数理解析研究所講究録
第 1818 巻 2012 年 17-32

17First Passage Time in Real Options*

Keiichi Tanaka
Graduate School of Social Sciences
Tokyo Metropolitan University

1

Introduction

We consider irreversible investment problems with regime switching feature under a monopoly
setting. Several parameters describing the economic environment vary according to a regime
switching with general number of states.
firm seeks an optimal timing to invest an irreversible
project while observing the potential profits. We present a systematic procedure to derive
the value function via solving a system of simultaneous ordinary differential equations withknowledge of linear algebra. It will enable us to investigate a comparative analysis of theinvestment problem. The contribution of this paper is a natural extension of Guo and Zhang(2004) and Jobert and Rogers (2006) to a real option problem with the general number of regime
states. Furthermore, we obtain an analytical expression of an expectation of a payoff at the firstpassage time to the stop region by applying the Dirichlet problem and the aforementioned
technique.
In the literature, the value function of a typical real option problem is usually calculated
by first guessing the form and then solving the unknown coefficients as in Dixit and Pyndick$A$(1994), Guo et al. (2005) for $S=2$ , whereis the number of regimes, and Grenadier andWang (2007) for general $S\geq 2$ . For American put options, Guo and Zhang (2004) take similar
approach for $S=2$ . Apparently one of the drawbacks is that there are no clues why such a form
is taken. Jobert and Rogers (2006), Jiang and Pistorius ( $200S$ ) utilize Wiener-Hopf factorization
to obtain the value function for $S\geq 2$ . They present explicit forms of the value function up
to exponential matrix. Similar problems with the regime switching features are discussed in
Boyarchenko and Levendorski (2008, 2009) who also make use of the Wiener-Hopf factorization
in a different form. Due to the complicated form, further analysis including comparative
analysis seems difficult without an insightful expression of the value function.
Our approach is simple and straightforward: solve a system of simultaneous ordinary differential equations with appropriate conditions directly on each interval of the thresholds. Due tothe form, knowledge of linear algebra helps a lot in the derivation. We do not rely on a “guess
functional form” nor the Wiener-Hopf factorization that is more technical. It is found that thevalue function is represented by the eigenvalues and the eigenvectors of a coefficient matrix.
With the same technique we can carry out a calculation of the expected and discounted valueof a payoff at the first passage time by applying the Dirichlet problem in the two-dimensional
space. The original value function can be decomposed with these expectations. The smooth
pasting conditions at the boundary between the continuation region and the stop region arerecovered by the optimality conditions of the thresholds. The detailed derivation is provided for
$S$

$g$

*Financial support by MEXT Grand-in-Aid and the Zenginkyo Foundation are acknowledged.

18
the case of two regimes. These results will help us to deepen our understanding of the investment
problem.

Setup

2

on infinite time horizon. Let $J=\{J(t)\}$ be a
We work on a probability spacecontinuous-time Markov chain on a finite state space $E=\{1,2, \cdots, S\}.$ $J(t)$ is interpreted
as a regime or a state of the economy at time . The intensity matrix of the regime switching isgiven by $Q$$(\Omega, \mathcal{F}, \mathbb{P})$$t$$Q=(q_{ij})_{i,j\in E}, q_{ii}=-\sum_{j\in E\backslash \{i\}}q_{ij}$

.

(2.1)

The process $X=\{X(t)\}$ satisfies$dX(t) = \mu_{J(t)}X(t)dt+\sigma_{J(t)}X(t)dW_{t}, X(O)=x$ ,(2.2)

are finite constants for each
andwhere $W=\{W_{t}\}$ is a standard Brownian motion,with $\mathcal{F}_{t}=\sigma(W_{S}, J(s), 0\leq s\leq t)$ .
$j\in E$ . Denote the filtration generated by $(W, J)$ as
The firm has a chance to start a project to make a product as a monopoly of the productwhose revenue depends on the state variables $(X(t), J(t))$ of the economy. We assume that thewhen the regime state
firm has a technology to enter into the project by paying the costis , and after the investment the firm obtains the instant revenue of $D_{J(t)}X(t)$ at time fromthe project, where $D_{i}K_{i}(i\in E)$ are positive constants. One may interpret that $X(t)$ is thebe the (potential) demand quantity for theunit price of the products from the project and$\sigma_{j}>0$$\mu_{j}$$\{\mathcal{F}_{t}\}$$K_{i}$$t$$i$$D_{i}$

products in the economy.
denotes the zero matrix of order andIn this paper a matrix is represented in bold.
denotes the identity matrix of order . An element of a matrix $A$ $=(a_{ij})$ is denoted by$a_{ij}=\{A\}_{ij}$ . Let us denote vectors, matrices and functions$n$$O_{n}$$n$$I_{n}$$e_{i} = (0, \cdots, 0,1,0, \cdots, 0)^{T}\in \mathbb{R}^{S}, 1_{S}=(1, \cdots, 1)^{T}\in \mathbb{R}^{S},$

$D$

$=$

$(D_{1}, \cdots, D_{S})^{T}$$M=$ diag $[\mu_{1}, \cdots, \mu_{S}],$$\Lambda(\beta) = (\begin{array}{llll}g_{1}(\beta) q_{12} \cdots q_{1S}q_{21} g_{2}(\beta) \cdots q_{2S}\vdots \vdots \ddots \vdots q_{S1} q_{S2} \cdots g_{S}(\beta)\end{array})$

(2.3)

,

.

$g_{i}( \beta) = \frac{1}{2}\sigma_{i}^{2}\beta^{2}+(\mu_{i}-\frac{1}{2}\sigma_{i}^{2})\beta+q_{ii}-r$(2.4), which has the L\’evy exponent. Then $\ln X(t)=L_{t}^{(J(t))}$ evolves like a diffusion
while the regime does not switch, and the matrix (2.3) is expressed as

For each $i\in E$ , consider a L\’evy process$L_{t}^{(i)}=( \mu_{i}-\frac{1}{2}\sigma_{i}^{2})t+\sigma_{i}W_{t}$$\theta^{(i)}(z)=(\mu_{i}-\frac{1}{2}\sigma_{i}^{2}-r)z+\frac{1}{2}\sigma_{i}^{2}z^{2}=g_{i}(z)-q_{ii}+r$$\Lambda(\beta)=$diag $[\theta^{(k)}(\beta);k\in E]+Q-rI_{S}.$19
For a simple notation it is convenient to introduce a “truncating” operator
square matrices
defined by$H_{n}$

on $S\cross S$$(a_{ij})_{1\leq i,j\leq S}$$H_{n}((a_{ij})_{1\leq i,j\leq S})=(a_{ij})_{1\leq i,j\leq n}$

The truncating operator
reflects our focusing onamong regimes as discussed later.
We assume the following properties;$H_{n}$

$n$

.

(2.5)

regime states in a continuation region$S$

Assumption 1

1. $Q$ has a property that$\forall i\in E,$$\exists j\in E,$$j\neq i,$$(q_{ij}, q_{ji})\neq(0,0)$.

2. $r> \max\{\max\mu_{i},$ $0\}.$3. The matrices $H_{n}(rI_{S}-M-Q)$ and $H_{n}(rI_{S}-Q)$ are invertible for all $n\in E.$
The property 1 of the above assumption is to restrict our analysis to non-redundant cases ofregime switching. Each regime has (non-zero) chance to move to another regime and $/or$ tobe transferred from another regime\dagger . The property 2 guarantees a convergence of total revenue
and other properties. The property 3 is due to a technical reason to make our discussion simple.
Note that the matrices mentioned in the above are expressed withas$a$$\Lambda(\beta)$$rI_{S}-M-Q=-A(1) , rI_{S}-Q=-\Lambda(0)$ .

3

Value function

The firm seeks the optimal timing of the investment. When the current regime state is , thevalue function
is defined by$i$$V_{i}$$V_{i}(x) = \max_{\tau}\mathbb{E}[l^{\infty}e^{-ru}D_{J(u)}X(u)du-e^{-r\tau}K_{J(\tau)}|X(0)=x, J(0)=i].$

The following lemma gives useful expressions for the calculation of the value function.

Lemma 1(1)$\mathbb{E}[e^{-rT}X(T)^{\beta}|\mathcal{F}_{t}]=X(t)^{\beta}e_{J(t)}^{T}\exp(\Lambda(\beta)(T-t))1_{S},$(2)$\mathbb{E}[l^{\infty}e^{-ru}D_{J(u)}X(u)du|\mathcal{F}_{t}]=e^{-rt}\alpha_{J(t)}D_{J(t)}X(t)$$\beta\in \mathbb{R},$

,

where
$\alpha_{i}D_{i}=e_{i}^{T}(rI_{S}-M-Q)^{-1}D.$$i\neq j$ , that is, each regime has a non-zerois called to be irreducible if
forchance to moveto another regime and to be transferred from another regime. The property 1 is weaker than the irreducibility.
It follows that some reducible matrices satisfy the property 1.
$\uparrow Q$$q_{ij}\neq 0$$\forall i,j\in E,$

20
By Lemma 1, the value function at the regime is reduced to$i$$V_{i}(x) = \max_{\tau}E[e^{-r\tau}(\alpha_{J(\tau)}D_{J(\tau)}X(\tau)-K_{J(\tau)})|X(0)=x, J(0)=i].$As discussed in Jobert and Rogers (2006) and Guo and Zhang (2004), the candidate of the
optimal stopping timemust be in a form of$\tau^{*}$$\tau^{*}=\min_{j\in E}\tau_{j}, \tau_{j}=\inf\{t>0:X(t)\geq x_{j}, J(t)=j\}$with some positive $x_{j}(j\in E)$ . We will obtain the explicit form of the value function by assumingthat the order of the thresholds is(3.1)$x_{S}\leq x_{S-1}\leq\cdots\leq x_{2}\leq x_{1}$in what follows. Namely the regime state is the best and the regime state 1 is the worst forstarting the project. In case that (3.1) is not satisfied, the following procedure must be carriedout after the regime indices are interchanged appropriately.
(continuation regimes) are in the conOn the n-th interval $(x_{n+1}, x_{n})$ the regimes 1,(stop regimes) are in the stop region. Hence,tinuation region while the regimes $n+1,$$S$$\cdots,$ $n$$\cdots,$$S$

by thethe state will enter into a stop region either if $X(t)$ moves upward gradually beyond
Brownian motion without regime switches or if the regime $J(t)$ is suddenly switched to eitherof the stop regimes by a regime switch.
The value function will take a different functional form on each interval of the thresholds as$x_{J(t)}$$V_{i}(x)=\{\begin{array}{ll}V_{i}^{(0)}(x) if x\in[x_{1}, \infty) , V_{i}^{(n)}(x) if x\in[x_{n+1}, x_{n}) , (n=1,2, \cdots, S-1) ,V_{i}^{(S)}(x) if x\in(O, xs) .

\end{array}$

We will calculate
for each on n-th interval by starting from $n=0$ and moving on to$n=S$ . When $x\in[x_{1}, \infty)$ , the state is in the stop region at any regime and it is optimal for thefirm to start the project immediately since the price $X(t)$ is high enough. Hence we have$V_{i}^{(n)}(x)$

$i$

(3.2)

$V_{i}^{(0)}(x)=\alpha_{i}D_{i}x-K_{i}, i\in E$

For $x\in[x_{n+1}, x_{n})(n=1,2, \cdots, S-1)$ , the firm will enter into the project if the regime is either, otherwise she should wait. Thus, the value function $V_{i}^{(n)}(1\leq i\leq n)$ satisfies
of $n+1,$$\cdots,$$S$$\frac{1}{2}x^{2}\sigma_{i}^{2}\frac{d^{2}}{dx^{2}}V_{i}^{(n)}(x)+x\mu_{i}\frac{d}{dx}V_{i}^{(n)}(x)-rV_{i}^{(n)}(x)+\sum_{j\in E\backslash \{i\}}q_{ij}(V_{j}^{(n)}(x)-V_{i}^{(n)}(x))=0$, (3.3)obeys the same $ODE$
and $V_{i}^{(n)}(x)=\alpha_{i}D_{i}x-K_{i}$ for $n+1\leq i\leq S$ . Finally, for $x\in(O, xs),$
as (3.3) with $n=S$ . The first three terms of (3.3) represents a change of the value function
due to a movement of Brownian motion and the last term represents a change due to a regime
switching. The optimality condition requires that the sum of these changes must be zero.
$V_{i}^{(S)}$21
In summary, we must solve simultaneous ODEs on an interval $[x_{n+1}, x_{n}),$ $(n=1,2,$$\cdots,$

$S-$

1 ,
$)$

$\mathcal{A}_{1}V_{1}^{(n)}(x)+\sum_{1\leq j\leq n,j\neq 1}q_{1j}V_{j}^{(n)}(x)=-\sum_{n+1\leq j\leq S}q_{1j}V_{j}^{(n)}(x)$$\mathcal{A}_{2}V_{2}^{(n)}(x)+\sum_{1\leq j\leq n,j\neq 2}q_{2j}V_{j}^{(n)}(x)=-\sum_{n+1\leq j\leq S}q_{2j}V_{j}^{(n)}(x)$:$\mathcal{A}_{n}V_{n}^{(n)}(x)+\sum_{1\leq j\leq n,j\neq n}q_{nj}V_{j}^{(n)}(x)=-\sum_{n+1\leq j\leq S}q_{nj}V_{j}^{(n)}(x)$,with the value matching condition and the smooth pasting conditions at $x=x_{n},$ $x_{n+1}$ , whereis a differential operator defined by$\mathcal{A}_{i}$$\mathcal{A}_{i}f(x)=\frac{1}{2}x^{2}\sigma_{i}^{2}\frac{d^{2}}{dx^{2}}f(x)+x\mu_{i}\frac{d}{dx}f(x)+(q_{ii}-r)f(x)$.

It is symbolic to represent them in a form of matrix as$(\begin{array}{llll}\mathcal{A}_{1} q_{12} \cdots q_{1n}q_{21} \mathcal{A}_{2} \cdots q_{2n}| | \ddots |q_{n1} q_{n2} \cdots \mathcal{A}_{n}\end{array})(\begin{array}{l}V_{1}^{(n)}(x)V_{2}^{(n)}(x)|V_{n}^{(n)}(x)\end{array})=-(\begin{array}{llll}q_{1,n+1} q_{1,n+2} \cdots q_{1S}q_{2,n+1} q_{2,n+2} \cdots q_{2S}| | \ddots |q_{n,n+1} q_{n,n+2} \cdots q_{nS}\end{array})(\begin{array}{l}V_{n+2}^{(n)}(x)V_{n+1}^{(n)}(x)|V_{S}^{(n)}(x)\end{array})$

.

(3.4)

The functions on the LHS are unknown and to be solved while ones on the RHS $V_{j}^{(n)}(x)=$$\alpha_{j}D_{j}x-K_{j}(j\in\{n+1, \ldots, S\})$

are known. As for an interval $(0, x_{S})$ a similar system of ODEs

must be solved$(\begin{array}{llll}\mathcal{A}_{1} q_{12} \cdots q_{1S}q_{21} \mathcal{A}_{2} \cdots q_{2S}| | \ddots |q_{S1} q_{S2} \cdots \mathcal{A}_{S}\end{array})(\begin{array}{l}V_{2}^{(S)}(x)V_{1}^{(S)}(x)|V_{S}^{(S)}(x)\end{array})=0_{S}.$A set of ODEs to be solved is dependent of the interval of . We study the equations (3.4)
ondefined on the interval $(x_{n+1}, x_{n})(n=1,2, \cdots, S)$ by modifying
the RHS and the interval of appropriately in the case of $n=S$ . Since we know the solution$x$$V_{i}^{(n)}(x)(i=1,2, \cdots, n)$$x$

for $i=n+1,$, the equations of the remaining
for $1\leq i\leq n$
are reduced to simultaneous second-order linear ODEs. It follows that the solution
isdecomposed with the general solution
and the special solution$n.$for each $i=1,2,$
The special solution
is easily found to be a linear function $v_{i}^{(n)}(x)=a_{i}^{(n)}x+b_{i}^{(n)}$ , where$V_{i}^{(n)}(x)=\alpha_{i}D_{i}x-K_{i}$$\cdots,$$S$$V_{i}^{(n)}$$V_{i}^{(n)}$$\tilde{V}_{i}^{(n)}$$v_{i}^{(n)}$$\cdots,$$v_{i}^{(n)}$22the coefficients$a^{(n)}=(a_{1}^{(n)}, \cdots, a_{n}^{(n)})^{T},$ $b^{(n)}=(b_{1}^{(n)}, \cdots, b_{n}^{(n)})^{T}$are given by$a^{(n)} = H_{n}(rI_{S}-M-Q)^{-1}\ovalbox{\tt\small REJECT}_{\sum_{=n+1}^{j=n+1}q_{nj}\alpha_{j}D_{j}}^{\sum q_{2j}\alpha_{j}D_{j}}j=sn\sum^{S}S+1q_{1j}\alpha_{j}D_{j}]$$b^{(n)} = -H_{n}(rI_{S}-Q)^{-1}\ovalbox{\tt\small REJECT}_{\sum_{=n+1^{q_{nj}K_{j}}}^{\sum.q_{1j}K_{j}}}^{j=n+1}j=ns\sum_{S}^{S}+1.q_{2j}K_{j})$

,

(3.5)

,

where the inverse matrices are guaranteed to exist by Assumption 1.
Now we tum our eyes to the general solution. Let us change the variable $y=\ln x$ andintroduce auxiliary functions. Then the equations forthe general solution part of (3.4) can be rewritten as a system of first-order ODEs,$\tilde{V}_{i}^{(n)}$$\overline{V}_{i}^{(n)}(y)=\tilde{V}_{i}^{(n)}(e^{y}),$$\overline{W}_{i}^{(n)}(y)=\frac{d}{dy}\overline{V}_{i}^{(n)}(y)$$\frac{d}{dy}(\overline{\frac{V}{W}}((nn))^{(y)}(y))=\Gamma_{n}(\overline{\frac{V}{W}}((nn))^{(y)}(y))$

,

(3.6)

where
$\Gamma_{n}$$=$$(\begin{array}{ll}O_{n} I_{n}R_{n} C_{n}\end{array})\in \mathbb{R}^{2n\cross 2n},$$\Sigma_{n}=\frac{1}{2}$diag$[\sigma_{1}^{2}, \cdots, \sigma_{n}^{2}]\in \mathbb{R}^{n\cross n},$$R_{m} = \Sigma_{n}^{-1}H_{n}(rI_{S}-Q))=-2(\begin{array}{llll}\frac{q_{11}-r}{\frac{q_{21}\sigma_{1}^{2}}{\sigma_{2}^{2}}}\cdots \frac{q_{22}^{\frac{q_{12}}{\sigma_{\underline{l}}^{2}}}r}{\sigma_{2}^{2}}\cdots \cdots \frac{}{}\frac{q_{2n}q_{1n}\sigma_{1}^{2}}{\sigma_{2}^{2}}| | \ddots |\frac{q_{nl}}{\sigma_{n}^{2}} \frac{q_{n2}}{\sigma_{n}^{2}} \cdots \frac{q_{nn}-r}{\sigma_{n}^{2}}\end{array})\in \mathbb{R}^{n\cross n},$$C_{n}$$=$$\Sigma_{n}^{-1}H_{n}(\Sigma_{S}-M)=$diag$[1- \frac{2\mu_{1}}{\sigma_{1}^{2}},$$\cdots,$$1- \frac{2\mu_{n}}{\sigma_{n}^{2}}]\in \mathbb{R}^{n\cross n}.$Thus, the solution is given by$(\overline{\frac{V}{W}}((nn))^{(y)}(y))=\exp((y-y_{0})\Gamma_{n})(\begin{array}{l}\overline{V}^{(n)}(y_{0})\overline{W}^{(n)}(y_{0})\end{array})$

with some $y0$ from the boundary conditions when the exponential matrix $\exp((y-y_{0})\Gamma_{n})$ isavailable. If the coefficient matrix
is diagonalizable, it is straightforward to solve and obtain$\Gamma_{n}$23an explicit representation of the solution of the system of ODEs (3.6). Otherwise, one canproceed similarly by making use of the Jordan normal form that is guaranteed to exist for any
square matrix by the theory.
By the knowledge of linear algebra the characteristic function of ofis obtained as$\Gamma_{n}$$\det(\begin{array}{ll}O_{n}-\beta I_{n} I_{n}R_{n} C_{n}-\beta I_{n}\end{array}) = f_{n}( \beta)\prod_{j=1}^{n}(\frac{1}{2}\sigma_{J}^{2}\prime)^{-1}$where$f_{n}(\beta) = \det(\Sigma_{n}(I_{n}\beta^{2}-C_{n}\beta-R_{n}))=detH_{n}(\Lambda(\beta))$

.

(3.7)

Thus, the eigenvalues are the solutions of $f_{n}(\beta)=0$ . In this paper we make the following
assumption for simple and useful results.

Assumption 22.

1. For $n=1,2,$$\cdots,$$S-1,$

has $2S$ distinct eigenvalues such thatare strictly negative.

$\Gamma_{S}$$\Gamma_{n}$has $2n$ distinct eigenvalues$\beta_{1}^{(S)},$$\beta_{S}^{(S)}$$\cdots,$$\beta_{1}^{(n)},$are strictly positive and$\beta_{2n}^{(n)}.$$\cdots,$$\beta_{S+1}^{(S)},$$\beta_{2S}^{(S)}$$\cdots,$

If the eigenvalues become complex numbers or duplicated so that the above assumption is notsatisfied, the following discussion can be accordingly modified by considering the Jordan normal
form as mentioned before.
By Assumption 2 there exist distinct eigenvalues $\beta_{j}^{(n)}(1\leq j\leq 2n)$ . Since the upper rightblock ofis , the eigenvector for the eigenvalue
must be in the form$\ddagger$$\Gamma_{n}$$I_{n}$$\beta_{j}^{(n)}$$\tilde{u}_{j}^{(n)}=(_{\beta_{j}^{(n)^{j}}u_{j}^{(n)}}u^{(n)})\in \mathbb{R}^{2n},$with some non-zero vector$u_{j}^{(n)}\in \mathbb{R}^{n}$satisfying$H_{n}(A(\beta))u_{j}^{(n)}=0_{n}$Such a vector$(n)$$u_{j}$

exists for each

$j$

(3.8)

because the determinant of the coefficient matrix on the

LHS of (3.8) is equal to $f_{n}(\beta_{j}^{(n)})=0$ by definition of$\Gamma_{n}=(\begin{array}{l}U^{(n)}U^{(n)}B^{(n)}\end{array})$

.

diag

$[\beta_{1}^{(n)},$$\cdots,$$\beta_{j}^{(n)}$. Thus,$\Gamma_{n}$is diagonalized as$\beta_{2n}^{(n)}](\begin{array}{l}U^{(n)}U^{(n)}B^{(n)}\end{array})$

where$U^{(n)}=$$(u_{1}^{(n)}$$u_{2}^{(n)}$...

$u_{2n}^{(n)})\in \mathbb{R}^{n\cross 2n},$$B^{(n)}=$diag$[\beta_{1}^{(n)},$$\cdots,$$\beta_{2n}^{(n)}]\in \mathbb{R}^{2n\cross 2n}.$Due to the duplicated eigenvalues, a Jordan normal form appears in the value function in Grenadier andWang (2007) in a context of hyperbolic discounting which may be able to be modeled with a regime switchingand our discussion may be applicable to.
$\ddagger$

24
The matrix$\beta_{2n}^{(n)}u_{2n}^{(n))=}u_{2n}^{(n)}(\tilde{u}_{1}^{(n)}$$\beta_{2}^{(n)}u_{2}^{(n)}u_{2}^{(n)}$$(\begin{array}{l}U^{(n)}U^{(n)}B^{(n)}\end{array})=(_{\beta_{1}^{(n)}u_{1}^{(n)}}u_{1}^{(n)}$$\tilde{u}_{2}^{(n)}$...

$\tilde{u}_{2n}^{(n)})\in \mathbb{R}^{2n\cross 2n}$$\ldots$

is invertible since the eigenvalues of$\Gamma_{n}$are distinct so that the corresponding eigenvectors$\tilde{u}_{j}^{(n)}$are linearly independent.
Then we can solve the system of ODEs (3.6) as$(\overline{\frac{V}{W}}((nn))^{(y)}(y))=(\begin{array}{l}U^{(n)}U^{(n)}B^{(n)}\end{array})$diag$[e^{\beta_{1}^{(n)}y},$$\cdots,$$e^{\beta_{2n}^{(n)}y}]A^{(n)},$. By adding the special solutions, we have the vector ofwith some constant vector$V^{(n)}(x)=(V_{1}^{(n)}(x), \cdots, V_{n}^{(n)}(x))^{T}$ at each regime on the interval $[x_{n+1}, x_{n})$the value functions
given as$A^{(n)}\in \mathbb{R}^{2n}$(3.9)$V^{(n)}(x) = U^{(n)}X^{(n)}(x)A^{(n)}+v^{(n)}(x)$ ,where$X^{(n)}(x)$

$=$

diag

$[x^{\beta_{1}^{(n)}},$$\cdots,$$x^{\beta_{2n}^{(n)}}],$$v^{(n)}(x)=a^{(n)}x+b^{(n)}.$,will be determined
and unknown vectors
Unknown boundaries
by the value matching conditions, the smooth pasting conditions and the values at $x=0$ . We
first and moving downward to $xs$ as follows.
will investigate them by looking at
are
The value matching conditions at $x=x_{n},$ $V_{i}^{(n)}(x_{n})=V_{i}^{(n-1)}(x_{n})$ for $i=1,$
represented by -dimensional vectors as$A^{(1)},$$x_{S}\leq\cdots\leq x_{1}$$A^{(S)}$$\cdots$$x_{1}$$\cdots,$ $n$$n$. (3.10)$U^{(n)}X^{(n)}(x_{n})A^{(n)}+v^{(n)}(x_{n})=(\begin{array}{ll}U^{(n-1)}X^{(n-1)}(x_{n})A^{(n-1)}+ v^{(n-1)}(x_{n})\alpha_{n}D_{n}x_{n}-K_{n} \end{array})$

Similarly, the smooth pasting conditionsquire$x_{n} \frac{d}{dx}V_{i}^{(n)}(x_{n})=x_{n}\frac{d}{dx}V_{i}^{(n-1)}(x_{n})$for $i=1,$$U^{(n)}dX^{(n)}(x_{n})A^{(n)}+a^{(n)}x_{n}=(\begin{array}{ll}U^{(n-l)}dX^{(n-1)}(x_{n})A^{(n-1)}+ a^{(n-1)_{X_{n}}}\alpha_{n}D_{n}x_{n} \end{array})$$\cdots,$ $n$,(3.11)

where$dX^{(n)}(x)=$ diag$[\beta_{1}^{(n)}x^{\beta_{1}^{(n)}},$$\cdots,$$\beta_{2n}^{(n)}x^{\beta_{2n}^{(n)}}]=B^{(n)}X^{(n)}(x)$,$a^{(S)}=0_{S}.$By coupling these conditions (3.10), (3.11) into one vector and making use of a relationship$(\begin{array}{l}U^{(n)}X^{(n)}(x)U^{(n)}dX^{(n)}(x)\end{array})=(\begin{array}{l}U^{(n)}U^{(n)}B^{(n)}\end{array})X^{(n)}(x)$

,

re-

25
is represented with a function of$A^{(n)}$$x_{n}$

and $A^{(n-1)}$ as$A^{(n)} = X^{(n)}(x_{n}^{-1})(\begin{array}{l}U^{(n)}U^{(n)}B^{(n)}\end{array})$(3.12)$\cross[(\begin{array}{ll}U^{(n-1)}X^{(n-1)}(x_{n})A^{(n-1)}+ v^{(n-1)}(x_{n})\alpha_{n}D_{n}x_{n}-K_{n} U^{(n-1)}dX^{(n-1)}(x_{n})A^{(n-1)}+ a^{(n-1)}x_{n}\alpha_{n}D_{n}x_{n} \end{array})-(\begin{array}{l}v^{(n)}(x_{n})a^{(n)}x_{n}\end{array})]$for $n=2,$$A^{(1)}$$A^{(S)}$$\cdots$, $S-1$ , and,(3.13)$=$$X^{(1)}(x_{1}^{-1})(\begin{array}{l}U^{(1)}U^{(1)}B^{(1)}\end{array})(\begin{array}{l}\alpha_{1}D_{1}x_{1}-K_{1}-v^{(1)}(x_{1})\alpha_{1}D_{1}x_{1}-a^{(1)_{X_{1}}}\end{array})$$=$$X^{(S)}(x_{S^{1}})(\begin{array}{l}U^{(S)}U^{(S)}B^{(S)}\end{array})(\begin{array}{ll}U^{(S-1)}X^{(S-1)}(x_{S})A^{(S-1)}+ v^{(S-1)}(x_{S})\alpha s^{D_{S^{X}S}-K_{S}} U^{(S-1)}dX^{(S-1)}(x_{S})A^{(S-1)}+ a^{(S-1)}x_{S}\alpha_{S}D_{S}x_{S} \end{array})$.

(3.14)

Therefore, we can represent unknown vectors
as functions ofrecursively.
Furthermore, on $(0, x_{S}], we want to$ impose another condition $\lim_{xarrow 0}V_{i}^{(S)}(x)=0$ for all
in order to make the value function finite. It implies that the coefficient ofcorresponding
to negative eigenvalues
must be zero,$A^{(1)},$$\cdots,$$A^{(S)}$$x_{1},$$\cdots,$ $x_{S}$$i$$A^{(S)}$$\beta_{S+1}^{(S)},$$\beta_{2S}^{(S)}$$\cdots,$$(0_{S} I_{S})A^{(S)}=0_{S}$.

(3.15)

This is a set of equations that
unknown constants
must satisfy. Apparently(3.15) is a system of complicated algebraic equations, hence they must be solved numerically.
In case that the numerical solution doesn’t satisfy the order condition (3.1), the indices of theregimes must be interchanged.
Then, by noting that $v^{(S)}(x)=0$ , the value function on $(0, x_{S})$ can be expressed with termswith positive eigenvalues as
$S$

$S$

$x_{1},$$V^{(S)}(x) = U_{S}X_{S}(x)A_{S}$$\cdots,$ $x_{S}$,(3.16)

where$U_{S}=(u_{1}^{(S)}\cdots u_{S}^{(S)}) , X_{S}(x)=diag[x^{\beta_{1}^{(S)}}, \cdots, x^{\beta_{S}^{(S)}}], A_{S}=(I_{S} 0_{S})A^{(S)}.$As a summary, we obtain the main result.
Proposition 1 Suppose that Assumption 1 and 2 hold, andThen the value function is given by (3.9) and (3.16).

$x_{1},$$\cdots,$ $x_{S}$satisfy (3.1) and (3.15).

26
4First passage time

With the same technique we can carry out a calculation of the expected and discounted valueof a payoff at the first passage time by applying the Dirichlet problem in the $tWf^{f}dimensiona1$
space.
Once the optimal stopping time is determined and the thresholds $xs\leq\cdots\leq x_{1}$ are fixed,
the value function can be decomposed by functions related to the regime at the first passage$\tau^{*}$

time as$V_{i}(x) = \max \mathbb{E}[e^{-r\tau}(\alpha_{J(\tau)}D_{J(\tau)}X(\tau)-K_{J(\tau)})|X(0)=x, J(0)=i]$$\tau$$= \sum_{k\in E}E[1_{\{\tau=\tau_{k}\}}e^{-r\tau}.

$=$

where

(\alpha_{J(\tau)}D_{J(\tau)}X_{\tau}*-K_{J(\tau)})|X(0)=x, J(0)=i]$$\sum_{k\in E}[\alpha_{k}D_{k}F_{i}^{k}(x)-K_{k}G_{i}^{k}(x)]$$\tau^{*}=\min_{k\in E}\tau_{k},$

(4.1)

,

$\tau_{k}=\inf\{t>0:X(t)\geq x_{k}, J(t)=k\}$and$F_{i}^{k}(x)$$=$$\mathbb{E}[e^{-r\tau^{*}}M_{F}^{k}(X_{\tau}\cdot, J(\tau^{*}))|X(O)=x,$$J(O)=i],$$M_{F}^{k}(x,j)=x\delta_{jk}$$G_{i}^{k}(x)$$=$$E[e^{-r\tau^{*}}M_{G}^{k}(X_{\tau}*, J(\tau^{*}))|X(O)=x,$$J(O)=i],$$M_{G}^{k}(x,j)=\delta_{jk}$

.

,

(4.2)

(4.3)

after starting from $(X$ (0) $, J(0))=$is the discounted expected value of the payoff of$(x, i)$ . Similarly,
is one of the payoff of
The purpose of this section is to obtain the explicit form of these decomposing functions. Our plan and idea are as follows. The functions defined by the expectation in (4.2)
and (4.3) are applicable to the Dirichlet problem thanks to the functional form in the definition.
However, the domain must be set appropriately in the two-dimensional space. Then we obtaina set of ODEs which can be solved with the same technique as in the previous section. Finally,
we verify the decomposition (4.1) and the smooth pasting conditions at the boundary, which
are not imposed in the Dirichlet problem.
has the regime-dependent thresholds, we need to rename them
Since the stopping time
$F_{i}^{k}(x)$$X_{\tau}*1_{\{J(\tau)=k\}}$$G_{i}^{k}(x)$$F_{i}^{k},$$1_{\{J(\tau^{*})=k\}}.$$G_{i}^{k}$$\tau^{*}$

on each relevant interval as$F_{i}^{(k,0)}(x)$$F_{i}^{k}(x)$

$=$

$\{$

$F_{i}^{(k,n)}(x)$$F_{i}^{(k,S)}(x)$$G_{i}^{k}(x)$

$=$

,
,
,

$x\in[x_{1}, \infty)$$x\in[x_{n+1}, x_{n})$,$1\leq n\leq S-1$

$x\in(0, xs)$$\{\begin{array}{l}G_{i}^{(k,0)}(x) , x\in[x_{1}, \infty)G_{i}^{(k,n)}(x) , x\in[x_{n+1}, x_{n}) , 1\leq n\leq S-1\end{array}$$G_{i}^{(k,S)}(x)$,$x\in(0, xs)$

On the n-th interval, the regimes $i>n$ are in the stop region so that
$F_{i}^{(k,n)}(x)=\delta_{ik}x\equiv M_{F}^{k}(x, i)$,$G_{i}^{(k,n)}(x)=\delta_{ik}\equiv M_{G}^{k}(x, i)$

In order to obtain the explicit expression of the functions
following Dirichlet problem.

,$F_{i}^{(k,n)}$$(i>n)$ .

and$G_{i}^{(k,n)}$(4.4), we apply the27Lemma 2 Consider an open set$\tau_{\mathcal{D}}=\inf\{t :. Define the first passage timeawith $Y_{0}=x\in C$ , and define $F(x)=$of Markov process on
$M$ :
a
given
continuous. Then solves the Dirichletforfunction$C\subset \mathbb{R}^{d}$

and
$Y$

Y_{t}\in \mathcal{D}\}$$\mathbb{E}[e^{-r\tau_{\mathcal{D}}}M(Y_{\tau_{\mathcal{D}}})]$$\mathcal{D}=\mathbb{R}^{d}\backslash C$$\mathbb{R}^{d}$$\partial Carrow \mathbb{R}$$F$

problem$\mathbb{A}_{Y}F=rF$

where$\mathbb{A}_{Y}$is the infinitesimal opemtor

in

$C,$

$F|_{\partial C}=M$$\mathbb{A}_{Y}F(x)=\lim_{t\downarrow 0}\frac{\mathbb{E}[F(Y_{t})|Y_{0}=x]-F(x)}{t}.$Proof.

See, for example, Peskir and Shiraev (2006).
For a regime switching diffusion $dX(t)=\mu_{J(t)}X(t)dt+\sigma_{J(t)}X(t)dW_{t}$, it is known that$\mathbb{A}_{X}F_{i}(x) = \frac{1}{2}\sigma_{i}^{2}x^{2}\frac{d^{2}F_{i}}{dx^{2}}(x)+\mu_{i}x\frac{dF_{i}}{dx}(x)+\sum_{j=1,j\neq i}^{s}q_{ij}(F_{j}(x)-F_{i}(x))$.

The continuation region of our problem and the relevant sets are given by$\mathcal{B} = (0, x_{1})\cross(0, S+1)\subset \mathbb{R}_{++}^{2}, C=\mathcal{B}\backslash \bigcup_{i=1}^{s}\{(x, i)\in \mathbb{R}_{++}^{2}|x_{i}\leqx<x_{1}\},$$\mathcal{D} = \mathbb{R}_{++}^{2}\backslash \mathcal{C}, \partial C=\partial \mathcal{B}\cup[\bigcup_{i=1}^{S}\{(x, i)\in \mathbb{R}_{++}^{2}|x_{i}\leq x<x_{1}\}].$

When we fix
with

$k$

$M_{F}^{k}(x, i)$and consider $F_{i}^{k}(i\in E)$ , the continuous function $M$ on the boundary is defined$(or M_{G}^{k}(x, i))$$M=\{\begin{array}{ll}M_{F}^{k}(x, i) , y=i\in E, x_{i}\leq x<x_{1},([y]+1-y)M_{F}^{k}(x_{1}, [y]) +(y-[y])M_{F}^{k}(x_{1}, [y]+1) , [y]\in E, x=x_{1},0, otherwise.\end{array}$

The function $M$ when considering $G_{i}^{k}(i\in E)$ can be constructed similarly by replacingwithin the above definition. Note that the domain is expanded to a dense subset in, though our interest is in a subset in, in order to apply the Dirichlet problem directly.
The regimes take values in only so that we don’t need to pay attention to the function values
in the non-integer area in the regimes. Thus, the interpolated values of $M$ at $(x_{1}, y)(y\not\in E)$ aresufficient to make $M$ continuous on the boundary
Then by Dirichlet problem, each ofandsatisfies a set of certain ODEs. We mustsolve simultaneous ODEs on each interval such as$M_{F}^{k}(x, i)$$M_{G}^{k}(x, i)$$\mathbb{R}^{2}$$\mathbb{R}\cross \mathbb{Z}$$\mathbb{Z}$$\partial C.$$F_{i}^{k}$$G_{i}^{k}$$(\begin{array}{llll}\mathcal{A}_{1} q_{12} \cdots q_{1n}q_{21} \mathcal{A}_{2} \cdots q_{2n}| | \ddots |q_{n1} q_{n2} \cdots \mathcal{A}_{n}\end{array})(\begin{array}{l}F_{2}^{(k,n)}(x)F_{1}^{(k,n)}(x)|F_{n}^{(k,n)}(x)\end{array})=-(\begin{array}{llll}q_{1,n+1} q_{1,n+2} \cdots q_{1S}q_{2,n+1} q_{2,n+2} \cdots q_{2S}| | \ddots |q_{n,n+1} q_{n,n+2} \cdots q_{nS}\end{array})(\begin{array}{l}F_{n+2}^{(k,n)}(x)F_{n+1}^{(k,n)}(x)|F_{S}^{(k,n)}(x)\end{array})$

on $x\in[x_{n+1}, x_{n}),$ $(n=1,2, \cdots, S-1)$ for each $k=1,2$ , where$\mathcal{A}_{i}f(x)=\frac{1}{2}x^{2}\sigma_{i}^{2}\frac{d^{2}}{dx^{2}}f(x)+x\mu_{i}\frac{d}{dx}f(x)+(q_{ii}-r)f(x)$

.

(4.5)

28
. The smooth pasting conin both and
The value matching conditions are imposed atin only since a smooth pasting condition at the boundary
ditions, however, are imposed at
is meaningless in the Dirichlet problem.
on the RHS of (4.5) are zero by (4.4). On
When $n\leq k-1,$ $F_{k}^{(k,n)}(x)=x$ but otheron the RHS of (4.5) are zero. Hence, when $n\leq k-1,$
the other hand, when $n\geq k$ , all of(4.5) becomes$C$$x_{j}$$\partial C$$C$$x_{j}$$F_{i}^{(k,n)}$$F_{i}^{(k,n)}$$(\begin{array}{llll}\mathcal{A}_{1} q_{12} \cdots q_{1n}q_{21} \mathcal{A}_{2} \cdots q_{2n}| | \ddots |q_{n1} q_{n2} \cdots \mathcal{A}_{m}\end{array})(\begin{array}{l}F_{1}^{(k,n)}(x)F_{2}^{(k,n)}(x)|F_{n}^{(k,n)}(x)\end{array})=-x(\begin{array}{l}q_{1k}q_{2k}|q_{nk}\end{array}),$and when $n\geq k,$ $(4.5)$ is reduced to$(\begin{array}{llll}\mathcal{A}_{1} q_{12} \cdots q_{1n}q_{21} \mathcal{A}_{2} \cdots q_{2n}| | \ddots |q_{n1} q_{n2} \cdots \mathcal{A}_{n}\end{array})(\begin{array}{l}F_{2}^{(k,n)}(x)F_{1}^{(k,n)}(x)|F_{n}^{(k,n)}(x)\end{array})=0_{n}.$In summary, (4.5) is equivalent to$(\begin{array}{llll}\mathcal{A}_{1} q_{12} \cdots q_{1n}q_{21} \mathcal{A}_{2} \cdots q_{2n}| | \ddots |q_{n1} q_{n2} \cdots \mathcal{A}_{m}\end{array})(\begin{array}{l}F_{1}^{(k,n)}(x)F_{2}^{(k,n)}(x)|F_{n}^{(k,n)}(x)\end{array})=-x1_{\{n<k\}}(\begin{array}{l}q_{1k}q_{2k}|q_{nk}\end{array}).$Similarly we have equations for$G_{i}^{k},$$(\begin{array}{llll}\mathcal{A}_{1} q_{12} \cdots q_{1n}q_{21} \mathcal{A}_{2} \cdots q_{2n}| | \ddots |q_{n1} q_{n2} \cdots \mathcal{A}_{m}\end{array})(\begin{array}{l}G_{1}^{(k,n)}(x)G_{2}^{(k,n)}(x)|G_{n}^{(k,n)}(x)\end{array})=-1_{\{n<k\}}(\begin{array}{l}q_{1k}q_{2k}|q_{nk}\end{array}).$For the following calculation, it is worth of noting that a special solution appears only when$n\leq k-1$ , especially $(k, n, i)=(2,1,1)$ if $S=2.$
We can obtain explicit representations for the case of $S=2$ by applying the same technique
as discussed in the previous section.
Let us change notations as $-q_{11}=q_{12}=q_{1}\geq 0,$ $-q_{22}=q_{21}=q_{2}\geq 0$ and define thefollowing quantities,$p_{1}= \frac{q_{1}}{r-\mu_{1}+q_{1}},$$p_{2}= \frac{q_{1}}{r+q_{1}},$$l_{1}=-g_{2}(\beta_{1}^{(2)})/q_{2},$$d_{1}=q_{2}[-(-\beta_{2}^{(1)}+\beta_{1}^{(2)})l_{1}-(\beta_{2}^{(1)}-\beta_{2}^{(2)})l_{2}],$

The following proposition shows the results in case ofcase of $q_{1}q_{2}=0.$$l_{2}=-g_{2}(\beta_{2}^{(2)})/q_{2},$$d_{2}=q_{2}[-(\beta_{1}^{(1)}-\beta_{1}^{(2)})l_{1}+(\beta_{1}^{(1)}-\beta_{2}^{(2)})l_{2}]$$q_{1}q_{2}\neq 0$. One can obtain the results in29
Proposition 2 Suppose that

$F_{1}^{(1,0)}(x)$$F_{1}^{(1,2)}(x)$

$=$

$x,$

$q_{1}q_{2}\neq 0$. Then$F_{i}^{(k,n)},$ $G_{i}^{(k,n)}$are given as follows.

$F_{1}^{(1,1)}(x)=x_{1} \frac{d_{1}(\frac{x}{x_{2}})^{\beta_{1}^{(1)}}+d_{2}(\frac{x}{x_{2}})^{\beta_{2}^{(1)}}}{d_{1}(^{x}x_{2}\lrcorner)^{\beta_{1}^{(1)}}+d_{2}(_{x_{2}}^{x}-\perp)^{\beta_{2}^{(1)}}},$$=$$x_{1} \frac{\beta_{1}^{(1)}-\beta_{2}^{(1)}}{d_{1}(\begin{array}{l}\lrcorner xx_{2}\end{array})+d_{2}(\begin{array}{l}\lrcorner xx_{2}\end{array})}[g_{2}(\beta_{1}^{(2)})(\frac{x}{x_{2}})^{\beta_{1}^{(1)}}-g_{2}(\beta_{2}^{(2)})(\frac{x}{x_{2}})^{\beta_{2}^{(1)}}],$$F_{2}^{(1,0)}(x)$$F_{2}^{(1,2)}(x)$

$=$

$0,$

$F_{2}^{(1,1)}(x)=0,$$=$$x_{1} \frac{\beta_{1}^{(1)}-\beta_{2}^{(1)}}{d_{1}(\begin{array}{l}\lrcorner xx2\end{array})+d_{2}(\begin{array}{l}\lrcorner xx_{2}\end{array})}[-q_{2}(\frac{x}{x_{2}})^{\beta_{1}^{(1)}}+q_{2}(\frac{x}{x_{2}})^{\beta_{2}^{(1)}}],$$F_{1}^{(2,0)}(x)$$F_{1}^{(2,2)}(x)$$F_{2}^{(2,0)}(x)$

$=$

$=$

$=$

$0,$

$F_{1}^{(2,1)}(x)=B_{1}( \frac{x}{x_{1}})^{\beta_{1}^{(1)}}+B_{2}(\frac{x}{x_{1}})^{\beta_{2}^{(1)}}+p_{1^{X}},$$g_{2}( \beta_{1}^{(2)})B_{3}(\frac{x}{x_{2}})^{\beta_{1}^{(2)}}+g_{2}(\beta_{2}^{(2)})B_{4}(\frac{x}{x_{2}})^{\beta_{2}^{(2)}}$$x,$$F_{2}^{(2,1)}(x)=x,$$F_{2}^{(2,2)}(x)=-q_{2}B_{3}( \frac{x}{x_{2}})^{\beta_{1}^{(2)}}-q_{2}B_{4}(\frac{x}{x_{2}})^{\beta_{2}^{(2)}}$

and$G_{1}^{(1,0)}(x)$$G_{1}^{(1,2)}(x)$

$=$

1,

$G_{1}^{(1,1)}(x)= \frac{d_{1}(\frac{x}{x_{2}})^{\beta_{1}^{(1)}}+d_{2}(\frac{x}{x_{2}})^{\beta_{2}^{(1)}}}{d_{1}(_{x}x_{2}\lrcorner)^{\beta_{1}^{(1)}}+d_{2}(_{x}x_{2}\lrcorner)^{\beta_{2}^{(1)}}},$$=$$\frac{\beta_{1}^{(1)}-\beta_{2}^{(1)}}{d_{1}(\begin{array}{l}\lrcorner xx_{2}\end{array})+d_{2}(\begin{array}{l}\lrcorner xx_{2}\end{array})}[g_{2}(\beta_{1}^{(2)})(\frac{x}{x_{2}})^{\beta_{1}^{(1)}}-g_{2}(\beta_{2}^{(2)})(\frac{x}{x_{2}})^{\beta_{2}^{(1)}}],$$G_{2}^{(1,0)}(x)$$=$$G_{2}^{(1,2)}(x)$

$=$

$0,$

$G_{2}^{(1,1)}(x)=0,$$\frac{\beta_{1}^{(1)}-\beta_{2}^{(1)}}{d_{1}(\begin{array}{l}\lrcorner xx_{2}\end{array})+d_{2}(\begin{array}{l}\lrcorner xx_{2}\end{array})}[-q_{2}(\frac{x}{x_{2}})^{\beta_{1}^{(1)}}+q_{2}(\frac{x}{x_{2}})^{\beta_{2}^{(1)}}],$$G_{1}^{(2,0)}(x)$$=$$G_{1}^{(2,2)}(x)$$=$$G_{2}^{(2,0)}(x)$

$=$

$0,$

$G_{1}^{(2,1)}(x)=b_{1}( \frac{x}{x_{1}})^{\beta_{1}^{(1)}}+b_{2}(\frac{x}{x_{1}})^{\beta_{2}^{(1)}}+p_{2},$$g_{2}( \beta_{1}^{(2)})b_{3}(\frac{x}{x_{2}})^{\beta_{1}^{(2)}}+g_{2}(\beta_{2}^{(2)})b_{4}(\frac{x}{x_{2}})^{\beta_{2}^{(2)}}$

1,$G_{2}^{(2,1)}(x)=1,$$G_{2}^{(2,2)}(x)=-q_{2}b_{3}( \frac{x}{x_{2}})^{\beta_{1}^{(2)}}-q_{2}b_{4}(\frac{x}{x_{2}})^{\beta_{2}^{(2)}}$

30
where
$B_{1}$$=$$B_{3}$$=$$-p_{1}x_{1} \frac{d_{1}\lambda^{-\beta i^{1)}}}{d_{1}\lambda^{-\beta_{1}^{(1)}}+d_{2}\lambda^{-\beta_{2}^{(1)}}}+q_{2}x_{2}\frac{l_{1}l_{2}(\beta_{1}^{(2)}-\beta_{2}^{(2)})-p_{1}(l_{1}(\beta_{1}^{(2)}-1)-l_{2}(\beta_{2}^{(2)}-1))}{d_{1}\lambda^{\beta_{2}^{(1)}}+d_{2}\lambda^{\beta_{1}^{(1)}}},$$-p_{1}x_{1}(\beta_{1}^{(1)}-\beta_{2}^{(1)})+l_{2}x_{2}((\beta_{2}^{(1)}-\beta_{2}^{(2)})\lambda^{-\beta_{1-}^{(1)}}(\beta_{1}^{(1)}-\beta_{2}^{(2)})\lambda^{-\beta_{2}^{(1)}})$$d_{1}\lambda^{-\beta_{1}^{(1)}}+d_{2}\lambda^{-\beta_{2}^{(1)}}$$-p_{1}x_{1} \frac{(\beta_{2}^{(1)}-1)\lambda^{-\beta_{1}^{(1)}}-(\beta_{1}^{(1)}-1)\lambda^{-\beta_{2}^{(1)}}}{d_{1}\lambda^{-\beta_{1}^{(1)}}+d_{2}\lambda^{-\beta_{2}^{(1)}}}$$B_{2}$$=$$b_{1}$$=$$b_{3}$$=$$b_{2}$$=$$-p_{1}x_{1}-B_{1},$$B_{4}=- \frac{x_{2}}{q_{2}}-B_{3},$$-p_{2} \frac{d_{1}\lambda^{-\beta_{1}^{(1)}}}{d_{1}\lambda^{-\beta_{1}^{(1)}}+d_{2}\lambda^{-\beta_{2}^{(1)}}}+q_{2}\frac{l_{1}l_{2}(\beta_{1}^{(2)}-\beta_{2}^{(2)})-p_{2}(l_{1}\beta_{1}^{(2)}-l_{2}\beta_{2}^{(2)})}{d_{1}\lambda^{\beta_{2}^{(1\rangle}}+d_{2}\lambda^{\beta_{1}^{(1)}}}$$\frac{-p_{2}(\beta_{1}^{(1)}-\beta_{2}^{(1)})+l_{2}((\beta_{2}^{(1)}-\beta_{2}^{(2)})\lambda^{-\beta_{1-}^{(1\rangle}}(\beta_{1}^{(1)}-\beta_{2}^{(2)})\lambda^{-\beta_{2}^{(1)}})}{d_{1}\lambda^{-\beta_{1}^{(1)}}+d_{2}\lambda^{-\beta_{2}^{(1)}}}-p_{2}\frac{\beta_{2}^{(1)}\lambda^{-\beta_{1}^{(1)}}-\beta_{1}^{(1)}\lambda^{-\beta_{2}^{(1)}}}{d_{1}\lambda^{-\beta_{1}^{(1)}}+d_{2}\lambda^{-\beta_{2}^{(1)}}}$$-p_{2}-b_{1},$$b_{4}=- \frac{1}{q_{2}}-b_{3}.$Note that Proposition 2 is valid for any. We don’t impose “smooth pasting conditions” ofat the boundary of the continuation region when solving the Dirichlet problem
since “smooth pasting conditions at the boundary” are meaningless in the problem. It follows
that, for arbitrary given, a function constructed with these obtained$x_{2}\leq x_{1}$$F_{i}^{k},$$G_{i}^{k}$$F_{i}^{k},$ $G_{i}^{k}$$x_{1},$ $x_{2}$$W_{i}^{(n)}(x)= \sum_{k\in E}(\alpha_{k}D_{k}F_{i}^{(k,n)}(x)-K_{k}G_{i}^{(k,n)}(x))$

does not necessary satisfy the smooth pasting conditionat the boundaryof the continuation region. The optimality condition involved inplays an importantrole in veryifying (4.1) and the smooth pasting conditions at the boundaries.
$\frac{d}{dx}W_{n}^{(n)}(x)=\alpha_{n}D_{n}$$x_{n}$$\lambda=x_{1}/x_{2}$Lemma 3 Suppose that$r> \max(\mu_{1}, \mu_{2},0)$,$q_{2}\neq 0$

and$(r-\mu_{1}+q_{1})(r-\mu_{2}+q_{2})-q_{1}q_{2}\neq 0,$$(r+q_{1})(r+q_{2})-q_{1}q_{2}\neq 0$

Then,(1) Assumption 1 and 2 hold.
(2) There exist threshholds $0<x_{2}\leq x_{1}$ if and only if there exists unique solution$0<\lambda\leq 1$ and,$\frac{-d_{11}\lambda^{\beta_{1}^{(1)}}+k_{K}d_{21}}{c_{11}\lambda^{\beta_{1}^{(1)}-1}-k_{K}k_{\alpha}c_{21}}=\frac{-d_{12}\lambda^{\beta_{2}^{(1)}}+k_{K}d_{22}}{c_{12}\lambda^{\beta_{2}^{(1)}-1}-k_{K}k_{\alpha}c_{22}}>0$$\lambda$

.

(4.6)

satisfying

(4.7)

31
where
$(\begin{array}{l}c_{11}c_{12}\end{array}) = (1-k_{K}p_{1}k_{\alpha})(_{\beta_{1}^{(1)}-1}^{1-\beta_{2}^{(1)}}) , (\begin{array}{l}d_{11}d_{12}\end{array})=(1-k_{K}p_{2})(\begin{array}{l}\beta_{2}^{(1)}-\beta_{1}^{(1)}\end{array}),$$(\begin{array}{l}c_{21}c_{22}\end{array}) = -(\begin{array}{ll}\beta_{2}^{(1)} -1-\beta_{1}^{(1)} 1\end{array})[ \frac{1}{\beta_{1}^{(2)}-\beta_{2}^{(2)}}(\begin{array}{ll}l_{1} l_{2}l_{1}\beta_{1}^{(2)} l_{2}\beta_{2}^{(2)}\end{array})(_{\beta_{1}^{(2)}-1}^{1-\beta_{2}^{(2)}})-p_{1} (\begin{array}{l}11\end{array})],$$(\begin{array}{l}d_{21}d_{22}\end{array}) = -(\begin{array}{ll}\beta_{2}^{(1)} -1-\beta_{1}^{(1)} 1\end{array})[ \frac{1}{\beta_{1}^{(2)}-\beta_{2}^{(2)}}(_{l_{1}\beta i^{2)}}l_{1} l_{2}\beta_{2}^{(2))}l_{2}(\begin{array}{l}\beta_{2}^{(2)}-\beta_{1}^{(2)}\end{array})+p_{2}(\begin{array}{l}10\end{array})].$

When the above condition (4. 7) is satisfied, the solution$(x_{1}, x_{2})$$(\begin{array}{l}x_{1}x_{2}\end{array})=\frac{1}{\tilde{\alpha}_{1}\lambda}\frac{-d_{11}\lambda^{\beta_{1}^{(1)}}+k_{K}d_{21}}{c_{11}\lambda^{\beta_{1}^{(1)}-1}-k_{K}k_{\alpha}c_{21}}(\begin{array}{l}1\lambda\end{array})$

The following lemma shows that

Lemma 4$\lambda$$\lambda$

is given by

.

(4.8)

is given by a solution of equations.

must satisfy the following vector equation,$\frac{(1-\beta_{2}^{(1)})(\alpha_{1}D_{1}-p_{1}\alpha_{2}D_{2})x_{1}+\beta_{2}^{(1)}(K_{1}-p_{2}K_{2})}{\beta_{1}^{(1)}-\beta_{2}^{(1)}}\lambda^{\beta_{1}^{(1)}}(\begin{array}{l}1\beta_{1}^{(1)}\end{array})$$- \frac{(1-\beta_{1}^{(1)})(\alpha_{1}D_{1}-p_{1}\alpha_{2}D_{2})x_{1}+\beta_{1}^{(1)}(K_{1}-p_{2}K_{2})}{\beta_{1}^{(1)}-\beta_{2}^{(1)}}\lambda^{\beta_{2}^{(1)}}(\begin{array}{l}1\beta_{2}^{(1)}\end{array})$$= l_{1} \frac{(1-\beta_{2}^{(2)})\alpha_{2}D_{2}x_{2}+\beta_{2}^{(2)}K_{2}}{\beta_{1}^{(2)}-\beta_{2}^{(2)}}(\begin{array}{l}1\beta_{1}^{(2)}\end{array})-l_{2}\frac{(1-\beta_{1}^{(2)})\alpha_{2}D_{2}x_{2}+\beta_{1}^{(2)}K_{2}}{\beta_{1}^{(2)}-\beta_{2}^{(2)}}(\begin{array}{l}l\beta_{2}^{(2)}\end{array})$$-p_{1}\alpha_{2}D_{2}x_{2}(\begin{array}{l}11\end{array})+p_{2}K_{2}(\begin{array}{l}10\end{array}).$By making use of Lemma 3 and 4, we take the optimality condition into consideration sothat we obtain the following result which is seemingly trivial but non-trivial.
Proposition 3 Suppose that$q_{1}q_{2}\neq 0$and$x_{1},$ $x_{2}$are given by$(4\cdot 8)$. Then it holds that

$V_{i}^{(n)}(x)=W_{i}^{(n)}(x) , n=0,1,2, i=1,2,$

where

5

$V_{i}^{(n)}$are given in Proposition 1.

Concluding remarks

Our technique with linear algebra provides an insight of the functional form of the value function
and it would help us in analysis of the value function more explicitly under many regime states.
Especially, the results on the first passage time are remarkable for futher research. Numericalcalculation is relatively easy due to the expression with eigenvalues and eigenvectors.

32
in order toAs remarks on parameters, we made explicit assumptions on $M,$ andand convergence of income flow multiplier . However,obtain appropriate eigenvalues
satisfying theare implicitly involved in calculation of thresholds
andconditions on
thougheigenvalues
order of $x_{S}\leq x_{S-1}\leq\cdots\leq x_{2}\leq x_{1}$ . Our assumptions assure the distinct
they can be relaxed. The cases of duplicate eigenvalues and an entry-exit problem are left for
$r,$

$Q$

$\Sigma$$\beta_{i}^{(n)}$$D$$\alpha_{i}$$K$$x_{n}$

future research.

References[1] Asmussen, S. (2003) Applied Probabilities and Queues. Springer.

[2] Dixit, A., Pyndick, $R.(1994)$ Investment Under Uncertainty. Princeton University.

[3] Grenadier, S. and Wang, $N.(2007)$ “Investment under uncertainty and time-inconsistentpreferences,” Joumal of Financial Economics, 84, 2-39.
[4] Guo, X. (2001) “When the “bull” meets the “bear” –A first passage time problem for ahidden Markov process,” Methodology and Computing in Applied Probability, 3, 135-143.
[5] Guo, X., Zhang, Q. (2004) “Closed-Form Solutions for Perpetual American Put Options
with Regime Switching,” SIAM Joumal on Applied Mathematics, 64, 2034-2049.
[6] Guo, X, Miao, J., and Morellec $E.(2005)$ “Irreversible investment with regime shifts,” Journal of Economic Theory, 122, 37-59.

[7] Jiang, Z., Pistorius, M. (2008) “On perpetual American put valuation and first-passage ina regime-switching model with jumps,” Finance and Stochastics, 12, 331-355.
[8] Jobert, A., Rogers, C. (2006) “Option pricing with Markov-modulated dynamics,” SIAM
Joumal on Control and optimization, 44, 2063-2078.
[9] Peskir, G., Shiraev, A. (2006) Optimal Stopping and Free-Boundary Problems, Birkh\"auser
Verlag.

Graduate School of Social Sciences
Department of Business Administration

Tokyo Metropolitan University
1-1 Minamiohsawa Hachiouji Tokyo 192-0397 Japan-mail address: tanaka-keiichi@tmu.ac.jp$E$$\Xi ffl\star\not\cong\ovalbox{\tt\small REJECT}\hat{F_{\backslash }}.

7\pm_{=}^{\Delta}\cdot H\not\cong\Re_{J\iota}^{R}H^{\backslash }\backslash H^{[\dagger]}\Re-$