# Optimal information acquisition for eliminating estimation risk

Zongxia Lianga, Qi Yeb Department of Mathematical Sciences, Tsinghua
University, Beijing 100084, China 111
$a$ email<EMAIL_ADDRESS>
$b$ Corresponding author, email<EMAIL_ADDRESS>
Funding: This work was funded by National Natural Science Foundation of China
(Grant Nos.12271290)

###### Abstract.

This paper diverges from previous literature by considering the utility
maximization problem in the context of investors having the freedom to
actively acquire additional information to mitigate estimation risk. We derive
closed-form value functions using CARA and CRRA utility functions and
establish a criterion for valuing extra information through certainty
equivalence, while also formulating its associated acquisition cost. By
strategically employing variational methods, we explore the optimal
acquisition of information, taking into account the trade-off between its
value and cost. Our findings indicate that acquiring earlier information holds
greater worth in eliminating estimation risk and achieving higher utility.
Furthermore, we observe that investors with lower risk aversion are more
inclined to pursue information acquisition.

Keywords: bayesian learning, estimation risk, filtering, information
inquisition, the value of information, variation method

JEL classification: G11, C11, C61

##  1\. Introduction

In the financial market, investors are often assumed to possess complete
information, allowing them to develop strategies that maximize their utility.
However, in reality, obtaining complete information, especially regarding
asset returns, is often uncertain, leading to estimation risk (Kumar et al.
(2008)). Gennotte (1986) and Karatzas and Xue (1991) considers optimal
consumption in an incomplete market setting, in Lakner (1995), Lakner (1998)
and Zohar (2001) the optimal terminal wealth is derived and the optimal
strategy determined for linear Gaussian dynamics of the returns. Karatzas and
Zhao (2001) solves the problem if the return is a fixed random variable with
known distribution. Runggaldier and Zaccaria (2000) consider the optimization
of the asymptotic growth rate for an infinite time horizon. Sass and Haussmann
(2004) considers the return follows continuous time Markov chain. To adapt to
changing circumstances, investors continuously update their beliefs about
asset returns over time as new asset prices arrive, employing Bayesian
learning (Liptser and Shiriaev (1977)).

However, the estimation of asset returns is not solely updated based on asset
prices. Investors have access to various additional sources of information,
such as corporate earnings reports, macroeconomic indicators, political news
or expert opinion (Frey et al. (2012)). With the introduction of this extra
information, it becomes intriguing to explore its impact on investment
decisions and objective utility. While, most papers assume the information is
given and usually it is of the same correlation of unknown part (Xiong and Yan
(2010)). Here, inspired by Banerjee and Breon-Drish (2020), we assume the
investors can actively and dynamically to acquire the information in the
market. While, this paper is originated by the classical paper Kyle (1985)
where the information is used to determine the ultimate payoffs’s expectation
and their object is to maximize the expected payoffs rather than the utility.
But we still absorb the setting that we can acquire any information over time
which can help to erase the estimation risk where the parameter is uncertain.

In this paper, we adopt the framework in Karatzas and Zhao (2001) and equipped
with extra information. We demonstrate that the correlated part of the extra
information with the unobserved Brownian motion $W$ (the diffusion term of the
risky asset’s log price) can be considered as a small component of $W$.
Importantly, this part is observed and can be utilized in return estimation.
Consequently, the presence of extra information reduces estimation risk at a
faster pace, leading to improved objective utility. Thus, this paper not only
addresses the maximization of expected utility with any given extra
information but also investigates the nature of the extra information itself,
ultimately determining optimal information acquisition strategies.

We reveal that the extra information with higher correlation at any time can
actually achieve higher utility. However, comparing utilities equipped with
any extra information still requires solving the value function. To facilitate
this analysis, we introduce the concept of an “informative clock”, which
represents the ratio of the conditional variance of estimation to the risky
asset’s volatility. The informative clock quantifies the total volume of
information and provides an intuitive index to assess the quality of the extra
information. This perspective enhances the precision of estimation and allows
for a concise presentation of the value function. It provides a more intuitive
understanding of how extra information influences investment decisions and
yields fruitful insights from various perspectives, such as attention,
inattention, and information quality (Huang and Liu (2007) and Veronesi
(2000)).

We propose a criterion to quantify the value of extra information. Inspired by
Cabrales et al. (2013) and Kadan and Manela (2019), we adopt the concept of
equivalence of certainty. It means that taking endowment along with extra
information can achieve the same utility as the case of taking endowment
adding the value of information along with no extra information. Our
calculations demonstrate that the value of extra information is relatively
independent of market conditions, including interest rates, volatility rates,
and return estimates. Instead, it primarily depends on risk aversion and the
informative clock. We show that the value of extra information remains
constant across different initial wealth levels in the case of Constant
Absolute Risk Aversion (CARA), providing a fixed value regardless of the
endowment. Conversely, in the case of Constant Relative Risk Aversion (CRRA),
investors focus on the multiple of wealth, where the value of information
contributes a fixed proportional increase to the endowment. However,
information acquisition and processing can incur significant costs in terms of
time, effort, or expenses. To address this, we propose a penalty function for
information acquisition based on the informative clock.

At last, we assume that any information characterized by an informative clock
can be obtained in the market. Investors can strategically determine the
optimal effort required to acquire such information for their investment
decisions. We employ the variation method to solve the functional associated
with the informative clock and derive its necessary condition. Our findings
indicate that investors pay greater attention to information acquisition in
the early stages rather than later stages and investors with less risk
aversion tend to pay greater attention.

In summary, this paper makes three primary contributions. First, we establish
a framework based on the concept of the ”informative clock,” enabling the
concise solution of the value function. Second, we employ the concept of
equivalence of certainty to determine the value of information. Lastly, we
solve the optimal information acquisition strategy.

The remainder of this paper is structured as follows: Section 2 describes the
market model and formulates the admissible space for investment strategies.
Section 3 derives the value function and corresponding strategy under the
assumption of given extra information, utilizing the concept of the
informative clock, presents the core analysis of the impact of extra
information on investment decisions, evaluating its value and cost. Section 4
complements the content of information’s source and informative clock’s
explosion. Lastly, the conclusion is drawn in the final section, with
technical proofs and calculations provided in the appendix.

##  2\. Market Model and Problem Formulation

Suppose that $(\Omega,\mathbb{F},\mathbb{P})$ is a probability space equipped
with the filtration $\mathbb{F}=\\{\mathcal{F}_{t}:t\geq 0\\}$ satisfying the
usual conditions. We consider a financial market with one risk-free asset and
one risky asset. The price of the risk-free asset $S_{0}=\\{S_{0}(t),\ t\geq
0\\}$ is given by

$\displaystyle dS_{0}(t)=rS_{0}(t)dt,\ \ S_{0}(0)=1,$

and the price of the risky asset $S=\\{S(t),\ t\geq 0\\}$ follows the
stochastic differential equation (abbr. SDE):

$\displaystyle dS(t)=S(t)\left[\mu dt+\sigma dW_{t}\right],\ \ S(0)=1,$

where $W=\\{W_{t}:{t\geq 0}\\}$ is a Brownian motion on
$(\Omega,\mathbb{F},\mathbb{P})$, $r$ and $\sigma$ are nonnegative constants.
The drift term $\mu$ is an $\mathcal{F}_{0}$-measurable Gaussian random
variable satisfying $\mu\sim\mathcal{N}(\mu_{0},\sigma_{0}^{2})$, and $\mu$
and $W$ are independent. The investor allocates the amounts $\pi_{t}$ of her
wealth into the risky asset at time $t$. Then the investor’s total wealth as
the self-financing process $X=\\{X_{t},\ t\geq 0\\}$ follows the following
SDE:

$\displaystyle dX_{t}=rX_{t}+\pi_{t}(\mu-r)dt+\pi_{t}\sigma dW_{t},\ \ t\geq
0,\ \ X_{0}=x_{0}.$ (2.1)

Traditionally, the investor formulates her investment strategy by observing
the price of risky asset, i.e., the strategy $\pi=\\{\pi_{t}:\ t\geq 0\\}$
must be only adapted to the filtration
$\mathbb{F}^{S}=\\{\mathcal{F}^{S}_{t}:t\geq 0\\}$, where
$\mathcal{F}^{S}_{t}=\sigma\\{S_{u}:0\leq u\leq t\\}$, which is strictly
smaller than $\mathcal{F}_{t}$, $t\geq 0$. To capture the cross-sectional
feature of unobserved process $W$, we introduce an extra process
$m=\\{m_{t}:t\geq 0\\}$ in our model. It has to have some properties: it is
observable, it is both Markov process and martingale from the background
information which is independent to random variable $\mu$. The most valuable
thing is that the process has correlation with the Brownian motion $W$:

$\displaystyle\frac{d\langle m,W\rangle_{t}}{\sqrt{d\langle
m\rangle_{t}d\langle W\rangle_{t}}}=\rho(t),$ (2.2)

where $\rho(t)\in(-1,1)$ is determined function of time $t$. it is easy to see
that $\rho(t)$ refers the correlation coefficient of two random variable
$dW_{t}$ and $dm_{t}$ in the slight time from $t$ to $t+dt$. For (2.2) is
well-defined, we set $\rho(t)=0$ when $\frac{d\langle m\rangle_{t}}{dt}=0$.
Now the investor can no longer only settle the strategy based on the
observation of $S$, the process $m$ will be likely to provide more information
for the more accurate estimation of $\mu$ where the ambiguity comes from.
Mathematically speaking, we amplify the space of our investment strategy that
$\pi$ can be adapted to the filtration $\mathbb{F}^{S}\vee\mathbb{F}^{m}$
rather than only adapted to the filtration $\mathbb{F}^{S}$. And we denote the
space of admissible strategy as
$\mathcal{A}_{m}:=\\{\pi|\int_{0}^{T}\pi_{t}^{2}dt<\infty,\pi\ \mbox{is
adapted to }\mathbb{F}^{S}\vee\mathbb{F}^{m}\\}$, where $T>0$ refers the
objective investment terminal time. The previous article’s objective is to
maximize the expectation utility of the terminal wealth at time $T$ over
$\mathcal{A}_{m}$:

$\displaystyle\sup_{\pi\in\mathcal{A}_{m}}EU(X_{T}),$ (2.3)

with the given extra information.

Our primary objective is to determine the optimal information to acquire in
the market. To achieve this, we first need to establish what we mean by
“optimal”. We must define a criterion to value the extra information, denoted
as $Value(m)$, and formulate a cost function, denoted as $Cost(m)$, to
quantify the expenses associated with acquiring this extra information.
Consequently, our objective is to solve the optimal information acquisition
strategy that maximizes the net value, defined as $Net(m)=Value(m)-Cost(m)$.

However, determining the criterion to value $Value(m)$ is not a
straightforward task. One possible approach is to use equation (2.3) to
represent the value $Value(m)$. However, this approach has limitations. For
instance, it may lead to negative values for $Value(m)$, which is
counterintuitive. Additionally, comparing the value of information for
investors with different risk aversions becomes challenging, as different
utility functions cannot be directly compared. Despite these challenges, we
still need to utilize the insights from equation (2.3) to establish a
criterion for valuing $Value(m)$, as a larger value in equation (2.3)
indicates greater value for the information. The specific criterion will be
provided in a later section.

##  3\. Extra information’s value, cost and its optimal acquisition

###  3.1. The effect of correlation coefficient

The correlation coefficient of the extra information can be comprehended as a
part of the unobserved $W$. A natural idea comes about that higher correlation
coefficient means more valuable information of $W$. Then it can be conjectured
that if two different sources of extra information have the same correlation
coefficient with $W$, two corresponding optimal investment would achieve the
same expected utility of terminal wealth. Similarly, if one extra information
has the bigger correlation coefficient than the other one, then its
corresponding expected utility would be bigger than the other. We now show the
two conjectures in the following Propositions 3.1 and 3.2.

######  Proposition 3.1.

If there are two sources of extra information $m^{1}$ and $m^{2}$ sharing the
same correlation coefficient that $\rho^{1}(t)=\rho^{2}(t)$ for all $t$, where
$\rho^{1}(t):=\frac{d\langle m^{1},W\rangle_{t}}{\sqrt{d\langle
m^{1}\rangle_{t}d\langle W\rangle_{t}}}$ and $\rho^{2}(t):=\frac{d\langle
m^{2},W\rangle_{t}}{\sqrt{d\langle m^{2}\rangle_{t}d\langle W\rangle_{t}}}$.
Denote $V_{m^{1}}:=\sup\limits_{\pi\in\mathcal{A}_{m^{1}}}EU(X_{T})$ and
$V_{m^{2}}:=\sup\limits_{\pi\in\mathcal{A}_{m^{2}}}EU(X_{T})$, similarly, the
superscript $1,2$ are indicating corresponding content in the case with
information $m^{1}$ and $m^{2}$ (see (2.3)). Then $V_{m^{1}}=V_{m^{2}}$.

###### Proof.

See Appendix A. ∎

Proposition 3.1 tells us that not the exact form of the extra information but
its correlation with $W$ indeed affect the objective function, thus it
provides us some innate feeling that it is fine to ignore the exact form of
extra information.

######  Remark 3.1.

Here we can assume $d\langle m\rangle_{t}=dt$. It is the reason that the class
of the process $\left\\{(H\cdot m)_{t}=\int_{0}^{t}H(s)dm_{s}\right.$, $H$ is
a certain determined function with respect to time t and $H(t)\neq 0$ almost
everywhere$\big{\\}}$ shares the same information as $m$, i.e.,
$\mathbb{F}^{m}=\mathbb{F}^{(H\cdot m)}$. Moreover, this transformation makes
sure (2.2) still valid. Then we can treat $m$ is a Brownian motion and (2.2)
is reduced into a simplified form as $d\langle m,W\rangle_{t}=\rho(t)dt$. In
addition, we can assume $\rho(t)\geq 0$ if we replace new process
$m^{\prime}=\left\\{m^{\prime}_{t}:=\int_{0}^{t}\left[1_{\\{\rho(t)\geq
0\\}}\\!\\!-\\!\\!1_{\\{\rho(t)<0\\}}\right]dm_{s}\right\\}$ with the process
$m$ to assure the nonnegative and this setting is applied in the later
content.

######  Proposition 3.2.

If there are two sources of extra information $m^{1}$ and $m^{2}$ sharing the
correlation coefficient with the relation that $\rho^{1}(t)\geq\rho^{2}(t)$
for $\forall$ $t$, then $V_{m^{1}}\geq V_{m^{2}}$.

###### Proof.

See Appendix B. ∎

Proposition 3.2 implies that higher correlation coefficient means the extra
information is more valuable. Beyond that, we first put forward the idea about
information dilution as the form $m^{3}$ being the mixture of the original
information $m^{1}$ and a noise $W^{noise}$, indeed, $W^{noise}$ has no use at
all. Because $m^{2}$ contributes the same estimation of $\mu$ as $m^{3}$, it
contributes less than that $m^{1}$ contributes. So does it in the investment
strategy.

######  Remark 3.2.

The method proving Proposition 3.2 corresponds the setting that $\rho(t)=0$ if
$\frac{d\langle m\rangle_{t}}{dt}=0$, which is explained as follows: When the
equation holds, it means $dm_{t}=0$ at that instant. This implies that no
valuable information is available at that instant. If we define the process
$m^{\prime}$: $m^{\prime}_{t}=\int_{0}^{t}[1_{\\{\frac{d\langle
m\rangle_{t}}{dt}\neq 0\\}}(s)dm_{s}+1_{\\{\frac{d\langle
m\rangle_{t}}{dt}=0\\}}(s)dW^{noise}_{s}]$, as that both no information is
equal to noise itself benefit nothing to strategy, then $m$ and $m^{\prime}$
have the same effect to the process and investment.

The extra information would bring about huge advantage for our strategy as
what the former content reveal. But we still need to objectively measure the
value of the extra information by a criterion rather than comparing two
different extra information. Therefore, it is necessary to calculate Problem
(2.3) for preparation.

###  3.2. Optimal investment Problem (2.3) with given extra information

The solution of control problem with ambiguity is usually based on bayesian
learning method that we find the posterior probability distribution evolves
over time by the data available up to each instant. Filtering method serves as
an instrumental appliance to transform the estimation of parameter into the
evolution of stochastic differential equation, which assists us in further
analysis (Liptser and Shiriaev (1977)). Among them, the most widespread
filtering is Kalman Bucy filtering, however, it can not be used directly. With
the extra process $m$, the estimation of $\mu$ over time can not be given
directly by two equations of conditional mean and variance as there must be
aggregation of two sources of information. It is necessary to find the
sufficiency statistic generated by $S$ and $m$ to fully estimate $\mu$. In
order to assure the deduction below carried out successfully, we need the
following assumption on the extra observable process $m$.

######  Assumption 3.1.

$\displaystyle\int_{0}^{T}\frac{1}{1-\rho^{2}(s)}ds<\infty.$ (3.4)

In the later subsection, we will illustrate the meaning of Assumption 3.1 and
discuss what happens in the absence of the assumption. We now present the
following conditional expectation
$E[\mu|\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}]$.

######  Theorem 3.3.

$\displaystyle
E[\mu|\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}]=\frac{1}{2}\sigma^{2}+\frac{y_{0}+\int_{0}^{t}q(t)^{2}\left[dY_{s}-\sigma\rho(s)dm_{s}\right]}{t_{0}+\int_{0}^{t}q(s)^{2}ds},$
(3.5)

where $t_{0}:=\frac{\sigma^{2}}{\sigma_{0}^{2}}$,
$y_{0}:=(\mu_{0}-\frac{1}{2}\sigma^{2})t_{0}$ and
$q(s):=\frac{1}{\sqrt{1-\rho(s)^{2}}}$, $Y$ is defined by (0.25) in Appendix
C.

###### Proof.

See Appendix C. ∎

We find from the form of (0.27) in Appendix C that the conditional
distribution of $\mu$ given $\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}$ is
still a Gaussian, i.e.,

$\displaystyle\ \ \ \mu|\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}$ (3.6)
$\displaystyle\sim\mathcal{N}\left(\frac{1}{2}\sigma^{2}+\frac{y_{0}+\int_{0}^{t}q(s)^{2}[dY_{s}-\sigma\rho(s)dm_{s}]}{t_{0}+\int_{0}^{t}q(s)^{2}ds},\right.\left.\frac{\sigma^{2}}{t_{0}+\int_{0}^{t}q(s)^{2}ds}\right).$

To solve Problem (2.3), based on Theorem 3.3, we introduce a new process
$Z=\\{Z_{t},\ t\geq 0\\}$ and the innovation process $\bar{W}=\\{\bar{W}_{t},\
t\geq 0\\}$ as follows:

$\displaystyle Z_{t}$
$\displaystyle:=\frac{1}{2}\sigma^{2}+\frac{y_{0}+\int_{0}^{t}q(s)^{2}[dY_{s}-\sigma\rho(s)dm_{s}]}{t_{0}+\int_{0}^{t}q(s)^{2}ds},$
$\displaystyle\bar{W}_{t}$
$\displaystyle:=\int_{0}^{t}\sigma^{-1}q(s)\left\\{dY_{s}-\sigma\rho(s)dm_{s}-[E[\mu|\mathcal{F}_{s}^{S}\vee\mathcal{F}_{s}^{m}]-\frac{1}{2}\sigma^{2}]ds\right\\}.$

Then $Z$ is adapted to the filtration
$\\{\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}:t\geq 0\\}$ and a sufficient
statistic for estimating $\mu$ in this filtration, and $\bar{W}$ is a Brownian
motion adapted to the filtration
$\\{\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}:t\geq 0\\}$. Moreover, it is
tested that $\bar{W}$ is independent to the Brownian motion $m$.

Based on the definition of Brownian motion $\bar{W}$ and the explicit form of
$E[\mu|\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}]$, we obtain that the
processes $X$, $Y$ and $Z$ satisfy the following SDEs:

$\displaystyle dX_{t}$
$\displaystyle=rX_{t}dt+\pi_{t}(Z_{t}-r)dt+\pi_{t}\sigma\rho(t)dm_{t}+\pi_{t}\sigma
q(t)^{-1}d\bar{W}_{t}$ (3.7) $\displaystyle dY_{t}$
$\displaystyle=(Z_{t}-\frac{1}{2}\sigma^{2})dt+\sigma\rho(t)dm_{t}+\sigma
q(t)^{-1}d\bar{W}_{t},$ (3.8) $\displaystyle dZ_{t}$
$\displaystyle=\frac{\sigma q(t)d\bar{W}_{t}}{t_{0}+\int_{0}^{t}q(s)^{2}ds}.$
(3.9)

Using the Markov property of the processes $X$ and $Z$, and the fact that $Z$
is a sufficient statistic of $\mu$, we establish the following expected
utility maximization problem given extra information:

$\displaystyle
V(t,x,z)=\sup_{\pi\in\mathcal{A}_{m}}E[U(X_{T})|X_{t}=x,Z_{t}=z].$ (3.10)

It is easy to see that Problem (2.3) is equivalent to Problem (3.10) with the
initial state $(0,x_{0},\mu_{0})$.

######  Remark 3.3.

If we define
$V(t,x,y,m):=\sup\limits_{\pi\in\mathcal{A}_{m}}E[U(X_{T})|X_{t}=x,Y_{t}=y,m_{t}=m]$.
The state value of $Y_{t}$ and $m_{t}$ can not give the estimation of $\mu$ as
the entire process of $Y$ and $m$ has capability. Then the dynamic programming
no longer hold for each instant the state’s failure to estimate $\mu$. What’s
more that the value function $V(t,x,y,m)$ is not well-defined. That is why we
must construct the process $Z$ to overcome this obstacle.

To solve Problem (3.10), based on the definition of $V(t,x,z)$ and dynamic
programming principle, we start with analyzing the Hamilton-Jacobi-Bellman
(HJB) equation satisfied by the value function $V(t,x,z)$ as follows:

$\displaystyle\sup_{\pi}\mathcal{L}^{\pi}V(t,x,z)=0,\ \ \ V(T,x,z)=U(x),$
(3.11)

where

$\displaystyle\mathcal{L}^{\pi}V(t,x,z)$
$\displaystyle:=V_{t}+[rx+\pi(z-r)]V_{x}+\frac{1}{2}\sigma^{2}\pi^{2}V_{xx}$
$\displaystyle+\sigma^{2}\pi\frac{1}{t_{0}+\int_{0}^{t}q(s)^{2}ds}V_{xz}$
$\displaystyle+\frac{1}{2}q(t)^{2}\sigma^{2}\frac{1}{[t_{0}+\int_{0}^{t}q(s)^{2}ds]^{2}}V_{zz}.$

We define a functional

$\displaystyle\tau(\cdot):=t_{0}+\int_{0}^{\cdot}q(s)^{2}ds.$ (3.12)

The explanation of the functional (3.12) will be given later. By using
$\tau(\cdot)$, the infinitesimal generator can be rewritten as the following
concise form:

$\displaystyle\mathcal{L}^{\pi}V(t,x,z)$
$\displaystyle:=V_{t}+[rx+\pi(z-r)]V_{x}+\frac{1}{2}\sigma^{2}\pi^{2}V_{xx}$
$\displaystyle+\sigma^{2}\pi\frac{1}{\tau(t)}V_{xz}+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)V_{zz}.$
(3.13)

Now we derive the closed-form solutions of Problem (3.10) given extra
information with CARA and CRRA utility functions, which is the foundation of
the further discussion about the extra information and the optimal information
acquisition in Section 4.

######  Theorem 3.4.

(1) If $U(x)=-\frac{1}{\beta}e^{-\beta x}$ (CARA case), then we have

$\displaystyle V\left(t,x,z\right)=U\left(e^{r(T-t)}x+\psi(t,z)\right),$

where

$\displaystyle\psi(t,z)=\frac{1}{2}a(t)(z-r)^{2}+c(t),$ $\displaystyle
a(t)=\frac{1}{\beta\sigma^{2}}\frac{\tau(t)(T-t)}{\tau(t)+(T-t)},$
$\displaystyle
c(t)=\frac{1}{2\beta}\int_{t}^{T}\frac{\tau^{\prime}(s)(T-s)}{\tau(s)[\tau(s)+(T-s)]}ds,$

and the optimal investment strategy is

$\displaystyle\pi^{*}=e^{-r(T-t)}\frac{1}{\beta}\frac{\tau(t)}{\tau(t)+T-t}\frac{Z_{t}-r}{\sigma^{2}}.$

(2) If $U(x)=\frac{x^{1-\gamma}}{1-\gamma}$ (CRRA case) where $\gamma$
satisfies the condition: $\gamma>1$ or $0<\gamma<1$ with
$\frac{t_{0}}{T}>\frac{1-\gamma}{\gamma}$, then we have

$\displaystyle V(t,x,z)=U(e^{r(T-t)}x)\exp\\{\gamma\psi(t,z)\\},$

where

$\displaystyle\psi(t,z)=\frac{1}{2}a(t)(z-r)^{2}+c(t),$ $\displaystyle
a(t)=\frac{1}{\gamma\sigma^{2}}\frac{\tau(t)\frac{1-\gamma}{\gamma}(T-t)}{\tau(t)-\frac{1-\gamma}{\gamma}(T-t)},$
$\displaystyle
c(t)=\frac{1}{2\gamma}\int_{t}^{T}\frac{\tau^{\prime}(s)\frac{1-\gamma}{\gamma}(T-s)}{\tau(s)[\tau(s)-\frac{1-\gamma}{\gamma}(T-s)]}ds,$

and the optimal investment strategy is

$\displaystyle\pi^{*}=\frac{\tau(t)}{\gamma\tau(t)-(1-\gamma)(T-t)}\frac{Z_{t}-r}{\sigma^{2}}X_{t}.$

(3) If $U(x)=\frac{x^{1-\gamma}}{1-\gamma}$ (CRRA case) and $\gamma$ satisfies
the condition $\frac{t_{0}}{T}\leq\frac{1-\gamma}{\gamma}$, then the value
function will be infinite, i.e., the problem become meaningless. The
occurrence of meaningless is not bring about by the extra information but
bayesian problem itself.
(4) If $U(x)=\ln(x)$, the case with $\gamma$ approaching $1$, then

$\displaystyle V(t,x,z)=U(e^{r(T-t)}x)+\psi(t,z),$

where

$\displaystyle\psi(t,z)=\frac{1}{2}a(t)(z-r)^{2}+c(t),$ $\displaystyle
a(t)=\frac{1}{\sigma^{2}}(T-t),$ $\displaystyle
c(t)=\frac{1}{2}\int_{t}^{T}\frac{\tau^{\prime}(s)(T-s)}{\tau^{2}(s)}ds,$

and the optimal investment strategy is

$\displaystyle\pi^{*}=\frac{Z_{t}-r}{\sigma^{2}}X_{t}.$

###### Proof.

See Appendix D. ∎

Especially, if no extra information is involved, Problem (3.10) reduces to
most viewed ambiguity case, which corresponds the situation the value function
with $\tau(\cdot)=t_{0}+\cdot$; If no ambiguity is considered here, the value
function will be given in the sense of limitation that $t_{0}=\infty$,
$\tau(t)=\infty$ and $Z_{t}=\mu$. Then Problem (3.10) reduces to the classical
problem.

Moreover, noticing the fact $\frac{\tau(t)}{\tau(t)+T-t}<1$, we can find with
ambiguity, the investment strategy will be conservative than the classical
case and less conservative if more information is involved in CARA case.
Similarly, in CRRA case, it depends on the value of relative risk aversion
$\gamma$. When $\gamma>1$, the investment strategy will be also conservative
and less conservative if more information is involved. But when
$\frac{T}{t_{0}+T}<\gamma<1$, the investment strategy will be aggressive and
less aggressive if more information is involved. At last, when $\gamma=1$, the
investment strategy remains same as the classical case.

###  3.3. Advantageous perspective: informative clock

Now we discuss the conditional variance
$var(t):=\frac{\sigma^{2}}{t_{0}+\int_{0}^{t}q(s)^{2}ds}$ at time $t$, which
is strictly decreasing over time, and introduce the concept of informative
clock as $\tau(t):=\frac{\sigma^{2}}{var(t)}=t_{0}+\int_{0}^{t}q(s)^{2}ds$, an
index measuring the volume of the information, where
$q(s):=\frac{1}{\sqrt{1-\rho(s)^{2}}}$ defined in Theorem 3.3.

To comprehend this concept, we take an analogy. If we want to obtain a normal
distribution’s mean where the variance $\sigma^{2}$ is known. We get a sample
as $X=(X_{1},\cdots,X_{N})$. Then the conditional distribution of $\mu$ given
$X$ is $\mathcal{N}(\bar{X},\frac{\sigma^{2}}{N})$, denoting this fact by
$\mu|X\sim\mathcal{N}(\bar{X},\frac{\sigma^{2}}{N})$. It can be seen that
$N=\frac{\sigma^{2}}{var(\mu|X)}$, We make a similar comparison as the size of
sample in this case and the informative clock in this article. It can be said
that informative clock is actually an index measuring the volume of the
information just like the size of the sample.

The natural case is that $\rho(t)\equiv 0$ among where $m$ can be treated as
noise. Then the estimate of $\mu$ over time is only estimated by the
information of the price of risky asset itself. Correspondingly, $q(s)=1$ and
$\tau(t)=t_{0}+t$. Informative clock keeps pace as the real time with a fixed
distance as $t_{0}$. In fact, $t_{0}$ is the informative clock at real time
$t=0$ which is actually the information volume hidden in the prior
distribution. Imagining that there exists a duration of history, if the
duration is of value $t_{0}$, the distribution of $\mu$ would be estimated
with variance of $\sigma_{0}^{2}$ by information in this duration. Reversely
thinking, that is why $t_{0}$ defined as
$t_{0}:=\frac{\sigma^{2}}{\sigma_{0}^{2}}$ to match the duration of history.

Now the extra information intrudes into our system. It is natural to predict
that it would bring about larger volume of information. This is obvious as
$q(t)\geq 1$ and $\tau(t)=t_{0}+\int_{0}^{t}q(s)^{2}ds$ where bigger
correlation coefficient can bring higher increasing speed of the informative
clock. The extra information serves as an acceleration of informative clock.
It accelerates to reduce the variance of the estimation of $\mu$ than the real
time, i.e., the extra information can eliminate estimation risk in fast pace.

The concept of informative clock is a more useful perspective for analyzing
the value of information rather than correlation coefficient. If there are two
sources of information, where the one has the information advantage than the
other, i.e., with respect to the informative clock, the one is bigger than the
other. The investor can make better decision to obtain higher utility.
Moreover, Proposition 3.5 is a more strong conclusion than Proposition 3.2 as
it covers more cases.

######  Proposition 3.5.

If there are two sources of extra information $m^{1}$ and $m^{2}$ with
informative clock relation that $\tau^{1}(t)\geq\tau^{2}(t)$ for $\forall
t\geq 0$, then $V_{m^{1}}\geq V_{m^{2}}$.

###### Proof.

See Appendix E. ∎

From the procedure of the comparison of $c$, we must broaden our mind.
$\tau^{1}(t)\geq\tau^{2}(t)$ implies that in any real time the first
information has more informative clock. However, if we take the perspective
with $(\tau^{1})^{-1}(u)\leq(\tau^{2})^{-1}(u)$, it tells that in any same
informative clock the first information has more remaining time to invest.
Then we see that the informative clock contributes the value function with two
ways: the whole larger informative clock just as the first term in (0.36)
shows and the adequate time to prepare in any same informative clock just as
the second term shows.

Another useful effect of informative clock is that HJB equation can be reduced
into a representation with only informative clock as (3.2). In the
infinitesimal generator, the term
$V_{t}+[rx+\pi(z-r)]V_{x}+\frac{1}{2}\sigma^{2}\pi^{2}V_{xx}$ is the most
classical HJB equation in the investment problem where $z$ is the certain rate
of drift of the risky asset. Then $\sigma^{2}\pi\frac{1}{\tau(t)}V_{xz}$ is
comprehended as the adjustment of investment by the ambiguity of drift. The
last term $\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)V_{zz}$ depicts
the evolution of the drift over time. Then HJB equation is intuitive under
this perspective.

Due to the close connection between $\rho$ and $\tau$ :
$\tau(t):=\frac{\sigma^{2}}{var(t)}=t_{0}+\int_{0}^{t}q(s)^{2}ds$ and
$q(s):=\frac{1}{\sqrt{1-\rho(s)^{2}}}$. For the convenience and vividness of
expression, we use the functional $\tau$ to refer the information which has
this volume of information. With it, all the value can be represented in
concise form. And we assume the continuum of the $\rho(t)\in[0,1)$, i.e., any
value in this interval can be obtained from the market. Thus we can assume any
$\tau$ as the continuous function with derivative larger than one is obtained
in the market. Finally, Proposition 3.5 has told us that the utility is
irrelevant to the actual form of the source of information but what
informative clock it can bring about for estimating $\mu$. Thus we can ignore
the form of the extra information and only focus on the problem via the
perspective of functional $\tau$.

###  3.4. Value and cost of extra information

Now we solve the Problem (2.3) and are equipped with the perspective of
informative clock. To objectively measure the value of the extra information
by a criterion, we introduce the concept of certainty equivalence of utility.
As we have posed, we define the value function $V_{\tau}(t,x,z)$ as the value
function in the case that extra information can bring about the informative
clock as the functional $\tau$. In the case without any extra information, the
functional has the structure $\tau(\cdot)=t_{0}+\cdot$. Here we define it as
the natural case and we denote that $\mathbb{0}$ as the functional in this
case. It is the value function that we achieve by price itself. Then we define
the value functional $Value(\cdot)$ by

$\displaystyle
V_{\tau}(0,x_{0},\mu_{0})=V_{\mathbb{0}}\left(0,x_{0}+Value(\tau),\mu_{0}\right).$
(3.14)

It means that taking initial value $x_{0}$ along with extra information with
informative clock $\tau$, we can achieve the same utility as the case of
dealing with initial value $x_{0}+Value(\tau)$ along with no extra
information. Here $V_{\mathbb{0}}$ is comprehended as our objective criterion.

In the case of CARA utility function, comparing two value functions with the
relationship
$V_{\tau}(0,x_{0},\mu_{0})=V_{\mathbb{0}}\left(0,x_{0}+Value(\tau),\mu_{0}\right)$,
we obtain

$\displaystyle
Value(\tau)=\int_{0}^{T}\frac{1}{2\beta}\frac{(T-t)\tau^{\prime}(t)}{\tau(t)\left[\tau(t)+T-t\right]}dt-
C_{1},$ (3.15)

where $C_{1}:=\int_{0}^{T}\frac{1}{2\beta}\frac{(T-t)}{(t+t_{0})(T+t_{0})}dt$
is a constant. This is actually the difference of $c(0)$ in two value
functions with corresponding informative clock and it is more than zero just
as Proposition 3.5 shows(see(0.36)). We see that it has nothing to do with the
endowment $x_{0}$. This is derived from the property of CARA utility. People
are caring about the absolute wealth in the utility function. Therefore the
extra information will bring about the certain value. In the case of CRRA
utility function, comparing two value function with the relation
$V_{\tau}(0,x_{0},\mu_{0})=V_{\mathbb{0}}(0,x_{0}+Value(\tau),\mu_{0})$, we
obtain

$\displaystyle\ \ \ \ Value(\tau)$ (3.16)
$\displaystyle=\left[\exp\left\\{\int_{0}^{T}\frac{1}{2(1-\gamma)}\frac{\tau^{\prime}(t)\frac{1-\gamma}{\gamma}(T-t)}{\tau(t)[\tau(t)-\frac{1-\gamma}{\gamma}(T-t)]}dt\right\\}/C_{2}-1\right]x_{0}.$

where
$C_{2}:=\exp\left\\{\int_{0}^{T}\frac{1}{2(1-\gamma)}\frac{\frac{1-\gamma}{\gamma}(T-t)}{(t+t_{0})[(t+t_{0})-\frac{1-\gamma}{\gamma}(T-t)]}dt\right\\}$
is a constant (The case $\gamma=1$ still holds in the sense of limit).
Distinct from the CARA utility, people are caring about the multiple of the
wealth with CRRA utility. The investors are surging to doubling their initial
value. And we know from Eq.(3.16) and Proof of Proposition 3.5 that the extra
information will bring about the multiple increase of the endowment and the
multiple number is the function of quotient of $c(0)$ with two informative
clocks which coincides our experience.

No matter in CARA utility or CRRA utility, we find that the value of the extra
information is independent with the market condition. It has nothing to do
with the interest rate, risky asset’s volatility and even the estimation of
return. It only concerns about the informative clock which represents
precision of the estimation and investor’s preference for risk.

Fig. 1. The corresponding value of the information with respect to informative
clock’s derivative in CARA case. Fig. 2. The corresponding value of the
information with respect to informative clock’s derivative in CRRA case.

Here are two pictures of the value of information in CARA and CRRA case where
the parameter is given with $t_{0}=4$, $T=2$, $\beta=0.001$, $x_{0}=1000$ and
most importantly, $\tau(t)=t_{0}+kt$ where $k$ is a constant represent the
derivative of the informative clock. For convenience, here we only use the
linear informative clock for representation. It is obvious the information
will bring huge value if the investor is less risk aversion both in CARA case
and CRRA case where the value is inversely proportional to the absolute risk
aversion in CARA case.

The other thing need to regard is that the marginal of volume of the
information contributes less value when the information is plenty. Moreover,
it has its bound which will be given in Subsection 4.2.

Conversely, the gleaning of huge volume of information is costly. There must
be a restriction of behavior that collecting information as much as the
investor is not allowed. We define the punishment functional $Cost(\cdot)$ as
follows:

$\displaystyle Cost(\tau)=\int_{0}^{T}cost\left(\tau^{\prime}(t)\right)dt,$
(3.17)

where the function $cost:[1,\infty)\to\mathrm{R}^{+}$ is an increasing and
convex function. As we know,
$\tau^{\prime}(t)=\frac{1}{\sqrt{1-\rho(t)^{2}}}\geq 1$ and when
$\tau^{\prime}(t)=1$, it means $\rho(t)=0$ and no useful extra information is
involved. Thus, we set the boundary condition: $cost(1)=0$. The monotone
property and convexity originate from the fact that the effort to glean the
marginal information ($\tau^{\prime}(t)$) is positive and increases w.r.t.the
volume of information.

###  3.5. The most worthwhile information to acquire

The value and cost of information are well defined, see (3.14) and (3.17). In
this subsection the most important thing we consider is to balance the value
and cost to determine the most worthwhile information to acquire, i.e.,
solving the following functional maximization problem:

$\displaystyle\sup_{\tau}Net(\tau),$ (3.18) $\displaystyle
Net(\tau):=Value(\tau)-Cost(\tau),$

where the informative clock $\tau$ is differentiable and its derivative is
larger than one everywhere. Now we assume $cost(x)=\lambda(x-1)^{2}$ for an
instance, where $\lambda>0$ is the relative ability of acquiring information,
and solve Problem (3.18) as follows.

######  Theorem 3.6.

In the case of CARA utility function, the optimal informative clock satisfies
the condition

$\displaystyle\tau^{\prime\prime}(t)+\frac{1}{4\beta\lambda}\frac{1}{[\tau(t)+T-t]^{2}}=0.$
(3.19)

In the case of CRRA utility function, the optimal informative clock satisfies
the condition

$\displaystyle\tau^{\prime\prime}(t)+\frac{y}{4\gamma\lambda}\frac{1}{[\tau(t)-\frac{1-\gamma}{\gamma}(T-t)]^{2}}=0,$
(3.20)

where $y$ is a positive number.

###### Proof.

See Appendix F. ∎

######  Remark 3.4.

In fact, using the same method as in Theorem 3.6 we can solve Problem (3.18)
for other punishment function. Theorem 3.6 just provides a template to find
extreme point of Problem (3.18). And this template is the most widespread for
describing the punishment function.

The last work to find the optimal informative clock is to search all the
functional satisfying the necessary condition with the different initial
condition $\tau^{\prime}(0)$ and $y$. This part is trivial and we omit it
here. Based on Eq.(3.19) and Eq.(3.20), we have

$\tau^{\prime\prime}(t)+g(t)=0,$

where $g(t)>0$ (it holds for any punishment functional). It follows that

$\displaystyle\tau^{\prime\prime}(t)<0.$

Fig. 3. The optimal informative clock in CARA case. Fig. 4. The optimal
informative clock in CRRA case.

That is to say the investor will devote less effort to glean the information
as the time goes. In fact, this conclusion can also be derived from
Proposition 3.5. If there is a information with
$\tau^{\prime}(t_{1})<\tau^{\prime}(t_{2})$ where $t_{1}<t_{2}$ having the
most net value, we reshuffle the order of $\tau^{\prime}$ as
$\hat{\tau}^{\prime}$ and define a new
$\hat{\tau}(t)=\int_{0}^{t}\hat{\tau}^{\prime}(s)ds$ to achieve
$\hat{\tau}(t)\geq\tau(t)$ for all $t$. Then $V^{\hat{\tau}}>V^{\tau}$ and
$Value(\hat{\tau})>Value(\tau)$. However, their cost is the same. Then we find
a new informative clock with more net value and there is a contradiction. Thus
we must glean the information as soon as possible. Here are two pictures of
the optimal informative clock in CARA and CRRA case where the parameter is
given the same as $t_{0}=4$, $T=2$, $x_{0}=1000$, $\lambda=1$. As we can see,
the optimal informative is really increasing and concave. Furthermore, if we
compare the optimal informative clock for the investor with different risk
aversion in CARA case or CRRA case, we will see investor with lower risk
aversion are more inclined to glean more information. In other words, gleaning
information benefits more to the investor with lower risk aversion.

##  4\. Complements of information itself

###  4.1. Sources of information

We provide a discussion around the extra information itself. In the financial
market, a wide range of information can be gathered and utilized. One commonly
considered type of information is systemic risk, which we assume is generated
by the process $W^{\text{systemic}}$. For instance, we can consider $m$ to be
equal to $W^{\text{systemic}}$. However, it is important to note that any
observed risk can be represented by the process $m$. Whenever a risky asset
exists in the market, various risk factors contribute to the price
fluctuations and can be observed. Furthermore, the phenomenon of co-movement
in asset prices offers us a perspective to identify the process $m$.

One potential concern might arise regarding the calculation of $\rho(t)$ as
$W$ itself is not directly observable. However, since we have the observation
$Y$ (defined in (0.25)), and $\rho(t)=\frac{d\langle
m,Y\rangle_{t}}{\sqrt{d\langle m\rangle_{t}d\langle Y\rangle_{t}}}$, we can
effectively calculate their correlation coefficient without direct observation
of $W$. This equation provides a feasible way to compute the correlation
coefficient, which empowers us to test different sources of information.
Therefore, we have the ability to assess the correlation between $Y$ and
different sources of information, allowing us to identify the ones with the
highest correlation. Furthermore, we can even combine multiple sources of
information through signal engineering techniques to achieve a higher
correlation with $Y$.

###  4.2. the explosion of informative clock

Assumption 3.1 (see (3.1)) can be rewritten as $\tau(T)<\infty$. It is the
case that the informative clock is finite from $0$ to $T$. It arouses our
curiosity that what would happen when the informative clock become infinite.
In fact, at that time, the ambiguity hidden in $\mu$ is totally erased. As the
conditional variance of $\mu$ becomes zero and the distribution of $\mu$ has
collapsed into a single point’s distribution. While this phenomenon has been
supposed as there is a ‘insider’ investor, who can observe both the drift and
driving Brownian motion $W$ (Zhao (1999)) . The informative clock becomes
infinite means that the investor becomes ‘insider’ investor knowing
everything.

It is natural to put forward some problems around that the informative clock
evolve from finite time to the infinite. The main one is that how it happens.
In fact, if we extend the range of $\rho$ with $\rho(t)\in[0,1]$. It will
happen when the fact $\rho(t)$ approaches or equals to one (for example,
$m=W$) with enough time which contributes infinite value for the informative
clock. Another one may be around the function of informative clock becomes
singular as it tends infinite in finite time. There is no worry as in the HJB
equation, we can observe that $\tau$ only exists with the form
$\frac{1}{\tau}$ which can be continuous from $\frac{1}{\tau(0)}$ to $0$ where
two ends are both finite. Thus, it really bothers nothing when there is a
surge of infinite information if we assure the continuity of $\frac{1}{\tau}$.
Moreover, all the deduction in Section 3 still hold if we use limitation as
there is always informative clock exist in the dominator to assure anything
well-posedness. Above all, everything remains alright if we just take
$(\frac{1}{\tau})(t)=0$ when $\tau(t)=0$.

Ultimately, we take $\tau(0^{+})=\infty$ for consideration (can be
comprehended as the limit $\tau(t)=t_{0}+nt$ where $n\to\infty$), that is, the
investor is assumed as ‘insider’ investor at the beginning (time is $0^{+}$).
This corresponds the value of $V(0^{+},x_{0},Z_{0^{+}})$ which is the
classical case with $\mu=Z_{0^{+}}$. And $\mu$ has the prior distribution
$\mathcal{N}(\mu_{0},\sigma^{2}_{0})$. Then we have that
$\int_{\mathrm{R}}V(0^{+},x_{0},\mu_{0}+u)\mathcal{N}_{\sigma_{0}^{2}}(u)du$
is actually the objective value what we want. In fact, this value can be
gotten directly in the value function if we allow the generalized function
calculation in $c(0)$ in the HJB equation. Taking CARA for example,

$\displaystyle c(0)$
$\displaystyle=\frac{1}{2\beta}\int_{0}^{T}\frac{\tau^{\prime}(s)(T-s)}{\tau(s)[\tau(s)+(T-s)]}ds$
$\displaystyle=\frac{1}{2\beta}\int_{0}^{0^{+}}\left[\frac{\tau^{\prime}(s)}{\tau(s)}-\frac{[\tau(s)+T]^{\prime}}{\tau(s)+T}\right]ds$
$\displaystyle=\frac{1}{2\beta}\left[\ln(\tau(s))-\ln(\tau(s)+T)\right]|_{0}^{0^{+}}$
$\displaystyle=\frac{1}{2\beta}\ln(\frac{t_{0}+T}{t_{0}}).$

Similarly, we obtain the value

$\displaystyle
c(0)=\frac{1}{2\gamma}\ln(\frac{t_{0}}{t_{0}-\frac{1-\gamma}{\gamma}T})$

in CRRA case if $\gamma\neq 1$. And

$\displaystyle c(0)=\frac{1}{2}\frac{T}{t_{0}}$

in CRRA case when $\gamma=1$. As regards to Proposition 3.5 and content of
information’s value, it is obvious that the value of any extra information has
its bound. And the bound is finite and can be approached if we pay enough
cost.

##  5\. Conclusions

In this paper, we show that, with the extra information, the estimation of the
risky asset’s is preciser and the utility can be improved in addition. We
first put forward the new concept of “informative clock” to describe the
precision of the estimation and use it to represent everything in the concise
form. The value and cost are well defined for any extra information. From the
process of trading off, it tells us to glean the extra information earlier
will be great and we find investors with less risk aversion are more inclined
to glean information. Finally, it bothers us nothing to permit the informative
clock infinite and find the bound of the extra information’s value.

## Appendix A. Proof of Proposition 3.1

###### Proof.

Without loss of generality, $m^{1}$ and $m^{2}$ are regarded as Brownian
motion (see in the Remark). As two processes $m^{1}$ and $m^{2}$ sharing the
same correlation coefficient with $Y$, by using Ito formula, the
characteristic functions of the two pairs $(Y,m^{1})$ and $(Y,m^{2})$ are
equal, as such, they have the same probability law. In the case containing
$m^{1}$, as the optimal $\pi^{1}$ is adapted to the filtration
$\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m^{1}}$, there must exist the
functional $f$ s.t. $\pi^{1}_{t}=f(Y_{u},m^{1}_{u},0\leq u\leq t)$. We choose
$\pi^{2}_{t}=f(Y_{u},m^{2}_{u},0\leq u\leq t)$ adapted to the filtration
$\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m^{2}}$. Then the process classes
$(Y,m^{1},\pi^{1},X^{1})$ and $(Y,m^{2},\pi^{2},X^{2})$ have the same
probability law. As such, $X_{T}^{1}$ and $X_{T}^{2}$ have also the same
probability law, and their expected utilities are equal. As $\pi^{1}$ is the
optimal one and $\pi^{2}$ is an selected one, $V_{m^{1}}\geq V_{m^{2}}$.
Similarly, we have $V_{m^{2}}\leq V_{m^{1}}$. Thus, $V_{m^{1}}=V_{m^{2}}$. ∎

## Appendix B. Proof of Proposition 3.2

Before proving the second conjecture, we prepare a lemma as follows:

######  Lemma .1.

Let $(\Omega,\mathcal{F},\mathcal{P})$ be the probability space with
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ being the sub-$\sigma$-algebras of
$\mathcal{F}$. For any random variable $X\in
L^{1}(\Omega,\mathcal{F},\mathcal{P})$, if $X$ and $\mathcal{G}_{1}$ are
independent of $\mathcal{G}_{2}$, then we have

$\displaystyle E[X|\mathcal{G}_{1}\vee\mathcal{G}_{2}]=E[X|\mathcal{G}_{1}].$
(0.21)

###### Proof.

As both sides of (0.21)are $\mathcal{G}_{1}\vee\mathcal{G}_{2}$-measurable,
and the nonempty collection $\mathfrak{F}_{0}\triangleq\\{A_{1}\cap
A_{2}|A_{1}\in\mathcal{G}_{1},A_{2}\in\mathcal{G}_{2}\\}$ of subsets of
$\Omega$ is a $\pi$-system generating $\mathcal{G}_{1}\vee\mathcal{G}_{2}$,
Eq.(0.21) is equivalent to

$\displaystyle\int_{A_{1}\cap
A_{2}}E[X|\mathcal{G}_{1}\vee\mathcal{G}_{2}]dP=\int_{A_{1}\cap
A_{2}}E[X|\mathcal{G}_{1}]dP$

for $\forall$ $A_{1}\cap A_{2}\in\mathfrak{F}_{0}$ . As
$E[X1_{A_{1}}|\mathcal{G}_{1}]$ and $1_{A_{2}}$ are independent, and
$X1_{A_{1}}$ and $1_{A_{2}}$ are independent, we have

$\displaystyle\ \ \ RHS$ $\displaystyle=\int_{\Omega}1_{\\{A_{1}\cap
A_{2}\\}}E[X|\mathcal{G}_{1}]dP=\int_{\Omega}1_{A_{1}}1_{A_{2}}E[X|\mathcal{G}_{1}]dP$
$\displaystyle=\int_{\Omega}E[X1_{A_{1}}|\mathcal{G}_{1}]1_{A_{2}}dP=E[E[X1_{A_{1}}|\mathcal{G}_{1}]]E[1_{A_{2}}]$
$\displaystyle=E[X1_{A_{1}}]E[1_{A_{2}}]=\int_{\Omega}X1_{A_{1}}1_{A_{2}}dP$
$\displaystyle=\int_{\Omega}1_{A_{1}\cap
A_{2}}E[X|\mathcal{G}_{1}\vee\mathcal{G}_{2}]dP=\int_{A_{1}\cap
A_{2}}E[X|\mathcal{G}_{1}\vee\mathcal{G}_{2}]dP.$

∎

###### Proof.

We can construct artificially a pure noise Brownian motion $W^{noise}$ which
is assumed to be on the probability space $(\Omega,\mathbb{F},\mathbb{P})$ and
adapted to the filtration $\mathbb{F}$. If not, there is an extension
$(\widetilde{\Omega},\widetilde{\mathbb{F}},\widetilde{\mathbb{P}})$ of
$(\Omega,\mathbb{F},\mathbb{P})$ which can be defined by the cartesian product
of the original one and the probability space of $W^{noise}$. Additionally,
the extension of space makes no difference of investing and objective function
as the restriction of $\pi$. Thus the probability space can be arbitrarily
fertile for our need. Similarly, $m^{1}$ and $m^{2}$ are still regarded as
Brownian motions.

Next we claim that the observed noise contributes nothing to our strategy for
any given extra information, that is,

$\displaystyle\sup\limits_{\pi\in\mathcal{A}_{m}}EU(X_{T})=\sup\limits_{\pi\in\mathcal{A}_{m,W^{noise}}}EU(X_{T}),$
(0.22)

where
$\mathcal{A}_{m,W^{noise}}:=\\{\pi|\int_{0}^{T}\pi_{t}^{2}dt<\infty,\pi\text{
adapted to }\mathbb{F}^{S}\vee\mathbb{F}^{m}\vee\mathbb{F}^{W^{noise}}\\}$.

Let us review the estimation of $\mu$. In fact, if we let $X=\mu$,
$\mathcal{G}_{1}=\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}$ and
$\mathcal{G}_{2}=\mathcal{F}_{t}^{noise}$ in Eq.(0.21), we have

$\displaystyle
E[\mu|\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}\vee\mathcal{F}_{t}^{W^{noise}}]=E[\mu|\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}].$
(0.23)

Eq.(0.23) tells us that the noise give totally no information of $\mu$. Using
the same method, we can construct innovation process $\bar{W}$ and sufficient
process $Z$, in which no differences exist with or without $W^{noise}$. Thus,
(0.22) holds. Taking $m=m^{1}$, we obtain

$\displaystyle\sup\limits_{\pi\in\mathcal{A}_{m^{1}}}EU(X_{T})=\sup\limits_{\pi\in\mathcal{A}_{m^{1},W^{noise}}}EU(X_{T}).$

Define the process $m^{3}:=\\{m^{3}_{t},t\geq 0\\}$:

$\displaystyle
m^{3}_{t}=\int_{0}^{t}\left\\{\frac{\rho^{2}(s)}{\rho^{1}(s)}dm^{1}_{s}+\sqrt{1-\left[\frac{\rho^{2}(s)}{\rho^{1}(s)}\right]^{2}}dW^{noise}_{s}\right\\}.$
(0.24)

where $\frac{\rho^{2}(s)}{\rho^{1}(s)}:=0$ if $\rho^{1}(s)=0$,
$\rho^{2}(s)=0$. Then $m^{3}$ is a Brownian motion with the property that
$\rho^{3}(t):=d\langle m,W\rangle_{t}=\rho^{2}(t)$. Using Proposition
LABEL:Proposition41, we have

$\displaystyle\sup\limits_{\pi\in\mathcal{A}_{m^{2}}}EU(X_{T})=\sup\limits_{\pi\in\mathcal{A}_{m^{3}}}EU(X_{T}).$

As $\mathcal{A}_{m^{3}}\subset\mathcal{A}_{m^{1},W^{noise}}$, based on the
definition of supreme, we obtain

$\displaystyle\sup\limits_{\pi\in\mathcal{A}_{m^{3}}}EU(X_{T})\leq\sup\limits_{\pi\in\mathcal{A}_{m^{1},W^{noise}}}EU(X_{T}).$

Thus

$\displaystyle\sup\limits_{\pi\in\mathcal{A}_{m^{1}}}EU(X_{T})\geq\sup\limits_{\pi\in\mathcal{A}_{m^{2}}}EU(X_{T}),$

that is, $V_{m^{1}}\geq V_{m^{2}}$. ∎

## Appendix C. Proof of Theorem 3.3

###### Proof.

$\displaystyle dY_{t}=(\mu-\frac{1}{2}\sigma^{2})dt+\sigma dW_{t},\ t\geq 0,\
\ Y_{0}=0.$ (0.25)

Define $n=\\{n_{t}:=\int_{0}^{t}q(s)[dW_{s}-\rho(s)dm_{s}],\ 0\leq t\leq
T\\}$. Assumption 3.1 guarantees that the process $n$ is well-defined.
Furthermore, $n$ is a Brownian motion and orthogonal to the process $m$. And
we have the following orthogonal decomposition of $W$:

$\displaystyle dW_{t}=\rho(t)dm_{t}+q(t)^{-1}dn_{t}.$

Using SDE(0.25), we obtain

$\displaystyle\left[dY_{t}-\sigma\rho(t)dm_{t}+\frac{1}{2}\sigma^{2}dt\right]=\mu
dt+\sigma q(t)^{-1}dn_{t}.$

Define the posterior distribution:

$\displaystyle p(u,t)du:=P(\mu\in
du|\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}),\ \forall u\in\mathrm{R}.$

Using the same detailed calculation as in Wonham (1964) , we have

$\displaystyle\ \ \ p(u,t)du$ (0.26)
$\displaystyle=p(u,0)du\exp\left\\{-\frac{1}{2}u^{2}\int_{0}^{t}\sigma^{-2}q(s)^{2}ds\right.$
$\displaystyle\left.+u\int_{0}^{t}\sigma^{-2}q(t)^{2}\left[dY_{s}-\sigma\rho(s)dm_{s}+\frac{1}{2}\sigma^{2}ds\right]\right\\}$
$\displaystyle\bigg{/}\int_{\mathrm{R}}p(u,0)du\exp\left\\{-\frac{1}{2}u^{2}\int_{0}^{t}\sigma^{-2}q(s)^{2}ds\right.$
$\displaystyle\left.+u\int_{0}^{t}\sigma^{-2}q(t)^{2}\left[dY_{s}-\sigma\rho(s)dm_{s}+\frac{1}{2}\sigma^{2}ds\right]\right\\}.$

The prior distribution of $\mu$ yields

$\displaystyle
p(u,0)du=\frac{1}{\sqrt{2\pi\sigma_{0}^{2}}}e^{-\frac{(u-\mu_{0})^{2}}{2\sigma_{0}^{2}}}.$

Putting it in (0.26), we obtain

$\displaystyle\hskip 8.5359ptp(u,t)du$ (0.27)
$\displaystyle=\frac{1}{\sqrt{2\pi\frac{\sigma^{2}}{t_{0}+\int_{0}^{t}q(s)^{2}ds}}}\times$
$\displaystyle\exp\left\\{-\frac{\left[u-\left(\frac{1}{2}\sigma^{2}+\frac{y_{0}+\int_{0}^{t}q(t)^{2}(dY_{s}-\sigma\rho(s)dm_{s})}{t_{0}+\int_{0}^{t}q(s)^{2}ds}\right)\right]^{2}}{2\frac{\sigma^{2}}{t_{0}+\int_{0}^{t}q(s)^{2}ds}}\right\\}.$

Therefore

$\displaystyle\ \ \
E[\mu|\mathcal{F}_{t}^{S}\vee\mathcal{F}_{t}^{m}]=\int_{\mathrm{R}}up(u,t)du$
(0.28)
$\displaystyle=\frac{1}{2}\sigma^{2}+\frac{y_{0}+\int_{0}^{t}q(t)^{2}\left[dY_{s}-\sigma\rho(s)dm_{s}\right]}{t_{0}+\int_{0}^{t}q(s)^{2}ds}.$

∎

## Appendix D. Proof of Theorem 3.4

###### Proof.

(1) CARA case: utility function $U(x)=-\frac{1}{\beta}e^{-\beta x}$. We use
the ansatz:

$\displaystyle V(t,x,z)=U(e^{r(T-t)}x+\psi(t,z)).$

Then the HJB equation corresponding to $V(t,x,z)$ becomes the following PDE of
$\psi$:

$\displaystyle 0=$ $\displaystyle-\beta
V\left[\psi_{t}+\pi^{*}(z-r)e^{r(T-t)}-\beta\frac{1}{2}\sigma^{2}\pi^{*2}e^{2r(T-t)}\right.$
(0.29)
$\displaystyle\left.-\beta\sigma^{2}\pi^{*}\frac{1}{\tau(t)}e^{r(T-t)}\psi_{z}-\beta\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)\psi_{z}^{2}+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)\psi_{zz}\right],\
$

where

$\displaystyle\pi^{*}=e^{-r(T-t)}\frac{(z-r)-\beta\sigma^{2}\frac{1}{\tau(t)}\psi_{z}}{\beta\sigma^{2}}.$

Putting $\pi^{*}$ into (0.29) yields

$\displaystyle\psi_{t}+\frac{\beta\left[\frac{1}{\beta}(z-r)-\sigma^{2}\frac{1}{\tau(t)}\psi_{z}\right]^{2}}{2\sigma^{2}}+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)[-\beta\psi^{2}_{z}+\psi_{zz}]=0.$
(0.30)

We guess that $\psi$ has the following form:

$\displaystyle\psi(t,z)=\frac{1}{2}a(t)(z-r)^{2}+b(t)(z-r)+c(t).$

Substituting the form of $\psi$ into Eq.(0.30), we have

$\displaystyle\frac{1}{2}a^{\prime}(t)+\frac{\beta\left[\frac{1}{\beta}-\sigma^{2}\frac{1}{\tau(t)}a(t)\right]^{2}}{2\sigma^{2}}$
(0.31)
$\displaystyle+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)(-\beta)a(t)^{2}=0,\
\ a(T)=0,$ $\displaystyle b^{\prime}(t)+(\cdots)b(t)=0,\ \ \ \ b(T)=0,$ (0.32)
$\displaystyle
c^{\prime}(t)+(\cdots)b(t)+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)(a(t))=0,\
\ c(T)=0.$ (0.33)

It follows that $\psi(T,z)=0$ and $b(t)=0$ for $t\in[0,T]$. To get $a$, we
define

$\displaystyle m(t)=\frac{1}{\beta}-\sigma^{2}\frac{1}{\tau(t)}a(t).$

Using the ODE of $a(t)$,

$\displaystyle\tau(t)m^{\prime}(t)+\beta\left[\tau^{\prime}(t)-1\right]m(t)^{2}-\tau^{\prime}(t)m(t)=0,$

which is a Riccati equation. Let $d(t)=\frac{1}{m(t)}$, the Riccati equation
is transformed into the following ODE:

$\displaystyle
d^{\prime}(t)=-\frac{\tau^{\prime}(t)}{\tau(t)}d(t)+\frac{\beta[\tau^{\prime}(t)-1]}{\tau(t)}.$

Thus

$\displaystyle d(t)$ $\displaystyle=\beta+\beta(T-t)\frac{1}{\tau(t)},$
$\displaystyle m(t)$
$\displaystyle=\frac{1}{\beta+\beta(T-t)\frac{1}{\tau(t)}},$ $\displaystyle
a(t)$
$\displaystyle=\frac{\tau(t)}{\sigma^{2}}\left[\frac{1}{\beta}-\frac{1}{\beta+\beta(T-t)\frac{1}{\tau(t)}}\right]$
$\displaystyle=\frac{1}{\beta\sigma^{2}}\frac{\tau(t)(T-t)}{\tau(t)+(T-t)},$
$\displaystyle c(t)$
$\displaystyle=\int_{t}^{T}\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(s)(a(s))ds$
$\displaystyle=\frac{1}{2\beta}\int_{t}^{T}\frac{\tau^{\prime}(s)(T-s)}{\tau(s)[\tau(s)+(T-s)]}ds.$

(2) As CRRA case is intricacy, we show the proof in three different
categories: The first one is $U(x)=\frac{x^{1-\gamma}}{1-\gamma}$ with the
necessary condition that $\frac{t_{0}}{T}>\frac{1-\gamma}{\gamma}$, the second
one with $\frac{t_{0}}{T}\leq\frac{1-\gamma}{\gamma}$ and the last one is
$U(x)=\ln(x)$.

(i) If $U(x)=\frac{x^{1-\gamma}}{1-\gamma}$ with
$\frac{t_{0}}{T}>\frac{1-\gamma}{\gamma}$, we use the ansatz:

$\displaystyle V(t,x,z)=U(e^{r(T-t)}x)\exp\\{\gamma\psi(t,z)\\}.$

The HJB equation corresponding to $V(t,x,z)$ is equivalent to the following:

$\displaystyle 0=\gamma V$
$\displaystyle\left[\psi_{t}+\frac{1-\gamma}{\gamma}\frac{\pi^{*}}{x}(z-r)-\frac{1}{2}\sigma^{2}(\frac{\pi^{*}}{x})^{2}(1-\gamma)\right.$
$\displaystyle\left.+\sigma^{2}\frac{\pi^{*}}{x}\frac{1-\gamma}{\tau(t)}\psi_{z}+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)\left[\gamma\psi^{2}_{z}+\psi_{zz}\right]\right],$

where

$\displaystyle\pi^{*}=x\frac{\sigma^{2}\frac{1}{\tau(t)}\psi_{z}+\frac{1}{\gamma}(z-r)}{\sigma^{2}}.$

Then

$\displaystyle\psi_{t}+\frac{(1-\gamma)\left[\sigma^{2}\frac{1}{\tau(t)}\psi_{z}+\frac{1}{\gamma}(z-r)\right]^{2}}{2\sigma^{2}}$
$\displaystyle+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)\left[\gamma\psi^{2}_{z}+\psi_{zz}\right]=0.$

We guess

$\displaystyle\psi(t,z)=\frac{1}{2}a(t)(z-r)^{2}+b(t)(z-r)+c(t).$

Similar to that of CARA case,

$\displaystyle\frac{1}{2}a^{\prime}(t)+\frac{(1-\gamma)\left[\sigma^{2}\frac{1}{\tau(t)}a(t)+\frac{1}{\gamma}\right]^{2}}{2\sigma^{2}}$
$\displaystyle+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)\gamma
a(t)^{2}=0,\ a(T)=0,$ $\displaystyle b(t)\equiv 0,\ b(T)=0,$ $\displaystyle
c^{\prime}(t)+\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(t)(a(t))=0,\
c(T)=0.$

As for the equation of $a$, first making a substitution:

$\displaystyle m(t)=\sigma^{2}\frac{1}{\tau(t)}a(t)+\frac{1}{\gamma},$

we get a Riccati equation of $m$ as follows:

$\displaystyle\tau(t)m^{\prime}(t)+\left[(1-\gamma)+\gamma\tau^{\prime}(t)\right]m(t)^{2}-\tau^{\prime}(t)m(t)=0.$
(0.34)

Using a substitution: $d(t)=\frac{1}{m(t)}$, the Riccati equation is the
following solvable form:

$\displaystyle
d^{\prime}(t)=-\frac{\tau^{\prime}(t)}{\tau(t)}d(t)+\frac{(1-\gamma)+\gamma\tau^{\prime}(t)}{\tau(t)}.$

Solving the last ODE,

$\displaystyle d(t)=\gamma-(1-\gamma)(T-t)\frac{1}{\tau(t)},$ $\displaystyle
m(t)=\frac{1}{\gamma-(1-\gamma)(T-t)\frac{1}{\tau(t)}}.$

Then

$\displaystyle a(t)$
$\displaystyle=\frac{\tau(t)}{\sigma^{2}}\left[\frac{1}{\gamma-(1-\gamma)(T-t)\frac{1}{\tau(t)}}-\frac{1}{\gamma}\right]$
$\displaystyle=\frac{1}{\gamma\sigma^{2}}\frac{\tau(t)\frac{1-\gamma}{\gamma}(T-t)}{\tau(t)-\frac{1-\gamma}{\gamma}(T-t)},$
$\displaystyle c(t)$
$\displaystyle=\int_{t}^{T}\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(s)(a(s))ds$
$\displaystyle=\frac{1}{2\gamma}\int_{t}^{T}\frac{\tau^{\prime}(s)\frac{1-\gamma}{\gamma}(T-s)}{\tau(s)[\tau(s)-\frac{1-\gamma}{\gamma}(T-s)]}ds.$

Thus we obtain that closed-form of the value function is solved in the first
situation. Moreover, if $\gamma>1$, the value function will always be bounded.
But if $0<\gamma<1$, the necessary condition that
$\frac{t_{0}}{T}>\frac{1-\gamma}{\gamma}$ assures the boundedness of the value
function.
(ii) If $U(x)=\frac{x^{1-\gamma}}{1-\gamma}$ with
$\frac{t_{0}}{T}\leq\frac{1-\gamma}{\gamma}$, the value function will be
infinite, the above method of “ansatz” does not work. Here we take the
strategy that $\pi_{t}=kX_{t}$. We will show that if $k\to\infty$, the utility
expectation will tend to infinite.

Indeed, if $\pi_{t}=kX_{t}$, the terminal state is

$\displaystyle
X_{T}=e^{rT}x_{0}\exp\left\\{\left[k(\mu-r)-\frac{1}{2}\sigma^{2}k^{2}\right]T+\sigma
kW_{T}\right\\}.$

As $\mu$ and $W_{T}$ are independent, we have

$\displaystyle E\left\\{U(X^{k}_{T})/U(e^{rT}x_{0})\right\\}$ $\displaystyle=$
$\displaystyle
E\left\\{\exp\left[(1-\gamma)(k(\mu-r)-\frac{1}{2}\sigma^{2}k^{2})T+(1-\gamma)\sigma
kW_{T}\right]\right\\}$ $\displaystyle=$
$\displaystyle\exp\left\\{(1-\gamma)k(\mu_{0}-r)T-\frac{1}{2}(1-\gamma)\sigma^{2}k^{2}T\right.$
$\displaystyle\ \ \ \ \ \
+\left.\frac{1}{2}(1-\gamma)^{2}\sigma^{2}k^{2}T+\frac{1}{2}(1-\gamma)^{2}k^{2}\sigma^{2}T^{2}\frac{1}{t_{0}}\right\\}$
$\displaystyle=$
$\displaystyle\exp\left\\{(1-\gamma)k(\mu_{0}-r)T+\frac{1}{2}(1-\gamma)^{2}\sigma^{2}k^{2}T\left[\frac{T}{t_{0}}-\frac{\gamma}{1-\gamma}\right]\right\\}.$

It follows that $EU(X^{k}_{T})\to\infty$ as $k\to\infty$. Thus, the original
problem is meaningless no matter the extra information is included or not. The
phenomenon happens when the investor has less relative risk aversion and the
ambiguity hidden in drift term is really too much that the extreme situation
where the drift term is far beyond the normal life weighted a lot in
probability which cause the expectation utility infinite. Moreover, if
$\frac{T}{t_{0}}-\frac{\gamma}{1-\gamma}<0$, then

$\displaystyle\ \ \ \sup_{k}E\\{U(X^{k}_{T})\\}$
$\displaystyle=U(e^{rT}x_{0})\exp\left\\{\frac{1}{2}\frac{T}{\frac{\gamma}{1-\gamma}-\frac{T}{t_{0}}}\frac{(\mu_{0}-r)^{2}}{\sigma^{2}}\right\\}$
$\displaystyle=U(e^{rT}x_{0})\exp\left\\{\gamma\cdot\frac{1}{2}a(0)(\mu_{0}-r)^{2}\right\\}.$

From which we see that $a(t)$ is a important function to weight the
informative clock and remaining time to the maturity. That is why with any
extra informative $a(0)$ is the same which will benefit the discussion of
value of information later. In this perspective, the calculation of $a(t)$ in
fact is not only by trick. (iii) If $U(x)=\ln(x)$, we use the ansatz:

$\displaystyle V(t,x,z)=U(e^{r(T-t)}x)+\psi(t,z).$

Similar to that of (i), we have

$\displaystyle\psi_{t}+(z-r)(\frac{\pi^{*}}{x})-\frac{1}{2}\sigma^{2}(\frac{\pi^{*}}{x})^{2}+\frac{1}{2}(-\frac{1}{\tau})^{\prime}\psi_{zz}=0,$
$\displaystyle\pi^{*}=x\frac{z-r}{\sigma^{2}}.$

And

$\displaystyle\psi(t,z)=\frac{1}{2}a(t)(z-r)^{2}+c(t),$ $\displaystyle
a(t)=\frac{1}{\sigma^{2}}(T-t),$ $\displaystyle
c(t)=\int_{t}^{T}\frac{1}{2}\sigma^{2}(-\frac{1}{\tau})^{\prime}(s)(a(s))ds=\frac{1}{2}\int_{t}^{T}\frac{\tau^{\prime}(s)(T-s)}{\tau^{2}(s)}ds.$
(0.35)

∎

## Appendix E. Proof of Proposition 3.5

###### Proof.

If we want to compare the value of $V_{m^{1}}$ and $V_{m^{2}}$, we only need
to compare the value of $V^{1}(0,x_{0},\mu_{0})$ and $V^{2}(0,x_{0},\mu_{0})$.
In fact, $a(0)$ is the value determined by $\tau(0)$ and $T$, which is
irrelevant to any extra information in both CARA case and CRRA case (see Proof
of Theorem 3.4(3)).The only comparison of value function will focus on the
value of $c(0)$. We make the comparison of $c(0)$ with different informative
clock by some transformations. In the case of CARA for example, based on
Theorem 3.4, we have

$\displaystyle c(0)$
$\displaystyle=\frac{1}{2\beta}\int_{0}^{T}\frac{\tau^{\prime}(s)(T-s)}{\tau(s)\left[\tau(s)+(T-s)\right]}ds$
$\displaystyle=\frac{1}{2\beta}\int_{0}^{T}\left[\frac{1}{\tau(s)}-\frac{1}{\tau(s)+T-s}\right]\tau^{\prime}(s)ds$
$\displaystyle=\frac{1}{2\beta}\int_{0}^{\tau(T)}\left[\frac{1}{u}-\frac{1}{u+T-\tau^{-1}(u)}\right]du,$

where the last equation is by the variable transformation $u=\tau(s)$. Then

$\displaystyle\ \ \ \ \ c^{1}(0)-c^{2}(0)$ (0.36)
$\displaystyle=\frac{1}{2\beta}\int_{\tau^{2}(T)}^{\tau^{1}(T)}\left[\frac{1}{u}-\frac{1}{u+T-(\tau^{1})^{-1}(u)}\right]du$
(0.37) $\displaystyle\ \ \ \
+\frac{1}{2\beta}\int_{0}^{\tau^{2}(T)}\left[\frac{1}{u+T-(\tau^{2})^{-1}(u)}-\frac{1}{u+T-(\tau^{1})^{-1}(u)}\right]du$
$\displaystyle\geq 0+0=0,$

where $c^{1}(0)$ and $c^{2}(0)$ are in the value function $V^{1}$ and $V^{2}$
respectively. The second term is deduced from the fact
$(\tau^{1})^{-1}(u)\leq(\tau^{2})^{-1}(u)$, where $\tau^{-1}$ is the reverse
function of $\tau$. Then

$\displaystyle\ \ \ \ V^{1}\left(0,x_{0},\mu_{0}\right)$
$\displaystyle=U\left(e^{rT}x_{0}+\frac{1}{2}a(0)(\mu_{0}-r)^{2}+c^{1}(0)\right)$
$\displaystyle\geq
U\left(e^{rT}x_{0}+\frac{1}{2}a(0)(\mu_{0}-r)^{2}+c^{2}(0)\right)$
$\displaystyle=V^{2}\left(0,x_{0},\mu_{0}\right).$

This is what we desire. Similarly, the conclusion can still be proven by the
same way as in CRRA case. When $\gamma\neq 1$.

$\displaystyle c(0)$
$\displaystyle=\frac{1}{2\gamma}\int_{0}^{T}\frac{\tau^{\prime}(s)\frac{1-\gamma}{\gamma}(T-s)}{\tau(s)\left[\tau(s)-\frac{1-\gamma}{\gamma}(T-s)\right]}ds$
$\displaystyle=\frac{1}{2\gamma}\int_{0}^{T}\left[\frac{1}{\tau(s)-\frac{1-\gamma}{\gamma}(T-s)}-\frac{1}{\tau(s)}\right]\tau^{\prime}(s)ds$
$\displaystyle=\frac{1}{2\gamma}\int_{0}^{\tau(T)}\left[\frac{1}{u-\frac{1-\gamma}{\gamma}[T-\tau^{-1}(u)]}-\frac{1}{u}\right]du.$

Similarly, If $\gamma<1$, we have $c^{1}(0)\geq c^{2}(0)$. But if $\gamma>1$,
we have $c^{1}(0)\leq c^{2}(0)$. But it doesn’t matter for the comparison of
value function if we notice that the utility functions $U(x)$ are positive or
negative with $\gamma<1$ or $\gamma>1$ . Hence, we obtain

$\displaystyle\ \ \ \ V^{1}\left(0,x_{0},\mu_{0}\right)$
$\displaystyle=U(e^{rT}x_{0})\exp\left\\{\gamma\cdot\left[\frac{1}{2}a(0)(\mu_{0}-r)^{2}+c^{1}(0)\right]\right\\}$
$\displaystyle\geq
U(e^{rT}x_{0})\exp\left\\{\gamma\cdot\left[\frac{1}{2}a(0)(\mu_{0}-r)^{2}+c^{2}(0)\right]\right\\}$
$\displaystyle=V^{2}\left(0,x_{0},\mu_{0}\right).$

At last, when $\gamma=1$,

$\displaystyle c(0)$
$\displaystyle=\frac{1}{2}\int_{0}^{T}\frac{\tau^{\prime}(s)(T-s)}{\tau^{2}(s)}ds$
$\displaystyle=\frac{1}{2}\int_{0}^{\tau(T)}\frac{T-\tau^{-1}(u)}{u^{2}}du$

It is easy to see the relation with $c^{1}(0)\geq c^{2}(0)$, we also have

$\displaystyle\ \ \ \ V^{1}\left(0,x_{0},\mu_{0}\right)$
$\displaystyle=rT+\ln(x_{0})+\frac{1}{2}a(0)(\mu_{0}-r)^{2}+c^{1}(0)$
$\displaystyle\geq rT+\ln(x_{0})+\frac{1}{2}a(0)(\mu_{0}-r)^{2}+c^{2}(0)$
$\displaystyle=V^{2}\left(0,x_{0},\mu_{0}\right).$

Thus, we end the proof. ∎

## Appendix F. Proof of Theorem 3.6

###### Proof.

In the case of CARA utility function, if we define

$\displaystyle\mathfrak{L}(t,\tau,\tau^{\prime}):=\frac{1}{2\beta}\frac{(T-t)\tau^{\prime}(t)}{\tau(t)[\tau(t)+T-t]}-\lambda[\tau^{\prime}(t)-1]^{2},$

then Problem (3.18) is equivalent to solving
$\sup\limits_{\tau}\int_{0}^{T}\mathfrak{L}(t,\tau(t),\tau^{\prime}(t))dt$. In
fact, based on the method of variations, we obtain that the optimal functional
has to satisfy the necessary condition as the Euler-Lagrange equation:

$\displaystyle\frac{d}{dt}\frac{\partial}{\partial\tau^{\prime}}\mathfrak{L}(t,\tau(t),\tau^{\prime}(t))=\frac{\partial}{\partial\tau}\mathfrak{L}(t,\tau(t),\tau^{\prime}(t)).$

It is simplified as

$\displaystyle\tau^{\prime\prime}(t)+\frac{1}{4\beta\lambda}\frac{1}{[\tau(t)+T-t]^{2}}=0.$
(0.38)

In the case of CRRA utility function with $\gamma\neq 1$, it will become a
little complex because the Euler-Lagrange method will become invalid due to
the nonlinear part of the integration. We use the method of duality to make
the integration linear. Noticing that $\frac{x_{0}}{C_{2}}\exp(x)$ has the
duality function $y-y\ln(\frac{C_{2}y}{x_{0}})$, we have

$\displaystyle\frac{x_{0}}{C_{2}}\exp(x)=\sup_{y>0}\left\\{y-y\ln(\frac{C_{2}y}{x_{0}})+xy\right\\}.$
(0.39)

By using Eq.(3.16) and Eq.(0.39), the objective function $Net(\tau)$ is

$\displaystyle Net(\tau)$
$\displaystyle=\sup_{y>0}\left\\{y-y\ln(\frac{C_{2}y}{x_{0}})-x_{0}\right.$
$\displaystyle\left.+y\left[\int_{0}^{T}\frac{1}{2(1-\gamma)}\frac{\tau^{\prime}(t)\frac{1-\gamma}{\gamma}(T-t)}{\tau(t)[\tau(t)-\frac{1-\gamma}{\gamma}(T-t)]}dt\right]\right.$
$\displaystyle\left.-\int_{0}^{T}\lambda[\tau^{\prime}(t)-1]^{2}dt\right\\}.$

If we define the

$\displaystyle\ \ \mathfrak{L}^{y}(t,\tau,\tau^{\prime})$
$\displaystyle:=\frac{y}{2(1-\gamma)}\frac{\tau^{\prime}(t)\frac{1-\gamma}{\gamma}(T-t)}{\tau(t)[\tau(t)-\frac{1-\gamma}{\gamma}(T-t)]}-\lambda[\tau^{\prime}(t)-1]^{2},$

then

$\displaystyle
Net(\tau)=\sup_{y>0}\left\\{y-y\ln(\frac{C_{2}y}{x_{0}})-x_{0}+\int_{0}^{T}\mathfrak{L}^{y}(t,\tau,\tau^{\prime})dt\right\\}.$

As such,

$\displaystyle\ \ \ \ \sup_{\tau}Net(\tau)$
$\displaystyle=\sup_{\tau}\sup_{y>0}\left\\{y-y\ln(\frac{C_{2}y}{x_{0}})-x_{0}+\int_{0}^{T}\mathfrak{L}^{y}(t,\tau,\tau^{\prime})dt\right\\}$
$\displaystyle=\sup_{y>0}\sup_{\tau}\left\\{y-y\ln(\frac{C_{2}y}{x_{0}})-x_{0}+\int_{0}^{T}\mathfrak{L}^{y}(t,\tau,\tau^{\prime})dt\right\\}.$

Then we fix $y>0$ and solve

$\displaystyle\sup_{\tau}\left\\{y-y\ln(\frac{C_{2}y}{x_{0}})-x_{0}+\int_{0}^{T}\mathfrak{L}^{y}(t,\tau,\tau^{\prime})dt\right\\}$

by Euler-Lagrange equation with

$\displaystyle\frac{d}{dt}\frac{\partial}{\partial\tau^{\prime}}\mathfrak{L}^{y}(t,\tau(t),\tau^{\prime}(t))=\frac{\partial}{\partial\tau}\mathfrak{L}^{y}(t,\tau(t),\tau^{\prime}(t)).$

It follows that

$\displaystyle\tau^{\prime\prime}(t)+\frac{y}{4\gamma\lambda}\frac{1}{[\tau(t)-\frac{1-\gamma}{\gamma}(T-t)]^{2}}=0.$
(0.40)

Moreover, for $U(x)=\ln(x)$, as the calculation is similar and the outcome
coincide (3.20) if we treat $\gamma=1$, we do not need to carefully
distinguish the case whether $\gamma$ equals to one. ∎

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