数理解析研究所講究録
第 1557 巻 2007 年 107-117

107

Super-replication cost for a single large investor
Takashi Adachi
Fukushima Medical University,
Fukushima City, Fukushima 960-1295, JAPAN(E-mail: ada@fmu.ac.jp)

March 31, 2007
Abstract
We consider the cost of hedging contingent claims in a financial market where
the trades of a large investor can move market prices. We provide a characterizationof a super-replication cost in terms of an associated stochastic control problem. We
also prove that the $super$-replication cost is a viscosity solution of a corresponding
dynamic programming equation in the case of a Markov market model.
Key words: large investor, super-replication cost, dynamic programming equation.
JEL Classification: G12, G13.

1

Introduction

Our concern is to examine the cost of hedging contingent claims in a financial market
where the trades of a large investor can move market prices, and the purpose of thispaper is to provide a characterization of a super-replication cost in terms of an associatedstochastic control problem.
1.1

General large investor problem

Let $T>0$ be a finite time horizon and $\{W(t), 0\leq t\leq T\}$ a standard d-dimensional
Brownian motion on a complete probability space, endowed with a filtration$F=\{\mathcal{F}_{t}, 0\leq t\leq T\}$ which is the P-augmentation of thefiltration generated by the
Brownian motion $W$ . Let denote the set of all -valued, F-progressively measurable
processes
such thata.s. Here $n\leq d$ .
$(\Omega, \mathcal{F}, P)$$\mathcal{P}$$p(\cdot)$$\int_{0}^{T}|p(t)|^{2}dt<\infty$$R^{\mathfrak{n}}$

108

l.l.a

$Pri$

ce dynamics in presence of large investor

We assume that there is a single large investor in a financial market where one bankaccount and stocks are traded continuously up to the time . Let
be a set ofadmissible trading strategiesof the investor . We assume $0\in\Pi$ .
Consider a model for price fluctuations as follows: If the investor starts at timewith an initial capital $x\in R$ and holds
shares of the j-th stock at time $t\in[0, T]$ ,, then the price processes$j=1,$of the bank account andof the stocks
evolve according to the stochastic differential equation (SDE, for short)
$I$

$T$

$n$

$\Pi\subset \mathcal{P}$$I$$\pi(\cdot)$

$I$

$0$

$\pi_{j}(t)$$\ldots,$$n$$\hat{S}^{\pi}(\cdot)$$B^{\pi}(\cdot)$$dB(t)=B(t)r^{\pi}(t)dt$ ,$B(O)=1$ ,$d\hat{S}(t)=diag[\hat{S}(t)]\{b^{\pi}(t)dt+\sigma^{\pi}(t)^{T}dW(t)\}$and the discounted wealth process$X^{x}$,“$(\cdot)$,$\hat{S}(0)=s\in(0, \infty)^{n}$of the investor$X(t)=x+ \int_{0}^{t}\pi(u)^{T}dS^{\pi}(u)$

,

$I$

,

is given as$t\in[0, T]$

,

(1.1)

where
denotes the transpose operation; $diag[s]$ is the $n\cross n$-diagonal matrix withdiagonal elements
is the discounted price process ofstocks; $\{r^{\pi}(t), 0\leq t\leq T\},$ $\{b^{\pi}(t)=(b_{1}^{\pi}(t), \ldots, b_{n}^{\pi}(t))^{T}, 0\leq t\leq T\}$ and$0\leq t\leq T$ } are bounded F-progressively measurable processes taking, respectively. Here the superscript means that theandvalues in,process( $h=r,$for instance) with the superscript depends on the path$\{\pi(u, \omega), 0\leq u\leq t\}$ for a.e. $(t,\omega)\in[0, T]\cross\Omega$ and $\pi\in\Pi$ . Therefore the price dynamics are influenced by the actions of the investor . It is for this reason that is called
the large investor. We also remark that the integral in (1.1) is well-defined by means ofa.s. and of the boundedness of the coefficients of market.
$T$$s_{1},$$\ldots,$$s_{n};S^{\pi}(\cdot)$$:=B^{\pi}(\cdot)^{-1}\hat{S}^{\pi}(\cdot)$$\{\sigma^{\pi}(t)=$$(\sigma_{1}^{\pi}(t)\cdots\sigma_{n}^{\pi}(t)),$$R_{+},$$R^{d}\otimes R^{n}$$R^{n}$$h^{\pi}(t,\omega)$

$b,$

$\pi$

$\pi$

$\sigma$$I$$I$$\int_{0}^{T}|\pi(t)|^{2}dt<\infty$

l.l.b

Contingent claim and super-replication cost

A contingent claim,
consists of an F-adapted, non-negative process$0\leq t\leq T\}$ and some classof F-stopping times. We assume. Let us consider
now the following situation: At time $t=0$ , two agents (the “buyer” and (seller’) enter
into an agreement. The seller agrees to provide the buyer with the random payment, where is an element ofat timeand at the disposalof the buyer.
The objective of the seller is to find a portfolio strategy $\pi\in\Pi$ which enables him
to fulfill his obligation whenever the buyer decides to ask for the payment. Hence thesuper-replication costis defined as$\{C^{\pi}(t)$$\{B^{\pi}C^{\pi}, \mathcal{T}^{\pi}\}$$T\in \mathcal{T}^{\pi}$$\mathcal{T}^{\pi}$$I$$t=\tau(\omega)$$B^{\pi}(\tau(\omega),\omega)C^{\pi}(\tau(\omega),\omega)$$\tau$$\mathcal{T}^{\pi}$$I$$h_{up}$$h_{up}$$:= \inf\{x\geq 0|\exists\pi\in\Pi$s.t.

$X^{x,\pi}(\tau)\geq C^{\pi}(\tau)$a.s.,$\forall\tau\in\tau^{\pi}.\}$

.

109

1.2

Existent studies dealing with analogous models

In a standard market of a small investor model (the coefficientson ), there exists an F-progressively measurable process :$\pi$$b(t,\omega)-r(t,\omega)1_{n}=-\sigma(t,\omega)^{T}\theta(t,\omega)$

$a.e$

$r,$

$b$

and

$\sigma$do not depend
such that$[0, T]\cross\Omegaarrow R^{d}$$\theta$. $(t,\omega)\in[0, T]\cross\Omega$(1.2)

and the stochastic exponential process$Z_{\theta}(t)$$:= \exp\{\int_{0}^{t}\theta(u)^{T}dW(u)-\frac{1}{2}\int_{0}^{t}|\theta(u)|^{2}du\}$

,

(1.3)

$0\leq t\leq T$

is a martingale, where $1_{n}=(1, \ldots, 1)^{T}\in R^{n}$ . Further, if the standard market is complete,by the martingale representation theorem and the Bayes’ rule, we then have,

$E[\frac{Z_{\theta}(T)}{Z_{\theta}(t)}C(T)|\mathcal{F}_{t}]=E[Z_{\theta}(T)C(T)]+\int_{0}^{t}\pi(u)^{T}dS(u)$(14)$t\in[0, T]$

for some hedging portfolio . Therefore the super-replication cost1 of European contingent
claim $\{B(T)C(T), \{T\}\}$ is given by $h_{up}=E[Z_{\theta}(T)C(T)]$ .
In some special cases, we can also use the martingale duality approach to study the
replication of European contingent claims by the large investor. Cuoco &Liu[6] hasprovided the dual formulation for the case thatandare independent of the
trading strategy , and satisfies$\pi$$r^{\pi},$$\pi$$C^{\pi}$$\sigma^{\pi}$$b^{\pi}$$q(t,\omega)^{T}b^{\pi}(t,\omega)=q(t,\omega)^{T}\mu(t,\omega)+h(t, q(t,\omega),\omega)$

$a.e$

.

$(t,\omega)\in[0, T]\cross\Omega$,where $q(t)=X(t)^{-1}diag[S(t)]\pi(t),$ is a bounded F-progressively measurable process
taking values in, and a function $h(t, q,\omega)$ on
satisfies:
is anoptional process for each $q\in R^{n};h(t, \cdot,\omega)$ is Lipschitz uniformly in $(t,\omega)\in[0, T]\cross\Omega$ ;is concave and upper semicontinuous for all $(t,\omega)\in[0, T]\cross\Omega;h(t, 0,\omega)=0$
for all $(t,\omega)\in[0, T]\cross\Omega.$ Bank&Baum[3] dealt with a general semimartingale market
model with the large investor. They presented a characterization of the super-replication
cost for European contingent claims in terms of an associated stochastic control problem
under the condition that (1.2) was satisfied for some process which did not depend on
the large investor’s position despite of the dependence ofand upon .
In the case of the general large investor model, however, it is difficult to use themartingale duality approach in order to show the existence of a $po$ rtfolio satisfying(1.4) because of the dependence of
upon . Hence the martingale duality
andapproach has not been successful to solve the general large investor problem. Therefore
the previous studies have provided several treatments of this problem which avoid thepassage from the dual formulation. These studies dealt with Markov market models withthe large investor as follows:$\mu$$R^{n}$$[0, T]\cross R^{n}\cross\Omega$$h(\cdot, q, \cdot)$$h(t, \cdot,\omega)$$\theta$

$\pi$

$r,$

$b$

$\sigma$

$\pi$

$\pi$

$\theta,$

$C$

$S$

$\pi$

1For a standard as set pricing theory, see the usual textbooks, e.g. Duffie[9], Karatzas[12] and Karatzas&Shreve[13].

110(i) Cvitani\v{c}&Ma[8]: For $h=b,$ ,$\sigma$$h^{\pi}(t)=h(t, S(t),$ $\pi(t),$ $X(t))$(ii) Soner&Touzi[18]: For $h=b,$$h^{\pi}(t)=h(t, S(t),$ $q(t))$$\sigma$,$r^{\pi}(t)=r(t, diag[S(t)]\pi(t), X(t))$.

and $q(t)=X(t)^{-1}diag[S(t)]\pi(t)$ ,,$r^{\pi}(t)\equiv 1$.

$(1i1)Rey[10]$ : In one-dimensional case $(n=d=1)$ ,$S^{\pi}(t)=\psi(t, Z_{\eta}(t),$ $\pi(t))$,$r^{\pi}(t)\equiv 1$,$\pi(t)=\phi(t, Z_{\eta}(t))$,

where is a smooth reaction function, the stochastic exponential
defined as (1.3)
with a constant is a fundamental state variable process, and the trading strategy
is selected from among smooth functions. In Platen&Schweizer[14] and Frey&
Stremme[ll] the state variable
and the reaction function have been obtained
from equilibrium considerations.
&Ma [8] characterized the cost and portfolio of hedging European option$B(T)C(T)=g(S(T))$ as asolution of aforward-backward SDE corresponding to their
Markov model, and proved the existence and uniqueness of the $so1_{t1}tion$ of this equation
under regularity conditions onand . $FYey[10]$ characterized the hedging portfolio of European option $C(T)=g(S(T))$ as asolution ofas sociated quasi-linear
partial differential equation and provided results on existenoe
uniqueness of thelution to this equation under regularity conditions on and Soner&Touzi[18] used a
new dynamic programming principle established in Soner&Touzi[16]to characterize thesuper-replication cost for European option $C(T)=g(S(T))$ as aviscosity solution of acorresponding dynamic programming equation under suitable conditions onand .
in our model do not depend on the value of the large $inve8tor’ s$
Since
and
wealth $X$ , our model does not include those of Cvitanic’ &Ma[8]and Soner&Touzi[18].
However we can apply our approach to the study of the replication in the model of Soner&Touzi[18], and we can treat Example 5.1 of Cvitani\v{c}’ &Ma[8] in our framework (seeAppendix in the $author[1]$ ). Extending the set of admissible portfolios in the model ofRey[10]to the set of controlled semimartingales$\psi$$Z_{\eta}$$\eta$$\phi$$Z_{\eta}$$\psi$$Cvitani\acute{c}$$r,$ $b,$$\sigma$$g$$\phi$$\bm{t}$$\bm{t}d$$St\succ$$g.$$\psi$

$b,$

$r,$

$b$

$\sigma$$g$$\sigma$$B$$d\pi(t)=\alpha(t)dt+\beta(t)dW(t)$(where$\alpha$and$\beta$are controls),

we also have the application of our approach to the study of the replication in the modelof Frey[10].
The remainder of this paper is organized in the following way: In the next section wecharacterize the $super$-rePlication cost in terms of associated stochastic control problem.
In \S 3, we derive a corresponding dynamic programming equation from the representation
obtained in \S 2, and characterize the super-replication cost as a viscosity solution of thisequation in the case of a Markov market model. For the proofs of assertions stated in \S 2
and \S 3, see the author’s paper[l].

111

2

Main result

In order to characterize the super-replication cost in terms of the stochastic control problem, we need the notion of the change of measure. Let
be the class of all-valued,$R^{d}$$\mathcal{D}_{m}$F-progressively measurable processes
such that
Then the stochastic exponential process$\nu(\cdot)$$Z_{\nu}(t)$$|\nu(t,\omega)|\leq m$$:= \exp\{\int_{0}^{t}\nu(u)^{T}dW(u)-\frac{1}{2}\int_{0}^{t}|\nu(u)|^{2}du\}$a.e., and,$\mathcal{D}$$:= \infty\bigcup_{m=1}\mathcal{D}_{m}$.

$0\leq t\leq T$

is a martingale for each $\nu\in D$ , and$\mathbb{P}_{\nu}(\Lambda):=E[Z_{\nu}(T)\bm{1}_{\Lambda}]$,$\Lambda\in \mathcal{F}_{T}$is a probability measure, where 1 is the indicator function.

When the seller receives the amount $x>h_{up}$ from the buyer, he can cover his
obligation at any time
without risk, i.e.
$I$$\tau\in \mathcal{T}^{\pi}$$inf\sup_{\pi\in\Pi_{\tau\in T\nu\in \mathcal{D}}}\sup E_{\nu}[(C^{\pi}(\tau)-X^{x,\pi}(\tau))^{+}]=0$

wheredenotes the expectation operator undercalculate$E_{\nu}$

$0=$
$=?,$

$P_{\nu}$,

and $a^{+}:= \max\{a, 0\}$ . Formally, we$inf\sup_{\pi\in\Pi_{\tau\in \mathcal{T}^{r}\nu\in D}}\sup E_{\nu}[(C^{\pi}(\tau)-X^{x,\pi}(\tau))^{+}]$,

inf sup $supE_{\nu}[C^{\pi}(\tau)-X^{x,\pi}(\tau)]$$\pi\in\Pi_{\tau\in \mathcal{T}^{\pi}\nu\in \mathcal{D}}$$= \inf_{\pi\in\Pi}\sup_{\tau\in \mathcal{T}^{\pi}}\sup_{\nu\in \mathcal{D}}E_{\nu}[C^{\pi}(\tau)-X^{0,\pi}(\tau)]-x$$= \inf_{\pi\in\Pi}\sup_{\tau\in \mathcal{T}^{n}}\sup_{\nu\in \mathcal{D}}E_{\nu}?[(C^{\pi}(\tau)-X^{0,\pi}(\tau))^{+}]-x$

Letting$x\downarrow h_{up}$.

, we conjecture$h_{up}= \inf_{\pi\in D}\sup_{\tau\in \mathcal{T}^{\pi}}\sup_{\nu\in \mathcal{D}}E_{\nu}[(C^{\pi}(\tau)-X^{0,\pi}(\tau))^{+}]$

.

(2.1)

Indeed, we have

Theorem 2.1

The super-replication costis erpressed as (2.1). Moreover, if$\pi\in\Pi,$any
and some constant $p=p(\pi, \tau)>1$ , thenfor

$E[|X^{0,\pi}(\tau)|^{p}]<\infty$$h_{up}$$\tau\in \mathcal{T}^{\pi}$$h_{up}=( \inf_{\pi\in\Pi}\sup_{\tau\in \mathcal{T}^{\pi}}\sup_{\nu\in \mathcal{D}}E_{\nu}[C^{\pi}(\tau)-X^{0,\pi}(\tau)])^{+}$

.

(2.2)

112

Remark 2.2 When we defer to the suggestion of Bank&Baum[3], the discounted wealthprocessshould be replaced by$X^{\pi}$$\tilde{X}^{\pi}(t)=X^{\pi}(t)-\frac{L^{\pi}(t)}{B^{\pi}(t)}$,$0\leq t\leq T$,

where $\{L^{\pi}(t), 0\leq t\leq T\}$ is a right-continuous, F-adapted increasing process with$L^{\pi}(O)=0$ .
Here
has the interpretation of the cumulative cost of the liquidity
risk up to time $t\in[0, T]$ . As seen in the proof stage, however, it is clear that if we replacewithin the equations $(2.1)-(2.2)$ , the assertions in the previous theorem remain
to be true without additional assumptions on .
$L^{\pi}(t)$$X^{\pi}$$\tilde{X}^{\pi}$$L^{\pi}$

In order to obtain further sharp results, we are now in a position to make someassumptions:

Assumption 2.3(i) For all $\pi\in\Pi$ there exists$\theta^{\pi}\in D$such that$b^{\pi}(t,\omega)-r^{\pi}(t,w)1_{n}=-\sigma^{\pi}(t,\omega)^{T}\theta^{\pi}(t,\omega)$

$a.e$

.

$(t,\omega)\in[0,T]\cross\Omega$

,

(2.3)

where $1_{n}=(1, \ldots, 1)^{T}\in R^{n}$ .
(ii) For all $\pi\in\Pi$ there exists a constant $p>1$ such that
$E[\int_{0}^{T}|\pi(t)|^{2p}dt]<\infty$

.

(2.4)

In the case of a small investor model, the $process-\theta$ of (2.3) is called the market price ofrisk process and the risk-neutral equivalent martingale measure plays an important role
for the pricing theory, as stated in \S 1.2. Moreover the conditions $(2.3)-(2.4)$ guarantees
that there is no arbitrage opportunity in a standard market of the small investor model.
Therefore it seems natural to as sume $(2.3)-(2.4)$ .
$P_{\theta}$

Corollary 2.4 Under Assumption 2.3, we have$h_{up}=$inf sup $supE_{\nu}[C^{\pi}(\tau)-X^{0,\pi}(\tau)]$ .
$\pi\in\Pi_{\mathcal{T}\in \mathcal{T}^{\pi}\nu\in \mathcal{D}}$

(2.5)

113

3

Dynamic programming equation

3.1

Markov market model

In order to adapt the arguments developed by Soner&Touzi[16]-[18] and Bensoussan$al.[4]$ to our large investor model, we now focus on the Markov case:$h^{\pi}(t)=h(t, B(t),$ $S(t),\pi(t))$

where

and

are

for $h=r,$

,

$b,$

$\sigma$$et$,-valued, bounded functions defined on $[0, T]\cross$. We further assume thatand are LIpschitz functions in thevariable, uniformly in . We consider the special case of European contingent claim:
$r,$

$b$

$\sigma$$R_{+},$ $R^{n}$and$R^{d}\otimes R^{\mathfrak{n}}$$R_{+}\cross R_{+}^{n}\cross R^{n}$

$r,$

$b$

$(\beta, s, \pi)$$\sigma$$t$

and$C^{\pi}(T)=g(B(T), S(T))$

where a non-negative function on$g$$(0, \infty)\cross R_{+}^{n}$$g(\beta, s)\leq c_{0}(\beta^{-l}+\beta^{l}+|s|^{l})$$\mathcal{T}^{\pi}=\{T\}$,satisfies the polynomial growth condition:,$(\beta, s)\in(0, \infty)\cross R_{+}^{n}$for certain constants$l>0$ .
Let
be a compact convex subset which contain the origin. We assume thatis the set of all processes
such that $\pi(t,w)\in K$ a.e. Let denote the support$q\in R^{n}$ . Define$:= \sup_{p\in K}(p^{T}q),$function$c_{0},$$K\subset R^{n}$$\Pi$$\pi\in \mathcal{P}$$\delta$$\delta(q)$$\mathcal{H}(p):=\inf\{\delta(q)-q^{T}p$:$|q|=1\}$ ,$\wedge h(\beta, s):=\sup_{q\in R_{+}^{n}}\{h(\beta,q)-\delta(q-s)\}$,$p\in R^{n}$,$(\beta, s)\in(0, \infty)xR_{+}^{\mathfrak{n}}$,for each function $h:(0, \infty)\cross R_{+}^{n}arrow R$. It is well known that the support function
non-negative, convex and positively homogeneous, and($p\in K$$\Leftrightarrow$$\mathcal{H}(p)\geq 0$

IFMrthermore we assume that$\sigma$$|\sigma(y, \pi)\xi|\geq c|\xi|$

“

and

“

$p\in intK$$\Leftrightarrow$$\mathcal{H}(p)>0$“.

$\delta$

is

(3.1)

satisfies the uniform ellipticity condition:,$y\in[0, T]\cross R_{+}^{n+1},$$\pi\in K,$$\xi\in R^{n},$$(3.2)$

for some constant $c>0$ . This condition guarantees that there exists a bounded function$\theta:[0,T]\cross R_{+}^{n+1}\cross Karrow R^{d}$such that$-\sigma(y,\pi)^{T}\theta(y,\pi)=b(y, \pi)-r(y,\pi)1_{n}$and hence the condition (2.3) holds.

,

114

3.2

Stochastic control problem and dynamic programming equation

Thanks to Girsanov’s theorem, we have
$E_{\nu+\theta}[X^{0,\pi}(T)]=E_{\nu+\theta}[\int_{0}^{T}\pi(u)^{T}diag[S(u)]\sigma(Y(u), \pi(u))^{T}\nu(u)du]$,$\pi\in\Pi,$ $\nu\in \mathcal{D}$,where $Y(u)$ $:=(u, B(u),$ $S(u))^{T}$ and $\theta(u)=\theta(Y(u), \pi(u))$ . Flrom (2.5), therefore, we canderive the stochastic control problem:$U(y)$ $:= \inf_{\pi\in\Pi}\sup_{\nu\in \mathcal{D}}E^{y}[g(B^{\pi}(T), S^{\pi,\nu}(T))-\int^{T}\pi(u)^{T}diag[S^{\pi,\nu}(u)]\sigma(a(u))^{T}\nu(u)du]$for $y=(t, \beta, s)\in[0, T]\cross(0, \infty)\cross(0, \infty)^{n}$ , where $a(u)=(Y^{\pi,\nu}(u), \pi(u))^{T},$solution of the equation$dS(u)=diag[S(u)]\sigma(a(u))^{T}\{\nu(u)du+dW(u)\}$ ,$S^{\pi,\nu}$(3.3)

is a unique,$t\leq u\leq T$

and the suffix $y=(t,\beta, s)$ of means that we have specifled the data $(B^{\pi}(t), S^{\pi,\nu}(t))=$.
Then, since
is linearly independent by means of (3.2), the dynamic programming equation (DPE, for short) for (3.3) is given as follows:$E$$(\beta, s)$$\{\sigma_{i}(y, \pi)\}_{i}$$0=U_{t}(y)+ \inf_{\pi\in K}\sup_{\nu\in R^{d}}\{r(y, \pi)\beta U_{\beta}(y)+(DU(y)-\pi)^{T}diag[s|\sigma(y, \pi)^{T}\nu$$+ \frac{1}{2}n[\{diag[s]\sigma^{T}\sigma(y, \pi)diag[s]\}D^{2}U(y)]\}$(3.4)$=\{\begin{array}{ll}\mathcal{G}^{DU(y)}U(y), if DU(y)\in K,+\infty , if DU(y)\not\in K,\end{array}$for $y=(t, \beta, s)\in[0,T)\cross(0, \infty)^{n+1}$ , whereanddifferentials of with respect to the variable ,$D\varphi$$\varphi$$D^{2}\varphi$

are the first and second order$s$$\mathcal{G}^{\pi}\varphi(y)=\varphi_{t}(y)+r(y, \pi)\beta\varphi_{\beta}(y)+\frac{1}{2},n[\{diag[s]\sigma^{T}\sigma(y, \pi)diag[s]\}D^{2}\varphi(y)]$and Tr[MN]
for symmetric matrices $M=[m_{jj}],$ $N=[n_{ij}]$ . Combining(3.4) with (3.1), we have the DPE$= \sum_{1j=1}^{n}m_{ij}n_{ij}$$\min\{-\mathcal{G}^{DU(y)}U(y),$$\mathcal{H}(DU(y))\}=0$,$y\in[0, T$ )$\cross(O, \infty)^{n+1}$

.

(3.5)

115

We are now in the position to provide some conditions on the payoff function
convex set $K$ .

and

$g$

Assumption 3.1(i) There are constants$c_{0},$$l>0$ andsuch that$\gamma_{0}\in K$$g(\beta, s)\leq c_{0}(\beta^{l}+\beta^{-l})+\gamma_{0}^{T}s$,$(\beta, s)\in(0, \infty)\cross R_{+}^{n}$. $(3.6)$(ii) Either one of the following conditions holds:$\bullet$$g$

is continuous,$or$$\bullet$$\wedge g$

is continuous and$g\wedge=\overline{g_{l}}$

,

(3.7)

where
$g_{*}(z):= \lim_{e\downarrow 0}\inf$$(\iota i\iota)$

For any$q,$ $q’\in R^{n}${ $g(z’)$ : $z’\in(0,$$\infty)\cross R_{+}^{n}$satisfying $q’-q\in R_{+}^{n}$ andand $|z-z’|\leq\epsilon$ },$|q_{k}|=|q_{k}’$$\delta(q)\geq\delta(q’)$, $k=1,$$\ldots$$z\in R_{+}^{n+1}$.

, , we have
$n$

.

(38)

Example 3.2 Let us consider the following two examples.
(i)$K.isthe(38)$closed ball

$B_{\rho}(O)$.

centered at$0$with radius $\rho>0$ . Then $\delta(q)=\rho|q|$ satisfies(ii) (Rectangular constraints) $K=J_{1}\cross\cdots\cross J_{n}$ with
Thensatisfies (3.8).

$J_{k}=[-\eta_{k}, \xi_{k}],$$0\leq\xi_{k}\leq\eta_{k}<\infty$.

$\delta(q)=\sum_{k=1}^{n}(\xi_{k}q_{k}^{+}+\eta_{k}q_{k}^{-})$

The following theorem characterizes the value function
as a viscosity solution of
the DPE (3.5). For the notion and general theory of viscosity solutions, we recommend
readers to refer to the User’s Guide by Crandall et $al.[5]$ .
$U$

Theorem 3.3 Let (3.2) and (3.6) hold. Then$U$

satisfies the following expressions:(i) (Growth condition) For all $(t,\beta, s)\in[0, T]\cross(0, \infty)\cross(0, \infty)^{n}$ ,$0\leq U(t,\beta, s)\leq c_{0}(\beta^{l}+\beta^{-l})e^{l||r||_{\infty}T}+\gamma_{0}^{T}s$.

(ii) (Supersolution) For any smooth test function and local minimizer $y=(t,\beta, s)\in$on, we haveof$\varphi$$[0, T)xR_{+}^{\mathfrak{n}+1}$$(U_{*}-\varphi)$$[0, T]\cross R_{+}^{\mathfrak{n}+1}$$\min\{-\mathcal{G}^{D\varphi(y)}\varphi(y),\sup_{p\in R^{n}}\mathcal{H}(D^{s,p}\varphi(y))\}\geq 0$

.

116

$(1i_{1})$(Subsolution) For any smooth test function$[0, T)\cross R_{+}^{n+1}$

of$(U^{*}-\varphi)$

on$[0, T]\cross R_{+;}^{n+1}$

and local maximizer $y=(t, \beta, s)\in$
we have$\varphi$$\min\{-\mathcal{G}^{D\varphi(y)}\varphi(y))\tilde{\mathcal{H}}(D\varphi(y):s)\}\leq 0$.

(iv) (Terminal condition)$z\in(O, \infty)^{n+1}$ .
,Moreover if in (3.6) is an element of int
and $(3.7)-(3.8)$ are satisfied,$z\in(O,\infty)^{n+1}$$U_{*}(T,z)=U^{*}(T,z)=\wedge
g(z)$
then
,
.
$U_{l}(T, z)\geq\overline{g_{*}}(z)$$(K\cap R_{+}^{n})$$\gamma_{0}$Here the upper (resp. lower) semicontinuous envelope
defined as$U^{*}(y)$$D^{s,p}\varphi$

and$:= \lim_{\epsilon\downarrow}\sup_{0}\{U(y’) :$:=(D_{1}^{s,p}\varphi, \ldots, D_{n}^{s,p}\varphi)^{T}$$\tilde{\mathcal{H}}(p:s)$$U^{*}$(resp.

|y-y’|\leq\epsilon, y’\in[0, T)\cross(0, \infty)^{n+1}\}$with$:=\mathcal{H}(p)1_{\{s\in(0,\infty)^{n}\}}+\infty 1_{ts\in\partial R}\dotplus$of$U_{*}$$:=-(-U)^{*}$ ),$y\in[0, T]\cross R_{+}^{n+1}$$D_{j}^{s,p}\varphi:=D_{\epsilon_{j}}\varphi 1_{\{\epsilon_{j}>0\}}+p_{j}1_{\{\epsilon_{j}=0\}}$

$U$

is

,

,

}.

We conclude the paper with mention of a verification theorem for the DPE (3.5).
Corollary 3.4 (Verification Theorem) Let (3.2) and (3.6) be satisfied, and assume. Let $u\in C^{1,1,2}([0, T)\cross(O, \infty)\cross(0, \infty)^{n})\cap C([0, T]\cross(0, \infty)^{n+1})$ be solution of$g\leq\overline{g_{*}}$$\mathcal{G}^{Du(y)}u(y)=0$

,

)
,
$x(0,
\infty)^{n+1}$$y\in[0,T$ ),$z\in(0, \infty)^{n+1}$ ,$y\in[0, T$$Du(y)\in K$,$u(T, z)=\overline{g_{*}}(z)$,$u(t, z) \leq c_{0}(1+|z|^{l}+\prod_{j=0}^{n}z_{j}^{-\downarrow)},$where$c_{0},$$l>0$ are constants.

Then $u=U$ on $[0, T$ )$\cross(0, \infty)^{n+1}$$z\in(0, \infty)^{n+1}$$\cross(O, \infty)^{n+1}$

.

,

117

References[1] Adachi, T.: Hedging costs for two large investors. Preprint, (2007).
(available from http: $//www.fmu.ac.jp/home/mathema/mathema-e/index.html$ )[2] Adachi, T.: Option on a unit-type closed-end investment fund. Adv. Math. Econ. 9, 1-23
(2006).
[3] Bank, P., Baum, D.: Hedging and portfolio optimization in financial markets with a large
trader. Math. Finance 14, 1-18 (2004).
[4] Bensoussan, A., Touzi, N., Menaldi, J.-L.: Penalty approximation and analytical characterization of the problem of super-replication under portfolio constraints. Asymptotic Analysis
41, 311-330 (2005).

[5] Crandall, M.G., Ishii, H., Lioms, P.-L.: User’s guide to viscosity solutions of second orderpartial differential equations. Bull. Amer. Math. Soc. 27, no.1, 1-67 (1992).
[6] Cuoco, D., Liu, H.: A martingale characterization of consumption choices and hedging
costs with margin requirements. Math. Finance 10, $355arrow 385$ (2000).

[7] Cvitani\v{c}, J., Karatzas, I., Soner, H.M.: Backward stochastic differentlal equations withconstraints on the gains-process. Ann. Probab. 26, 1522-1551 (1999).
[8] Cvitani\v{c}, J., Ma, J.: Hedging options for a large investor and forward-backward SDE’s.
Ann. Appl. Probab. 6, no.2, 370-398 (1996).

[9] Duffie, D.: Dynamic asset pricing thmry. 3rd ed., Princeton Univ. Press 2001.
[10] Frey, R.: Perfect option hedging for a large trader. Finance&Stochast. 2, 115-141 (1998).
[11] Rey, R., Stremme, A.: Market volatillty and feedback effects from dynamic hedging. Math.
Finance 7, 351-374 (1997).

[12] Karatzas, I.: Lectures on the mathematics of finance. CRM Monographs 8, AMS 1997.
[13] Karatzas, I., Shreve, S.E.: Methods of mathematical finance. Springer-Verlag, NY 1998.
[14] Platen, E., Schweizer, M.: On feedback effects kom hedging derivatives. Math. Finance 8,67-84 (1998).

[15] Sekine, J.: On superhedging under delta constraints. Appl. Math. Finance 9, 103-121(2002).

[16] Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. European Math. Soc. 4, 201-236 (2002).
[17] Soner, H.M., Touzi, N.: Stochastic target problems, dynamic programming and viscosity
solutions. SIAM J. Control&Opt. 41, 404-424 (2002).
[18] Soner, H.M., Touzi, N.: The problem of super-replication under constraints. Paris-Princeton
Lectures on Mathematical Finance 2002, LNM 1814, Springer-Verlag, 133-172 (2003).