011): finite horizon with a priori hyperbolic discounting.

“Endogeneous” Hyperbolic Discounting

§1. Introduction ⊲ Key Finding ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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For Bayesian agents,

■ characterize the critical discount rate:

ρ(t) := δ + β 1 + αt, to ensure the solvability, i.e., |U (γ)(x)| < ∞ iff one of the following conditions is satisfied, (a) δ > δ, or (b) δ = δ and β > β.

Plan

§1. Introduction ⊲ Key Finding ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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• 2. Setup: Bayesian CRRA-utility maximization
• 3. Preliminary: Non-Bayesian Result
• 4. Bayesian CRRA-optimal Utility

◆ Karatzas and Zhao (2001).

• 5. Long-time Asymptotics

◆ hyperbolic growth ◆ cost of uncertainty

• 6. Lifetime Optimal Consumption

◆ “endogeneous” hyperbolic discount rate (Miyata, 2012)

Market Model(1)

§1. Introduction §2. Setup ⊲ Market Model(1) ⊲ Market Model(2) ⊲ Bayes. Util. Max. ⊲ Related works §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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■ S0

t := ert: riskless,

■ S := (S1, . . . , Sn)⊤, Si := (Si

t)t≥0: n-risky,

dSt = diag(St)

• r1dt + σ(t, St)d ˜

Wt

• ,

S0 ∈ Rn

++

• n (Ω, F, ˜

P, (Ft)t≥0), Ft := σ( ˜ Wu; u ∈ [0, t]), 0 < ∃c1I ≤ σσ⊤(t, y) ≤ ∃c2I.

■ (Ω, F, ˜

P) : “reference” prob. space,

◆ n-dim. BM. ˜

W := ( ˜ W 1, . . . , ˜ W n)⊤, ˜ W i := ( ˜ W i

t )t≥0,

◆ n-dim. r.v. λ ⊥

⊥ ˜

• W. ν(dx) := ˜

P(λ ∈ dx).

■ Ft = σ(Su; u ∈ [0, t]): “public” information.

Market Model(2)

§1. Introduction §2. Setup ⊲ Market Model(1) ⊲ Market Model(2) ⊲ Bayes. Util. Max. ⊲ Related works §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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■ (Gt)t≥0, Gt := Ft ∨ σ(λ), ■ real-world prob. measure:

P on (Ω, ∨t≥0Gt) by dP|Gt = Ztd˜ P|Gt, Zt := exp

• λ⊤( ˜

Wt − ˜ W0) − |λ|2 2 t

• .

W := (Wt)t≥0, Wt := ˜ Wt − λt: (P, Gt)-BM.

■ P-dynamics:

dSt =diag(St) {µ(t, St)dt + σ(t, St)dWt} , µ(t, y) :=r1 + σ(t, y)λ λ := σ−1(µ − r1) : unobservable market price of risk.

■ W ⊥

⊥ λ under P, P(λ ∈ dx) = ˜ P(λ ∈ dx) = ν(dx).

Bayesian CRRA-Utility Maximization

§1. Introduction §2. Setup ⊲ Market Model(1) ⊲ Market Model(2) ⊲ Bayes. Util. Max. ⊲ Related works §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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U (T,γ)(x) := sup

π∈AT

Eu(γ)

• Xx,π

T

• with CRRA-utility,

u(γ)(x) :=    x1−γ 1 − γ if γ > 0, = 1, log x if γ = 1, where dXx,π

t

= Xx,π

t

n

• i=1

πi

t

dSi

t

Si

t

+

• 1 −

n

• i=1

πi

t

• dS0

t

S0

t

• ,

Xx,π = x, and AT :=

• (ft)t∈[0,T]
• n-dim. Ft-prog. m’ble,

T

0 |ft|2dt < ∞ a.s.

• .

Related works

§1. Introduction §2. Setup ⊲ Market Model(1) ⊲ Market Model(2) ⊲ Bayes. Util. Max. ⊲ Related works §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Similar market models and problems with unobservable random market price of risks :

■ Brennan and Xia (2001), Cvitani´

c et. al. (2006), Karatzas (1997), Karatzas and Zhao (2001), Lakner (1995), Pham and Quenez (2001), Rieder and B¨ auerle (2005), Xia (2001), and Zohar (2001), for example.

Constant λ: Merton (1969, 1971)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result ⊲ Constant λ ⊲ LT Opt. (1) ⊲ LT Opt. (2) §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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(A) The optimal investment strategy ˆ π(γ) := (ˆ π(γ)

t

)t∈[0,T] ∈ AT : ˆ π(γ)

t

:= 1 γ (σσ⊤)−1(µ − r1)(t, St) = 1 γ (σ⊤)−1(t, St)λ, : T-independent ! (B) The optimal wealth process ( ˆ X(γ)

t

)t∈[0,T] : ˆ X(γ)

t

:= Xx,ˆ

π(γ) t

= x exp 1 γ λ⊤ (Wt + λt) +

• r −

1 2γ2 |λ|2

• t
• .

(C) The optimal utility: U (T,γ)(x) = Eu(γ) ˆ X(γ)

T

• = u(γ)
• xe
• r+ 1

2γ |λ|2

T

• .

Long-term Optimalities (1)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result ⊲ Constant λ ⊲ LT Opt. (1) ⊲ LT Opt. (2) §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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■ When γ = 1, for ∀π,

lim

T→∞

1 T log Xx,π

T

≤ 1 T log ˆ X(1)

T

= −Γ′(1) a.s..

■ When 0 < γ < 1, for ∀π,

lim

T→∞

1 T log Eu(γ)(Xx,π

T ) ≤ lim T→∞

1 T log Eu(γ)( ˆ X(γ)

T ) = Γ(γ).

■ When γ > 1, for ∀π,

lim

T→∞

1 T log Eu(γ)(Xx,π

T ) ≥ lim T→∞

1 T log Eu(γ)( ˆ X(γ)

T ) = Γ(γ).

where Γ(γ) := (1 − γ)

• r + 1

2γ |λ|2

• ,

−Γ′(1) := r + 1 2|λ|2.

Long-term Optimalities (1)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result ⊲ Constant λ ⊲ LT Opt. (1) ⊲ LT Opt. (2) §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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■ When γ = 1, for ∀π, (ˆ

π(1): Kelly portfolio) lim

T→∞

1 T log Xx,π

T

≤ 1 T log ˆ X(1)

T

= −Γ′(1) a.s..

■ When 0 < γ < 1, for ∀π,

lim

T→∞

1 T log Eu(γ)(Xx,π

T ) ≤ lim T→∞

1 T log Eu(γ)( ˆ X(γ)

T ) = Γ(γ).

■ When γ > 1, for ∀π, (ˆ

π(γ) (γ > 1): fractional Kelly portfolio) lim

T→∞

1 T log Eu(γ)(Xx,π

T ) ≥ lim T→∞

1 T log Eu(γ)( ˆ X(γ)

T ) = Γ(γ).

where Γ(γ) := (1 − γ)

• r + 1

2γ |λ|2

• ,

−Γ′(1) := r + 1 2|λ|2.

Long-term Optimalities (2)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result ⊲ Constant λ ⊲ LT Opt. (1) ⊲ LT Opt. (2) §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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■ Let 0 < γ < 1. With k(γ) > −Γ′(1), for ∀π,

lim

T→∞

1 T log P

• Xx,π

T

≥ ek(γ)T ≤ lim

T→∞

1 T log P

• ˆ

X(γ)

T

≥ ek(γ)T .

■ Let γ > 1. With r < k(γ) < −Γ′(1), for ∀π,

lim

T→∞

1 T log P

• Xx,π

T

≤ ek(γ)T ≥ lim

T→∞

1 T log P

• ˆ

X(γ)

T

≤ ek(γ)T , where k(γ) := −Γ′(γ) = r + 1 2γ2 |λ|2.

■ Pham (2003), Hata, Nagai and Sheu (2010).

Log-utility case (1)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Let F(t, y) :=

• Rn exp
• z⊤y − |z|2

2 t

• ν(dz).

Recall ˜ E[Zt|Ft] = F(t, ˜ Wt − ˜ W0) and the expression for the Bayesian estimator ˆ λt := E[λ|Ft] = ˜ E[Ztλ|Ft] ˜ E[Zt|Ft] = ∇ log F(t, ˜ Wt − ˜ W0).

Log-utility case (2)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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• 1. The optimal wealth process ˆ

X(T,1) := ( ˆ X(T,1)

t

)t∈[0,T], ˆ X(T,1)

t

= xertF(t, ˜ Wt − ˜ W0).

• 2. The optimal strategy ˆ

π(T,1) := (ˆ π(T,1)

t

)t∈[0,T] s.t. ˆ X(T,1)

t

= Xx,ˆ

π(T,1) t

t ∈ [0, T], ˆ π(T,1)

t

:= (σ(t, St)⊤)−1 ∇F F

• (t, ˜

Wt − ˜ W0) = (σ(t, St)⊤)−1ˆ λt.

• 3. The optimal expected utility,

U(T,1)(x) = log x + rT + ˜ EF(T, ˜ WT − ˜ W0) log F(T, ˜ WT − ˜ W0) = log x + rT + 1 2E T |ˆ λt|2dt.

Power-utility case (1)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Let G(T,γ)(t, y) :=˜ E

• F(T, y + ˜

Wt − ˜ W0)

1 γ

• =
• Rn F(T, y +

√ tz)

1 γ

1 (2π)

n 2 e− |z|2 2 dz,

∇G(T,γ)(t, y) =

• Rn(∇F · F

1−γ γ )(T, y +

√ tz) 1 (2π)

n 2 e− |z|2 2 dz,

(0 ≤ t ≤ T < ∞), assuming the integrals have finite values.

Power-utility case (1)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Let G(T,γ)(t, y) :=˜ E

• F(T, y + ˜

Wt − ˜ W0)

1 γ

• =
• Rn F(T, y +

√ tz)

1 γ

1 (2π)

n 2 e− |z|2 2 dz,

∇G(T,γ)(t, y) =

• Rn(∇F · F

1−γ γ )(T, y +

√ tz) 1 (2π)

n 2 e− |z|2 2 dz,

(0 ≤ t ≤ T < ∞), assuming the integrals have finite values.

■ “γ > 1” + “E|λ| < ∞” is a sufficient condition.

Power-utility case (2)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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• 1. The optimal wealth process ˆ

X(T,γ) := ( ˆ X(T,γ)

t

)t∈[0,T], ˆ X(T,γ)

t

= xert G(T,γ)(T − t, ˜ Wt − ˜ W0) G(T,γ)(T, 0) ,

• 2. The optimal strategy ˆ

π(T,γ) := (ˆ π(T,γ)

t

)t∈[0,T] s.t. ˆ X(T,γ)

t

= Xx,ˆ

π(T,γ) t

t ∈ [0, T], ˆ π(T,γ)

t

:= (σ(t, St)⊤)−1 ∇G(T,γ)(T − t, ˜ Wt − ˜ W0) G(T,γ)(T − t, ˜ Wt − ˜ W0) .

• 3. The optimal expected utility,

U(T,γ)(x) = u(γ)(xerT )

• G(T,γ)(T, 0)

γ .

Sketch of Proof (1): HJB Approach

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Eu(γ)(Xx,π

T ) = ˜

EZT u(γ)(Xx,π

T ) = ˜

E ˜ ZT u(γ)(Xx,π

T ),

where ˜ Zt := ˜ E[Zt|Ft] = F(t, ˜ Wt − ˜ W0). Let αt := σ⊤(t, St)πt to see

• Xx,π

T

1−γ = x1−γe(1−γ)rT M((1−γ)α)

T

exp

• −γ(1 − γ)

2 T |αt|2 dt

• ,

where M(α)

t

:= exp t α⊤

u d ˜

Wu − 1 2 t |αu|2 du

• .

Introduce ˜ P(α)

T : prob. measure on (Ω, FT ) and (˜

P(α)

T , Ft)-BM,

d˜ P(α)

T

d˜ P

• Ft

:= M ((1−γ)α)

t

, ˜ W (α)

t

:= ˜ Wt − (1 − γ) t αudu.

Sketch of Proof (2): HJB Approach

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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With α := σ⊤π, rewrite ET u(γ)(Xπ

T ) = u(γ)(xerT )

× ˜ E(α)

T

exp

• log F(T, ˜

WT − ˜ W0) − γ(1 − γ) 2 T |αt|2 dt

• .

Let γ > 1. Consider, for 0 ≤ t ≤ T < ∞, e

¯ V (T )(t,y) :=

inf

α

˜ E(α)

T

exp

• log F(T, Y (y)

T−t − ˜

W0) + γ(γ − 1) 2 T−t |αs|2ds

• ,

where dY (y)

s

= (1 − γ)αsds + d ˜ W (α)

s

, Y (y) = y.

Sketch of Proof (3): HJB Approach

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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The HJB equation, −∂tV =1 2

• ∆V + |∇V |2

+ inf

α∈Rn

• (1 − γ)α⊤∇V + γ(γ − 1)

2 |α|2

• ,

V (T, y) = log F(T, y − ˜ W0). Here, (1−γ)α⊤∇V +γ(γ − 1) 2 |α|2 = γ(γ − 1) 2

• α − 1

γ ∇V

• 2

−γ − 1 2γ |∇V |2 . So, rewritten as −∂tV =1 2∆V + 1 2γ |∇V |2, V (T, y) = log F(T, y − ˜ W0).

Sketch of Proof (4): HJB Approach

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao ⊲ Log-utility case (1) ⊲ Log-utility case (2) ⊲ Power-util. case (1) ⊲ Power-util. case (2) ⊲ HJB Approach (1) ⊲ HJB Approach (2) ⊲ HJB Approach (3) ⊲ HJB Approach (4) §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Note L := e

1 γ V satisfies

−∂tL = 1 2∆L, L(T, y) = F(T, y − ˜ W0)

1 γ .

So, we deduce eV (t,y) =

• ˜

E

• F(T, ˜

WT − ˜ W0)

1 γ

˜ Wt = y γ . After verification-steps, we obtain U(T,γ)(x) = u(γ)(xerT )

• ˜

E

• F(T, ˜

WT − ˜ W0)

1 γ

γ .

Log Case

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics ⊲ Log Case ⊲ Power Case (1) ⊲ Power Case (2) ⊲ Cost of Uncert. (1) ⊲ Cost of Uncert. (2) §6. Bayesian Lifetime Optimal Consumption §7. Remarks

22 / 30

ˆ π(1)

t

:= (σ⊤)−1(t, St)ˆ λt is T-independent. So, re-define ˆ X(1) := ( ˆ X(1)

t

)t≥0, ˆ π(1) := (ˆ π(1)

t )t≥0

to see, for any π ∈ A , lim

T→∞

1 T E log Xx,π

T

≤ lim

T→∞

1 T E log ˆ X(1)

T

= r + 1 2E|λ|2 and lim

T→∞

1 T log Xx,π

T

≤ lim

T→∞

1 T log ˆ X(1)

T

= r + 1 2|λ|2 P-a.s..

Power Case (1)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics ⊲ Log Case ⊲ Power Case (1) ⊲ Power Case (2) ⊲ Cost of Uncert. (1) ⊲ Cost of Uncert. (2) §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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Assume that γ > 1 and that ν(dz) = fν(z)dz with fν ∈ L∞(Rn), which is continuous at 0 ∈ Rn. It holds that U (T,γ)(x) =u(γ)(xerT ) 2π T

• γ

γ − 1 γ n

2

Ψ(γ)(T, 0)γ ∼u(γ)(xerT ) 2π T

• γ

γ − 1 γ n

2

fν(0) as T → ∞ and that ∂T log U (T,γ)(x) = (1 − γ)r − n 2 1 T + ǫ(T),

Power Case (2)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics ⊲ Log Case ⊲ Power Case (1) ⊲ Power Case (2) ⊲ Cost of Uncert. (1) ⊲ Cost of Uncert. (2) §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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where Ψ(γ)(T, y) :=

• Rn

γ − 1 2πγT n

2

exp

• −γ − 1

2γT

• z −

1 γ − 1y

• 2

× ψ

• T, y + z

T 1

γ

dz, ψ(t, x) := t 2π n

2

Rn exp

• − t

2 |x − z|2

• ν(dz).

■

lim

T→∞ Ψ(γ)(T, y) = fν(0)

1 γ for each y ∈ Rn.

■ The residual ǫ(T) is “smaller” than 1/T as T → ∞ s.t.

lim

T→∞

• T

1

ǫ(t)dt

• =
• log fν(0) − log Ψ(γ)(1, 0)
• < ∞.

Cost of Uncertainty of λ (1)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics ⊲ Log Case ⊲ Power Case (1) ⊲ Power Case (2) ⊲ Cost of Uncert. (1) ⊲ Cost of Uncert. (2) §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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■ Consider an “inside” investor, whose available information flow

is Gt := Ft ∨ σ(λ) (t ≥ 0).

■ Insider’s utility maximization,

sup

π∈A G

T

E

• u(γ)(Xx,π

T )|G0

• .

■ Expected optimal utility for the insider,

¯ U (T,γ)(x) :=E

• sup

π∈A G

T

E

• u(γ)(Xx,π

T )|G0

• =E
• u(γ)
• xe
• r+ 1

2γ |λ|2

T

• .

Cost of Uncertainty of λ (2)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics ⊲ Log Case ⊲ Power Case (1) ⊲ Power Case (2) ⊲ Cost of Uncert. (1) ⊲ Cost of Uncert. (2) §6. Bayesian Lifetime Optimal Consumption §7. Remarks

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It holds that U(T,1)(x) ∼ ¯ U(T,1)(x) as T → ∞ (cost of uncertainty becomes negligible as T → ∞ for log-investor) and that, for γ > 1, U (T,γ)(x) ∼ ¯ U (T,γ)(x)

• γ

γ − 1 (γ−1)n

2

as T → ∞. (cost of uncertainty does not disappear even when T → ∞ for power-investor).

Setup

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption ⊲ Setup ⊲ Lifetime Consump. ⊲ Min. Discount §7. Remarks

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■ Similar model with a hidden, unobservable λ and stochastic

interest rates. On the reference prob. space (Ω, F, ˜ P, (Ft)t≥0), dS0

t =r(Yt)S0 t dt,

S0

0 = 1,

dSt =diag(St)

• r(Yt)1dt + σ(t, St)d ˜

Wt

• ,

S0 ∈ Rn

++,

dYt =KYtdt + d ˜ Wt, Y0 ∈ Rn, r(y) := a0 + a⊤

1 y, a0 ∈ R, a1 ∈ Rn.

■ real-world prob. measure:

P on (Ω, ∨t≥0Gt) by dP|Gt = Ztd˜ P|Gt, Zt := exp

• λ⊤( ˜

Wt − ˜ W0) − |λ|2 2 t

• .

With Wt := ˜ Wt − λt: (P, Gt)-BM., the market dynamics under P is described.

Bayesian Lifetime Optimal Consumption

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption ⊲ Setup ⊲ Lifetime Consump. ⊲ Min. Discount §7. Remarks

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Let dXx,π,c

t

=

n

• i=1

πi

t

dSi

t

Si

t

+

• Xx,π,c

t

−

n

• i=1

πi

t

• dS0

t

S0

t

− ctdt, Xx,π,c = x, where (π, c) is Ft-adapted. Consider U (γ)(x) := sup

(π,c)

E ∞ exp

• −

t ρ(u)du

• u(γ)(ct)dt,

ρ(t) :=δ + β 1 + αt subject to Xx,π,c ≥ 0.

■ Minimal discount rate ρ(t) to ensure |U (γ)(x)| < ∞ ?

Minimal Discount Rate (Miyata, 2012)

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption ⊲ Setup ⊲ Lifetime Consump. ⊲ Min. Discount §7. Remarks

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Let γ > 1. |U (γ)(x)| < ∞ iff one of the following holds (a) δ > δ, (b) δ = δ and β > γ − n

2 ,

where δ = lim

T→∞

1 T log |U (T,γ)(x)| = lim

T→∞ inf π

1 T log E(Xx,π,0

T

)(1−γ) =(1 − γ)

• a0 − 1

2

• K⊤a1
• 2

.

■ Hyperbolic term is endogenously derived.

Remarks and Open problems

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks ⊲ Rem. & Open

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■ “Good” T-independent strategy (˜

πt)t≥0 for long-term investor, who is more risk-averse than log-investor ?

◆ Fractional Bayesian Kelly : ˇ

π(γ)

t

:= 1 γ (σ⊤)−1(t, St)ˆ λt ??

Remarks and Open problems

§1. Introduction §2. Setup §3. Preliminary: Non-Bayesian Result §4. Bayesian CRRA-optimal Utility: Karatzas and Zhao §5. Long-time Asymptotics §6. Bayesian Lifetime Optimal Consumption §7. Remarks ⊲ Rem. & Open

30 / 30

■ “Good” T-independent strategy (˜

πt)t≥0 for long-term investor, who is more risk-averse than log-investor ?

◆ Fractional Bayesian Kelly : ˇ

π(γ)

t

:= 1 γ (σ⊤)−1(t, St)ˆ λt ??

■ Bayesian lifetime consumption with a general stochastic interest

rate model.