数理解析研究所講究録 1215 巻 2001 年 127-142

127

An Asymptotic Expansion Scheme for the Optimal
Portfolio for Investment*

Akihiko Takahashi and Nakahiro Yoshida
Graduate School of Mathematical Sciences, University of Tokyo,
3-8-1 Komaba, MegurO-ku, Tokyo 153, Japan.

January 29, 2001

Abstract
We shall propose anew computational scheme for the evaluation of the
optimal portfolio for investment. Our method is based on an extensionof the asymptotic expansion approach which has been recently developed
for the pricing problems of the contingent claims’ analysis by KunitomOTakahashi(1992,1995,1998), Yoshida(1992), Takahashi $(1995,1999)$ and others. In particular, we will explicitly derive the formula of the optimal portfolio associated with maximizing utility from terminal wealth for apower
utility function in afinancial market with Markovian coefficients.

’We thank N.Kunitomo and S.Kusuoka for some discussions on the related issues andtheir helpful comments on the previous versions

128
1Introduction
We shall propose anew computational scheme for the evaluation of the
optimal portfolios for investment. Our method is based on the asymp
totic expansion approach, aunified method of efficient computation justified
by Malliavin-Watanabe(1987) theory, which has been recently developed
for the pricing problems of the contingent claims’ analysis by KunitomOTakahashi(1992,1995,1998), Yoshida(1992), Takahashi $(1995,1999)$ , Kunitomo and Kim(1999), Sorensen and Yoshida(1998) and Kashiwakura andYoshida(2001). They have developed the method through deriving formulasfor practical examples such as average options, basket options, and options
with stochastic volatility and with stochastic interest rates in aMarkovian
setting, as well as bond options(swaptions), average options on interest rates,
and average options on foreign exchange rates with stochastic interest rates
in the Heath-Jarrow-Morton(1992) framework. In this paper, we extend themethod for portfolio problems. In particular, we will explicitly derive the formula of the optimal portfolio associated with maximizing utility from terminal wealth for apower utility function in afinancial market with Markovian
coefficients. In general, it is quite difficult to compute an optimal portfolio
explicitly when the investment oppotunity is stochastic in amultiperiod setting. The stochastic control approach initiated by Merton $(1969,1971)$ gives
asolution in terms of the derivatives of the value function: While the s0lution can be evaluated numerically based on the Hamilton-Jacobi-Bellman
equation, the implementation is not easy especially for the case of multiple
assets. In the martingale approach initiated by Karazas et alQ987) andCox and Huang(1989), Ocone and Karatzas(1991) proposed the representation of optimal portfolios by utilizing the Clark formula. Although theirrepresentation formulas were derived in general setting, explicit evaluation
was obtained only for logarithmic utility functions or afinancial market withdeterministic coefficients, which were already known without their formulas.
Starting with the Clark formula, we will present an explicit expression forthe optimal portfolio in afinancial market with Markovian coefficients which
is more concrete but practically sufficient setting. Moreover, our method
can be easily extended to the optimal portfolios associated with maximizing utility from both consumption and terminal wealth, and to the hedging
portfolios associated with contingent claims. The organization of this paper
is as follows. In Section 2we explain the problem of the optimal portfolio forinvestment, and in Section 3we restate our problem in aMarkovian setting.
In Section 4we derive the second order scheme explicitly for acase of power
utility function through explaining the asymptotic expansion approach, andin the appendix we show the result of the third order scheme.

2The Representation of Optimal Portfolio
We will breifly describe the financial market, and introduce the representation of the optimal portfolio for investment derived by Ocone and Karatazas
$($

129
Let
probability space, $w(t)=(w^{\alpha}(t), \cdots, w^{r}(t))^{*}$ for $0\leq t\leq T$ ,-valued Brownian motion defined on
andfor $0\leq t\leq T$$P_{t}^{p}=\sigma(w(s);0\leq
s\leq t)$ . Here,-augmentation of the natural filtration,
we use the notation of as the transpose of .
, $i=1$ , , anddenote the prices at time $t\in[0,T]$ of the risky asset and of the risless asset
respectively. The prices are assumed to follow the stochastic processes: For$t\in[0,T]$ ,$(\Omega, \mathcal{F}, P)$$R^{r}$$(\Omega, \mathcal{F}, P)$$\{\mathcal{F}_{t}\}$$\mathrm{P}$$x^{*}$$x$$S_{\dot{1}}(t)$$\cdots$$S_{0}(t)$

$r$

$i$

$dS_{i}$$=$$S(t)[b_{\dot{1}}(t)dt+ \sum_{j=1}^{r}\sigma j(t)dw_{j}(t)];S_{\dot{l}}(0)=s:i=1$

$dS0$

$=$

$r(t)S_{0}(t)dt;S_{0}(0)=1$,$\cdots$,$r$

where we suppose that $r(t),b:(t)$ and, $i,j=1$ , , are bounded
and progressively measurable with respect to. We also assume thenondegeneracy condition; for thematrixthere
exists areal number $>0$ such that$\sigma_{j}(t)$$r$$\cdots$$\{\mathcal{F}_{t}\}$$r\mathrm{x}$$\sigma(t)\equiv\{\sigma ij(t)\}_{1\leq i,j\leq r}$$r$$\epsilon$$\xi^{*}\sigma(t,\omega)\sigma(t,\omega)^{*}\xi\geq\epsilon|\xi|^{2}$;$\forall\xi\in \mathrm{R}^{r}$,$(t,\omega)\in[0, T]\mathrm{x}\Omega$.

Then, the stochastic process of an investor’s wealth denoted by $W(t)$ areexpressed as$dW(t)=[r(t)W(t)-c(t)]dt+\pi(t)^{*}[(b(t)-r(t)1)dt+\sigma(t)dw(t)]$where $W(0)=W>0$ is the initial capital, 1denotes the vector inwith all elements equal to 1, $c(t)$ denotes the consumption rate, and $\pi(t)=$satisfy the integrability
denotes the portfolio. $c(t)$ andcondition;$\mathrm{R}^{r}$$\pi(t)$$\{\pi_{i}(t)\}_{i=1,\cdots,r}^{*}$$\int_{0}^{T}\{|\pi(t)|^{2}+c(t)\}dt<\infty a.s$.

Next, let $A(W)$ denote the set of stochastic processes$W(t)\geq 0$ for all $t\in[0, T]$ given $W(0)=W$ . We call
$W$ if $(\pi, c)\in A(W)$ .

which generate
is admissible for$(\pi, c)$$(\pi, c)$

The problem of maximizing utility from terminal wealth is formulated as,follows: With$c\equiv 0$$( \pi,c)\in A(W)\sup E[U(W(T))]$

denotes the expecdenotes autility function, andwhere :
tation operator under . We assume is astrictly increasing, strictly con, $U’(0+)\equiv$
cave function of class , with
.
and
$U$

$E[\cdot]$$(0, \infty)arrow_{1}\mathrm{R}$

$U$

$P$

$\mathrm{C}^{2}$$\lim_{c\downarrow 0}U’(c)=\infty$$U(0+) \equiv\lim_{\mathrm{c}\downarrow 0}U(c)\in[-\infty, \infty)$$U^{l}( \infty)\equiv\lim_{\mathrm{c}arrow\infty}U’(c)=0$, $t\in[0, T]$ an
Let the market price of risk
measurable bounded process defined by$\theta(t)$$\mathrm{w}\{\mathrm{t})=\sigma(t)^{-1}[b(t)-r(t)1]$$R^{r}$.

-valued progressively

130Then, the martingale measure denoted byfor all$A\in \mathcal{F}_{T}$$P_{0}$

whereis defined: Po{A)$Z(t)= \exp(-\int_{0}^{t}\theta(s)^{*}dw(s)-\frac{1}{2}\int_{0}^{t}|\theta(s)|^{2}ds)$$=\mathit{1}$; $0\leq t\leq$’

We note that under $P_{0}w_{0}(t) \equiv w(t)+\int_{0}^{t}\theta(u)du$ , $0\leq t\leq T$ is a
Brownian motion.
Regarding the problem of maximizing utility from terminal wealthknown that the optimal wealth level of terminal wealth given by, and that the value function $V(W):= \sup_{(\pi,c)\in A(w}$can be computed as $V(W)=G(\mathcal{Y}(W))$ , where $G(y):=E[U(I(yH_{0}$$y<\infty$ . (See for instance Theorem 7.6 in Karatzas and Shreve(1998
Here, $I\in C^{1}((0, \infty);(0, \infty))$ denotes the inverse of, and
the inverse of the continuous, decreasing function:$I(\mathcal{Y}(W)H\mathrm{o}(T))$$U’(\cdot)$$\mathcal{Y}($$\mathcal{X}(y)=\mathrm{E}_{0}[\beta(T)I(yH_{0}(T))]=\mathrm{E}[H_{0}(T)I(yH_{0}(T))];0<y<$which we assume maps
, where $\beta(t)\equiv 1/S_{0}(t)$intodenotes the state price density at , anddenotes thetion operator under .
$(0, \infty)$$(0, \infty)$$\beta(t)Z(t)$$t$$E_{0}[\cdot]$$P_{0}$

Ocone and Karazas(91) gives the following theorem by utilizing
formula regarding the problem of the optimal portfolio for investr
ciated with mctirnizing utility from teminal wealth.
Theorem 1Suppose that
$I(y)+|I’(y)|\leq K(y^{\alpha}+y^{-\beta})$, $0<y<\infty$

holds for some real, positive, constants , , $K$ .
Then the optimal portfolio admits the repesenatation$\alpha$$\pi^{*}(t)\sigma(t)$$\beta$:$=$$- \frac{1}{\beta(t)}\{\theta^{*}(t)\mathrm{E}_{0}[\beta(T)\mathcal{Y}(W)H_{0}(T)I’(\mathcal{Y}(W)H_{0}(T))|$$+$$\mathrm{E}_{0}[\beta(T)\phi’(\mathcal{Y}(W)H_{0}(T))($ $\int_{t}^{T}D_{t}r(u)du+\sum_{\alpha=1}^{r}\int_{t}^{\mathrm{I}}$where $\phi(y)\equiv yI(y),0<y<\infty$ , and $D_{t}r(u)$ and.
deonote the Malliavin derivatives of $r(u)$ and$D_{t}\theta_{\alpha}(u)$,$\alpha=$$\theta_{\alpha}(u)$Here we suppose that$\theta$and r satisfy the following conditions:-valued progressivlely measurable process is bounded;$[0, T]$ $r(s, \cdot)\in D_{1,1}$ where
denotes the Sobolev space$(p, s)=(1,1)$ , $(s,\omega)arrow Dr(s,\omega)\in(L^{2}([0,T]))’$ admits pro
measurable version, and$r$$\mathrm{R}$$\bullet$$fo_{\acute{1}}$$D_{1,1}$$||r||_{1,1}^{a}= \mathrm{E}[(\int_{0}^{T}|r(s)|^{2}ds)\frac{1}{2}+(\int_{0}^{T}||Dr(s)||^{2}ds)\frac{1}{2}]$where$||\cdot||$

denotes the $L^{2}([0, T])$ norm, $and||Dr(s)||^{2}= \sum_{=1}^{r}$

131$\bullet$-valued progressively measurable process is bounded; for $a.e$ . $s\in$$[0, T]$, $(s,\omega)arrow D\theta(s,\omega)\in(L^{2}([0,T]))^{r^{2}}$ admits
progressively measurable version, and$\mathrm{R}^{r}$$\theta$$\theta(s, \cdot)\in(\mathrm{D}_{1,1})^{r}$$a$$|| \theta||_{1,1}^{a}=\mathrm{E}[(\int_{0}^{T}|\theta(s)|^{2}ds)^{1}\tau+(\int_{0}^{T}||D\theta(s)||^{2}ds)^{1}\tau]<\infty$where$\bullet$$||D \theta(s)||^{2}=\sum_{\dot{l}j=1}^{r},||D^{:}\theta_{j}(s)||^{2}$.

For some $p>1$ we have$\mathrm{E}[(\int_{0}^{T}||Dr(s)||^{2}ds)^{2}]\epsilon<\infty$,$\mathrm{E}[(\int_{0}^{T}||D\theta(s)||^{2}ds)^{2}]\epsilon<\infty$.

We note that the optimal portfolio is also expressed under $P$ :$\pi^{*}\sigma(t)=-\mathrm{E}[\frac{H_{0}(T)}{H_{0}(t)}\mathcal{Y}(W)H_{0}(T)I’(\mathcal{Y}(W)H_{0}(T))|\mathcal{F}_{t}]\theta^{*}(t)$$- \mathrm{E}[\frac{H_{0}(T)}{H_{0}(t)}\phi’(\mathcal{Y}(W)H_{0}(T))\cross$($\int_{t}^{T}D_{t}r(u)du+\sum_{\alpha=1}^{r}\{\int_{t}^{T}\{D_{t}\theta_{\alpha}(u)\}dw^{\alpha}(u)+\int_{t}^{T}\{D_{t}\theta_{\alpha}(eg)\}\theta_{\alpha}(u)du\}$$=W(t) \theta^{*}(t)-\mathrm{E}[\frac{H_{0}(T)}{H_{0}(t)}\phi^{J}(\mathcal{Y}(W)H_{0}(T))|\mathcal{F}_{t}]$)$|\mathcal{F}_{t}]$HO (t)$- \mathrm{E}[\frac{H_{0}(T)}{H_{0}(t)}\phi’(\mathcal{Y}(W)H_{0}(T))\cross$$( \int_{t}^{T}D_{t}r(u)du+\sum_{\alpha=1}^{r}\{\int_{t}^{T}\{D_{t}\theta_{\alpha}(u)\}dw^{\alpha}(u)+\int_{t}^{T}\{D_{t}\theta_{\alpha}(u)\}\theta_{\alpha}(u)du\})|\mathcal{F}_{t}]$where $W(t)$ denotes the optimal wealth at time , and$t$$W(t)= \mathrm{E}[\frac{H_{0}(T)}{H_{0}(t)}I(\mathcal{Y}(W)H_{0}(T))|\mathcal{F}_{t}]$.

is easily derived for two
It is well known that the optimal portfolio
simple cases: (See for instance chapter 3in Karatzas and Shreve(1998).)
utility function $U(x)=\log x$ ,For the case of a$\pi(t)$$\log$$\pi^{*}(t)=\theta^{*}(t)\sigma(t)^{-1}W(t)$where $\theta(t)=\sigma(t)^{-1}[b(t)-r(t)1]$ . For the case of apower utility function
are deterministic,
, if
and
,
,
$\mathrm{U}(\mathrm{x})=-x_{\Gamma}^{\delta}$$\delta<1$$\delta\neq 0$$\mathrm{r}(-)$$\theta(\cdot)$$\pi^{*}(t)=\frac{1}{(1-\delta)}\theta^{*}(t)\sigma(t)^{-1}W(t)$.

are not deterministic, it is difficult to evaluate
andHowever, if
explicitly for apower utility function$r(\cdot)$$\theta(\cdot)$$\pi(t)$132
3The Optimal Portfolio Problem for Investment
in aMarkovian Setting
In the spirit of Ocone and Karazas (1991), we will aconsider more concrete
but sufficiently general setting for practical purpose in the sequel.

be a -dimensional diffusion process defined by the stochastic differLet
ential equation:, $X_{t}‘=x$ ,$\mathrm{d}$$X_{u}^{\epsilon}$$dX_{\mathrm{u}}^{e}=V_{0}(X_{u}^{\epsilon},\epsilon)du+V(X_{u}^{\epsilon},\epsilon)dw_{u}$$\{$$dS_{u}^{\epsilon}=Isb(X_{u}^{\epsilon})du$ $+Is\sigma(X_{u}^{\epsilon})dw_{u}$$dS_{0u}‘=S_{0u}r(X_{u}^{\epsilon})du$,,$S_{t}^{\epsilon}=s$$S_{0t}‘=s0$

for $u\in[t, T]$ where Is denotes theelement of . Here we suppose

diagonal matrix with -th diagonal
,
and
where
denotes a$(0,1]arrowE$
whose derivatives
class of smooth mappings :. Note that timesuch thatandare all bounded for
dependent-coefficient diffusion processes are included in the above equation
if we enlarge the process to ahiger-dimensional one. We also assume that,andare bounded,
is non-singular. Then, is defined as
and that$r\mathrm{x}$$r$$\epsilon\in(0,1]$$S_{\dot{1}}$$\mathrm{i}$$V_{0}\in C_{b}^{\infty}(\mathrm{R}^{d}\mathrm{x} (0,1];\mathrm{R}^{d})$$C_{b}^{\infty}(R^{d}\mathrm{x}(0,1];E)$$V=(V\rho)_{\beta=1}^{r}\in C_{b}^{\infty}(\mathrm{R}^{d}\mathrm{x}(0,1];\mathrm{R}^{d}\otimes \mathrm{R}^{r})$$f$$\mathrm{R}^{d}\mathrm{x}$$\mathrm{n}\in \mathrm{Z}_{+}^{d}$$b\in C_{b}^{\infty}(\mathrm{R}^{d};\mathrm{R}^{r})$$\partial_{x}^{\mathrm{n}}\partial_{e}^{m}f(x, \epsilon)$$|\mathrm{n}|\geq 1$$r\in C_{b}^{\infty}(\mathrm{R}^{d};\mathrm{R}_{+})$$m\in \mathrm{Z}_{+}$$\sigma\in C_{b}^{\infty}(\mathrm{R}^{d};\mathrm{R}^{f}\otimes \mathrm{R}^{r})$$\sigma\in C_{b}^{\infty}(\mathrm{R}^{d};\mathrm{R}^{f}\otimes \mathrm{R}^{r})$$\theta$$\theta(X_{u}^{\epsilon})=\sigma(X_{u}^{\epsilon})^{-1}[b(X_{u}‘)-r(X_{u}^{\epsilon})1]$

and,is bounded.

$\theta\in C_{b}^{\infty}(\mathrm{R}^{d};\mathrm{R}^{r})$be aunique solution of theLetequation:$\mathrm{Y}_{t,u}^{\epsilon}$-matrix valued stochastic differential$d\mathrm{x}d$$d \mathrm{Y}_{t,u}^{\epsilon}=\sum_{\alpha=0}^{r}\partial_{x}V_{\alpha}(X_{u}^{\epsilon}, \epsilon)\mathrm{Y}_{t,u}^{e}dw_{u}^{\alpha}$$\{$$\mathrm{Y}_{t,t}^{\epsilon}=\underline{I}$It is then known that
$D_{t}X_{u}^{\epsilon}$$=$$\mathrm{Y}_{t,u}^{\epsilon}V(X_{t}^{\epsilon}, \epsilon)=\mathrm{Y}_{t,u}‘ V(x_{t}, \epsilon)$, $u\geq t$ .

(See for instance pp.109 of Nualart(1995).)

Let f$\in C_{b}^{\infty}(\mathrm{R}^{d};\mathrm{R})$and utilizing the fact
, $u\geq t$$D_{t}f(X_{u}‘)=\partial f(X_{u}‘)[D_{t}X_{u}^{\epsilon}]=\partial f(X_{u}^{\epsilon})\mathrm{Y}_{t,u}‘ V(x_{t}, \epsilon)$

we give the represehtation of the optimal portfolio$\pi(t)$in our Markovian

setting:$\pi^{*}(t)\sigma(x)=W\theta^{*}(x)-\mathrm{E}[H_{0,t,T}\phi’(\mathcal{Y}H_{0,t,T})]\theta^{*}(x)$$-\mathrm{E}[H_{0,t,T}\phi’(\mathcal{Y}H_{0,t,T})($ $\int_{t}^{T}\partial r(X_{u}^{\epsilon})\mathrm{Y}_{t,u}^{\epsilon}V(x, \epsilon)du+\sum_{\alpha=1}^{f}\int_{t}^{T}\partial\theta_{\alpha}(X_{u}^{\epsilon})\mathrm{Y}_{t,u}^{\epsilon}1$$+ \sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}(X_{u}^{\epsilon})\partial\theta_{\alpha}(X_{u}^{\epsilon})\mathrm{Y}_{t,u}^{\epsilon}V(x, \epsilon)du)]$133
where W is the wealth at time t,$H_{0,t,T}$$\equiv$$\frac{H_{0}(T)}{H_{0}(t)}$$=$$\exp(-\int_{t}^{T}\theta(X_{u}^{\epsilon})^{*}dw(u)-\frac{1}{2}\int_{t}^{T}|\theta(X_{u}^{\epsilon})|^{2}du-\int_{t}^{T}r(X_{u}^{\epsilon})du)$and the relation between $W$ and$\mathcal{Y}$is given by the equation:$W=\mathrm{E}[H_{0,t,T}I(\mathcal{Y}H_{0,t,T})]$$X_{u}^{\epsilon}$,.

for $u\in[t,T]$ is generated by the SDE:
$dX_{u}^{\epsilon}=V_{0}(X_{u}^{\epsilon}, \epsilon)du+V(X_{u}^{\epsilon}, \mathrm{e})\mathrm{d}\mathrm{w}\mathrm{u}$with the initial value$X_{t}^{\epsilon}=x$,.

explicitly. In the present article, we will
Our objective is to evaluate
propose apractical and effcient scheme for computing the optimal portfolio
by utilizing the asymptotic expansion approach.
$\pi(t)$4An Asymptotic Expansion Scheme
Preparations

4.1

First, we will summarize the basic tools for the asymptotic expansion approach. We assume the deterministic limit condition:$[A1]V(\cdot, 0)\equiv 0$.

It follows from $[A1]$ that the limit process
istic) solution of the ordinary differential equation:$(X_{u}^{0})_{u\in[t,T|}$$X_{u}^{0}$$=$$x+ \int_{t}^{u}V_{0}(X_{s}^{0}, \mathrm{O})ds$

is aunique (determin-.

is non-singular for all $u\in[t, T]$ . Next, put
We further assume. Clearly,
is aunique (deterministic) solution of the ordinary differential equation:$\sigma(X_{u}^{0})$$Y_{t,s}:=Y_{t,s}^{0}$$Y_{t,s}$$d\mathrm{Y}_{t,s}$$‘ Y_{t,t}$and$\mathrm{Y}_{t,s}\in GL(d, \mathrm{R})$

$=$

$=$

$\partial_{x}V_{0}(X_{s}^{0},0)\mathrm{Y}_{t,s}ds$$\underline{I}$$s\in[t,T]$.

. Next, let$D(t;u)=\neq_{\epsilon}^{\partial X^{\epsilon}}|_{\epsilon=0}$. Then $D(t;u)$ , $E(t,u)$ andby the following stochastic differential equations:,$E(t;u)= \frac{\partial^{2}}{\partial\epsilon}P^{e}|_{\epsilon=0}X$$\mathrm{Y}_{t,u}^{[1]}(u\in[t,T])$$Y_{t,u}^{[1]}= \frac{\partial Y_{t.u}^{\epsilon}}{\partial\epsilon}|_{\epsilon=0}$$dD(t;u)$$=$$D(t;t)$$=$are determined$\partial_{x}V_{0}(X_{u}^{0}, \mathrm{O})D(t;u)du+\sum_{\alpha=0}^{r}\partial_{\epsilon}V_{\alpha}(X_{u}^{0}, \mathrm{O})dw^{\alpha}$

$\{$

0,

and

134
dE $(t; u)$$\partial_{x}V_{0}(X_{u}^{0}, \mathrm{O})E(t;u)du+\partial_{x}^{2}V_{0}(X_{u}^{0}, \mathrm{O})[D(t;u), D(t;u)]du$

$=$

+2

$\sum_{\alpha=0}^{r}\partial_{x}\partial_{\epsilon}V_{\alpha}(X_{u}^{0}, \mathrm{O})D(t;u)dw^{\alpha}$$\{$$+ \sum_{\alpha=0}^{r}\partial_{\epsilon}^{2}V_{\alpha}(X_{u}^{0},0)dw^{\alpha}$$E(t;t)$

0

$=$

and
$d\mathrm{Y}_{t,s}^{[1]}$

$=$

$\{$

$\partial_{x}V_{0}(X_{s}^{0},0)\mathrm{Y}_{t,s}^{[1]}ds+\partial_{x}^{2}V_{0}(X_{s}^{0}, \mathrm{O})[D(t;s)]\mathrm{Y}_{t,s}ds$$+ \sum_{\alpha=0}^{\mathrm{r}}\partial_{\epsilon}\partial_{x}V_{\alpha}(X_{s}^{0},0)\mathrm{Y}_{t,s}dw_{f}^{\alpha}$$\mathrm{Y}_{t,t}^{[1]}$

$=$

0.

Here we used the fact that
aconventions $dw^{0}=du$ ,$\partial_{x}V_{\alpha}(\cdot, 0)=0$$\partial i=\partial^{:}/\partial(X_{u}^{\epsilon})^{:}$,

for a $=1$ ,, . Moreover, we use, and notations:$\ldots$$\partial_{\epsilon}.\cdot=\partial^{:}/\partial\epsilon^{\dot{l}}$$r$,$\partial_{x}^{2}V_{0}(X_{u}^{0}, \mathrm{O})[D(t;u), D(t;u)]=\sum_{:\mathrm{j}=1}^{d}\partial_{x_{2}}\partial_{x_{f}}V_{0}(X_{u}^{0},0)D^{(:)}(t;u)D^{(j)}(t;u)$and$\partial_{x}^{2}V_{0}(X_{s}^{0}, \mathrm{O})[D(t;s)]\mathrm{Y}_{t,s}ds$$=$$\sum_{:,j=1}^{d}\partial_{x^{t}}\partial_{x}rV_{0}(X_{s}^{0},0)D^{(j)}(t;s)(\mathrm{Y}_{t,s})^{(:,\cdot)}ds$

where $D^{(:)}(t;s)$ denotes the -th element of $D(t;s)$ and. We will use the following abbreviations:-th row of
$i$

$i$

$(\mathrm{Y}_{t,s})^{(:,\cdot)}$.

denotes the$Y_{t,s}$$X_{u}$$=$$X_{u}^{0}$,$\mathrm{Y}_{u}=\mathrm{Y}_{u}^{0}$,$V_{\alpha u}=V_{\alpha u}^{[0]}=V_{\alpha}(X_{u},0)$

We then have representations of $D(t;u)$ , $E(t;u)$ andof stochastic differential equations:$D(t;u)$$=$$\mathrm{Y}_{t,u}\int_{t}^{u}Y_{t,s}^{-1}\sum_{\alpha=0}^{r}\partial_{\epsilon}V_{\alpha s}dw_{s}^{\alpha}$$E(t;u)$$=$$\mathrm{Y}_{t,u}\int_{t}^{u}\mathrm{Y}_{t,s}^{-1}\{\partial_{x}^{2}V_{0s}[D(t;s), D(t;s)]ds$+2$\mathrm{Y}_{t,u}^{[1]}$$=$, $\alpha=0,1$ ,$\mathrm{Y}_{t,u}^{[1]}$$\cdots$, .
$r$ffom the above set$\sum_{\alpha=0}^{r}\partial_{x}\partial_{\epsilon}V_{\alpha s}D(t;s)dw^{\alpha}+\sum_{\alpha=0}^{r}\partial_{\epsilon}^{2}V_{\alpha s}dw^{\alpha}\}$$\mathrm{Y}_{t,u\int_{t}^{u}\epsilon x\alpha s}(\mathrm{Y}_{t,s})^{-1}[\partial_{x}^{2}V_{0s}[D(t;s)]\mathrm{Y}_{t,s}ds+\sum_{\alpha=0}^{r}\partial\partial V\mathrm{Y}_{t,s}dw_{s}^{\alpha}]$.

Next, we will illustrate our approach by using an example of apower utility
function

135
The Case of aPower Utility Function

4.2

We assume autility function to be so called apower function, that is $U(x)=$,,.
$-x_{\Gamma}^{\delta}$$\delta<1$$\delta\neq 0$

Then, $I(y)$ and$\frac{-\delta}{(1-\delta)}I(y)$$\phi(y)$.

are given by,$I(y)=y^{\frac{-1}{(1-\delta)}}$$\phi(y)=y^{1-}\overline{\varpi}^{\delta}7$, and $\phi’(y)=$Hence,$\pi^{*}(t)\sigma(x)$$=$$\frac{1}{(1-\delta)}W\theta(x)^{*}+\frac{\delta}{(1-\delta)}(\mathcal{Y})^{(_{\overline{1-}r}^{-1})}\mathrm{E}[(H_{0,t,T})^{\mathrm{t}_{\overline{1}-\Pi}^{-\delta})}$$( \int_{t}^{T}\partial r(X_{u}^{\epsilon})\mathrm{Y}_{t,u}^{\epsilon}V(x, \epsilon)du+\sum_{\alpha=1}^{r}\int_{t}^{T}\partial\theta_{\alpha}(X_{u}^{\epsilon})\mathrm{Y}_{t,u}^{\epsilon}V(x, \epsilon)dw^{\alpha}(u)$$+ \sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}(X_{u}^{\epsilon})\partial\theta_{\alpha}(X_{u}^{\epsilon})\mathrm{Y}_{t,u}^{\epsilon}V(x, \epsilon)du)]$

where

.

$W=(\mathcal{Y})^{(\frac{-1}{1-\delta})}\mathrm{E}[(H_{0,t,T})^{(_{\overline{1-}t}^{-\delta})]}$

Here, we use the abbreviations $r(u)=r(X_{u}^{\epsilon})$ and$\theta_{\alpha}(u)=\theta_{\alpha}(X_{u}^{\epsilon})$.

We set$E$$\equiv$$\frac{\delta}{(1-\delta)}(\mathcal{Y})^{(\frac{-1}{1-\delta})_{\mathrm{E}}}[(H_{0,t,T})^{(_{\overline{1-}f}^{-\delta})}$$( \int_{t}^{T}\partial r(X_{u}^{\epsilon})\mathrm{Y}_{t,u}^{\epsilon}V(x, \mathrm{e})\mathrm{d}\mathrm{u}+\sum_{\alpha=1}^{r}\int_{t}^{T}\partial\theta_{\alpha}(X_{u}^{\epsilon})Y_{t,u}^{\epsilon}V(x, \epsilon)dw^{\alpha}(u)$.

$+ \sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}(X_{u}^{\epsilon})\partial\theta_{\alpha}(X_{u}^{\epsilon})Y_{t,u}^{\epsilon}V(x, \epsilon)du)]$We start with slightly general setting. Define$\zeta_{t,u}^{\epsilon}$$:=$$\exp(\int_{t}^{u}a_{0}(X_{s}^{\epsilon})ds+\int_{t}^{u}a(X_{s}^{\epsilon})dw_{s})$. Here,andwhere:denotes aclass of smooth functions
derivatives are of polynomial growth orders.
$a_{0}\in C_{\uparrow}^{\infty}(\mathrm{R}^{d};\mathrm{R})$$a\in C_{\uparrow}^{\infty}(\mathrm{R}^{d};\mathrm{R}^{r})$$f$$C_{\uparrow}^{\infty}(\mathrm{R}^{d};\mathrm{R})(C_{\uparrow}^{\infty}(\mathrm{R}^{d};\mathrm{R}^{r}))$$\mathrm{R}^{d}arrow \mathrm{R}$We assume the following integrability condition for[A2] For any $p\in(1, \infty)$ ,$\sup_{\epsilon\in(0,1]}||\zeta_{t,T}^{\epsilon}||_{p}<\infty$

Under Condition [A2], it is easily seen that$(f : \mathrm{R}^{d}arrow \mathrm{R}^{r})$$\zeta_{t,T}^{\epsilon}$

whose

.

.

$\zeta_{t,T}^{\epsilon}\mathrm{h}\mathrm{s}$astochastic expansion$\zeta_{t,T}^{0}+\epsilon\zeta_{t,T}^{[1]}+\frac{\epsilon^{2}}{2}\zeta_{t,T}^{[2]}+\cdots$$\zeta_{t,T}^{\epsilon}$

,

136
in

$\mathrm{D}^{\infty}$as$\epsilon\downarrow 0$. The first three coefficients are given by$=$$\zeta_{t,T}^{0}$$\zeta_{t,T}^{[1]}$$=$,$\exp(\int_{t}^{T}a\mathrm{o}(X_{s})ds+\int_{t}^{T}a(X_{s})dw_{s)}$$\zeta_{t,T}^{0}(\int_{t}^{T}\partial_{x}a\mathrm{o}(X_{s})D(t;s)ds+\int_{t}^{T}\partial_{x}a(X_{s})D(t;s)dw_{S)}$and$\zeta_{t,T}^{[2]}$$=$$\zeta_{t,T}^{0}\{(\int_{t}^{T}\partial_{x}a\mathrm{o}(X_{s})D(t;s)ds+\int_{t}^{T}\partial_{x}a(X_{s})D(t;s)dw_{s})^{2}$$+ \int_{t}^{T}\partial_{x}a_{0}(X_{s})E(t;s)ds+\int_{t}^{T}\partial_{ax}a(X_{s})E(t;s)dw_{s}$.

$+ \int_{t}^{T}\partial_{x}^{2}a_{0}(X_{s})[D(t;s), D(t;s)]ds+\int_{t}^{T}\partial_{x}^{2}a(X_{S})[D(t;s), D(t;s)]dw_{s}\}$For$f\in C_{\uparrow}^{\infty}(\mathrm{R}^{d};\mathrm{R})$$g^{\alpha,e}$, put$\int_{t}^{T}\partial f(X_{u}^{\epsilon})Y_{t,u}^{\epsilon}V(x_{t}, \epsilon)dw_{u}^{\alpha}$$=$,$\alpha=0,1$,$\cdots,r$Since $V(x, 0)\equiv 0$ from [A1], we see that$g^{\alpha,0}$

The first derivative

The second derivative$=$

0$g^{\alpha,[1]}\equiv*_{\epsilon}^{\alpha}\partial|-\cdot|_{\epsilon=0}$$g^{\alpha,[1]}$$g^{\alpha,[2]}$

$=$

2

$=$

$(\alpha=0,$

1, \ldots ,r).

ofis given by$g^{\alpha,\epsilon}$$\int_{t}^{T}\partial_{x}f(X_{u})[Y_{t,u}\partial‘ V(x_{t},0)]dw_{u}^{\alpha}$$g^{\alpha,|2]}\equiv*_{\epsilon}^{\partial^{2\alpha}’}\cdot|‘=0$of$g^{\alpha,\epsilon}$

(1)

.

(2)

is given by$\int_{t}^{T},\sum_{\dot{l}j=1}^{d}\partial_{\dot{l}}\partial_{\mathrm{j}}f(X_{u})D^{(j)}(t;u)Y_{t,u}^{(\dot{l},\cdot)}\partial_{\epsilon}V(x_{t}, 0)dw_{u}^{\alpha}$$+2 \int_{t}^{T}\sum_{\dot{|}=1}^{d}\partial_{\dot{l}}f(X_{u})\mathrm{Y}_{t,u}^{[1],(:,\cdot)}\partial_{\epsilon}V(x_{t}, 0)dw_{u}^{\alpha}$$+ \int_{t}^{T}\sum_{=1}^{d}\partial_{}f(X_{u})Y_{t,u}^{(\dot{1},\cdot)}\partial^{2}‘ V(x_{t}, 0)dw_{u}^{\alpha}$

After all, from (1) and (2), and by tedius routine work, we obtain the stochas:tic expansion of$g^{\alpha,\epsilon}$$\epsilon g^{\alpha,|1]}+\frac{\epsilon^{2}}{2}g^{\alpha,[2|}+\cdots$$g^{\alpha,\epsilon}$

137
in

$\mathrm{D}^{\infty}(\mathrm{R}^{d})$as$\epsilon\downarrow 0$.

Utilizing above results, we will derive an asymptotic expansion of the insideof the expectation of $E$ .
First, we directly apply the expression of the expansion for$\zeta_{t,u}^{\epsilon}=(H_{0,t,T})^{(\frac{-\delta}{1-\delta})}$where$a_{0}(X_{s}^{\epsilon})$and$a(X_{s}^{\epsilon})$$\zeta_{t,u}^{\epsilon}$

if we set

are specified by$a_{0}(X_{s}^{\epsilon})$$=$$( \frac{\delta}{1-\delta})r(X_{s}^{\epsilon})+\frac{\delta}{2(1-\delta)}|\theta(X_{s}^{\epsilon})|^{2}$$a(X_{s}^{\epsilon})$$=$$( \frac{\delta}{1-\delta})\theta(X_{s}^{\epsilon})$.

Here, we note that [A2] is satisfied in this case because of the boundedness.
assumptions ofand$r(\cdot)$$\theta(\cdot)$

Next, we show the expansions of$g_{r}^{\epsilon}$$\equiv$$\int_{t}^{T}D_{t}r(u)du=\int_{t}^{T}\partial r(u)Y_{t,u}^{\epsilon}V(x, \epsilon)\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{z}$$g_{\theta}^{\alpha,\epsilon}$$\equiv$$\int_{t}^{T}D_{t}\theta_{\alpha}(u)dw^{\alpha}(u)=\int_{t}^{T}\partial\theta_{\alpha}(u)\mathrm{Y}_{t,u}^{\epsilon}V(x, \epsilon)dw^{\alpha}(u)$$g_{\theta^{2}}^{\alpha,\epsilon}$$\equiv$$\int_{t}^{T}$

byReplacingthe expansions of$f(\cdot)$

DteQ$r(\cdot)$$g_{r}^{\epsilon}$

,

,

.

$( \mathrm{u})\mathrm{e}\mathrm{Q}(\mathrm{u})\mathrm{d}\mathrm{u}=\int_{t}^{T}\partial\theta_{\alpha}(u)\mathrm{Y}_{t,u}^{\epsilon}V(x, \epsilon)\theta_{\alpha}(u)du$

,
, and

$\theta_{\alpha}(\cdot)$$g_{\theta}^{\alpha,\epsilon}$$g_{r}^{\epsilon}$$=$$g_{\theta}^{\alpha,\epsilon}$$=$$g_{\theta^{2}}^{\alpha,\epsilon}$$=$$\frac{1}{2}\theta_{\alpha}^{2}(\cdot)$$g_{\theta^{2}}^{\alpha,\epsilon}$and, utilizing above result, we obtain:$\epsilon g_{r}^{[1]}+\frac{\epsilon^{2}}{2}g_{r}^{[2]}+o(\epsilon^{2})$$\epsilon g_{\theta}^{\alpha,[1]}+\frac{\epsilon^{2}}{2}g_{\theta}^{\alpha,[2]}+o(\epsilon^{2})$$\epsilon g_{\theta^{2}}^{\alpha,[1]}+\frac{\epsilon^{2}}{2}g_{\theta^{2}}^{\alpha,[2]}+o(\epsilon^{2})$where$g_{r}^{[1]}$$g_{\theta}^{\alpha,[1]}$$g_{\theta^{2}}^{\alpha,[1]}$$g_{r}^{[2]}$$=$$\int_{t}^{T}\partial r^{[0]}(u)Y_{t,u}\partial_{\epsilon}V(x, \mathrm{O})du$–$\int_{t}^{T}\partial\theta_{\alpha}^{[0]}(u)Y_{t,u}\partial_{\epsilon}V(x, 0)dw^{\alpha}(u)$$=$$\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}\partial_{\epsilon}V(x, \mathrm{O})du$

$=$

2

$+$

$( \int_{t}^{T}\partial r^{[0|}(u)\mathrm{Y}_{t,u}du)\partial_{\epsilon}^{2}V(x, 0)$$( \int_{t}^{T}\partial^{2}r^{[0]}(u)[D(t;u)]Y_{t,u}du+\int_{t}^{T}\partial r^{[0]}(u)\mathrm{Y}_{t,u}^{[1]}du)\partial_{\epsilon}V(x, 0)$138$g_{\theta}^{\alpha,[2]}$2$=$$g_{\theta^{2}}^{\alpha,[2]}$$( \int_{t}^{T}\partial^{2}\theta_{\alpha}^{[0]}(u)[D(t;u)]Y_{t,u}dw^{\alpha}(u)+\int_{t}^{T}\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}^{[1]}dw^{\alpha}(u))\delta$$+$$( \int_{t}^{T}\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}dw^{\alpha}(u))\partial_{\epsilon}^{2}V(x,0)$

$=$

2

$+$

$( \int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du)\partial_{\epsilon}^{2}V(x,0)$$( \int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial^{2}\theta_{\alpha}^{[0]}(u)[D(t;u)]Y_{t,u}du+\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}^{[1]}du$[The second order scheme(the asymptotic expansion upto theorder)]

We will obtain the asymptotic expansion of the optimal portfolio upto tl-order. In the appendix, we will also show the third order scheme.
$\epsilon$

Based on the previous expansions, we have$( \int_{t}^{T}\partial r(X_{u}^{\epsilon})Y_{t,u}^{\epsilon}V(x, \epsilon)du+\sum_{\alpha=1}^{r}\int_{t}^{T}\partial\theta_{\alpha}(X_{u}^{\epsilon})\mathrm{Y}_{t,u}^{\epsilon}V(x, \epsilon)dw^{\alpha}(u)$$+ \sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}(X_{u}^{\epsilon})\partial\theta_{\alpha}(X_{u}^{\epsilon})\mathrm{Y}_{t,u}‘ V(x,\epsilon)du)$$= \epsilon(g_{r}^{[1]}+\sum_{\alpha=1}^{r}g_{\theta}^{\alpha,[1]}+\sum_{\alpha=1}^{r}g_{\theta^{2}}^{\alpha,[1]})+o(\epsilon)$(u)\’e$= \epsilon(\int_{t}^{T}\partial r^{|0]}(u)Y_{t,u}du+\sum_{\alpha=1}^{r}\int_{t}^{T}\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}dw^{\alpha}(u)+\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}$$+o(\epsilon)$,and$ee^{2(1-\delta)} \mathrm{x}\mathrm{t}_{\overline{1}-\nabla}^{\iota})\int_{\mathrm{c}^{r(u)du\neg}}^{\tau\iota}\mathrm{l}0|\int_{t}^{\tau_{|\theta}}10|(u)|^{2}du$$(H_{0,t,T})^{(_{\overline{1}-\pi}^{-\delta})}$$=$$e^{-\frac{1}{2}(\frac{\delta}{1-\delta})^{2}\int_{t}^{\tau_{|\theta}}10|(u)|^{2}du+(\frac{\delta}{1-\delta})\int_{t}^{\tau_{\theta^{10|}}}(u)dw(u)}\mathrm{x}$$(1+ \epsilon(\frac{\delta}{1-\delta})\int_{t}^{T}\partial r^{[0|}(u)D(t;u)du+\epsilon(\frac{\delta}{1-\delta})\sum_{\alpha=1}^{r}\int_{t}^{T}\acute{\mathrm{c}}$$+ \epsilon(\frac{\delta}{1-\delta})\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0|}(u)D(t;u)du)+o(\epsilon)$.

Then, E’s expansion is obtained byE$=$$\frac{\delta}{(1-\delta)}(\mathcal{Y})(_{\overline{1-}\delta}^{-1})ee^{2(1-\delta)}\cross(_{\overline{1}-\nabla}^{\delta})\int_{t^{f}}T10|(u)du\neg\int_{l}^{T}\delta|\theta^{|0|}(u)|^{2}du$$\epsilon()^{T}\partial r^{[0]}(u)\mathrm{Y}_{t,u}du+\frac{1}{(1-\delta)}\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)Y_{t,u}du)$$\partial_{\epsilon}V($139
If we utilize the relation;$( \mathcal{Y})^{\mathrm{t}_{\overline{1-}t}^{-1})}=\frac{W}{\mathrm{E}[(H_{0,t,T})^{(\frac{-\delta}{1-\delta})}]}$,and the expansion;$\mathrm{E}[(H_{0,t,T})^{(\frac{-\delta}{1-\delta})}]=e^{(\frac{\delta}{1-\delta})\int_{t}^{T}r^{\iota 0|}(u)du}e^{\frac{\delta}{2(1-\delta)^{2}}\int_{t}^{T}|\theta 10|(u)|^{2}du}\mathrm{x}$$(1+ \epsilon(\frac{\delta}{1-\delta})\int_{t}^{T}\partial r^{[0]}(u)\hat{D}_{1}(t;u)du+\epsilon\frac{\delta}{(1-\delta)^{2}}\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\hat{D}_{1}(t;u)du)+o(\epsilon)$

where
,

$\hat{D}_{1}(t;u)\equiv Y_{t,u}\int_{t}^{u}\mathrm{Y}_{t,s}^{-1}\{\partial_{\epsilon}V_{0}^{[0]}(s)ds+(\frac{\delta}{1-\delta})\partial_{\epsilon}V^{[0]}(s)\theta^{[0]}(s)ds\}$$\mathrm{E}$$E$’s expression in terms of $W$ is given by$=$$\frac{\delta}{(1-\delta)}W\mathrm{x}$$\epsilon(\int_{t}^{T}\partial r^{[0]}(u)Y_{t,u}du+\frac{1}{(1-\delta)}\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0[}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du)$de $V(x, 0)+o(\epsilon)$ .

Then, we have the following theorem:

Theorem 2An asymptotic expansion of the optimal portfolio for investment for a $pow$ er utility function is given by$\pi^{*}(t)$$=$$\frac{1}{(1-\delta)}W[\theta^{*}(x)+$$\delta\epsilon(\int_{t}^{T}\partial r^{[0]}(u)Y_{t,u}du+\frac{1}{(1-\delta)}\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du)\partial_{\epsilon}V(x, 0)]\sigma^{-1}(x)+o(\epsilon)$

References[1] Cox, J. and Huang C.-H. (1989), “Optimal Consumption and Portfo
lio Policies When Asset Prices FIlow Diffusion Processes,” Journal of
Economic Theory, Vo1.49,33-83.
[2] Heath, D. Jarrow, R. and Morton, A. (1992), “Bond Pricing and theTerm Structure of Interest Rates :ANew Methodology for Contingent
Claims Valuation,” Econometrica, Vo1.60,77-105

140[3] Karazas, I. Lehoezky, J. and Shreve, S. (1987), “Optimal Portfolio
and Consumption Decisions for a‘small investor’ on aFinite Horizon,”
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[4] Karazas, I. and Shreve, S. (1998), Methods of Mathematical FinanceSpringer.
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[6] Kim, Y.J. and Kunitomo, N. (1999), “Pricing Options Under Stochastic
Interest Rates,” Asia Pacific Financial Markets, Vo1.6,49-70.

[7] Kunitomo, N. and Takahashi, A. (1992), “Pricing Average Options,”
Japan Financial Review, Vo1.14, $1- 20$ . (In Japanese.)[8] Kunitomo, N. and Takahashi, A. (1995), “The Aysmptotic Expansion
Approach to the Valuation of Interest Rate Contingent Claims,” Discussion Paper No. 95-F-l9, Faculty of Economics, University of Tokyo.
(to appear Mathematical Finance.)[9] Kunitomo, N. and Takahashi, A. (1998), “On Validity of the Asymp
totic Expansion Approach in Contingent Claim Analysis,” Discussion
Paper N0.98-F-6, Faculty of Economics, University of Tokyo.
[10] Ocone, D. and Karazas, 1.(1991), “A Generalized Clark Representation Formula, with Application to Optimal Portfolios,” Stochastics andStochastics Reports, Vo1.34,187-220.

[11] Merton, R.C. (1969), “Lifetime Portfolio Selection under Uncertanity: The Continuous-Time Case,” Revi ew of Economics and Statistics,
V01.51, 247-257.
[12] Merton, R.C. (1971), “Optimum Consumption and Portfolio Rules in aContinuous-Time Model,” Journal of Economic Theory, Vo1.3,373-413.

[13] Sorensen, M. and Yoshida, N. (1998), “Random limit expansion forsmall diffusion processes,” unpablished manuscript.
[14] Takahashi, A. (1995), “Essays on the Valuation Problems of Contingent Claims,” Unpublished Ph.D. Dissertation, Haas School of Business, University of California, Berkeley.
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Contingent Claims,” Preprint, forthcoming in Asia-Pacific Financial
MarketsVol.6,115-151.

[16] Watanabe, S. (1987), “Analysis of Wiener Functional (Malliavin Calculus) and its Applications to Heat Kernels,” The Annals of Probability,
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159.

141
5Appendix
In this appendix, we will show the result of the third order scheme of the
optimal portofolio for the case of apower utility function.
[The third order scheme(the asymptotic expansion upto theorder)]$\epsilon^{2_{-}}$$\pi^{*}(t)=\frac{W}{(1-\delta)}[\theta^{*}(x)+\delta C\{\epsilon A+\epsilon^{2}B-\epsilon^{2}AD\}]\sigma^{-1}(x)+o(\epsilon^{2})$

where

A

$\equiv(\int_{t}^{T}\partial r^{[0]}(u)\mathrm{Y}_{t,u}du+\frac{1}{(1-\delta)}\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du)\partial_{\epsilon}V(x,$C$\equiv\exp((\frac{\delta}{1-\delta})\int_{t}^{T}r^{[0]}(u)du+\frac{\delta}{2(1-\delta)^{2}}\int_{t}^{T}|\theta^{[0]}(u)|^{2}du)$

0),

D

,
,

$\equiv(\frac{\delta}{1-\delta})\int_{t}^{T}\partial r^{[0]}(u)\hat{D}_{1}(t;u)du+\frac{\delta}{(1-\delta)^{2}}\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\hat{D}_{1}(t;u)du$and B is the sum of the following terms:
1.
$( \frac{\delta}{1-\delta})\{\int_{t}^{T}\partial r^{[0]}(u)\hat{D}_{1}(t;u)du\}\{\int_{t}^{T}\partial r^{[0]}(u)\mathrm{Y}_{t,u}du\}\mathrm{d}\mathrm{f}\mathrm{V}(\mathrm{x},0)$$2$.
$( \frac{\delta}{1-\delta})^{2}\{\int_{t}^{T}\partial r^{[0]}(u)\hat{D}_{1}(t;u)du\}\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0|}(u)\partial\theta_{\alpha}^{[0]}(u)Y_{t,u}du\}$[et{x) 0)$+( \frac{\delta}{1-\delta})\{\sum_{\alpha=1}^{r}\int_{t}^{T}\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}(\int_{u}^{T}\partial r^{[0]}(s)\mathrm{Y}_{t,s}ds)\mathrm{Y}_{t,u}^{-1}\partial_{\epsilon}V_{u}^{[0],(\cdot\alpha)}\prime du\}\partial_{\epsilon}V(x, 0)$$3$.
$( \frac{\delta}{1-\delta})\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du\}\partial_{\epsilon}V(x, 0)\{\int_{t}^{T}\partial r^{[0]}(u)\hat{D}_{1}(t;u)du\}$4.
$( \frac{\delta}{1-\delta})\{\sum_{\alpha=1}^{r}1\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\hat{D}_{1}(t;u)du\}\{\int_{t}^{T}\partial r^{[0]}(u)\mathrm{Y}_{t,u}du\}$5.
$( \frac{\delta}{1-\delta})^{2}\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0|}(u)\partial\theta_{\alpha}^{[0|}(u)\hat{D}_{1}(tiu)du\}\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0|}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du\}$$( \frac{\delta}{1-\delta})\sum_{\alpha=1}^{r}\{\int_{t}^{T}(,\sum_{\alpha=1}^{r}\int_{u}^{T}\theta_{\alpha}^{[0]},(s)\partial\theta_{\alpha}^{[0]},(s)ds)\mathrm{Y}_{t,u}^{-1}\partial_{\epsilon}V_{u}^{[0],(\cdot\alpha)}’\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du\}\partial_{\epsilon}V$

142

(. ,)

$\ovalbox{\tt\small REJECT}$j$\mathrm{e}!’](\mathrm{u})\mathrm{t}?5?\mathrm{r}](u)\mathrm{W}_{u},du$$\mathrm{t}?.\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}(\mathrm{z},0)$$\mathrm{e};](\mathrm{u})\mathrm{C}?\mathrm{e};](u)\ovalbox{\tt\small REJECT}.(\mathrm{t}\ovalbox{\tt\small REJECT} \mathrm{u})\mathrm{d}\mathrm{t}\mathrm{Z}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{j}$

\yen

7.
$( \frac{\delta}{1-\delta})^{2}\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\hat{D}_{1}(t;u)du\}\{\int_{t}^{T}\partial r^{[0]}(u)\mathrm{Y}_{t,u}du\}\partial_{\epsilon}V(x, 0)$8.
$( \frac{\delta}{1-\delta})^{2}\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\hat{D}_{1}(t;u)du\}\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du\}\partial_{\epsilon}V(x, 0)$$9$.
$( \frac{\delta}{1-\delta})^{3}\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\hat{D}_{1}(t;u)du\}\{\sum_{\alpha=1}^{r}\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du\}\partial_{\epsilon}V(x,0)$$+( \frac{\delta}{1-\delta})\{\sum_{\alpha=1}^{r}\int_{t}^{T}\partial\theta_{\alpha}^{[0]}(u)\hat{D}_{1}(t;u)\partial\theta_{\alpha}^{[0]}(u)Y_{t,u}du\}\partial_{\epsilon}V(x,0)$$+( \frac{\delta}{1-\delta})^{2}\sum_{\alpha=1}^{r}\int_{t}^{T}(,\sum_{\alpha=1}^{r}\int_{u}^{T}\theta_{\alpha}^{[0]},(s)\partial\theta_{\alpha}^{[0]},(s)ds)\mathrm{Y}_{t,u}^{-1}\partial_{\epsilon}V_{u}^{[0],(\cdot,\alpha)}du$10.
$( \frac{1}{1-\delta})(\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial^{2}\theta_{\alpha}^{[0]}(u)[\hat{D}_{1}(t;u)]\mathrm{Y}_{t,u}du+\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)[\hat{Y}_{t,u}^{[1]}]du)\partial_{\epsilon}V(x, 0)$$+( \frac{1}{1-\delta})(\int_{t}^{T}\theta_{\alpha}^{[0]}(u)\partial\theta_{\alpha}^{[0]}(u)\mathrm{Y}_{t,u}du)\partial_{\epsilon}^{2}V(x,0)$

where$\hat{\mathrm{Y}}_{t,u}^{[1]}\equiv\int_{t}^{u}\mathrm{Y}_{t,u}Y_{t,s}^{-1}\{\partial_{x}^{2}V_{0s}[\hat{D}_{1}(t;s)]Y_{t,s}ds+\partial_{\epsilon}\partial_{x}V_{0s}\mathrm{Y}_{t,s}ds+(\frac{\delta}{1-\delta})\sum_{\alpha=1}^{r}\theta_{\alpha}(s)\partial_{\epsilon}\partial_{x}V_{\alpha s}\mathrm{Y}_{t,s}ds\}$

11.
$( \int_{t}^{T}\partial_{\mathrm{t}}^{2}r^{[0]}(u)[\hat{D}_{1}(t;u)]Y_{t,u}du+\int_{t}^{T}\partial r^{[0]}(u)\hat{\mathrm{Y}}_{t,u}^{[1]}du)\partial_{\epsilon}V(x,0)$$+( \int_{t}^{T}\partial r^{[0]}(u)Y_{t,u}du)\partial_{\epsilon}^{2}V(x, 0)$