数理解析研究所講究録 1443 巻 2005 年 144-150

144

ON THE VERIFICATION THEOREM OF CONTINUOUS-TIME
OPTIMAL PORTFOLIO PROBLEMS WITH STOCHASTIC
MARKET PRICE OF RISKTOSHIKI HONDA( 本多 俊毅) AND SHOJI KAMIMURA( 上村 昌司)
ABSTRACT. In this paper, we study a continuous-tim portfolio optimization
problem when the market price of risk is driven by linear Gaussian processes. We show sufficient conditions to verify that a solution derived fromthe Hamilton-Jacobi-Bellman equation is in fact an optimal solution to the$\mathrm{e}$portfolio selection problem.

1. INTRODUCTION

In this paper, we study a continuous-time portfolio optimization problem in Kim
and Omberg [7]. We show sufficient conditions to verify that a solution derived fromthe Hamilton-Jacobi-Bellman (HJB) equation is in fact an optim al solution to theportfolio selection problem.
Since Merton’s seminal work (Merton [10], [11], [12]), many studies have been
done on continuous-time portfolio optimization problems. In particular, there has
been increasing interest in finding an optimal portfolio strategy when investment
opportunities are stochastic, because many empirical works conclude that investment opportunities are time varying. In this paper, we study a continuous-time
power-utility maximization problem when the market price of risk is driven by linear Gaussian processes. Such a problem has been studied by many authors. See,for example, Kim and Omberg [7], Liu [9], Wachter [14], Bielecki and Pliska [2],Bielecki et al. [3], and Nagai [13]. In this paper, we concentrate on the Kim-Omberg
model [7], where the market price of risk is driven by an Ornstein-Uhlenbeck pro-cess.

There are two main approaches to solving the continuous-time portfolio optimization problem. One is the stochastic control approach and the other is themartingale approach. Since the market is incomplete in our model, the martingale approach is not applied directly. In this paper, we thus employ the former
approach. For an example of the latter approach, see Karatzas and Schreve [6]. In
the stochastic control approach, an optimal solution is conjectured by guessing asolution to the HJB equation. It is necessary to verify that the conjectured solution
is in fact a solution to the original problem. The solution conjectured from the HJB
equation could be an incorrect solution to the original problem. However, as Korn
and Kraft [8] pointed out, the verification is often skipped since it is mathematically demanding. For example, Kim and Omberg [7] examined the finiteness ofthe conjectured value function very carefully, but they did not provide verification
Key words and phrases. Optimal portfolios, stochastic market price of risk, verification

theorem

145
TOSHIKI HONDA(tJ Sit) AND SHOJI KAMIMU$\mathrm{R}\mathrm{A}(f:*\not\subset\ovalbox{\tt\small REJECT}\overline{\mathrm{Q}}\neg)$conditions. Therefore, in this paper, we will give sufficient conditions to verify thatthe conjectured solution is in fact the solution to the original problem.
2. FORMULATION OF THE PROBLEM

We fix a complete probability space $(\Omega, F, P)$ on which a two-dimensional stanis defined, and we also fix a time interval
dard Brownian motion$[0, T]$ . Let) be the augmentation of the filtration $\mathcal{F}^{B}(t):=\sigma(B(s);0<s<t)$ ,$B=(B^{1}, B^{2})^{\mathrm{T}}$$\mathrm{T}\{\mathrm{t}$$0<t<T$ .

Let $X$ be an Ornstein-Uhlenbeck process:(1)$dX(t)=\lambda(\overline{X}-X(t))dt+\sigma_{X}(\rho dB^{1}(t)+\sqrt{1-\rho^{2}}dB^{2}(t))$$X(0)=x_{0}\in$ R.

R. We call $X$ a state process, because it
determines an investment opportunity set in our portfolio problem.
of theThere is one riskless asset and one risky asset. Suppose the priceriskless asset satisfies$S_{0}(0)=1$ ,$dS_{0}(t)=rSo(t)dt$ ,where $r\geq 0$ is constant. The risky asset price satisfies the stochastic differential
equation$dS(t)=S(t)\mu(X(t))dt$ $+S(t)\sigma dB^{1}(t)$ , $S(\mathrm{O})=s>0$ ,(2)$\rho\in[-1,1]$, A $>0$ , $\sigma_{X}>0$ , and$\overline{X}\in$$S_{0}$

$S$

where

$\mu:\mathbb{R}arrow \mathbb{R}$satisfies ($\mu(x)$

-

$r$

}

$/\sigma=x$

for $x\in$ R. Then (2) can be written by$dS(t)=S(t)(r+\sigma X(t))dt+S(t)\sigma dB^{1}(t)$.

We consider an investor who can divide his wealth between the riskless asset andbe a set of $F(t)$ -progressively measurable processes
the risky assets. Let
such that$\mathcal{L}^{2}(t_{0}, t_{1})$$\phi:\Omega\rangle\zeta[t_{0}, t_{1}]arrow \mathbb{R}$(3)$P( \int_{t_{\mathit{0}}}^{t_{1}}\phi(t)^{2}dt<\infty)=1$.

as a fraction
a portfolio strategy. We regard
We call an element ofprocess
wealthinvestor’s
.
The
at
time
risky
asset
of the wealth invested in the$W^{\phi}(0)=w_{0}>0$
and
is given bycorresponding to$\phi_{i}(t)$$\mathcal{L}^{2}(t_{0},t_{1})$$t$$\phi\in \mathcal{L}^{2}(0, T)$$W^{\phi}$$dW(t)=W(t)[\phi(t)(\mu(X(t))-r)+r]dt$ $+W(t)\phi(t)\sigma dB^{1}(t)$(4)$=W(t)[\phi(t)\sigma X(t)+r]dt+W(t)\phi(t)\sigma dB^{1}(t)$.

The market is incomplete in the sense that there are some random processes thatare not replicated by the self-financing portfolio strategy .
The investor maximizes the expected utility of his wealth at terminal date . We
assume that the investor has a power utility function with a relative risk aversion
coefficient :$\phi$$T$$\gamma$(5)$E[ \frac{W^{\phi}(T)^{1-\gamma}}{1-\gamma}]$$\phi\in A_{\gamma}(0,T)\max$.

denotes the set of admissible portfolio strategies defined as follows. A
Hereifstochastic process is said to be an admissible portfolio strategy on, when $0<\gamma$ $<1$(i)$A_{\gamma}$$[t_{0}, t_{1}]$$\phi$$\phi\in \mathcal{L}^{2}(t_{0}, t_{1})$148
OPTIMAL PORTFOLIO PROBLEMS WITH STOCHASTIC MARKET PRICE OF RISK

satisfying the linear growth conditionl,: $[0, T]$$\gamma>1$, when.
on. The choice
is denoted by
The set of all admissible strategies onof our set of portfolio strategies seems to be restrictive. The reason why such a
restrictive definition is needed will be explained in the end of Section 4.
Since the market is incomplete, there is no unique equivalent martingale measure,and we cannot apply the so-called martingale approach directly. It is thus common
to apply the dynamic programming approach using the HJB equation. Let(ii) for some function$\mathrm{x}\mathbb{R}arrow \mathbb{R}$$\tilde{\phi}$$\mathrm{W}(\mathrm{t})=\overline{\phi}(t_{\}}\mathrm{X}(\mathrm{t})$$[t_{0}, t_{1}]$$A_{\gamma}(t_{0}, t_{1})$$[t_{0}, t_{1}]$$J(t, w, x; \phi)=E^{t,w,x}[\frac{W^{\phi}(T)^{1-\gamma}}{1-\gamma}]$,

Here and in the sequel, we use the notation $E^{t,w,x}[\cdot]=E[\cdot|W(t)=w, X(t)=x]$ .
Let$Q=[\mathrm{O}, T)\mathrm{x}$We then define$(0, \infty)\mathrm{x}$$\mathbb{R}$.

by$V:Qarrow \mathbb{R}$$V(t, w_{7}x)= \sup_{\gamma(t\tau)},J(t, w,x;\phi)\emptyset\in A^{\cdot}$

The functionlem ( is,$V$

is called a value function. The HJB equation related to the prob-$5^{\backslash }$(6)$\sup D^{\phi}G(t, w, x)=0$$\phi\in 1\mathrm{R}$with the boundary condition(7)$G(T, w, x)= \frac{w^{1-\gamma}}{1-\gamma}$

,

where
$D^{\phi}G(t, w, x)=G_{t}+w(\phi\sigma x+r)G_{w}+\lambda(\overline{X}-x)G_{x}$$+ \frac{1}{2}w^{2}\phi^{2}\sigma^{2}G_{ww}+\frac{1}{2}\sigma_{X}^{2}G_{xx}+\sigma_{X}w\phi\sigma\rho G_{wx}$$G$.

It is well-known from Kim and Omberg [7], Liu [9], and others that the function
is separable and has the following form:(8)$G(t, w,x)= \frac{w^{1-\gamma}}{1-\gamma}f(t,x)$

,

where
$f(t, x)= \exp\{a(t)+b(t)x+\frac{1}{2}c(t)x^{2}\}$

with the boundary conditions $a(T)=b(T)=c(T)$ $=0$ .
It follows from the first order condition for (6) that the candidate optimal portfolio strategy is given by.

(9)$\phi^{*}(t)=\frac{1}{\gamma}\frac{X(t)}{\sigma}+\frac{1}{\gamma}\frac{\rho\sigma_{X}}{\sigma}(b(t)+c(t)X(t))$

1A function$h:[0, T]\mathrm{x}$i said to satisfy the linear growth condition if $|h(t, x)|\leq$is the Euclidean norm$\mathrm{R}^{K}arrow \mathbb{R}^{L}$$k(1+|x|)$ for some $k>0$ , where$\mathrm{s}$$|\cdot|$

147
TOSHIKI HONDA$( 8 \ovalbox{\tt\small REJECT} \mathrm{a}\mathrm{e})$AND SHOJI KAMIMURA(4i\gamma$\ovalbox{\tt\small REJECT}\overline{\mathrm{Q}}\neg\rangle$

Substituting this conjectured solution into the HJB equation, we obtain the differas follows:, and,ential equation for$a(\cdot)$$b(\cdot)$$c(\cdot)$(10)$\dot{c}(t)=-\sigma_{X}^{2}(\frac{1-\gamma}{\gamma}\rho^{2}+1)c(t)^{2}-2(\frac{1-\gamma}{\gamma}\sigma x\rho-\lambda)c(t)-\frac{1-\gamma}{\gamma}$(11)$\dot{b}(t)=-\sigma_{X}^{2}(\frac{1-\gamma}{\gamma}\rho^{2}+1)b(t)c(t)-(\frac{1-\gamma}{\gamma}\sigma_{X}\rho-\lambda)b(t)-\lambda\overline{X}c(t)$(12)$\dot{a}(t)=-\frac{1}{2}\sigma_{X}^{2}(\frac{1-\gamma}{\gamma}\rho^{2}+1)b(t)^{2}-\frac{1}{2}\sigma_{X}^{2}c(t)-\lambda\overline{X}b(t)-(1-\gamma)r$.

The first term of (9) is the usual mean-variance portfolio in a continuous-timemodel. The second term is a so-called hedging portfolio, which is held by investors
in order to hedge against an unfavorable shift in the state variables. Both terms
turned out to be linear with respect to state variable $X$ . In general, it is difficult tosolve optimal portfolio problems when the market is incomplete. Portfolio (9) is aninteresting exception that solves the HJB equation when the market is incomplete.
In order to complete the whole story, we need to verify that $G=V$ and thatis indeed a solution to (5). In the next
the candidate optimal portfolio strategy
section, we will prove a verification theorem.
$\phi^{*}$

3. VERIFICATION THEOREM
Verification theorems, such as those in Fleming and Rishel [4] and Fleming andSoner [5], ensure that the solution to the HJB equation coincides with the valuefunction and the candidate portfolio is indeed the optimal portfolio strategy.
ever, since the wealth process (4) and the conjectured value function (8) do notsatisfy the usual assum ptions, such as the Lipschitz condition and the polynomial
growth condition, we cannot apply standard verification theorems directly to ourmodel. We therefore use the method employed in Nagai [13] and the references$\mathrm{H}\mathrm{o}\mathrm{w}arrow$

therein.
Theorem 1 (Verification Theorem). Assume that the solution to (10) exists on$\phi^{*}(t, X(t))$ ,. Then, the function defined by (8) satisfies $G=V$ . Further.
defined by (9), is an optimal portfolio strategy.
The following lemma is crucial to the proof of the verification theorem. This
result is proven essentially in Bensoussan [1, Lemma 4.1.1]. For a stochastic process, define$G$$[0_{2}T]$$g$$\xi(t,g)$ $:= \exp\{\oint_{0}^{t}g(u)^{\mathrm{T}}dB(u)-\frac{1}{2}\int_{0}^{t}|g(u)|^{2}du\}$Lemma 2. Let $g(t):=\tilde{g}(t, X(t))$ , where
growth condition. Then$\tilde{g}:[0, T]$$E[\xi(T, g)]=1$

and$\xi_{s}(t,g)$$\mathrm{x}\mathbb{R}arrow \mathbb{R}^{2}$$:= \frac{\xi(t,g)}{\xi(s,g)}$.

satisfies the linear.

Using this lemma, we can show the theorem.

Proof of Theorem 1. Using It6’s formula, we obtain(13)$dG(t, W^{\phi}(t),$ $X(t))$$=D^{\phi}G(t, W^{\phi}(t),X(t))dt+G(t, W^{\phi}(t),X(t))g^{\phi}(t)^{\mathrm{T}}dB(t)$148
OPTIMAL PORTFOLIO PROBLEMS WITH STOCHASTIC MARKET PRICE OF RISK

where$g^{\phi}(t):=(1-\gamma)\phi(t)(\sigma,0)^{\mathrm{T}}+\mathrm{c}\mathrm{r}\mathrm{x}(\mathrm{b}(\mathrm{t})+c(t)X(t))$

for all $t\in[0, T]$ and. Let $(t, w, x)$(6) andequationis the solution to the HJB
that$\phi\in A_{\gamma}(t, T)$

$G$

(14)

$G(T, W^{\phi^{*}}(T)$: $X(T))$$(\rho,$$\sqrt{1-\rho^{2}})^{\mathrm{T}}$be fixed. Sinceinis the maximizer (6), it follows$\in[0, T)\mathrm{x}[0, \infty)\mathrm{x}$$\phi^{*}$$\mathbb{R}$$=G(t,w, x)\xi_{t}(T,g^{\phi^{*}})$

Using (9), we have$g^{\phi^{*}}(t)=b(t)(( \frac{1-\gamma}{\gamma}+1)\rho\sigma_{X},$$\sqrt{1-\rho^{2}}\sigma_{X})^{\mathrm{T}}$$+c(t)X(t)( \frac{1-\gamma}{\gamma}+(\frac{1-\gamma}{\gamma}+1)\rho\sigma_{X},$

Then, it follows from Lemma 2 that the processfrom (14), we have(15)$\xi(t, g^{\phi^{*}})$$E^{t,w,x}[ \frac{W^{\phi^{*}}(T)^{1-\gamma}}{1-\gamma}]=E^{t,w,x}[G(T, W^{\phi^{*}}(T),$$\sqrt{1-\rho^{2}}\sigma_{X})^{\mathrm{T}}$is a martingale. Hence,$X(T))]=G(t,w,x)$ .

On the other hand, it follows from Lemma 2 and the definition of admissible
portfolio strategies that the process$H_{t}(u):=G(t,w,x)\xi_{t}(u,g^{\phi})$

is a supermartingale for all(16)$\phi\in A_{\gamma}(t, T)$,$t\leq u\leq T$. Then, using (6) and (13), we obtain$E^{t,w,x}[ \frac{W^{\phi}(T)^{1-\gamma}}{1-\gamma}]=E^{t,w,x}[G(T, W^{\phi}(T), X(T))]\leq E^{t,w,x}[H_{t}(T)]$$\leq G(t, w,x)$

for all.
Combining (15) and (16), we see that $G=V$ and $\phi^{*}(t, X(t))$ is an optimal
portfolio strategy.
$\phi\in A_{\gamma}(t, T)$$\square$Prom Theorem 1, we see that if a solution to the Riccati equation (10) exist,
then the conjectured function (8) is in fact the value function. In the following, we
can concentrate on if a solutions to the Riccati equation (10) exists, We howeveremphasize that the choice of portfolio trading strategies set plays an important role
here. A key property is if $\{H_{t}(u)\}$ is a martingale forand a supermartingale
for all. When $\gamma>1$ ,is a martingale for all
because of Lemma 2. Thus $\{H_{t}(u)\}$ is a (super)martingale for all
Ay $(t,T)$ .
However, for a broader set of portfolio strategies, say$\{H_{t}(u)\}$,may not be
a supermartingale even if
is supermartingale, since $G(t, w, x)$ is negative. This is why we restrict the set of admissible portfolio strategies when
when$\gamma>1$ .
Further, it is easy to see that another possible definition of admissible
portfolio strategies foris thatandis a martingale.
$\phi^{*}$$\phi\in A_{\gamma}(t,T)$$\{\xi_{t}(u,g^{\phi})\}$$\phi\in A_{\gamma}(t,T)$$\phi\in$$\phi\in \mathcal{L}^{2}$$\{\xi_{t}(u,g^{\phi})\}$$\gamma>1$$\gamma>1$$\phi\in \mathcal{L}^{2}$$\{\xi(t,g^{\phi})\}$

148
TOSHIKI HONDA($\mathrm{B}!$Si) AND SHOJI KAMIMURA(JJ$\ovalbox{\tt\small REJECT}$$=-\downarrow$)

4. THE RICCATI EQUATION

It follows from Theorem 1 that the solution to the Riccati equation (10), if itexists, give us the solution to the original problem. In this section, we discuss asufficient condition for the existence of the solution to the Riccati equation (10),The method for solving (10) is standard. See Kim and Omberg [7] for details.
Let$C_{0}= \frac{1-\gamma}{\gamma}$

(17)

,

$C_{1}=2( \frac{1-\gamma}{\gamma}\sigma_{X}\rho-\lambda)$,$C_{2}= \sigma_{X}^{2}(\frac{1-\gamma}{\gamma}\rho^{2}+1)$$q=C_{1}^{2}-4C_{0}C_{2}=4 \lambda^{2}\{1-\frac{1-\gamma}{\gamma}(\frac{\sigma_{X}^{2}}{\lambda^{2}}+\frac{2\rho\sigma_{X}}{\lambda})\}$

and$\sqrt{q}$$\eta=\{$,$\sqrt{-q}$,,$q\geq 0$,$q<0$ .

Then, the solution to (10) is given by$(q>0)$$\frac{2C_{0}(1-e^{-\eta(T-t)})}{2\eta-(C_{1}+\eta)(1-e^{-\eta(T-t)})}$$(q=0_{7}C_{1}\neq 0)$$- \frac{1}{C_{2}(T-t-\frac{2}{c_{1}})}-\frac{C_{1}}{2C_{2}}$(18)$c(t)=\{$$\frac{1-\gamma}{\gamma}(T-t)$$(q=0, C_{1}=0)$$\frac{\eta}{2C_{2}}\tan(\frac{\eta}{2}(T-t)+\tan^{-1}(\frac{C_{1}}{\eta}))-\frac{C_{1}}{2C_{2}}$$(q<0)$ .

Note that, by (11) and (12), $\sup_{t\in[0,T]}|c(t)|<\infty$ implies that.
andWe can easily show that if $\gamma>1$ , then the solution to (10) always exists on $[0, T]$
since $\gamma>1$ implies $q>0$ . Then we can obtain the following result.

$\sup_{\mathrm{t}\in[0,T]}|b(t)|<\infty$$\sup_{t\in \mathrm{I}0,T]}|a(t)|<\infty$

Proposition 3. If$\gamma>1$, then the solution to (5) exists.

If $0<\gamma<1$ , the solution to (10) may not exist. If $q<0$ and(19)$0< \frac{2}{\eta}(\frac{\pi}{2}+\tan^{-1}(\frac{C_{1}}{\eta}))<T$,

then (18) takes infinite value at some point on $[0, T)$ . Therefore, there is no solution
to (10) on $[0, T]$ in this case. However, we can obtain the following result.
Proposition 4. If $0<\gamma<1$ and q $>0$ , then the solution to (5) exists.

5. CONCLUSION
In this paper, we have derived sufficient conditions that confirm the conjectured
solution in Kim and Omberg [7] to be in fact a solution to the original problem.
We have shown that if the Riccati equation related to the HJB equation has asolution, then the conjectured solution is in fact a solution to the original problem.
If $0<\gamma<1$ , th en the related Riccati equation does not always have a solution. On
the other hand, if $\gamma>1$ , then the related Riccati equation always have a solution.
However, portfolio strategies are chosen from a rather restrictive set of stochastic
processes

150
OPTIMAL PORTFOLIO PROBLEMS WITH STOCHASTIC MARKET PRICE OF RISKREFERENCES[1] A. Bensoussan, Stochastic control of partially observable systems, Cambridge University
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(T. Honda) GRADUATE SChOOl OF INTERNATIONAL CORPORATE STRATEGY, HITOTSUBASHI UN1, TOKYO 101-S439, JAPAN (101-8439, 東京都千代田区橋 2-1-2, 一橋大学大学院国際企業戦略研究科)$E$ -mail addre6i,.: thonda@ics.hit-u.ac jp

VERSITY, 2-1-2 HITOTSUBASHI,$\mathrm{C}\mathrm{H}\mathrm{I}\mathrm{Y}\mathrm{O}\mathrm{D}\mathrm{A}\sim \mathrm{K}\mathrm{U}$$-\backslash \backslash J$(S. Kamimura) GRADUATE School OF INTERNATIONAL CORPORATE STRATEGY, HITOTSUBASHI
UNIVERSITY, 2-1-2 HITOTSUBASHI, CHIYODA-KU, Tokyo 101-8439, JAPAN (101-8439, 東京都千代田区一 橋 2-1-2, 一橋大学大学院国際企業戦略研究科)$E$-mail address: kamimura@ics hit-u. ac. jP$\backslash \backslash j$