Bayesian Optimal Power-utility Grows Hyperbolically in the Long Run Hideaki Miyata† and Jun Sekine‡ † Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan. e-mail: [email protected] ‡

Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka 560-8531, Japan. e-mail: [email protected] March 5, 2012 Abstract A Bayesian power-utility maximization is considered, where meanreturn-rates of risky assets (or, more precisely, the market price of risk) is an unobservable random vector and the Arrow-Pratt’s risk-aversion parameter is larger than 1. It is shown that the optimal expected utility grows hyperbolically in the long run if we omit the effect of the risk-free interest rate. This provides a sharp contrast to the results of “non-Bayesian” settings: for instance, in the case of constant market price of risk, the optimal expected power-utility grows exponentially in the long run. Keywords: Bayesian CRRA-utility maximization, partial information, long-term growth rate, hyperbolic growth, hyperbolic discount, Kelly portfolio, fractional Kelly portfolio.

1

Introduction

In the present article, we introduce major findings of Hayashi, Miyata and Sekine (2012), where the expected power-utility maximization of terminal wealth (1.1)

U (T,γ) (x) := sup Eu(γ) (XTx,π ) π

is considered in a continuous-time financial market, consists of one riskless asset and n-risky assets. Here, we use notation for the CRRA-utility function u(γ) (x) := 1

x1−γ , 1−γ

where Arrow-Pratt’s relative risk-aversion parameter is set as γ > 1, and we denote by XTx,π the wealth of a self-financing investor at the terminal date T ∈ R++ , where x ∈ R++ is an initial wealth and π := (πt )t∈[0,T ] is a dynamic investment policy. In particular, we assume that the so-called market price of risk vector λ is a hidden, unobservable random variable, which means that (1.1) is a partially-observable (or Bayesian) optimization problem (see Section 2 for the detail of the setup). For this problem, we are interested in the long-term growth rate of optimal expected utility, i.e., writing {∫ } T

U (T,γ) (x) = u(γ) (x) exp

∂t log U (t,γ) (x)dt , 0

we are interested in the asymptotic behaviour of ∂t log U (t,γ) (x) as t ≫ 1. We obtain the following hyperbolic long-term growth rate, (1.2)

∂T log U (T,γ) (x) = (1 − γ)r −

n1 + ϵ(T ) 2T

as T → ∞,

where r is the constant risk-free interest rate and ϵ(T ) is a function “smaller” than 1/T as T → ∞ (see Proposition 5.2 and Remark 5.2 for the details). It is interesting to see that (1.2) provides a sharp contrast to the result of “nonBayesian” case: if the market price of risk vector λ is constant, then, we have the exact expression, ( ) 1 (T,γ) 2 ∂T log U (x) = (1 − γ) r + |λ| , 2γ (see (3.3)), i.e., this optimal non-Bayesian power-utility grows exponentially with respect to T , and the (norm) of the market price of risk vector affects the growth rate. It is also interesting to see that the right-hand-side of (1.2) has a “universal” value: it is independent of the law of λ (except for the residual term ϵ(T )), and the hyperbolic term depends on the number n of risky-assets (= the dimension of driving Brownian motion) only. Remark 1.1 (“Endogenous” Hyperbolic Discounting). The above hyperbolic growth of optimal power-utility plays an interesting role in the lifetime consumption maximization problem, { ∫ t } ∫ ∞ (1.3) U (γ) (x) := sup E exp − ρ(u)du u(γ) (ct )dt, (π,c)

0

0

which is studied in Miyata (2012), [15]. To consider (1.3), a similar market model with partial information is employed and a Bayesian self-financing investor with the wealth process (Xtx,π,c )t≥0 is considered, where x ∈ R++ is an initial wealth, π := (πt )t≥0 is a dynamic investment policy, and c := (ct )t≥0 is 2

a dynamic consumption plan, and, for a given discount rate process (ρ(t))t≥0 , the maximization is considered with respect to both π and c. Actually, with the hyperbolic discount rate β , ρ(t) := δ + 1 + αt we can characterize the critical values (δ, β) to ensure the solvability of (1.3), i.e., |U (γ) (x)| < ∞ holds if and only if one of the following conditions are satisfied, (a) δ > δ, or (b) δ = δ and β > β (see [15] for details). Among mathematical finance literatures, Zervos (2008), Bjork and Murgoci (2010), Eckland, Mbodji and Pirvu (2011), and so on treat optimal consumption problems on finite horizon with hyperbolic discounting. They all consider the problems with a priori hyperbolic discounting rates. It is interesting to see that, in Bayesian setting, contrarily to the above-mentioned studies, hyperbolic discounting is derived as a natural consequence, i.e., the critical and “minimal” discounting rate ρ(t) := δ +

β 1 + αt

contains a hyperbolic term. The organization of the present article is as follows. In Section 2, we formulate our financial market model with partial information. After introducing the setup, in Section 3, we mention about the standard baseline results: Merton’s optimal power-utility result, which grows exponentially with respect to the terminal time T . In Section 4, we introduce the results on Bayesian CRRAutility maximization of terminal wealth, which is studied in Karatzas and Zhao (2001). In Section 5, we analyze the long-time asymptotics of the optimal Bayesian CRRA-utility, and observe its hyperbolic growth in power-utility case. In Appendix, we mention about a dynamic programming approach to solve our Bayesian CRRA-utility maximization.

2

Market Model

Consider a continuous-time financial market, consisting of one riskless asset and n-risky assets. The price process S 0 := (St0 )t≥0 of the riskless asset is given by (2.1)

St0 := ert ,

where r ∈ R≥0 is the constant risk-free interest rate. The price process S := (S 1 , . . . , S n )⊤ , S i := (Sti )t≥0 of n-risky assets, where (·)⊤ denotes the transpose ˜ be a of a vector or a matrix, is defined in the following way: Let (Ω, F, P) standard probability space, endowed with the n-dimensional Brownian motion ˜ := (W ˜ 1, . . . , W ˜ n )⊤ , W ˜ i := (W ˜ i )t≥0 , where W ˜ 0 ∈ Rn is constant, and the W t

3

˜ . The law of λ is n-dimensional random variable λ, which is independent of W denoted by ν, i.e., ˜ ∈ dx). ν(dx) := P(λ We alway assume ∫ (2.2) Rn

|z|ν(dz) < ∞

(or a stronger condition (4.1)). We call this probability space the reference ˜ (Ft )t≥0 ), where probability space. On (Ω, F, P, ˜ u ; u ∈ [0, t]), Ft := σ(W we define (2.3)

˜ t, dSt = diag(St )σ(t, St )dW

S0 ∈ Rn++ ,

where σ : R+ × Rn+ ∋ (t, y) 7→ σ(t, y) ∈ Rn×n satisfies c1 I ≤ σσ ⊤ (t, y) ≤ c2 I for any (t, y) ∈ R+ × Rn+ with some constants 0 < c1 < c2 and diag(x) denotes the diagonal matrix whose (i, i)-element is equal to the i-th element xi of x := (x1 , . . . , xn )⊤ ∈ Rn . The stochastic differential equation (abbreviated to SDE, hereafter) (2.3) has a unique strong solution, which implies that Ft ⊃ σ(Su ; u ∈ [0, t]). Moreover, we see that ∫ (2.4)

t

˜t = W ˜0 + W

{ } σ(u, Su )−1 diag(Su )−1 dSu − r1du ,

0

which implies that Ft ⊂ σ(Su ; u ∈ [0, t]). Hence, we deduce the relation (2.5)

Ft = σ(Su ; u ∈ [0, t])

for all t ≥ 0. We next define the filtration (Gt )t≥0 by (2.6)

Gt := Ft ∨ σ(λ),

and the probability measure P on (Ω, ∨t≥0 Gt ) that satisfies (2.7)

˜G dP|Gt = Zt dP| t

for each t ≥ 0, where { } 2 ˜t − W ˜ 0 ) − |λ| t . Zt := exp λ⊤ (W 2 We call P the real-world probability measure. By Cameron-Martin-MaruyamaGirsanov’s theorem, the process W := (Wt )t≥0 , given by (2.8)

˜ t − λt, Wt := W 4

is a (P, Gt )-Brownian motion. Combining (2.3) and (2.8), we obtain the Pdynamics of S, dSt = diag(St ) {µ(t, St )dt + σ(t, St )dWt } ,

S0 ∈ Rn++ ,

where we define the mean-return-rate vector µ(t, y) := r1 + σ(t, y)λ with 1 := (1, . . . , 1)⊤ ∈ Rn . Here, note that W and λ are independent under the real-world probability measure P since the increment Wt2 − Wt1 of the (P, Gt )Brownian motion W is independent of G0 = σ(λ). Also, note that P(λ ∈ A) = ˜ ∈ A) = ν(A) for any A ∈ B(Rn ). P(λ Remark 2.1. For agents having information (Ft )t≥0 , the market price of risk λ = {σ −1 (µ − r1)}(t, St ) of S, which is a G0 -measurable random variable, cannot be directly observed. It is a hidden variable, which has to be estimated. The law ν of λ is called the prior distribution of λ, and the conditional expectation ˆ t := E[λ|Ft ], λ

(2.9)

t ≥ 0,

where E[·] denotes expectation with respect to P, is called the Bayesian estimator of λ. From the Bayes rule, we see ˜ ˆ t = E[Zt λ|Ft ] , λ ˜ t |Ft ] E[Z

(2.10)

˜ ˜ The denominator of where we denote by E[·], expectation with respect to P. (2.10) can be expressed as ˜ t |Ft ] = F (t, W ˜t − W ˜ 0 ), E[Z

(2.11) where

( ) |z|2 ⊤ F (t, y) := exp z y − t ν(dz), 2 Rn ∫

(2.12)

˜ t ). So, we see that and the numerator of (2.10) is equal to ∇F (t, W ˆ t = ∇ log F (t, W ˜t − W ˜ 0 ). λ

(2.13)

˜ , which is expressed as (2.4), can be interpreted as “cumulative Remark 2.2. W Sharpe ratio of the market”: ∫ t { } ˜ t =W ˜0 + W σ(s, Ss )−1 diag(Ss )−1 dSs − r1dt 0 ∫ t 0 ˜0 + =W (“vol. matrix”)−1 s (“return of Ss ” − “return of Ss ”1). 0

5

Next, on this financial market, consider a self-financing investor whose available information flow is (Ft )t≥0 . The wealth process X x,π := (Xtx,π )t≥0 of the investor is defined by the SDE { n ( ) } n ∑ dS i ∑ dSt0 x,π x,π t i i (2.14) dXt = Xt πt i + 1 − πt , X0x,π = x, 0 S S t t i=1 i=1 where x ∈ R++ is the initial wealth of the investor and π := (π 1 , . . . , π n )⊤ , π i := (πti )t≥0 is a dynamic investment strategy of the investor, which is Ft adapted. For a given finite time horizon T ∈ R++ and initial wealth x ∈ R++ , consider the utility maximization of terminal wealth, U (T,γ) (x) := sup Eu(γ) (XTx,π ) ,

(2.15)

π∈AT

where

 1−γ x u(γ) (x) := 1 − γ  log x

(2.16)

if γ > 0, ̸= 1, if γ = 1

is the CRRA-utility function with Arrow-Pratt’s relative risk-aversion parameter γ and { } n-dimensional Ft -progressively measurable, AT := (ft )t∈[0,T ] ∫ T |ft |2 dt < ∞ a.s. 0 is the totality of admissible investment strategies. Remark 2.3. Similar market models with partial information, where unobservable random market price of risks are employed, are studied in 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.

3

Exponential Growth of Optimal Power-utility with Constant λ

Before analyzing Bayesian utility maximization (2.15), in this section, we consider a special “non-Bayesian” situation: let ν(dx) := δλ0 (dx) be the Dirac’s delta measure, i.e., let λ :≡ λ0 ∈ Rn be a constant vector. Then, the solution to (2.15), which has been originally investigated by Merton (1969, 1971), is now well-known and described as follows: (γ)

(A) The optimal investment strategy π ˆ (γ) := (ˆ πt )t∈[0,T ] ∈ AT is given by (3.1)

(γ)

π ˆt

:=

1 1 (σσ ⊤ )−1 (µ − r1)(t, St ) = (σ ⊤ )−1 (t, St )λ. γ γ 6

ˆ t(γ) )t∈[0,T ] is written as (B) The associated optimal wealth process (X { ( ) } (γ) ˆ t(γ) := Xtx,ˆπ = x exp 1 λ⊤ (Wt + λt) + r − 1 |λ|2 t . (3.2) X γ 2γ 2 (C) The optimal utility is computed as ) ( ( (γ) ) 2 1 ˆ U (T,γ) (x) = Eu(γ) X = u(γ) xe(r+ 2γ |λ| )T . T

(3.3)

In this constant λ case, since the optimal strategy (3.1) is T -independent, we (γ) can re-define the investment strategy π ˆ (γ) := (ˆ πt )t≥0 by (3.1) and the wealth (γ) ˆ (γ) := (X ˆ t )t≥0 by (3.2) on the time interval [0, ∞). Let process X { } A := (ft )t≥0 ; (ft )t∈[0,T ] ∈ AT for all T > 0 . ˆ (γ) . We note that, for We then see the following long-term optimalities of X obtaining (II)–(IV) below, the exponential growth of optimal power-utility (3.3) (with γ ̸= 1) with respect to T is essential. (I) (Maximizing long-term growth rate). We have that, for any π ∈ A , 1 1 ˆ (1) = −Γ′ (1) log XTx,π ≤ lim log X T T →∞ T T →∞ T lim

a.s..

where

1 −Γ′ (1) := r + |λ|2 . 2 For the proof, see Theorem 3.10.1 of Karatzas and Shreve (1998) [8].

(II) (Maximizing long-term growth rate of expected power-utility). When 0 < γ < 1, we see that, for any π ∈ A , lim

T →∞

1 1 ˆ (γ) ) = Γ(γ). log Eu(γ) (XTx,π ) ≤ lim log Eu(γ) (X T T →∞ T T

When γ > 1, we see that, for any π ∈ A , 1 1 ˆ (γ) ) = Γ(γ). log Eu(γ) (XTx,π ) ≥ lim log Eu(γ) (X T T →∞ T T →∞ T lim

Here, we set

) 1 2 |λ| . Γ(γ) := (1 − γ) r + 2γ (

(III) (Maximizing long-term upside-chance large deviation probability). Let 0 < γ < 1. We have that, for any π ∈ A , ( ( ) ) 1 1 1 1 (γ) x,π ˆ lim log P log XT ≥ k(γ) ≤ lim log P log XT ≥ k(γ) , T →∞ T T →∞ T T T 7

where the target growth rate k(γ) > −Γ′ (1) is defined by (3.4)

k(γ) := −Γ′ (γ) = r +

1 |λ|2 . 2γ 2

For the details, see Pham (2003). (IV) (Minimizing long-term downside-risk large deviation probability). Let γ > 1. We have that, for any π ∈ A , ( ) ( ) 1 1 1 1 (γ) x,π ˆ lim log P log XT ≤ k(γ) ≥ lim log P log XT ≤ k(γ) , T →∞ T T T T →∞ T where the target growth rate r < k(γ) < −Γ′ (1) is defined by (3.4). For the details, see Hata et. al. (2010). Remark 3.1 (Kelly and fractional Kelly portfolios). The log-optimal portfolio π ˆ (1) is sometimes called the GOP (Growth Optimal Portfolio) by the property (I), or the Kelly portfolio. The latter name comes from the pioneer work by Kelly (1956) [10]: the optimality in (I) can be interpreted as a corollary of the result obtained in [10]. Also, the power-optimal portfolio π ˆ (γ) with the risk-aversion parameter γ > 1 is sometimes called the fractional Kelly portfolio, which has been proposed to decrease the “risky” features of the “full” Kelly portfolio π ˆ (1) : see Chapter IV and Section 27 of Chapter III of Maclean et. al. (2011) [12]. Note that the above (II) and (IV) characterize long-term optimalities of the fractional Kelly portfolio π ˆ (γ) (γ > 1).

4

Bayesian CRRA Utility Maximization

In this section, we introduce the solution to our Bayesian CRRA-utility maximization (2.15), which has been obtained in Karatzas and Zhao (2001), [9], in an essential form. For γ = 1, i.e., log-utility case, we see the following. Theorem 4.1 (Theorem 3.2, Example 3.3 and 4.4 of [9]). Assume ∫ (4.1) |z|2 ν(dz) < ∞. Rn

For any T, x ∈ R>0 , the following are valid. ˆ (T,1) := (X ˆ t(T,1) )t∈[0,T ] is given by 1. The optimal wealth process X (4.2)

ˆ t(T,1) = xert F (t, W ˜t − W ˜ 0 ), X

where we use (2.12). (T,1) 2. The optimal strategy π ˆ (T,1) := (ˆ πt )t∈[0,T ] that satisfies ˆ t(T,1) = Xtx,ˆπ X

(T ,1)

8

t ∈ [0, T ]

is given by (4.3)

(T,1)

π ˆt

:= (σ(t, St )⊤ )−1

(

)

∇F F

ˆt, ˜t − W ˜ 0 ) = (σ(t, St )⊤ )−1 λ (t, W

where we use (2.9). 3. The optimal expected utility is expressed as ˜ (T, W ˜T − W ˜ 0 ) log F (T, W ˜T − W ˜ 0) U (T,1) (x) = log x + rT + EF ∫ T 1 ˆ t |2 dt. = log x + rT + E |λ 2 0 Proof. The first two assertions are derived directly from Example 3.3 of [9]. To see the third assertion, we first deduce ˜ (T, W ˜T − W ˜ 0 ) log F (T, W ˜T − W ˜ 0 ) = E log F (T, W ˜T − W ˜ 0 ). EF Use Itˆo’s formula and (2.13) to see that ∫ t ∫ 1 t ˆ 2 ˆ ⊤ dW ˜t − W ˜ 0) = ˜ (4.4) log F (t, W λ − |λu | du u u 2 0 0 ∫ t ∫ t ˆ ⊤ dBu + 1 ˆ u |2 du, = λ |λ u 2 0 0 where

∫

t

ˆ u du, λ

˜t − Bt := W

t≥0

0

is a (P, Ft )-Brownian motion by Cameron-Martin-Maruyama-Girsanov’s formula. Note that ∫ t ∫ t 2 ˆ u |2 du = E |λ duE |E[λ|Fu ]| ≤ tE|λ|2 < ∞ 0

0

for any t > 0. Hence, it follows that ˜t − W ˜ 0) = 1 E E log F (t, W 2

∫

T

ˆ t |2 dt. |λ 0

Next, we consider a power-utility, which is more risk-averse than the logutility, i.e., employ (2.16) with (4.5)

γ ∈ (1, ∞).

To treat this situation, we introduce, for 0 ≤ t ≤ T < ∞, [ ] ˜ F (T, y + W ˜t − W ˜ 0 ) γ1 (4.6) G(T,γ) (t, y) :=E ∫ √ 1 1 |z|2 − 2 = F (T, y + tz) γ dz n e (2π) 2 Rn 9

where we use (2.12). Here, recalling that ( )∫ ( 1 2 t F (t, y) = exp |y| exp − z − 2t 2 Rn we see that

∫

) ( ) y 2 1 2 |y| , ν(dz) ≤ exp t 2t

√ 1 tz) γ

|z|2 1 − 2 dz n e (2π) 2 Rn √ { } ∫ |z|2 |y + tz|2 1 − ≤ exp dz n 2γT 2 Rn (2π) 2 ( ) n2 { } γT |y|2 = exp , γT − t 2(γT − t)

F (T, y +

hence, the integral in (4.6) has a finite value. Moreover, we see that, for 0 ≤ t ≤ T < ∞, [ ] 1−γ ˜ (∇F · F γ )(T, y + W ˜t − W ˜ 0) ∇G(T,γ) (t, y) =E ∫ √ 1−γ |z|2 1 − 2 = (∇F · F γ )(T, y + tz) dz n e (2π) 2 Rn and that the above integral has a finite value. Indeed, when t = 0, these equalities are trivial, and, for 0 < t ≤ T , we deduce that ( )∫ ( ) ( ) 1 2 t y 2 1 2 ˜ |∇F (t, y)| ≤ exp |y| |z| exp − z − ν(dz) ≤ exp |y| E|λ| 2t 2 t 2t Rn and that

∫

√ 1 1−γ |z|2 − 2 dz (∇F · F γ )(T, y + tz) n e (2π) 2 Rn √ { } ∫ 1 |y + tz|2 |z|2 ˜ ≤E|λ| − dz n exp 2γT 2 Rn (2π) 2 ( ) n2 { } γT |y|2 ˜ =E|λ| exp . γT − t 2(γT − t) We now see the following. Theorem 4.2 (Theorem 3.2, Example 3.5 and 4.6 of [9]). Assume (2.2) and (4.5). For any T, x ∈ R>0 , the following are valid. ˆ (T,γ) := (X ˆ t(T,γ) )t∈[0,T ] is given by 1. The optimal wealth process X ˜t − W ˜ 0) (T − t, W , G(T,γ) (T, 0)

(T,γ)

(4.7)

ˆ t(T,γ) = xert G X

where we use (4.6). 10

(T,γ)

2. The optimal strategy π ˆ (T,γ) := (ˆ πt

ˆ t(T,γ) = Xtx,ˆπ X

)t∈[0,T ] that satisfies

(T ,γ)

t ∈ [0, T ]

is given by (4.8)

(T,γ)

π ˆt

:= (σ(t, St )⊤ )−1

˜t − W ˜ 0) ∇G(T,γ) (T − t, W . ˜t − W ˜ 0) G(T,γ) (T − t, W

3. The optimal expected utility is expressed as { }γ (4.9) U (T,γ) (x) = u(γ) (xerT ) G(T,γ) (T, 0) . In [9], the so-called martingale method is employed to solve this problem with partial information. In Appendix, we describe a different solution method, using a standard dynamic programming. Example 4.1 (Gaussian Prior). This example treats a slight generalization of computations demonstrated in Cvitani´c et. al. (2006). Let ν ∼ N (l, L), i.e., { } 1 1 ⊤ −1 √ exp − (z − l) L (z − l) dz, ν(dz) := 2 (2π)n/2 det(L) where L ∈ Rn×n is a symmetric and positive definite covariance matrix and l ∈ Rn is a mean vector. Then, the Bayesian estimator (2.9), which is expressed as (2.13), is computed as (4.10)

ˆ t = ∇ log F (t, W ˜ t ). = (L−1 + tI)−1 (W ˜ t + L−1 l). λ

Indeed, we see F (t, y), given by (2.12), is computed as { 1 F (t, y) = exp (y + L−1 l)⊤ (L−1 + tI)−1 (y + L−1 l) 2 } 1 1 − l⊤ L−1 l − log det(I + tL) . 2 2 Let

Kt := (L−1 + tI)−1 = L(I + tL)−1 = (I + tL)−1 L

and define, for θ ∈ (0, 1), (T )

Pt

{ } −1 (θ) = KT tθ (I − tθKT ) + KT−1 KT .

We recall that I − tθKT =I − tθ(I + T L)−1 L =(I + T L)−1 {I + (T − tθ)L} ≥(I + T L)−1 > 0 11

(T )

for all t ∈ [0, T ] and θ ∈ (0, 1). So, Pt (θ) is a well-defined, symmetric and (T ) positive definite matrix. Also, we may notice that (Pt (θ))t∈[0,T ] solves the differential Riccati equation d P = θP 2 , dt

P0 = KT .

Using these functions, we deduce that (4.6) is calculated as (4.11)

G(T,γ) (t, y) [ θ (T ) = exp (L−1 l + y)⊤ Pt (θ)(L−1 l + y) 2

] } 1 θ { ⊤ −1 − l L l + log det(I + T L) − log det (I − tθKT ) , 2 2

where we set θ :=

1 . γ

Indeed, defining kt :=Kt L−1 l ⊤

−1

κt :=l L

and

Kt L−1 l − l⊤ L−1 l − log det(I + tL),

we see G(T,γ) (t, y) { ( )} 1 ⊤ 1 ⊤ = exp θ y KT y + kT y + κT 2 2 ) { ( } ∫ √ 1 1 2 t ⊤ ⊤ × θ z KT z + t(kT + KT y) z − |z| dz n exp 2 2 Rn (2π) 2 [ ( ) 1 1 1 ⊤ y KT y + kT⊤ y + κT − log det (I − tθKT ) = exp θ 2 2 2 ] tθ2 −1 −1 ⊤ −1 + (L l + y) KT (I − tθKT ) KT (L l + y) , 2 hence, the expression (4.11) is obtained from this calculation. Inserting (4.11) into (4.8) in Theorem 4.2 and combining this with (4.10), we obtain (T,γ)

π ˆt

( ) 1 (T ) ˜t − W ˜ 0 + L−1 l) = (σ ⊤ )−1 (t, St )PT −t γ1 (W γ ( ) 1 (T ) ˆt. = (σ ⊤ )−1 (t, St )PT −t γ1 (L−1 + tI)λ γ

12

5 5.1

Long-time Asymptotics Log-optimal Case

First, consider the log-utility case, assuming (4.1). Then, both the optimal wealth (4.2) and the optimal investment strategy (4.3) are T -independent. So, we re-define ˆ (1) := (X ˆ t(1) )t≥0 X by (4.2) and (1)

π ˆ (1) := (ˆ πt )t≥0 ˜ u ; u ≤ t). by (4.3). We may assume that Ft (t ≥ 0) is the P-completion of σ(W We then see the following. Proposition 5.1 (Example 5.1 of [9]). For any π ∈ A , it holds that (5.1)

1 1 ˆ (1) = r + 1 E|λ|2 . E log XTx,π ≤ lim E log X T T →∞ T T →∞ T 2 lim

Proof. From Proposition 4.1, we can deduce that lim

T →∞

1 1 ˆ (1) E log XTx,π ≤ lim E log X T T →∞ T T

holds for any π ∈ A . Using (4.4) and Fubini’s theorem, we can see that the right-hand-side of the above is equal to 1 r + lim T →∞ 2T

(5.2)

∫

T

ˆ t |2 dt. E|λ 0

ˆ t , we can apply the martingale convergence Recalling the definition (2.9) of λ ˆ theorem to (λt )t≥0 to deduce that ˆ ∞ := lim λ ˆ t = E[λ|F∞ ] P-a.s., λ

(5.3)

t→∞

ˆ ∞ = λ, P-a.s.. Actually, from where F∞ := ∨t≥0 Ft . Moreover, we see that λ (2.8), we see that 1 1 ˜ Wt = Wt + λ. t t 1 ˜ From this, we deduce that limt→∞ t Wt = λ, P-a.s. since we have limt→∞ 1t Wt = 0, P-a.s. by the strong law of large numbers. Hence, the F∞ -measurability of λ follows. So, we deduce that (5.2) is equal to the right-hand-side of (5.1). Remark 5.1. We can also deduce that, for any π ∈ A , lim

T →∞

1 1 ˆ (1) = r + 1 |λ|2 log XTx,π ≤ lim log X T T →∞ T T 2

13

P-a.s..

Indeed, the above inequality follows from Theorem 3.1 of [8]. To derive the expression of the right-hand-side of the above, using (4.4), we write as 1 ˆ (1) = r + 1 MT + 1 ⟨M ⟩T , log X T T T 2T where we define

∫

t

Mt :=

ˆ ⊤ dBu . λ u

0

We first deduce that 1 1 lim ⟨M ⟩T = lim T →∞ T T →∞ T

∫

T

ˆ t |2 dt = |λ|2 |λ

P-a.s.,

0

ˆ t → λ, P-a.s. as t → ∞. We next deduce that where we recall λ lim

T →∞

MT ⟨M ⟩T 1 MT = lim = 0, T →∞ ⟨M ⟩T T T

where the strong law of large numbers for square-integrable martingales is applied, recalling limT →∞ ⟨M ⟩T = ∞ on {λ ̸= 0}. Remark 5.2. The portfolio π ˆ (1) ∈ A is sometimes called the Bayesian Kelly portfolio.

5.2

Power-optimal Case

We next consider power-utility case. In this subsection, in addition to (2.2) and (4.5), we assume that (5.4)

ν(dz) = fν (z)dz

with fν ∈ L∞ (Rn ), which is continuous at 0 ∈ Rn .

We define two functions, (

( ) |x|2 exp − , 2 ( ( ) n2 ∫ ) t t 2 ψ(t, x) := exp − |x − z| ν(dz), 2π 2 Rn ϕ(x) :=

1 2π

) n2

recalling that ( ˜ Eh

) ∫ ˜t W +λ = h(x)ψ(t, x)dx t Rn

14

for any bounded Borel measurable function h : R → R. Using these, we see that (5.5) G(T,γ) (t, y) √ √ )} γ1 ( ) {( ) n2 ( ∫ |y + tz|2 2π y + tz = exp ψ T, ϕ(z)dz 2γT T T Rn n ( ( ) 2γ ) n2 { } 2π γT |y|2 = exp T γT − t 2(γT − t) ( √ √ )1 2 ) ( ) n2 ∫ ( γT − t y + tz γ γT − t t × z− ψ T, exp − y dz. 2πγT 2γT γT − t T Rn Also, when t = T , we see that ( (5.6)

G(T,γ) (T, y) =

2π T

n ( ) 2γ

γ γ−1

) n2

{ exp

|y|2 2(γ − 1)T

} Ψ(γ) (T, y),

where we define Ψ(γ) (T, y) := ∫ ( Rn

γ−1 2πγT

) n2

(

2 ) ( )1 γ − 1 1 y+z γ exp − y dz. z− ψ T, 2γT γ−1 T

We deduce the following. 1

γ for (T, y) ∈ Lemma 5.1. (1) |ψ(t, y)| ≤ ∥fν ∥∞ and |Ψ(γ) (T, y)| ≤ ∥fν ∥∞ n R+ × R . (2) lim ψ(t, x) = fν (x) for each x ∈ Rn , if fν is continuous at x ∈ Rn .

t→∞

1

(3) lim Ψ(γ) (T, y) = fν (0) γ for each y ∈ Rn . T →∞

Proof. (1) The assertion is straightforward to see. (2) We see that ( lim ψ(t, x) = lim

t→∞

t→∞

1 2π

) n2 ∫

( ) ( ) 1 2 1 exp − |y| fν x − √ y dy = fν (x) 2 t Rn

by the dominated convergence theorem. (3) We see that 1 (γ) Ψ (T, y) − fν (0) γ ( 2 ) ( )n )1 ∫ ( 1 γ − 1 1 y+z γ γ−1 2 γ dz. y exp − z − ψ T, − f (0) ≤ ν 2πγT 2γT γ−1 T Rn By the dominated convergence theorem, the desired assertion follows. 15

With the help of this lemma, we obtain the following. Recall a notation in asymptotic analysis: we write “a(T ) ∼ b(T ) as T → ∞” when limT →∞ a(T )/b(T ) = 1 holds. Proposition 5.2. It holds that { (5.7)

U

(T,γ)

rT

(x) =u(γ) (xe

) {

∼u(γ) (xerT ) and that (5.8)

2π T 2π T

( (

γ γ−1 γ γ−1

)γ } n2 Ψ(γ) (T, 0)γ )γ } n2 fν (0)

as T → ∞

{

ˆ (T,γ) X T

( ) n 1 log 1 − + rT − log Ψ(γ) (T, 0) 2 γ )} ( ˜T − W ˜0 1 ˜ 1 W 2 ˜ + |WT − W0 | + log ψ T, . 2γT γ T

=x exp

Proof. (5.7) is derived from (4.9) and (5.6), using Lemma 5.1. (5.8) is computed from (4.7), (5.5), and (5.6). Remark 5.3 (Hyperbolic Growth). From (5.7), we see that ∂T log U (T,γ) (x) = (1 − γ)r −

n1 + ϵ(T ), 2T

where we set ϵ(T ) := γ∂T log Ψ(γ) (T, 0). The residual term ϵ(T ) is “smaller” than 1/T as T → ∞ in the sense that ∫ T lim ϵ(t)dt = log fν (0) − log Ψ(γ) (1, 0) < ∞, T →∞ 1 which is deduced from Lemma 5.1 (3). Remark 5.4 (Bayesian Fractional Kelly Portfolio). Let γ > 1. Define the (γ) Bayesian fractional Kelly portfolio π ˜ (γ) := (˜ πt )t≥0 ∈ A by (γ)

π ˜t

:=

1 ⊤ −1 ˆt. (σ ) (t, St )λ γ

As mentioned in Remark 3.1, from a practical point of view, this portfolio may be a candidate for long-term “risk-averse” investment. Write the associated ˜ (γ) := (X ˜ t(γ) )t≥0 , i.e., wealth process as X ˜ t(γ) := Xtx,˜π X

16

(γ)

t ≥ 0.

We can deduce that ˜ (γ) = log X ˆ (T,γ) + log G(T,γ) (T, 0) + γ − 1 log X T T 2γ 2 and that

∫

T

ˆ t |2 dt |λ 0

} 1{ ˜ (γ) − log X ˆ (T,γ) = γ − 1 |λ|2 log X T T T →∞ T 2γ 2 lim

5.3

P-a.s..

Cost of Uncertainty

In this subsection, we evaluate a “cost of uncertainty of λ” in the long run, which is proposed and discussed in [9]: We consider an “inside” investor, whose available information flow is (Gt )t≥0 , where we use (2.6). Note that, for this insider, the market price of risk vector λ is observable at time 0 and so, it can be regarded as a constant. The insider’s CRRA-utility maximization is described as [ ] sup E u(γ) (XTx,π )|G0 , π∈ATG

where maximization is considered over the space ATG of n-dimensional Gt -progressively ∫T measurable processes (ft )t∈[0,T ] so that 0 |ft |2 dt < ∞ a.s. From (3.3), we deduce that ( ) [ ] 2 1 sup E u(γ) (XTx,π )|G0 = u(γ) xe(r+ 2γ |λ| )T . π∈ATG

Let

[ ¯ (T,γ)

U

(x) :=E

] [ ] x,π sup E u(γ) (XT )|G0

π∈ATG

)] ( 2 1 =E u(γ) xe(r+ 2γ |λ| )T [

be the expected optimal utility for the inside investor, and we are interested in ¯ (T,γ) (x). evaluating the ratio of two optimized expected utilities U (T,γ) (x) and U We then see the following. Proposition 5.3. It holds that ¯ (T,1) (x) U (T,1) (x) ∼ U

(5.9)

as T → ∞

and that, for γ > 1, ( (5.10)

U

(T,γ)

(x) ∼ U

¯ (T,γ)

(x)

γ γ−1

) (γ−1)n 2

as T → ∞.

We may interpret that, for log-utility-investors, “cost of uncertainty of λ” becomes negligible as T → ∞, while for power-utility-investors who are risk-averse than log-utility-investors, the “cost of uncertainty of λ” does not disappears even when T → ∞. 17

Proof. We see that, from Theorem 4.1, ( ) ¯ (T,γ) (x) log x + r + 21 E|λ|2 T U ) ( = U (T,γ) (x) log x + r + 1 ∫ T E|λ ˆ t |2 dt T 2T 0 ( ) 1 x + r + 12 E|λ|2 T log ( ) → 1 as T → ∞ = ∫T 1 1 ˆ t |2 dt log x + r + E| λ T 2T 0 ∫T ˆ t |2 dt → E|λ|2 as we see in Proof of Proposition 5.1. since limT →∞ T1 0 E|λ Hence, (5.9) follows. To obtain (5.10) with γ > 1, we write the optimal expected power-utility of the “insider” as ∫ ( rT ) γ−1 2 (T,γ) ¯ U (x) =u(γ) xe e− 2γ T |z| fν (z)dz. Rn

By Laplace’s method, we see that { lim

T →∞

T (γ − 1) 2πγ

} n2 ∫

e−

γ−1 2 2γ T |z|

Rn

fν (z)dz = fν (0).

Hence, it follows that ( ) ¯ (T,γ) (x) ∼ u(γ) xerT U

{

2π T

(

γ γ−1

)} n2

fν (0) as T → ∞.

Combining it with Proposition 5.2, we complete the proof.

References ¨ rk, T. and A. Murgoci (2010). A general theory of Markovian time [1] Bjo inconsistent stochastic control problems, preprint. [2] Brennan, M. and Y. Xia (2001). Assessing asset pricing anomalies. Review of Financial Studies, 14, 905–945. ´, J., A. Lazrak, L. Martellini, and F. Zapatero (2006). [3] Cvitanic Dynamic portfolio choice with parameter uncertainty and the economic value of analysts’ recommendations. Review of Financial Studies 19 (4), 1113–1156. [4] Ekeland, I., O. Mbodji, and T.A. Pirvu (2011). Time consistent portfolio management, preprint. [5] Hata. H., H. Nagai and S.J. Sheu (2010). Asymptotics of the probability minimizing a “down-side” risk, Annals of Applied Probability 20 (1), 52–89.

18

[6] Hayashi, T., H. Miyata, and J. Sekine (2012). On hyperbolic growth of optimal Bayesian power-utility, preprint. [7] Karatzas, I. (1997). Adaptive control of a diffusion to a goal, and an associated parabolic Monge-Ampere-type equation. Asian J. Mathematics 1, 324–341. [8] Karatzas, I. and S. Shreve (1998). Methods of Mathematical Finance. Springer-Verlag, Berlin. [9] Karatzas, I. and X. Zhao (2001). Bayesian adaptive portfolio optimization. Handbook of Mathematical Finance, Optimization, Pricing, InterestRates, and Risk Management. Cambridge University Press, 632–669. [10] Kelly, J. L., Jr. (1956). A new interpretation of information rate. Bell System Technical Journal 35, 917–926. [11] Lakner, P. (1995). Utility maximization with partial information. Stoch. Proc. and their Appl. 56, 247–273. [12] Maclean, L.C., Thorp, E.O. and Ziemba, W.T. (2011). The Kelly capital growth investment criterion. World Scientific. [13] Merton, R. C., (1969). Lifetime portfolio selection under uncertainty: the continuous time case. Review of Economics and Statistics, 51, 247–257. [14] Merton, R. C., (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3, 373–413. [15] Miyata, H. (2012). Lifetime consumption problems under partial observations, preprint. [16] Pham, H. (2003). A large deviations approach to optimal long term investment. Finance and Stochastics, 7, 169–195. [17] Pham, H. and M.-C. Quenez (2001). Optimal portfolio in partially observed stochastic volatility models. Ann. Appl. Probab. 11(1), 210–238. ¨uerle (2005). Portfolio optimization with unob[18] Rieder, U. and N. Ba servable Markov-modulated drift process. Journal of Applied Probability, 42, 362–378. [19] Xia, Y. (2001). Learning about predictability: the effect of parameter uncertainty on dynamic asset allocation. Journal of Finance, 56 (1), 205– 246. [20] Zervos, M. (2008). Optimal consumption and investment with habit formation and hyperbolic discounting. Lecture in Workshop on “Finance and related mathematical and statistical issues”, CSFI, Osaka University. [21] Zohar, G. (2001). A generalized Cameron-Martin formula with applications to partially observed dynamic portfolio optimization. Mathematical Finance 11(4), 475–494. 19

A

Appendix: HJB Approach

In this appendix, we sketch a standard dynamic programming approach for solving Bayesian CRRA-utility maximization (2.15), which is different to the martingale method, employed in [9]. First, we reformulate (2.15), introducing a measure-change. Note that ˜ T u(γ) (X x,π ) = E ˜ Z˜T u(γ) (X x,π ), Eu(γ) (XTx,π ) = EZ T T where we write ˜ t |Ft ] = F (t, W ˜t − W ˜ 0) Z˜t := E[Z and use (2.7) and (2.11) since X x,π is Ft -adapted. Combining (2.1), (2.3) and (2.14), we see { } ˜ t , X x,π = x. dXtx,π = Xtx,π rdt + πt⊤ σ(t, St )dW 0 So,

] [ { 2 } γ ˜t . d(Xtx,π )1−γ = (Xtx,π )1−γ (1 − γ) r − σt⊤ πt dt + (1 − γ)πt⊤ σt dW 2 Hence, we have { } ∫ γ(1 − γ) T ⊤ 2 x,π 1−γ 1−γ (1−γ)rT ⊤ =x e MT ((1 − γ)σ π) exp − (XT ) σt πt dt , 2 0 where we define (A.1)

(∫ Mt (α) := exp

t

˜u − αu⊤ dW

0

1 2

∫

t

) 2 |αu | du .

0

(1) UT

Let be the totality of n-dimensional progressively measurable process p := ∫T (pt )t∈[0,T ] on the time-interval [0, T ] so that 0 |pt |2 dt < ∞ a.s. and that (1) ˜ (α) on EMT ((1 − γ)α) = 1. For α ∈ UT , we define the probability measure P T (Ω, FT ) by the formula ˜ (α) dP T := Mt ((1 − γ)α), t ∈ [0, T ]. ˜ F dP t (α)

˜t By Cameron-Martin-Maruyama-Girsanov’s theorem, the process (W defined by ∫ t (α) ˜ ˜ Wt := Wt − (1 − γ) αu du,

)t∈[0,T ] ,

0

˜ (α) , Ft )-Brownian motion. Recall that, when α := σ ⊤ π ∈ is an n-dimensional (P T (1) UT , we have (A.2)

log ET (XTπ )1−γ = (1 − γ)(log x + rT ) } { ∫ T γ(1 − γ) (α) 2 ˜ exp log F (T, W ˜T − W ˜ 0) − |αt | dt , + log E T 2 0

20

˜ (α) (·) denotes expectation with respect to P ˜ (α) . where E T T We now consider, for 0 ≤ t ≤ T < ∞, (A.3) V¯ (T ) (t, y) :=

{ } ∫ γ(γ − 1) T −t (α) (y) 2 ˜ ˜ inf log ET exp log F (T, YT −t − W0 ) + |αs | ds , α∈UT −t 2 0 (1)

(y)

where we set a suitable UT −t , a subset of UT −t , and define the process (Ys )s∈[0,T −t] by ˜ s(α) , Y (y) = y. dYs(y) = (1 − γ)αs ds + dW 0 The associated HJB equation is written down as { } ) γ(γ − 1) 2 1( |α| , −∂t V = ∆V + |∇V |2 + infn (1 − γ)α⊤ ∇V + α∈R 2 2 (A.4) ˜ V (T, y) = log F (T, y − W0 ). Here, we see

2 γ(γ − 1) 2 γ(γ − 1) 1 γ−1 2 (1 − γ)α ∇V + |α| = |∇V | . α − ∇V − 2 2 γ 2γ ⊤

So, the minimizer in (A.4) is given by α ¯ :=

1 ∇V γ

and (A.4) is rewritten as 1 1 −∂t V = ∆V + |∇V |2 , 2 2γ ˜ 0 ). V (T, y) = log F (T, y − W

(A.5) 1

Noting that L := e γ V satisfies 1 ˜ 0 ) γ1 , ∆L, L(T, y) = F (T, y − W 2 we deduce the expression for the solution to (A.5) [ ] ˜ F (T, W ˜T − W ˜ 0 ) γ1 W ˜t = y . V (t, y) = γ log E −∂t L =

From (A.2) and (A.3), we see that the relation { } ˜ 0) U (T,γ) (x) = u(γ) (xerT ) exp V¯ (T ) (0, W holds. So, we can deduce the expression ]}γ { [ ˜ F (T, W ˜T − W ˜ 0 ) γ1 U (T,γ) (x) = u(γ) (xerT ) E , which is nothing but the representation (4.9). After demonstrating the so-called verification steps, we can establish all assertions in Theorem 4.2. 21