---
abstract: 'The traditional difficulty about stochastic singular control is to characterize the regularities of the value function and the optimal control policy. In this paper, a multi-dimensional singular control problem is considered. We found the optimal value function and the optimal control policy of this problem via Dynkin game, whose solution is given by the saddle point of the cost function. The existence and uniqueness of the solution to this Dynkin game are proved through an associated variational inequality problem involving Dirichlet form. As a consequence, the properties of the value function of this Dynkin game implies the smoothness of the value function of the stochastic singular control problem. In this way, we are able to show the existence of a classical solution to this multi-dimensional singular control problem, which was traditionally solved in the sense of viscosity solutions, and this enables the application of the verification theorem to prove optimality. [^1]'
author:
- 'Yipeng Yang [^2]'
title: 'A Multi-dimensional Stochastic Singular Control Problem Via Dynkin Game and Dirichlet Form'
---

Dynkin game, Dirichlet form, Multi-dimensional diffusion, Stochastic singular control

49J40, 60G40, 60H30, 93E20

Introduction and Problem Formulation
====================================

The characterization of the regularities of value function and optimal policy in stochastic singular control remains a big challenge in stochastic control theory, especially the higher dimensional case, see, e.g., [@Soner89]. The traditional approach is to use the viscosity solution technique, see [@Fleming06] [@Bayraktar12] [@Bassan02], which usually yields a less regular solution. Another approach to solve singular control problems and characterize the regularity of value functions is through variational inequalities and optimal stopping or Dynkin game, see, e.g., Karatzas and Zamfirescu [@Karatzas05], Guo and Tomecek [@Guo08]. In [@Karatzas85] Karatzas and Shreve studied the connection between optimal stopping and singular stochastic control of one dimensional Brownian motion, and showed that the region of inaction in the control problem is the optimal continuation region for the stopping problem. In [@Baldursson97], the authors established and exploited the duality between the myopic investor’s problem (optimal stopping) and the social planning problem (stochastic singular control), where an integral form and change of variable formula were also presented on this connection. Ma [@ma92] dealt with a one dimensional stochastic singular control problem where the drift term is assumed to be linear and the diffusion term is assumed to be smooth, and he showed that the value function is convex and $C^2$ and the controlled process is a reflected diffusion over an interval. Guo and Tomecek [@Guo09] solved a one dimensional singular control problem via a switching problem [@Guo08], and showed, using the smooth fit property [@Pham07], that under some conditions the value function is continuously differentiable ($C^1$).

It is found that [@Fuku02] through the approach via game theory and optimal stopping, it is possible to show the existence of a smooth solution. The connection is the following: given a symmetric Markov process on a locally compact separable metric space, it is well known that the solution of an optimal stopping problem admits its quasi continuous version of the solution to a variational inequality problem involving Dirichlet form, e.g., see Nagai [@Nagai78]. Zabczyk [@Zab84] extended this result to a zero-sum game (Dynkin game). In the one dimensional case, the integrated form of the value function of the Dynkin game was identified to be the solution of an associated stochastic singular control problem, e.g., see Taksar [@Taksar85], Fukushima and Taksar [@Fuku02] where a more general one dimensional diffusion is assumed. As a result, the classical smooth solution ($C^2$) can be obtained for this singular control problem.

This paper extends the work by Fukushima and Taksar [@Fuku02] to multi-dimensional stochastic singular control problem. There are many difficulties in this extension. In the one dimensional singular control problem, each point in the space has a positive capacity [@Fuku02], hence the nonexistence of the proper exceptional set. However, this is no longer the case in multi-dimensional singular control problem. We overcome this difficulty using the absolute continuity of the transition function of the underlying process [@Fuku06]. Under some conditions, the optimal control policy of the one dimensional case is proved to be the reflection of the diffusion at two boundary points, but the form of the optimal control policy and the conditions on the regularity of the value function in multi-dimensional case are much more complicated. For instance, in the two dimensional case, the boundary of the continuation region can have various formats, e.g., bounded curves, unbounded curves, singular points, disconnected curves, line segments, etc. The difficulty in characterizing the continuation region is due to the fact that its boundary is a free boundary, and this paper investigates such issues.

In this paper, we are concerned with a multi-dimensional diffusion on ${\mathbb{R}}^n$: $$\label{omodel}
d{\bf X}_t={\bf \mu}({\bf X}_t)dt+{\bf\sigma}({\bf X}_t)d{\bf B}_t,$$ where $${\bf X}_t=\left(\begin{array}{c}X_{1t}\\
\vdots\\
X_{nt}
\end{array}\right), \mu=\left(\begin{array}{c}\mu_1\\
\vdots\\
\mu_n
\end{array}\right), \sigma=\left(\begin{array}{ccc} \sigma_{11} &\  \cdots & \sigma_{1m}\\
\vdots &  & \vdots\ \\
\sigma_{n1} &\ \cdots & \sigma_{nm}\end{array}\right), {\bf
B}_t=\left(\begin{array}{c}B_{1t}\\
\vdots\\
B_{mt}\end{array}\right),$$ in which $\mu_i=\mu_i({\bf X}_t)$ and $\sigma_{i,j}=\sigma_{i,j}({\bf X}_t)$ ($1\leqslant i\leqslant n,1\leqslant j\leqslant
m$) are continuous functions of $X_{1t},X_{2t},...,X_{(n-1)t}$, and ${\bf B}_t$ is $m$-dimensional Brownian motion with $m\geqslant n$. Thus we are given a system $(\Omega, \mathcal{F},\mathcal{F}_t, {\bf
X},\theta_t,P_{\bf x})$, where $(\Omega,\mathcal{F})$ is a measurable space, ${\bf X}={\bf X}(\omega)$ is a mapping of $\Omega$ into $C({\mathbb{R}}^n)$, $\mathcal{F}_t=\sigma({\bf X}_s,s\leqslant
t)$, and $\theta_t$ is a shift operator in $\Omega$ such that ${\bf
X}_s(\theta_t\omega)={\bf X}_{s+t}(\omega)$. Here $P_{\bf x}$(${\bf
x}\in{\mathbb{R}}^n$) is a family of measures under which $\{{\bf
X}_t,t\geqslant 0\}$ is an $n$-dimensional diffusion with initial state ${\bf x}$. We assume that $\mu$ and $\sigma$ satisfy the usual Lipschitz growth condition.

A control policy is defined as a pair $(A_t^{(1)},A_t^{(2)})=\mathcal{S}$ of $\mathcal{F}_t$ adapted processes which are right continuous and nondecreasing in $t$ and we assume $A_0^{(1)},A_0^{(2)}$ are nonnegative. Denote $\mathbb{S}$ the set of all admissible policies, whose detailed definition will be given in Section \[mdssc\].

Given a policy $\mathcal{S}=(A_t^{(1)},A_t^{(2)})\in\mathbb{S}$ we define the following controlled process: $$\begin{array}{l}
dX_{1t}=\mu_1dt+\sigma_{11}dB_{1t}+\cdots+\sigma_{1m}dB_{mt},\\
\vdots  \quad\quad \vdots \quad\quad\quad \vdots\\
dX_{nt}=\mu_ndt+\sigma_{n1}dB_{1t}+\cdots+\sigma_{nm}dB_{mt}+dA_t^{(1)}-dA_t^{(2)},\\
{\bf X}_0={\bf x},
\end{array}$$ with the cost function $$\begin{aligned}
\label{scost}
k_{\mathcal{S}}({\bf x})=E_{\bf x}\left(\int_0^\infty e^{-\alpha
t}h({\bf X_t})dt+\int_0^\infty e^{-\alpha t}\left(f_1({\bf
X}_t)dA_t^{(1)}+f_2({\bf X}_t)dA_t^{(2)}\right)\right),&&\\
f_1({\bf x}),f_2({\bf x})>0,\ \forall {\bf
x}\in{\mathbb{R}}^n.&&\nonumber\end{aligned}$$ Here we assume that $A_t^{(1)}-A_t^{(2)}$ is the minimal decomposition of a bounded variation process into a difference of two increasing processes.

A natural question is that why the control only applies on one dimension. The difficulty arises in the step where the value function of the zero-sum game is integrated (in one dimension) to obtain the value function of the singular control problem. If the control were applied to multi dimensions, no result so far is know on the choice of the direction of integration. This represents a traditional difficulty in multi-dimensional singular control problem. Interested readers are referred to [@Soner89] for a result on two dimensional singular control problem.

There are two types of costs associated with the process ${\bf X_t}$ for each policy $\mathcal{S}$. The first one is the holding cost $h({\bf X}_t)$ accumulated along time. The second one is the control cost associated with the processes $(A_t^{(1)},A_t^{(2)})$, and this cost increases only when $(A_t^{(1)},A_t^{(2)})$ increase.

One looks for a control policy $\mathcal{S}$ that minimizes $k_{\mathcal{S}}({\bf x})$, i.e., $$\label{mincostw}
W({\bf x})=\min_{\mathcal{S}\in\mathbb{S}} k_{\mathcal{S}}({\bf x}).$$

As an application of this model, a decision maker observes the expenses of a company under a multi-factor situation but only has control over one factor, yet she still wants to minimize the total expected cost. Analogously, by studying the associated maximization problem, i.e., taking the negative of $\min$, this model can be used to find the optimal investment policy where an investor observes the prices of several assets in a portfolio and manages the portfolio by adjusting one of them. Notice that every time there is a control action, it yields a certain associated cost, e.g., the transaction cost.

The rest of this paper is organized as follows: we first introduce some preliminaries on Dirichlet form and a variational inequality problem in Section \[DformDgame\]. In Section \[DgameFBP\] we identify conditions for the value function as well as the optimal policy of the associated Dynkin game. The integrated form of the value function of this Dynkin game is shown in Section \[mdssc\] to be the value of a multi-dimensional singular control problem, and the optimal control policy is also determined consequently. In the appendix we shall correct an error found in the paper by Fukushima and Taksar [@Fuku02].

Dirichlet Form and a Variational Inequality Problem {#DformDgame}
===================================================

Let $\mathbb{D}$ be a locally compact separable metric space, ${\mathbb m}$ be an everywhere dense positive Radon measure on $\mathbb{D}$, and $L^2(\mathbb{D},m)$ denotes the $L^2$ space on $\mathbb{D}$. We assume that the Dirichlet form $(\mathcal{E},\mathscr{F})$ on $L^2(\mathbb{D},m)$ is regular in the sense that $\mathscr{F}\cap C_0(\mathbb{D})$ is $\mathcal{E}_1$ dense in $\mathscr{F}$ and is uniformly dense in $C_0(\mathbb{D})$, where the $\mathcal{E}_1$ norm is defined as follows: $$\|u\|_{\mathcal{E}_1}=\left(\mathcal{E}(u,u)+\int_{\mathbb{D}}u({\bf
x})^2{\mathbb m}(d{\bf x})\right)^{1/2}.$$ Analogously we define $\mathcal{E}_\alpha(u,v)$ as $\mathcal{E}_\alpha(u,v)=\mathcal{E}(u,v)+\alpha(u,v)\ (\alpha>0)$, where $$(u,v)=\int_{\mathbb{D}}u({\bf x})v({\bf x}){\mathbb m}(d{\bf x}).$$

For this Dirichlet form, there exists an associated Hunt process ${\bf M}=({\bf X}_t,P_{\bf x})$ on $\mathbb{D}$, see [@Fuku11], such that

$$p_tf({\bf x}):=E_{\bf x}f({\bf X}_t),\quad {\bf x}\in\mathbb{D}$$

is a version of $T_tf$ for all $f\in C_0(\mathbb{D})$, where $T_t$ is the $L^2$ semigroup associated with the Dirichlet form $(\mathcal{E},\mathscr{F})$. Furthermore, the $L^2$-resolvent $\{G_\alpha,\ \alpha>0\}$ associated with this Dirichlet form satisfies $$\label{resol}
G_\alpha f\in\mathscr{F},\quad \mathcal{E}_\alpha(G_\alpha
f,u)=(f,u),\quad \forall f\in L^2(\mathbb{D};m),\ \forall
u\in\mathscr{F},$$ and the resolvent $\{R_\alpha,\ \alpha>0\}$ of the Hunt process ${\bf M}$ given by $$R_\alpha f({\bf x})=E_{\bf x}\left(\int_0^\infty e^{-\alpha t}f({\bf
X}_t)dt\right),\quad {\bf x}\in\mathbb{D},$$ is a quasi-continuous modification of $G_\alpha f$ for any Borel function $f\in
L^2(\mathbb{D};m)$.

For $\alpha>0$, a measurable function $f$ on $\mathbb{D}$ is called $\alpha$-excessive if $f({\bf x})\geqslant 0$ and $e^{-\alpha t}p_t
f({\bf x})\uparrow  f({\bf x})$ as $t\downarrow  0$ for any ${\bf
x}\in\mathbb{D}$. A function $f\in\mathscr{F}$ is said to be an $\alpha$-potential if $\mathcal{E}_\alpha(f,g)\geqslant 0$ for any $g\in\mathscr{F}$ with $g\geqslant 0$. For any $\alpha$-potential $f\in\mathscr{F}$, define $\hat{f}({\bf x})=\lim_{t\downarrow 0}p_t
f({\bf x})$, then $f=\hat{f}\ m$-a.e. and $\hat{f}$ is $\alpha$-excessive (see Section 3 in [@Fuku06]). $\hat{f}$ is called the $\alpha$-excessive regularization of $f$. Furthermore, any $\alpha$-excessive function is finely continuous (see Theorem A.2.7 in [@Fuku11]).

As related literature, Nagai [@Nagai78] considered an optimal stopping problem and showed that there exist a quasi continuous function $w\in\mathscr{F}$ which solves the variational inequality $$w\geqslant g,\quad \mathcal{E}_\alpha(w,u-w)\geqslant 0,\quad
\forall u\in\mathscr{F} {\rm \ with\ }u\geqslant g,$$ and a properly exceptional set $\mathbb{N}$ such that for all ${\bf x}\in\mathbb{D}/\mathbb{N}$, $$w({\bf x})=\sup_\sigma E_{\bf x}\left(e^{-\alpha\sigma}[g({\bf
X}_\sigma)]\right)=E_{\bf x}\left(e^{-\alpha\hat{\sigma}}[g({\bf
X}_{\hat{\sigma}})]\right),$$ where $g$ is a quasi continuous function in $\mathscr{F}$ and $$\hat{\sigma}=\inf\{t\geqslant 0;w({\bf X}_t)=g({\bf
X}_t)\}.$$ Moreover, $w$ is the smallest $\alpha$-potential dominating the function $g$ ${\mathbb m}$-a.e.

Zabczyk [@Zab84] then extended this result to the solution of the zero-sum game (Dynkin game) by showing that there exist a quasi continuous function $V({\bf x})\in\mathscr{F}$ which solves the variational inequality $$\label{vineqH0}
g\leqslant V\leqslant h\ \ {\mathbb m}\ {\rm a.e.},\
\mathcal{E}_\alpha(V,u-V)\geqslant 0,\quad \forall u\in\mathscr{F},\
g\leqslant u\leqslant h\ \ {\mathbb m}\ {\rm a.e.},$$ and a properly exceptional set $\mathbb{N}$ such that for all ${\bf
x}\in\mathbb{D}/\mathbb{N}$, $$\label{saddleH0}
V({\bf x})=\sup_\sigma \inf_\tau J_{\bf x}(\tau,\sigma)=\inf_\tau
\sup_\sigma J_{\bf x}(\tau,\sigma)$$ for any stopping times $\tau$ and $\sigma$, where $$\label{JH0}
J_{\bf x}(\tau,\sigma)=E_{\bf
x}\left(e^{-\alpha(\tau\wedge\sigma)}\left(I_{\sigma\leqslant\tau}g({\bf
X}_\sigma)+I_{\tau<\sigma}h({\bf X}_\tau)\right)\right),$$ and $g\leqslant h$ $\mathbb m$-a.e. are quasi-continuous functions in $\mathscr{F}$.

In these works, there always existed an exceptional set $\mathbb{N}$. Fukushima and Menda [@Fuku06] showed that, if the transition function of ${\bf M}$ satisfies an absolute continuity condition, i.e., $$\label{abscont} p_t({\bf x},\cdot)\ll {\mathbb m}(\cdot),$$ for all $t>0$ and ${\bf x}\in\mathbb{D}$, and $g,h$ satisfy the following separability condition:\
*There exist finite $\alpha$-excessive functions $v_1,v_2\in\mathscr{F}$ such that, for all ${\bf x}\in\mathbb{D}$, $$\label{sepcond}
g({\bf x})\leqslant v_1({\bf x})-v_2({\bf x})\leqslant h({\bf x}),$$*then Zabczyk’s result still holds and there does not exist the exceptional set $\mathbb{N}$. In what follows we shall introduce a version of Theorem 2 in [@Fuku06], where we used $-f_1,f_2$ in places of $g,h$ respectively for the convenience of later use.

Let $f_1,f_2\in\mathscr{F}$ be finely continuous functions such that for all ${\bf x}\in\mathbb{D}$ $$\label{bdcond}
-f_1({\bf x})\leqslant f_2({\bf x}),\ |f_1({\bf
x})|\leqslant\phi({\bf x}),\ |f_2({\bf x})|\leqslant \psi({\bf x}),$$ where $\phi,\psi$ are some finite $\alpha$-excessive functions, and $f_1,f_2$ are assumed to satisfy the following separability condition $$\label{sepcondf}
-f_1({\bf x})\leqslant v_1({\bf x})-v_2({\bf x})\leqslant f_2({\bf
x}).$$

We further define the set $$\label{dk}
K=\{u\in\mathscr{F}:\ -f_1\leqslant u\leqslant f_2,\ {\mathbb m}
{\rm -a.e.}\}.$$ Considering the variational inequality problem $$\label{vineqH0f}
V\in K,\ \mathcal{E}_\alpha(V,u-V)\geqslant 0,\quad \forall u\in K,$$ we have:

\[fuku2\] Assume conditions (\[abscont\]), (\[bdcond\]) and (\[sepcondf\]). There exists a finite finely continuous function $V$ satisfying the variational inequality (\[vineqH0f\]) and the identity $$V({\bf x})=\sup_\sigma\inf_\tau J_{\bf
x}(\tau,\sigma)=\inf_\tau\sup_\sigma J_{\bf x}(\tau,\sigma),\quad
\forall {\bf x}\in\mathbb{D},$$ where $\sigma,\tau$ range over all stopping times and $$\label{JH0f}
J_{\bf x}(\tau,\sigma)=E_{\bf
x}\left(e^{-\alpha(\tau\wedge\sigma)}\left(I_{\sigma\leqslant\tau}(-f_1({\bf
X}_\sigma))+I_{\tau<\sigma}f_2({\bf X}_\tau)\right)\right),$$

Moreover, the pair $(\hat{\tau},\hat{\sigma})$ defined by $$\hat{\tau}=\inf\{t>0:V({\bf X}_t)=f_2({\bf X}_t)\},\quad
\hat{\sigma}=\inf\{t>0:V({\bf X}_t)=-f_1({\bf X}_t)\},$$ is the saddle point of the game in the sense that $$J_{\bf x}(\hat{\tau},\sigma)\leqslant J_{\bf
x}(\hat{\tau},\hat{\sigma})\leqslant J_{\bf
x}(\tau,\hat{\sigma}),\quad \forall {\bf x}\in\mathbb{D},$$ for all stopping times $\tau,\sigma$.

For a given function $H\in L^2(\mathbb{D};m)$ one looks for a solution $V\in K$ to the following variational inequality problem $$\label{vi} \mathcal{E}_\alpha(V,u-V)\geqslant(H,u-V),\ \ \forall
u\in K.$$ Then we have the following proposition:

\[solVIeq\] There exists a unique finite finely continuous function $V\in K$ which solves (\[vi\]).

The proof is essentially identical to the proof of Proposition 2.1 in [@Fuku02] and is omitted here.

We assume further the following separability condition:

There exist finite $\alpha$-excessive functions $v_1,v_2\in\mathscr{F}$ such that, for all ${\bf x}\in\mathbb{D}$, $$\label{sepcondHneq0}
-f_1({\bf x})-G_\alpha H({\bf x})\leqslant v_1({\bf x})-v_2({\bf
x})\leqslant f_2({\bf x})-G_\alpha H({\bf x}),$$

then the following result holds:

\[DgameHneq0\] For any function $H\in L^2({\mathbb D};{\mathbb m})$ ) and any $f_1,f_2\in\mathscr{F}$ such that $f_1({\bf x})+G_\alpha H({\bf x})$ and $f_2({\bf x})-G_\alpha H({\bf x})$ are finely continuous and bounded by some finite $\alpha$-excessive functions, respectively. Assuming (\[abscont\])(\[sepcondHneq0\]), we put $$\begin{aligned}
\label{Jcost}
J_{\bf x}(\tau,\sigma)&=&E_{\bf x}\left(\int_0^{\tau\wedge\sigma}
e^{-\alpha t}H({\bf X}_t)dt\right)\\
&&+E_{\bf x}\left( e^{-\alpha
(\tau\wedge\sigma)}\left(-I_{\sigma\leqslant\tau}f_1({\bf
X}_\sigma)+I_{\tau<\sigma}f_2({\bf X}_\tau)\right)\right)\nonumber\end{aligned}$$ for any stopping times $\tau,\sigma$. Then the solution of (\[vi\]) admits a finite finely continuous value function of the game $$V({\bf x})=\inf_\tau\sup_\sigma J_{\bf
x}(\tau,\sigma)=\sup_\sigma\inf_\tau J_{\bf x}(\tau,\sigma),\quad
\forall {\bf x}\in\mathbb{D}.$$

Furthermore if we let $$E_1=\{{\bf x}\in
{\mathbb{D}}: V({\bf x})=-f_1({\bf x})\}, \quad E_2=\{{\bf x}\in
{\mathbb{D}}: V({\bf x})=f_2({\bf x})\},$$ then the hitting times $\hat{\tau}=\tau_{E_2}$, $\hat{\sigma}=\tau_{E_1}$ is the saddle point of the game $$\label{saddleHn0}
J_{\bf x}(\hat{\tau},\sigma)\leqslant J_{\bf
x}(\hat{\tau},\hat{\sigma})\leqslant J_{\bf x}(\tau,\hat{\sigma})$$ for any ${\bf x}\in \mathbb{D}$ and any stopping times $\tau,\sigma$. In particular, $$V({\bf x})=J_{{\bf x}}(\hat{\tau},\hat{\sigma}),\quad \forall {\bf
x}\in \mathbb{D}.$$

$E_1$ is the set of points where $V=-f_1$ and $E_2$ is the set of points where $V=f_2$. So $\hat{\tau}$ and $\hat{\sigma}$ in Theorem \[DgameHneq0\] can be defined in the same way as in Theorem \[fuku2\]. The proof of Theorem \[DgameHneq0\] is identical to Theorem 2.1 in [@Fuku02].

The Dynkin Game and Its Value Function {#DgameFBP}
======================================

Two players $P_1$ and $P_2$ observe a multi-dimensional underlying process ${\bf X}_t$ in (\[omodel\]) with accumulated income, discounted at present time, equalling $\int_0^\sigma e^{-\alpha
t}H({\bf X}_t)dt$ for any stopping time $\sigma$. If $P_1$ stops the game at time $\sigma$, he pays $P_2$ the amount of the accumulated income plus the amount $f_2({\bf X}_\sigma)$, which after been discounted equals $e^{-\alpha \sigma}f_2({\bf X}_\sigma)$. If the process is stopped by $P_2$ at time $\sigma$, he receives from $P_1$ the accumulated income less the amount $f_1({\bf X}_\sigma)$, which after been discounted equals $e^{-\alpha \sigma}f_1({\bf
X}_\sigma)$. $P_1$ tries to minimize his payment while $P_2$ tries to maximize his income. Let $\tau,\sigma$ be two stopping times, the value of this game is thus given by $$\label{vgame}
V({\bf x})=\inf_\tau\sup_\sigma J_{{\bf x}}(\tau,\sigma),\quad
\forall {\bf x}\in {\mathbb{R}}^n,$$ where $J_{\bf x}$ is given by (\[Jcost\]) on ${\mathbb{R}}^n$.

For the diffusion (\[omodel\]), define its infinitesimal generator $\mathcal{L}$ as $$\label{infgen}
\mathcal{L}:=\sum_{i=1}^n\mu_i\frac{\partial}{\partial
x_i}+\sum_{i,j=1}^n A_{ij}\frac{\partial^2}{\partial x_i\partial
x_j},$$ where $[A_{ij}]={\bf A}=\frac{1}{2}\sigma
\sigma^T$. We assume that ${\bf A}$ is non-degenerate.

Define the measure ${\mathbb m}(d{\bf x})=\rho({\bf x})d{\bf x}$, where $\rho({\bf x})$ satisfies the following condition: $$\label{rhocond}
{\mathbb A}\nabla\rho=\rho\cdot(\mu-{\bf b}),$$ where $b_i=\nabla\cdot{\bf A}_i,i=1,2,...,n$ in ${\bf
b}$. (Notice that when $\mu$ and ${\mathbb A}$ are constants, $\rho({\bf x})$ reduces to $\rho({\bf x})=\exp(({\mathbb
A}^{-1}\mu)\cdot {\bf x})$.) It can be seen that the absolute continuity condition (\[abscont\]) is satisfied.

We are unable to solve the case with a general multidimensional diffusion. Even in the case of one dimensional diffusion, conditions on $\mu$ and $\sigma$ should be made (see Appendix).

For the generator $\mathcal{L}$, its associated Dirichlet form $(\mathcal{E},\mathscr{F})$ densely embedded in $L^2({\mathbb{R}^n};{\mathbb m})$ is then given by $$\label{NDform}
\mathcal{E}(u,v)=\int_{\mathbb{R}^n}\nabla u({\bf x})\cdot{\bf
A}\nabla v({\bf x}){\mathbb m}(d{\bf x}), \quad u,v\in\mathscr{F},$$ where $$\mathscr{F}=\{u\in L^2({\mathbb{R}^n};{\mathbb m}):\ u {\rm\ is\
 continuous},\ \int_{{\mathbb{R}^n}}\nabla u({\bf
x})^T\nabla u({\bf x}){\mathbb m}(d{\bf x})<\infty\}.$$

For given functions $H,f_1,f_2$ satisfying the conditions of Theorem \[DgameHneq0\], and noticing that ${\bf X}_t$ is a non-degenerate Ito diffusion, we can conclude that $V({\bf x})$ in Eq.(\[vgame\]) is finite and continuous, and it solves (\[vi\]). Furthermore if we let $$\label{e12R} E_1=\{{\bf x}\in
{\mathbb{R}}^n: V({\bf x})=-f_1({\bf x})\}, \quad E_2=\{{\bf x}\in
{\mathbb{R}}^n: V({\bf x})=f_2({\bf x})\},$$ then the hitting times $\hat{\tau}=\tau_{E_2}$, $\hat{\sigma}=\tau_{E_1}$ is the saddle point of the game $$\label{saddle_Cont}
J_{\bf x}(\hat{\tau},\sigma)\leqslant J_{\bf
x}(\hat{\tau},\hat{\sigma})=V({\bf x})\leqslant J_{\bf
x}(\tau,\hat{\sigma})$$ for any ${\bf x}\in {\mathbb{R}}^n$ and any stopping times $\tau,\sigma$.

In the next section we shall give conditions on $H,f_1,f_2$ and characterize the regularities of $V({\bf x})$ and the form of the optimal control policy.

Optimal Stopping Regions
------------------------

In the one dimensional case, if the functions are defined over a bounded interval, a lot of properties are automatically satisfied [@Fuku02]. But in multi-dimensional case, this is much harder.

It is obvious that the conditions on $H,f_1,f_2$ are critical on the form of optimal control policy. For example, if $H\equiv 0$ and $-f_1({\bf x})<0<f_2({\bf x}),\ \forall {\bf x}$, then no party would ever stop the game and there is no optimal control.

\[assumpHf\] $f_1,f_2\in\mathscr{F}$ are smooth functions, $-M<-f_1({\bf
x})<0<f_2({\bf x})<M,\ \forall {\bf x}\in\mathbb{R}^n$ where $M$ is a constant, and $H\in L^2(\mathbb{R}^n;\mathbb{m})$ is everywhere continuous, and the separability condition (\[sepcondHneq0\]) holds. $H(\bar{\bf x},x_n)$ is strictly increasing in $x_n$, $f_1(\bar{\bf x},x_n)$ is nondecreasing in $x_n$, $f_2(\bar{\bf
x},x_n)$ is nonincreasing in $x_n$. Further more, $(\alpha-\mathcal{L})f_1(\bar{\bf x},x_n)+H(\bar{\bf x},x_n)$ is strictly increasing in $x_n$ and $(\alpha-\mathcal{L})f_2(\bar{\bf
x},x_n)-H(\bar{\bf x},x_n)$ is strictly decreasing in $x_n$. The (hyper)curves $a(\bar{\bf x})$, $b(\bar{\bf x})$ such that $$\begin{aligned}
(\alpha-\mathcal{L})f_1(\bar{\bf x},a(\bar{\bf x}))+H(\bar{\bf
x},a(\bar{\bf x}))&=&0,\\
(\alpha-\mathcal{L})f_2(\bar{\bf x},b(\bar{\bf x}))-H(\bar{\bf
x},b(\bar{\bf x}))&=&0,\end{aligned}$$ with $a(\bar{\bf x})<b(\bar{\bf
x})$, $\forall \bar{\bf x}\in{\mathbb{R}}^{n-1}$, are assumed to be bounded and uniformly Lipschitz continuous.

Then it is easy to see that

\[f1f2Hsign\] Assume Assumption \[assumpHf\]. For any $(\bar{\bf x},x_n)$ with $x_n<a(\bar{\bf
x})$, $$(\alpha-\mathcal{L})f_1(\bar{\bf x},x_n)+H(\bar{\bf x},x_n)<0,$$ and for any $(\bar{\bf x},x_n)$ with $x_n>a(\bar{\bf
x})$, $$(\alpha-\mathcal{L})f_1(\bar{\bf x},x_n)+H(\bar{\bf x},x_n)>0.$$ Similarly, for any $(\bar{\bf x},x_n)$ with $x_n<b(\bar{\bf x})$, $$(\alpha-\mathcal{L})f_2(\bar{\bf x},x_n)-H(\bar{\bf x},x_n)>0,$$ and for any $(\bar{\bf x},x_n)$ with $x_n>b(\bar{\bf
x})$, $$(\alpha-\mathcal{L})f_2(\bar{\bf x},x_n)-H(\bar{\bf x},x_n)<0.$$

Define the set $$\label{cregion}
E=\{{\bf x}\in{\mathbb{R}}^n:-f_1({\bf x})<V({\bf x})<f_2({\bf
x})\}.$$Since $P_2$ would stop the game once $V({\bf x})\leqslant -f_1({\bf x})$ and the instant payoff is $-f_1({\bf x})$, while $P_1$ would stop the game once $V({\bf
x})\geqslant f_2({\bf x})$ and the instant payoff is $f_2({\bf x})$, we could write ${\mathbb{R}}^n$ as a partition: $${\mathbb{R}}^n=E_1\cup E\cup E_2,$$ where $E_1,E_2$ were given in (\[e12R\]).

\[Jandf\] Assume Assumption \[assumpHf\]. For each ${\bf x}\in E_1$, $$(\alpha-\mathcal{L})f_1({\bf x})+H({\bf x})\leqslant0,$$ and for each ${\bf x}\in E_2$, $$(\alpha-\mathcal{L})f_2({\bf
x})-H({\bf x})\leqslant 0.$$

We only give proof to the first half. We know at the point ${\bf
x}\in E_1$ it must be true that $V({\bf x})\leqslant-f_1({\bf x})$, and it is optimal for $P_2$ to stop the game immediately. Suppose $$(\alpha-\mathcal{L})f_1({\bf x})+H({\bf x})>0,\ {\bf x}\in E_1,$$ then by the smoothness of $f_1$ and the continuity of $H$, we can find a small ball $B_r({\bf x})$ containing the point ${\bf x}$, such that for each ${\bf y}\in B_r({\bf x})$, $$(\alpha-\mathcal{L})f_1({\bf y})+H({\bf y})>0.$$ Consider a policy for $P_2$ to stop the game at the first exit time of $B_r({\bf x})$, denoted $\tau_{B_r}$. Then by Dynkin’s formula, the payoff would be $$\begin{aligned}
J_{{\bf x}}&=&E_{{\bf x}}\int_0^{\tau_{B_r}}e^{-\alpha t}H({\bf
X}_t)dt+E_{{\bf
x}}(e^{-\alpha \tau_{B_r}}(-f_1({\bf X}_{\tau_{B_r}})))\\
&=&E_{{\bf x}}\int_0^{\tau_{B_r}}e^{-\alpha t}H({\bf
X}_t)dt-f_1({\bf x})+E_{{\bf x}}\int_0^{\tau_{B_r}}e^{-\alpha
t}(\alpha-\mathcal{L})f_1({\bf X}_t)dt\\
&=&-f_1({\bf x})+E_{{\bf x}}\int_0^{\tau_{B_r}}e^{-\alpha
t}[(\alpha-\mathcal{L})f_1({\bf
X}_t)+H({\bf X}_t)]dt\\
&>&-f_1({\bf x}).\end{aligned}$$ This is a contradiction since $P_2$ tries to maximize his payoff but we assumed that the optimal policy at ${\bf
x}$ was to stop the game immediately.

Assume Assumption \[assumpHf\]. If ${\bf x}=(\bar{\bf x},x_n)\in E_1$, then for any point $(\bar{\bf x},y)$ with $y< x_n$, $$(\alpha-\mathcal{L})f_1(\bar{\bf x},y)+H(\bar{\bf x},y)<0.$$ If ${\bf x}=(\bar{\bf x},x_n)\in E_2$, then for any point $(\bar{\bf
x},y)$ with $y> x_n$, $$(\alpha-\mathcal{L})f_2(\bar{\bf x},y)-H(\bar{\bf x},y)<0.$$ Furthermore, $E_1\subseteq{\mathbb R}^{n-1}\times(-\infty,a]$ and $E_2\subseteq{\mathbb R}^{n-1}\times[b,\infty)$.

This can be easily seen from Proposition \[Jandf\] and the conditions on $f_1,f_2, H$ given in Assumption \[assumpHf\].

Further, noticing the conditions on the curves $a({\bar{\bf x}})$ and $b({\bar{\bf x}})$, $\bar{\bf x}\in{\mathbb R}^{n-1}$, we have the following:

Assume Assumption \[assumpHf\]. $E\supseteq {\mathbb R}^{n-1}\times(a,b)$ and hence $E$ is not empty. Furthermore, the value of this game $V$ is bounded by $M$, where $M$ is given in Assumption \[assumpHf\].

Take any point ${\bf x}=({\bar{\bf x}},x_n)\in E_1$, and denote $\sigma_a$ the hitting time to the curve $a(\cdot)$. Notice that the diffusion (\[omodel\]) is a conservative process by the given conditions, and also by noticing the conditions given on $a(\cdot)$, it can be concluded that $E_{\bf x}(e^{-\alpha \sigma_a})$ goes to zero as $x_n$ goes to $-\infty$. Similarly $E_{\bf x}(e^{-\alpha
\sigma_b})$, ${\bf x}=({\bar{\bf x}},x_n)\in E_2$, goes to zero as $x_n$ goes to $\infty$.

\[assumAB\] There exist functions $A(\bar{\bf x}), B(\bar{\bf x})$, $\bar{\bf
x}\in{\mathbb{R}}^{n-1}$ that are uniformly bounded and such that for any ${\bf x}\in{\mathbb R}^{n-1}\times(-\infty,A]$, $$E_{\bf x}\left(\int_0^{\sigma_a} e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha\sigma_a}M\right)<-f_1({\bf x}),$$ and for any ${\bf x}\in{\mathbb
R}^{n-1}\times[B,\infty)$, $$E_{\bf x}\left(\int_0^{\sigma_b} e^{-\alpha t}H({\bf
X}_t)dt-e^{-\alpha\sigma_b}M\right)>f_2({\bf x}).$$

Assume Assumptions \[assumpHf\] and \[assumAB\], then $A(\bar{\bf x})<a(\bar{\bf x})$ and $B(\bar{\bf
x})>b(\bar{\bf x})$, $\forall\bar{\bf x}\in{\mathbb{R}}^{n-1}$. Furthermore, on ${\mathbb R}^{n-1}\times(-\infty,A]$ player $P_2$ would stop the game immediately and $V=-f_1$, and on ${\mathbb
R}^{n-1}\times[B,\infty)$ player $P_1$ would stop the game immediately and $V=f_2$.

Suppose there is a point ${\bf x}=(\bar{\bf x},x_n)$ with $a(\bar{\bf x})\leq x_n\leq A(\bar{\bf x})$. Then by Dynkin’s formula, $$\begin{aligned}
&&E_{\bf x}\left(\int_0^{\sigma_a\wedge T} e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha(\sigma_a\wedge T)}M\right)\\
&& > E_{\bf x}\left(\int_0^{\sigma_a\wedge T} e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha(\sigma_a\wedge T)}(-f_1({\bf X}_{\sigma_a\wedge
T}))\right)\\
&&=-f_1({\bf x})+E_{\bf x}\left(\int_0^{\sigma_a\wedge T}e^{-\alpha
t}((\alpha-\mathcal{L})f_1({\bf X}_t)+H({\bf
X}_t))dt\right)>-f_1({\bf x})\end{aligned}$$ by Proposition \[f1f2Hsign\]. Taking $T\to\infty$ we get a contradiction.

Now suppose ${\bf x}\in{\mathbb R}^{n-1}\times(-\infty,A]$. For any stopping time $\sigma$ for player $P_2$ the payoff will be $$\begin{aligned}
&&E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf
x}[e^{-\alpha \sigma}(-f_1({\bf X}_{\sigma}))]\\
&&=\left(E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf
x}[e^{-\alpha \sigma}(-f_1({\bf X}_{\sigma}))]\right)P_{\bf
x}(\sigma\leq \sigma_a)\\
&&+\left(E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf
x}[e^{-\alpha \sigma}(-f_1({\bf X}_{\sigma}))]\right)P_{\bf
x}(\sigma>\sigma_a).\end{aligned}$$ When $\sigma\leq\sigma_a$, the following quantity $$\begin{aligned}
&&=E_{\bf x}\int_0^{\sigma\wedge T\wedge\sigma_a}e^{-\alpha t}H({\bf
X}_t)dt+E_{\bf
x}[e^{-\alpha (\sigma\wedge T\wedge \sigma_a)}(-f_1({\bf X}_{\sigma\wedge T\wedge\sigma_a}))]\\
&&=-f_1({\bf x})+E_{\bf x}\int_0^{\sigma\wedge
T\wedge\sigma_a}e^{-\alpha t}((\alpha-\mathcal{L})f_1({\bf
X}_t)+H({\bf X}_t))dt\end{aligned}$$ is less than $-f_1({\bf x})$, $\forall T>0$ by Proposition \[f1f2Hsign\].

When $\sigma>\sigma_a$, $$\begin{aligned}
 &&E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf x}[e^{-\alpha \sigma}(-f_1({\bf
X}_{\sigma}))]\\
&&\leq E_{\bf x}\left(\int_0^{\sigma_a}e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha\sigma_a}M\right)\end{aligned}$$ because $M$ is the bound of the payoff of each player. Hence by Assumption \[assumAB\], $$E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf
x}[e^{-\alpha \sigma}(-f_1({\bf X}_{\sigma}))]<-f_1({\bf x}).$$ As a summary, if ${\bf x}\in{\mathbb R}^{n-1}\times(-\infty,A]$, then for any stopping policy of $P_2$, the expected payoff is less than $-f_1({\bf x})$, and the optimal strategy is to stop the game immediately. The other half of this proposition can be proved in a similar way.

By the properties of $A(\cdot)$ (or $B(\cdot)$), we can choose a bounded and continuous curve below $A(\cdot)$ (or a bounded and continuous curve above $B(\cdot)$) which also has the properties as given in Assumption \[assumAB\]. Without loss of generality, we assume $A(\cdot)$ and $B(\cdot)$ are bounded and continuous.

Now it is easy to see that ${\mathbb
R}^{n-1}\times(-\infty,A]\subseteq E_1$ and ${\mathbb
R}^{n-1}\times[B,\infty)\subseteq E_2$. Since the functions $V,f_1,f_2$ are all continuous, the boundary of $E$ consists of continuous curves.

Let $\tilde{E}_1$ be the largest connected region in $E_1$ containing the set ${\mathbb R}^{n-1}\times(-\infty,A]$, and $\tilde{E}_2$ be the largest connected region in $E_2$ containing the set ${\mathbb R}^{n-1}\times[B,-\infty)$, then obviously $V({\bf
x})=-f_1({\bf x}),\forall {\bf x}\in\tilde{E}_1$, and $V({\bf
x})=f_2({\bf x}),\forall {\bf x}\in\tilde{E}_2$. Furthermore $\tilde{E}_1$ has a continuous boundary curve $\tilde{a}$ that is bounded by $A$ and $a$, $\tilde{E}_2$ has a continuous boundary curve $\tilde{b}$ that is bounded by $b$ and $B$.

$\tilde{E}_1,\tilde{E}_2$ are simply connected regions.

It suffices to prove that $\tilde{E}_1$ is simply connected. Suppose there is a point ${\bf x}_0$ such that $V({\bf x}_0)>-f_1({\bf
x}_0)$, and the boundary $\partial D_{\bf x_0}$ of the largest connected region containing ${\bf x}_0$ as well as the points ${\bf
x}$ with $V({\bf x})>-f_1({\bf x})$ belongs to $\tilde{E}_1$, i.e., $\partial D_{\bf x_0}\subset\tilde{E}_1$. Then for any ${\bf x}\in
\partial D_{\bf x_0}$, $V({\bf x})=-f_1({\bf x})$, and for any ${\bf x}\in D_{\bf
x_0}$, $(\alpha-\mathcal{L})f_1({\bf x})+H({\bf x})<0$. Consider any stopping strategy for player $P_2$ with the reward $$\begin{aligned}
&&E_{\bf x_0}\left(\int_0^\sigma e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha\sigma}(-f_1({\bf X}_\sigma))\right)\nonumber\\
&&=-f_1({\bf x}_0)+E_{\bf x_0}\int_0^\sigma e^{-\alpha
t}((\alpha-\mathcal{L})f_1({\bf X}_t)+H({\bf X}_t))dt\nonumber\\
&&<-f_1({\bf x}_0),\end{aligned}$$ since $\sigma\leqslant \tau_{\partial D_{\bf x_0}}$ a.s., where $\tau_{\partial D_{\bf x_0}}$ is the first hitting time to $\partial D_{\bf x_0}$, and on $\partial D_{\bf x_0}$ it is known that the game should be stopped by $P_2$ with the payoff $-f_1$. But this is a contradiction since we have assumed that $V({\bf
x}_0)>-f_1({\bf x}_0)$. Therefore there is no *hole* in $\tilde{E}_1$.

If there is any point $(\bar{\bf x},x_n)$ with $x_n<a(\bar{\bf x})$ such that $V(\bar{\bf x},x_n)>-f_1(\bar{\bf x},x_n)$, then the connected region containing this point with $V>-f_1$ is connected to the region ${\mathbb{R}}^{n-1}\times[a,b]$. A similar result holds on the curve $b$.

Suppose not, then the boundary of the connected region containing $(\bar{\bf x},x_n)$ with $V>-f_1$ is contained in ${\mathbb
R}^{n-1}\times(-\infty,a)$. For any stopping strategy for player $P_2$, the process is stopped before it hits the curve $a$. The expected payoff is less than or equal to $-f_1(\bar{\bf x},x_n)$ because in this region $(\alpha-\mathcal{L})f_1+H<0$, and this contradicts the assumption $V(\bar{\bf x},x_n)>-f_1(\bar{\bf
x},x_n)$.

Now it is clear that the region $E$ is connected. Recall that $\hat{\sigma}=\tau_{E_1},\hat{\tau}=\tau_{E_2}$ are the first hitting times to the sets $E_1,E_2$ respectively, and they are finite a.s., we can rewrite $V$ as $$V({\bf x})=E_{\bf
x}\left(\int_0^{\hat{\tau}\wedge\hat{\sigma}}e^{-\alpha t}H({\bf
X}_t)dt+I_{\hat{\sigma}<\hat{\tau}}e^{-\alpha\hat{\sigma}}(-f_1({\bf
X}_{\hat{\sigma}}))+I_{\hat{\tau}<\hat{\sigma}}e^{-\alpha\hat{\tau}}f_2({\bf
X}_{\hat{\tau}})\right).$$

But at this point we still can not tell that $E$ is simply connected. Let ${\bf U}$ be any connected region such that ${\mathbb
R}^{n-1}\times[a,b]\subset {\bf U}$ and ${\bf U}\subset {\mathbb
R}^{n-1}\times[A,B]$. Define $\tau_{\bf U}$ the first exit time of this region, then obviously $\tau_{\bf U}$ is finite a.s. Define the function $$\label{fdef}
F_{\bf U}({\bf x})=E_{\bf x}\left(\int_0^{\tau_{\bf U}}e^{-\alpha
t}H({\bf X}_t)dt+e^{-\alpha\tau_{\bf U}}R({\bf X}_{\tau_{\bf
U}})\right),\ \ \forall {\bf x}\in{\bf U},$$ where $R({\bf X}_{\tau_{\bf U}})=-f_1({\bf X}_{\tau_{\bf
U}})$ if ${\bf X}_{\tau_{\bf U}}\in{\mathbb R}^{n-1}\times[A,a]$, and $R({\bf X}_{\tau_{\bf U}})=f_2({\bf X}_{\tau_{\bf U}})$ if ${\bf
X}_{\tau_{\bf U}}\in{\mathbb R}^{n-1}\times[b,B]$.

We put an assumption jointly on the process ${\bf X}_t$ and the functions $f_1,f_2,h$. Consider any point $(\bar{\bf
x}_0,x_n)\in\bar{\bf U}$ with $x_n<a(\bar{\bf x}_0)$. Define the cone $$C_{(\bar{\bf x}_0,x_n)}=\left\{(\bar{\bf
x},y):x_n\leqslant y\leqslant a(\bar{\bf
x}),\frac{y-x_n}{\|<\bar{\bf x}-\bar{\bf x}_0,y-x_n>\|_2}\geqslant
\zeta\right\},$$ where $\zeta\in(0,1)$ is a constant. Construct the new connected region $\tilde{\bf U}={\bf U}\cup
C_{(\bar{\bf x}_0,x_n)}$ and define the function $F_{\tilde{\bf U}}$ similarly as in (\[fdef\]).

\[unif1\] For any point $(\bar{\bf x}_0,x_n)\in\bar{\bf U}$ with $x_n<a(\bar{\bf x}_0)$, $F_{\tilde{\bf U}}(\bar{\bf
x}_0,x_n)\geqslant F_{\bf U}(\bar{\bf x}_0,x_n)$.

Similarly, if we consider any point $(\bar{\bf x}_0,x_n)\in\bar{\bf
U}$ with $x_n>b(\bar{\bf x}_0)$, we may define the cone $$C_{(\bar{\bf x}_0,x_n)}=\left\{(\bar{\bf
x},y):x_n\geqslant y\geqslant b(\bar{\bf
x}),\frac{x_n-y}{\|<\bar{\bf x}-\bar{\bf x}_0,y-x_n>\|_2}\geqslant
\zeta\right\},$$ where $\zeta\in(0,1)$ is a constant. Construct the new connected region $\tilde{\bf U}={\bf U}\cup
C_{(\bar{\bf x}_0,x_n)}$ and define the function $F_{\tilde{\bf U}}$ similarly as in (\[fdef\]), then we put the following assumption.

\[unif2\] For any point $(\bar{\bf x}_0,x_n)\in\bar{\bf U}$ with $x_n>b(\bar{\bf x}_0)$, $F_{\tilde{\bf U}}(\bar{\bf
x}_0,x_n)\leqslant F_{\bf U}(\bar{\bf x}_0,x_n)$.

In the case of one dimensional diffusion, Assumptions \[assumpHf\] and \[assumAB\] are sufficient to imply that $E$ - the continuation region - is an interval, but in the multidimensional case, more conditions are needed to guarantee a regular property.

Under Assumptions \[unif1\],\[unif2\], the continuation region $E$ is simply connected. Furthermore, the lower and upper boundaries of $E$ (in $x_n$) are uniformly Lipschitz functions of $\bar{\bf
x}\in{\mathbb R}^{n-1}$.

We have shown that $E$ is connected. For any point $(\bar{\bf
x},x_n)\in\bar{E}$ with $x_n<a(\bar{\bf x})$, we consider the new continuation region $E\cup C_{(\bar{\bf x},x_n)}$, and have $F_{E\cup C_{(\bar{\bf x},x_n)}}(\bar{\bf x},x_n)\geqslant
V(\bar{\bf x},x_n)$ by Assumption \[unif1\]. Since in the region below the curve $a$, only player $P_2$ wants to stop the game who wants to maximize his payoff, the new continuation region $E\cup
C_{(\bar{\bf x},x_n)}$ is certainly better than $E$, if not identical, and this is a contradiction since $E$ is assumed to be optimal. Thus $E\cup C_{(\bar{\bf x},x_n)}=E$. Since this holds for any point $(\bar{\bf x},x_n)\in\bar{E}$ with $x_n<a(\bar{\bf x})$, we know that the region in $E$ below the curve $a$ is simply connected, and the lower boundary of $E$ is uniformly Lipshcitz continuous. The second half is proved in a similar manner.

Now we can claim that $\tilde{E}_1=E_1,\tilde{E}_2=E_2$. In what follows we shall still use $\tilde{a}$ as the upper boundary (in $x_n$) of the set $E_1$ (also the lower boundary of $E$), and $\tilde{b}$ as the lower boundary (in $x_n$) of the set $E_2$ (also the upper boundary of $E$), and they are uniformly Lipschitz continuous and bounded curves. We notice that the curve $a$ in Assumption \[assumpHf\] is not necessarily identical to the curve $\tilde{a}$, and the curve $b$ is not necessarily identical to the curve $\tilde{b}$ either. Figure \[region\] illustrates the continuation region $E$ in a two dimensional case.

Regularities of the Value Function
----------------------------------

\[prop2\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\]. If the curves $\tilde{a}=\tilde{a}(\bar{\bf x})$, $\tilde{b}=\tilde{b}(\bar{\bf x})$ are smooth, then $V$ is smooth on ${\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})$, and $$\begin{aligned}
\alpha V({\bf x})-\mathcal{L}V({\bf x})&=H({\bf x}),\quad {\bf
x}\in{\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}),\label{inAB}\\
\alpha V({\bf x})-\mathcal{L}V({\bf x})&>H({\bf x}),\quad {\bf
x}\in{\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}), \\
\alpha V({\bf x})-\mathcal{L}V({\bf x})&<H({\bf x}),\quad {\bf
x}\in{\mathbb{R}}^{n-1}\times(\tilde{b},\infty).\end{aligned}$$

It is still not clear under what conditions the free boundary curves are smooth. But it is reasonable to believe that any condition on this should involve jointly the underlying process and all the reward functions of this problem.

We notice that $V$ is $H$-$\alpha$ harmonic on ${\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})$, which by [@Fuku11] implies the validation of the following equation: $$\label{VIeq}
\mathcal{E}_\alpha(V,u)=(H,u),\quad \forall u\in
C_0^{1,\cdots,1}({\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})).$$ The continuity of $H$ implies that $V$ is smooth on the same region, and an integration by parts yields (\[inAB\]) on this region. The rest of the proof follows from the fact that $V({\bf
x})=-f_1({\bf x}),\forall {\bf
x}\in{\mathbb{R}}^{n-1}\times(-\infty,\tilde{a})$ and $V({\bf
x})=f_2({\bf x}),\forall {\bf x}\in(\tilde{b},\infty)$.

In what follows we will characterize the regularity of $V$ on the boundary curves.

Let $\partial_{\bf u} V$ denote the one-sided directional derivative along a unit vector ${\bf u}\in{\mathbb{R}}^n$ defined in the following manner $$\partial_{\bf u} V({\bf x})=\lim_{\lambda\to 0+}\frac{V({\bf x}+\lambda{\bf u})-V({\bf
x})}{\lambda}.$$ By Proposition \[prop2\], it can be seen that $\partial_{\bf u}
V({\bf x})$ is well defined at any point ${\bf x}$ in any direction ${\bf u}$, and $\partial_{\bf u} V({\bf x})$ is continuous. If ${\bf
x}$ is not on the curves $\tilde{a}(\cdot)$ or $\tilde{b}(\cdot)$, then obviously $\partial_{\bf u} V({\bf x})=\partial_{-\bf u} V({\bf
x})$. The following proposition will characterize the property of $\partial_{\bf u} V({\bf x})$ when ${\bf x}$ is on $\tilde{a}(\cdot)$ or $\tilde{b}(\cdot)$.

Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\]. If the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ are smooth, then $$\partial_{\bf u} V(\bar{\bf x},\tilde{a}(\bar{\bf
x}))=-\partial_{\bf u} f_{1}(\bar{\bf x},\tilde{a}(\bar{\bf x})), \
\partial_{\bf u} V(\bar{\bf x},\tilde{b}(\bar{\bf x}))=\partial_{\bf u} f_{2}(\bar{\bf x},\tilde{b}(\bar{\bf x})),\ \forall
\bar{\bf x}\in{\mathbb{R}}^{n-1},\ \forall {\bf u}\in{\mathbb{R}}^n.$$

-   Pick any point $(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$ on the curve $\tilde{a}(\cdot)$ and any unit vector ${\bf u}\in{\mathbb
    R}^n$ (similar result can be derived for the curve $\tilde{b}(\cdot)$), and construct a ball $B_\delta(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))$ centered at this point with radius $\delta$, and such that for any point $(\bar{\bf x},x_n)\in
    B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, we have $A({\bar{\bf x}})<x_n<b(\bar{\bf x})$. This can be easily done under Assumptions \[assumpHf\] and \[assumAB\]. Also by the smoothness and Lipschitz property of the curve $\tilde{a}(\cdot)$, we can choose $\delta$ small so that all the points $(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))+\lambda {\bf u}$ ($0<\lambda\leqslant\delta$) are either on the curve $\tilde{a}(\cdot)$, below the curve $\tilde{a}(\cdot)$ or above the curve $\tilde{a}(\cdot)$. In the first two cases, since we have showed that $V=-f_1$, the result automatically holds. Now we assume $(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\lambda {\bf u}$ ($0<\lambda\leqslant\delta$) are above the curve $\tilde{a}(\cdot)$, then it has been shown that $V((\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))+\lambda {\bf u})>-f_1((\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))+\lambda {\bf u})$, hence $\partial_{\bf u}V(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))\geqslant -\partial_{\bf
    u}f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$. Assume the equality does not hold, then there is $\epsilon>0$ such that $\partial_{\bf
    u}V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))= -\partial_{\bf
    u}f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\epsilon$. By the continuity of $\partial_{\bf u}V$ and the Lipschitz property of $\tilde{a}(\cdot)$, we can construct a cone $\mathcal{C}_{\bf u}$ containing ${\bf u}$ such that $\forall {\bf v}\in \mathcal{C}_{\bf
    u}$, the points $(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\lambda
    {\bf v}$ ($0<\lambda\leqslant\delta$) are above the curve $\tilde{a}(\cdot)$ and $\partial_{\bf v}V(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))>-\partial_{\bf v}f_1(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))+\epsilon/2$.

-   Define $\tau_{\tilde{a},\delta}$ the first exit time from $B_\delta(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))$. Firstly, by the fact that $\bf{X}_t$ is uniformly elliptic, we have $$E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}(\tau_{\tilde{a},\delta})=o(\delta),$$where $o(\delta)$ is a small quantity satisfying $\lim_{\delta\to 0}{o(\delta)}/{\delta}=0$, (see, e.g., [@Fuku02], [@Karatzas91]). Further, we notice that $$1-E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}\right)=\alpha
    E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}e^{-\alpha
    t}dt\right)\leqslant\alpha E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}(\tau_{\tilde{a},\delta})=o(\delta).$$

-   Recall that player $P_2$ would stop the game at the point $(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$ immediately, and the payoff is $-f_1(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))$. Consider the region $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, it is shown that play $P_1$ would not stop the game in this region. Since $\hat{\tau}=\tau_{\tilde{a},\delta}+\hat{\tau} \circ
    \theta_{\tau_{\tilde{a},\delta}}$ where $\theta_t$ is the shift operator, we get $$\hat{\tau}\wedge\sigma=\tau_{\tilde{a},\delta}+(\hat{\tau}\wedge\hat{\sigma})\circ\theta_{\tau_{\tilde{a},\delta}},$$ where $\sigma=\tau_{\tilde{a},\delta}+\hat{\sigma}\circ\theta_{\tau_{\tilde{a},\delta}}$. Therefore by (\[saddle\_Cont\]), $$\begin{aligned}
    \label{ineqVx0}
    &&-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))=V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\geqslant J_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(\hat{\tau},\sigma)\nonumber\\
    &&\ \ \ \ \ \ \ =E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}e^{-\alpha t}H({\bf
    X}_t)dt\right) +E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}V({\bf
    X}_{\tau_{\tilde{a},\delta}})\right).\end{aligned}$$ By Assumption \[assumpHf\], it is obvious that $H$ is bounded on $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, i.e., there exists $U>0$ such that $|H({\bf x})|\leqslant U,\ \forall {\bf x}\in
    B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$. Then we have $$\begin{aligned}
    &&0\leqslant \left|E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}e^{-\alpha t}H({\bf
    X}_t)dt\right)\right|\leqslant UE_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}e^{-\alpha
    t}dt\right)\nonumber\\
    &&= \frac{U}{\alpha}\left(1-E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}\right)\right)
    =o(\delta).\nonumber\end{aligned}$$

    The other part $$\begin{aligned}
    &&E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}V({\bf
    X}_{\tau_{{a},\delta}})\right)=E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left((1-\alpha\tau_{\tilde{a},\delta}+o(\alpha\tau_{\tilde{a},\delta}))V({\bf
    X}_{\tau_{\tilde{a},\delta}})\right)\nonumber\\
    &&=E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(V({\bf
    X}_{\tau_{\tilde{a},\delta}})\right)+E_{(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left((-\alpha\tau_{\tilde{a},\delta}+o(\alpha\tau_{\tilde{a},\delta}))V({\bf
    X}_{\tau_{\tilde{a},\delta}})\right).\nonumber\end{aligned}$$ Since $V$ is also bounded and $E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}(\tau_{\tilde{a},\delta})=o(\delta)$, we get $$E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}V({\bf
    X}_{\tau_{\tilde{a},\delta}})\right)=E_{(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))}\left(V({\bf
    X}_{\tau_{\tilde{a},\delta}})\right)+o(\delta).$$

-   By Dynkin’s formula, $$E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(-f_1({\bf
    X}_{\tau_{\tilde{a},\delta}})\right)=-f_1(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))+E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}\mathcal{L}(-f_1)({\bf
    X}_t)dt\right).$$ Since the function $-f_1$ is bounded on $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))$, and $E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}(\tau_{\tilde{a},\delta})=o(\delta)$, we get $$\label{f_1approx}
    E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(-f_1({\bf
    X}_{\tau_{\tilde{a},\delta}})\right)=-f_1(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))+o(\delta).$$ On the other hand, for each point ${\bf y}\in \partial
    B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, i.e., the boundary of $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, and by the smoothness of $-f_1$, $$-f_1({\bf y})=-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))+\partial_{\bf v}(-f_1)(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))\delta+o(\delta),$$ so $$\begin{aligned}
    \label{f_1approx1}
    &&E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(-f_1({\bf
    X}_{\tau_{\tilde{a},\delta}})\right)\nonumber\\
    &&=-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\int\partial_{\bf
    v}(-f_1)(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta p_{(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf y})+o(\delta),\end{aligned}$$ where $p_{(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf y})$ is the probability density function of the exit distribution. By comparing (\[f\_1approx\]) with (\[f\_1approx1\]), we get $$\int\partial_{\bf v}(-f_1)(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))\delta p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf
    y})=o(\delta).$$

-   On the boundary of $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, $V({\bf y})\geqslant -f_1({\bf y})$. Since $V$ is smooth a.e., $$V({\bf y})=V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\partial_{\bf
    v}V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta+o(\delta),$$ and $$\begin{aligned}
    &&E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(V({\bf
    X}_{\tau_{\tilde{a},\delta}})\right)\\
    &&=V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\int \partial_{\bf
    v}V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta p_{(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf y})+o(\delta)\nonumber\\
    &&=-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\int \partial_{\bf
    v}V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta p_{(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf y})+o(\delta)\nonumber\\
    &&\geqslant -f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\int
    \partial_{\bf v}(-f_1)(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta
    p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf
    y})\nonumber\\
    && + \int_{\mathcal{C}_{\bf u}\cap\partial B_\delta(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))} \frac{\epsilon}{2}\delta
    p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf
    y})-|o(\delta)|,\nonumber\end{aligned}$$ in view of the conclusion of Step 1, and $\mathcal{C}_{\bf u}\cap\partial B_\delta(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))$ is the part of $\partial
    B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$ in the cone $\mathcal{C}_{\bf u}$. Substitute in (\[ineqVx0\]) the results from Steps 3 and 4, we get $$\begin{aligned}
    &&-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))=V(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))\nonumber\\
    && \geqslant -f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))+\int_{\mathcal{C}_{\bf u}\cap\partial B_\delta(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))} \frac{\epsilon}{2}\delta
    p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf
    y})-|o(\delta)|.\nonumber\end{aligned}$$ Since ${\bf X}_t$ is uniformly elliptic, $\int_{\mathcal{C}_{\bf
    u}\cap\partial B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}
     p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
    x}_0))}(d{\bf y})\to \eta$ as $\delta\to 0$, where $0<\eta<1$ is a constant. By choosing $\delta$ small, we get $$\begin{aligned}
    -f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))=V(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf x}_0))\geqslant -f_1(\bar{\bf
    x}_0,\tilde{a}(\bar{\bf
    x}_0))+\frac{\epsilon\delta\eta}{4}-|o(\delta)|.\end{aligned}$$ If $\delta$ is sufficiently small, the part $\frac{\epsilon\delta\eta}{4}-|o(\delta)|>0$, and we get a contradiction.

Now the proof is complete.

As a summary we have the following theorem:

\[Vcond\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\]. If the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ are smooth, then $$\begin{aligned}
-f_1({\bf x})<&&V({\bf x})<f_2({\bf x}),\quad \forall{\bf x}\in
{\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}),\\
V({\bf x})&&=-f_1({\bf x}), \quad \forall{\bf x}\in {\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}],\\
V({\bf x})&&=f_2({\bf x}), \quad \forall{\bf x}\in
{\mathbb{R}}^{n-1}\times[\tilde{b},\infty),\end{aligned}$$ $$\partial_{\bf u} V(\bar{\bf x},\tilde{a}(\bar{\bf x}))=-\partial_{\bf u}
f_{1}(\bar{\bf x},\tilde{a}(\bar{\bf x}),\ \partial_{\bf u}
V(\bar{\bf x},\tilde{b}(\bar{\bf x}))=\partial_{\bf u}
f_{2}(\bar{\bf x},\tilde{b}(\bar{\bf x})),\ \forall \bar{\bf
x}\in{\mathbb{R}}^{n-1},$$ where ${\bf u}$ is any directional vector.

Furthermore $V$ is $C^{1,\cdots,1,1}$ on ${\mathbb{R}}^{n}$, $C^{2,\cdots,2}$ on ${\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})\cup{\mathbb{R}}^{n-1}\times(-\infty,\tilde{a})\cup{\mathbb{R}}^{n-1}\times(\tilde{b},\infty)$ and $$\begin{aligned}
\alpha V({\bf x})-\mathcal{L}V({\bf x})&=H({\bf x}),\quad
 &\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}), \\
\alpha V({\bf x})-\mathcal{L}V({\bf x})&>H({\bf x}),\quad
 &\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}), \\
\alpha V({\bf x})-\mathcal{L}V({\bf x})&<H({\bf x}),\quad
&\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(\tilde{b},\infty),\end{aligned}$$ where $\mathcal{L}$ is given in (\[infgen\]).

The Multi-Dimensional Stochastic Singular Control Problem {#mdssc}
=========================================================

Define $h({\bf x}),W({\bf x}),\ {\bf x}\in{\mathbb{R}}^n$, as follows: $$\begin{aligned}
h(\bar{\bf x},y)&=&\int_{\tilde{a}(\bar{\bf x})}^yH(\bar{\bf x},u)du+C(\bar{\bf x}),\label{hcost}\\
W(\bar{\bf x},y)&=&\int_{\tilde{a}(\bar{\bf x})}^yV(\bar{\bf
x},u)du,\quad \bar{\bf x}\in{\mathbb{R}}^{n-1},\
y\in{\mathbb{R}},\label{wval}\end{aligned}$$ where $C(\bar{\bf x})$ is a function of $\bar{\bf x}$ such that $$\lim_{y\to \tilde{a}(\bar{\bf x})+}\alpha W(\bar{\bf
x},y)-\mathcal{L}W(\bar{\bf x},y)-h(\bar{\bf x},y)=0,$$ then $h(\bar{\bf x},y)$ and $W(\bar{\bf x},y)$ satisfy the following:

\[Wcond\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\]. If the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ are smooth, then $W$ is $C^{2,\cdots,2}$ on ${\mathbb{R}}^n$ and $$\begin{aligned}
&&\alpha W({\bf x})-\mathcal{L}W({\bf x})=h({\bf x}),\quad
 \forall{\bf x}\in {\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}), \\
&&\alpha W({\bf x})-\mathcal{L}W({\bf x})<h({\bf x}),\quad
\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}) \cup
{\mathbb{R}}^{n-1}\times(\tilde{b},\infty),\\
&&-f_1({\bf x})<\frac{\partial }{\partial x_n}W({\bf x})<f_2({\bf x}),\quad \forall{\bf x}\in {\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}),\\
&&\frac{\partial}{\partial x_n}W({\bf x})=-f_1({\bf x}), \quad \forall{\bf x}\in {\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}],\\
&&\frac{\partial}{\partial x_n}W({\bf x})=f_2({\bf x}), \quad
\forall{\bf x}\in {\mathbb{R}}^{n-1}\times[\tilde{b},\infty),\end{aligned}$$ and $$\begin{aligned}
&&\frac{\partial^2}{\partial x_n\partial x_k}W(\bar{\bf
x},\tilde{a}(\bar{\bf x}))=-\frac{\partial f_1}{\partial
x_k}(\bar{\bf
x},\tilde{a}(\bar{\bf x})),\nonumber\\
&&\frac{\partial^2}{\partial x_n\partial x_k}W(\bar{\bf
x},\tilde{b}(\bar{\bf x}))=\frac{\partial f_2}{\partial
x_k}(\bar{\bf x},\tilde{b}(\bar{\bf x})),\ \forall\bar{\bf
x}\in{\mathbb{R}}^{n-1},\ 1\leqslant k\leqslant n.\nonumber\end{aligned}$$

We first need a lemma.

\[cont\_alpha\_L\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\] and that the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ being smooth. The function $\alpha W({\bf x})-\mathcal{L}W({\bf x})$ is continuous.

This result obviously holds for ${\bf
x}\in{\mathbb{R}}^{n-1}\times(-\infty,\tilde{a})\cup{\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})\cup{\mathbb{R}}^{n-1}\times(\tilde{b},\infty)$. On the curves $\tilde{a}(\bar{\bf x})$ and $\tilde{b}(\bar{\bf x})$, $W$ is twice continuously differentiable along the $x_n$ direction by (\[wval\]). The only term that seems not to be continuous in this function is $\mathcal{L}W({\bf x})$, which involves the first and second derivative with respect to each variable. Denote $\partial_{x_k} W$ the directional derivative along $x_k, 1\leqslant
k\leqslant n-1$, then by (\[wval\]) we have the following: $$\partial_{x_k}W(\bar{\bf x},y)=\int_{\tilde{a}(\bar{\bf
x})}^y \partial_{x_k}V(\bar{\bf x},u)du-V(\bar{\bf
x},\tilde{a}(\bar{\bf x}))\partial_{x_k}\tilde{a}(\bar{\bf x}).$$ Notice that $V$ is $C^{1,1,...,1}$ on ${\mathbb{R}}^n$, so $\partial_{x_k}W(\bar{\bf
x},y)=\partial_{x_k}W({\bf x})$ is continuous in ${\bf x}$. Now consider $$\begin{aligned}
\frac{\partial^2 W}{\partial x_k^2}(\bar{\bf
x},y)&=&\int_{\tilde{a}(\bar{\bf x})}^y\frac{\partial^2 V}{\partial
x_k^2}(\bar{\bf x},u)du-2\partial_{x_k}V(\bar{\bf
x},\tilde{a}(\bar{\bf x}))\cdot\partial_{x_k}\tilde{a}(\bar{\bf
x})\\
\quad && -\partial_{x_n}V(\bar{\bf x},\tilde{a}(\bar{\bf
x}))\cdot(\partial_{x_k}\tilde{a}(\bar{\bf x}))^2-V(\bar{\bf
x},\tilde{a}(\bar{\bf x}))\frac{\partial^2 \tilde{a}}{\partial
x_k^2}(\bar{\bf x}).\end{aligned}$$ Because the two curves $\tilde{a}(\bar{\bf x})$ and $\tilde{b}(\bar{\bf
x})$ have zero Lebesgue measure, and the functions $\partial_{x_k}V$, $\partial_{x_k}\tilde{a}$ and $\frac{\partial^2
\tilde{a}}{\partial x_k^2}$ are all continuous, we conclude that $\frac{\partial^2 W}{\partial x_k^2}$ is continuous, $1\leqslant
k\leqslant n-1$. The continuity of $\frac{\partial^2 W}{\partial
x_i\partial x_j},i\neq j,$ can be proved in a similar manner. Combined with previous argument that $\frac{\partial^2 W}{\partial
x_n^2}$ is continuous, this lemma is proved.

Since $(\alpha-\mathcal{L})W$ is continuous, and the functions $V$ and $H$ are continuous too, we know that the function $h({\bf x})$ in (\[hcost\]) is continuous, hence the continuity of $C(\bar{\bf
x})$ in (\[hcost\]).

For fixed $\bar{\bf x}$, consider the function $$U(y)=\alpha W(\bar{\bf x},y)-\mathcal{L}W(\bar{\bf x},y)-h(\bar{\bf
x},y)$$ with $$U'(y)=\alpha V(\bar{\bf x},y)-\mathcal{L}V(\bar{\bf x},y)-H(\bar{\bf
x},y),$$ and we know $U(\tilde{a}(\bar{\bf x}))=0$. Notice that $U'(y)=0$ for $\tilde{a}(\bar{\bf x})< y< \tilde{b}(\bar{\bf
x})$; $U'(y)>0$ for $y<\tilde{a}(\bar{\bf x})$; $U'(y)<0$ for $y>\tilde{b}(\bar{\bf x})$, and by Lemma \[cont\_alpha\_L\] the function $U(y)$ is continuous, it can be seen that $$\alpha W(\bar{\bf x},y)-\mathcal{L}W(\bar{\bf x},y)<h(\bar{\bf
x},y), {\rm\ for\ } y<\tilde{a}(\bar{\bf x}) {\rm\ or\ }
y>\tilde{b}(\bar{\bf x}).$$ The rest of the proof is obvious.

The result of Theorem \[Wcond\] gives conditions to the solution of the stochastic singular control problem (\[scost\]) and (\[mincostw\]) (see, e.g., [@Taksar85]), where the holding cost $h(\cdot)$ is given in (\[hcost\]) and the boundary penalty costs $f_1(\cdot),f_2(\cdot)$ are given in Assumption \[assumpHf\].

We call a quadruplet $\mathcal{S}=(S,{\bf X}_t,A_t^{(1)},A_t^{(2)})$ ($\mathcal{S}=(A_t^{(1)},A_t^{(2)})$ for simplicity) admissible policy if the following conditions are satisfied:

\[assumpgen\]

1.  $S$ is a compact region given in the form ${\mathbb{R}}^{n-1}\times[\beta,\gamma]$ where $\beta(\bar{\bf x}),
    \gamma(\bar{\bf x})$ are continuous functions of $\bar{\bf
    x}\in{\mathbb{R}}^{n-1}$ with $\beta(\bar{\bf x})< \gamma(\bar{\bf
    x})$.

2.  There is a filtered measurable space $(\Omega,\{\mathcal{F}_t\}_{t\geqslant
    0})$ subject to usual conditions and a probability measure $\{P_{\bf
    x}\}_{{\bf x}\in S}$ on it such that

    -   $\{{\bf X}_t\}_{t\geqslant 0}$ is an $\{\mathcal{F}_t\}$-adapted process, and

    -   $\{A_t^{(1)},A_t^{(2)}\}_{t\geqslant 0}$ are $\{\mathcal{F}_t\}$-adapted right continuous processes with bounded variation such that $$\label{Acond}
        E_{\bf x}\left(\int_{0^-}^\infty e^{-\alpha
        t}dA_t^{(1)}\right)<\infty, E_{\bf x}\left(\int_{0^-}^\infty
        e^{-\alpha t}dA_t^{(2)}\right)<\infty,\forall {\bf x}\in S,$$ and $A_t^{(1)}-A_t^{(2)}$ is the minimal decomposition of a bounded variation process into a difference of two increasing processes.

3.  There are $\{\mathcal{F}_t\}$-adapted independent Brownian motions $B_{1t},...,B_{mt}$ ($m\geqslant n$) starting at the origin under $P_{\bf x}$ for any ${\bf x}\in S$ such that the following controlled diffusion ${\bf X}_t=(X_{1t},...,X_{nt})$ $$\begin{aligned}
    \label{cdiff}
    dX_{1t}&=&\mu_1dt+\sigma_{11}dB_{1t}+\cdots+\sigma_{1m}dB_{mt},\\
    \vdots && \quad\quad \vdots \quad\quad\quad \vdots\nonumber\\
    dX_{nt}&=&\mu_ndt+\sigma_{n1}dB_{1t}+\cdots+\sigma_{nm}dB_{mt}+dA_t^{(1)}-dA_t^{(2)},\nonumber\\
    {\bf X}_0&=&{\bf x},\nonumber\end{aligned}$$ holds $P_{\bf x}$-a.s., $\forall{\bf x}\in
    S$. Furthermore we assume $$P_{\bf x}({\bf X}_t\in S,\forall t\geqslant 0)=1,\quad \forall {\bf
    x}\in S.$$

The probability space $\Omega$ with the filtration $\{\mathcal{F}_t\}$ is not fixed a priori. It is part of an admissible policy. The filtration $\{\mathcal{F}_t\}$ is assumed to be right continuous and $\mathcal{F}_0$ is assumed to contain every $P_{\bf x}$-negligible set for any ${\bf x}\in S$.

Both $A_t^{(1)}$ and $A_t^{(2)}$ are nontrivial in the sense that for any $T>0$, $$P_{\bf x}(A_t^{(i)}=A_0^{(i)},\ \ \forall t\in[0,T])=0,\quad \forall
{\bf x}\in S,\ \ i=1,2.$$

If both $A_t^{(1)}$ and $A_t^{(2)}$ are trivial, ${\bf X}_t$ will hit every open region of positive Lebesgue measure in ${\mathbb{R}}^n$ with positive probability, but this is a contradiction since ${\bf X}_t$ is concentrated on $S$. If either $A_t^{(1)}$ or $A_t^{(2)}$ is trivial, ${\bf X}_t$ can not be concentrated on $S$ which again is a contradiction.

Define the following notations: $$\begin{aligned}
\Delta A_t^{(i)}&=&A_t^{(i)}-A_{t^-}^{(i)},\quad t\geqslant 0,
i=1,2,\\
\Delta {\bf X}_t&=&{\bf X}_t-{\bf X}_{t^-},\quad t\geqslant
0,\\
 \Delta W({\bf X}_t)&=&W({\bf X}_t)-W({\bf X}_{t^-}),\quad
t\geqslant 0.\end{aligned}$$ Then due to the fact that $A_t^{(1)},A_t^{(2)}$ are the minimal decomposition of a bounded variation process into a difference of two increasing processes, $\Delta A_t^{(1)}\cdot\Delta
A_t^{(2)}=0,\ \forall t\geqslant 0$. By convention we let $$B_{1t}=\cdots=B_{nt}=0,\ A_{t}^{(1)}=A_{t}^{(2)}=0,\quad \forall
t<0,$$ so that $$\Delta A_0^{(i)}=A_0^{(i)},\quad i=1,2,\quad {\bf X}_0={\bf x}\quad
P_{\bf x} {\rm\ a.s.},\quad {\bf x}\in S.$$

Notice that the integrals in (\[Acond\]) involve the possible jumps at time $0$ so that they are the sum of the integrals over $(0,\infty)$ as well as $A_0^{(i)},i=1,2$. In particular, the jump only happens to the $x_n$ coordinate. In what follows, we use $A_t^{(i),c} (i=1,2)$ to denote the continuous part of the processes $A_t^{(i)},i=1,2$.

\[Veri\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\], \[unif2\], \[assumpgen\] and that the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ being smooth. Let $k_{\mathcal S}({\bf x})$ be given by the following $$\begin{aligned}
\label{kcostjump}
k_{\mathcal{S}}({\bf x})=&&E_{\bf x}\left(\int_0^\infty e^{-\alpha
t}h({\bf X_t})dt\right)\\
&&+E_{\bf x}\left(\int_0^\infty e^{-\alpha t}\left(f_1({\bf
X}_t)dA_t^{(1),c}+f_2({\bf X}_t)dA_t^{(2),c}\right)\right)\nonumber\\
&&+E_{\bf x}\left(\sum_{0\leqslant t<\infty}e^{-\alpha
t}\left(\int_{X_{nt^-}}^{X_{nt^-}+\Delta A_t^{(1)}} f_1({\bf
X}_t)dy\right.\right.\nonumber\\
&&\left.\left.+\int_{X_{nt^-}-\Delta A_t^{(2)}}^{X_{nt^-}} f_2({\bf
X}_t)dy\right)\right),\nonumber\end{aligned}$$ then

1.  For any admissible policy $\mathcal{S}$, $W({\bf x})\leqslant k_{\mathcal S}({\bf
    x}),\ \forall {\bf x}\in{\mathbb{R}}^n$.

2.  $W({\bf x})= k_{\mathcal S}({\bf
    x}),\ \forall {\bf x}\in{\mathbb{R}}^n$, if and only if $S={\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$, where $\tilde{a}(\bar{\bf x}),\tilde{b}(\bar{\bf x})$ are given in Theorem \[Wcond\], the process ${\bf X}_t$ is the reflecting diffusion on $S$, and $\mathcal{S}=(A_t^{(1)},A_t^{(2)})$ where $A_t^{(1)}$ increases only when ${\bf X}_t$ is on the boundary $(\bar{\bf
    x},\tilde{a}(\bar{\bf x}))$ and $A_t^{(2)}$ increases only when ${\bf X}_t$ is on the boundary $(\bar{\bf x},\tilde{b}(\bar{\bf
    x})),\ \forall\bar{\bf x}\in{\mathbb{R}}^{n-1}$.

The cost function consists of several parts. The first integral in (\[kcostjump\]) is the holding cost. The second integral is a control cost associated with the increment of controls $A_t^{(i)}$ ($i=1,2$) in the continuous part. The last integral is a control cost associated with the jumps in $A_t^{(i)},i=1,2$ (or equivalently jumps in ${\bf X}_t$). We further extend $k_{\mathcal{S}}({\bf x})$ outside the region ${\mathbb{R}}^{n-1}\times[\beta,\gamma]$ for two continuous functions $\beta(\bar{\bf x})<\gamma(\bar{\bf x}),\
\forall\bar{\bf x}\in{\mathbb{R}}^{n-1}$ as the following: $$\begin{aligned}
\label{extcost}
k_{\mathcal{S}}({\bf x})&=&k_{\mathcal{S}}(\bar{\bf
x},\beta(\bar{\bf x}))+\int_{x_n}^{\beta(\bar{\bf x})} f_1(\bar{\bf
x},u)du,\quad \forall{\bf
x}=(\bar{\bf x},x_n)\in{\mathbb{R}}^{n-1}\times(-\infty,\beta),\\
k_{\mathcal{S}}({\bf x})&=&k_{\mathcal{S}}(\bar{\bf
x},\gamma(\bar{\bf x}))+\int_{\gamma(\bar{\bf x})}^{x_n}f_2(\bar{\bf
x},u)du,\quad \forall{\bf x}=(\bar{\bf
x},x_n)\in{\mathbb{R}}^{n-1}\times(\gamma,\infty),\end{aligned}$$ and we are looking for an admissible control $\mathcal{S}$ such that $$\label{winf}
W^*({\bf x})=\inf_{\mathcal{S}\in\mathbb{S}}k_{\mathcal S}({\bf
x}),\quad \forall{\bf x}\in{\mathbb{R}}^n,$$ where $\mathbb{S}$ is the set of all admissible control policies.

1.  Consider the diffusion given in (\[cdiff\]) with ${\bf x}\in S$. Applying the generalized Ito formula to $e^{-\alpha t}W({\bf X}_t)$ (see [@Harrison83]) yields $$\begin{aligned}
    \label{gIto}
    e^{-\alpha t}W({\bf X_t})&=&W({\bf x})-\alpha\int_0^t e^{-\alpha
    s}W({\bf X}_s)ds+\int_0^t e^{-\alpha s}\mathcal{L}W({\bf
    X}_s)ds\nonumber\\
    &&+\int_0^t e^{-\alpha s}\nabla W({\bf
    X}_s) \cdot \sigma({\bf X}_s) d{\bf B}_{s}\\
    &&+\int_0^te^{-\alpha s}\frac{\partial }{\partial x_n}W({\bf
    X}_s)(dA_{s}^{(1),c}-dA_s^{(2),c})+\sum_{0< s\leqslant t}e^{-\alpha
    s}\Delta W({\bf X}_s).\nonumber\end{aligned}$$ Using the following identity $$W({\bf x})+\sum_{0< s\leqslant t}e^{-\alpha s}\Delta W({\bf
    X}_s)=W({\bf X}_{0^-})+\sum_{0\leqslant s\leqslant t}e^{-\alpha
    s}\Delta W({\bf X}_s),$$ and taking expectation of both sides of (\[gIto\]) with respect to $P_{\bf x}$ and let $t\to\infty$, we get the following: $$\begin{aligned}
    \label{widen}
    W({\bf x})&=&E_{\bf x}\left(\int_0^\infty e^{-\alpha
    t}\left(\alpha-\mathcal{L}\right)W({\bf X}_t)dt\right)\\
    &&-E_{\bf x}\left(\int_0^\infty e^{-\alpha
    t}\frac{\partial}{\partial x_n}W({\bf
    X}_t)(dA_{t}^{(1),c}-dA_t^{(2),c})\right)\nonumber\\
    &&-E_{\bf x}\left(\sum_{0\leqslant t <\infty}e^{-\alpha t}\Delta
    W({\bf X}_t)\right).\nonumber\end{aligned}$$ Therefore $$\begin{aligned}
    \label{kmW}
    &&\quad\quad \quad k_{\mathcal S}({\bf x})-W({\bf x})\\
    &&=E_{\bf x}\left(\int_0^\infty
     e^{-\alpha t}\left[h({\bf X}_t)-(\alpha-\mathcal{L})W({\bf
    X}_t)\right]dt\right)\nonumber\\
    &&+E_{\bf x}\left(\int_0^\infty e^{-\alpha t}\left[f_1({\bf
    X}_t)+\frac{\partial }{\partial x_n}W({\bf
    X}_t)\right]dA_t^{(1),c}\right)\nonumber\\
    &&+E_{\bf x}\left(\int_0^\infty e^{-\alpha t}\left[f_2({\bf
    X}_t)-\frac{\partial }{\partial x_n}W({\bf X}_t)\right]dA_t^{(2),c}\right)\nonumber\\
    &&+E_{\bf x}\left(\sum_{0\leqslant t<\infty}e^{-\alpha t}\Delta
    W({\bf X}_t)\right)\nonumber\\
    &&+E_{\bf x}\left(\sum_{0\leqslant t<\infty}e^{-\alpha
    t}\left(\int_{X_{nt^-}}^{X_{nt^-}+\Delta A_t^{(1)}} f_1({\bf
    X}_t)dy+\int_{X_{nt^-}-\Delta A_t^{(2)}}^{X_{nt^-}} f_2({\bf
    X}_t)dy\right)\right).\nonumber\end{aligned}$$ By Theorem \[Wcond\], the first three integrands in (\[kmW\]) are all nonnegative for the process ${\bf X}_t$ staying in the region $S$.

    Define the sets $$\Gamma_+=\{t\geqslant 0: \Delta
    A_t^{(1)}>0\},\quad \Gamma_-=\{t\geqslant 0: \Delta A_t^{(2)}>0\},$$ then $\Gamma_+\cap\Gamma_-=\phi$. Rewrite the last two expectations of (\[kmW\]) as $$\begin{aligned}
    &&E_{\bf x}\left(\sum_{t\in\Gamma_+}e^{-\alpha
    t}\int_{X_{nt^-}}^{X_{nt^-}+\Delta A_t^{(1)}}\left[
    \frac{\partial}{\partial x_n}W({\bf X}_t)+f_1({\bf
    X}_t)\right]dy\right)\\
    &&+E_{\bf x}\left(\sum_{t\in\Gamma_-}e^{-\alpha
    t}\int_{X_{nt^-}-\Delta A_t^{(2)}}^{X_{nt^-}}\left[
    -\frac{\partial}{\partial x_n}W({\bf X}_t)+f_2({\bf
    X}_t)\right]dy\right).\end{aligned}$$ By Theorem \[Wcond\] this quantity is nonnegative, and this shows $k_{\mathcal{S}}({\bf x})\geqslant
    W({\bf x}),\forall {\bf x}\in S$.

    Due to the extension (\[extcost\]), we proved $k_{\mathcal{S}}({\bf x})\geqslant W({\bf x}),\forall {\bf
    x}\in{\mathbb{R}}^n$.

2.  If $S={\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$ and the process ${\bf X}_t$ is the reflecting diffusion on $S$, then by Theorem \[Wcond\], the first integral in (\[kmW\]) is obviously zero. As to the second and third integrals in (\[kmW\]), because $dA_t^{(1)},dA_t^{(2)}$ are zero whenever ${\bf X}_t$ is in ${\mathbb{R}}^{n-1}\times({a},{b})$, while at the boundary where $A_t^{(1)},A_t^{(2)}$ increases, the integrands are zero, these two integrals are zero too. The last two expectations are also zero due to this construction hence $W({\bf x})= k_{\mathcal S}({\bf
    x}),\forall {\bf x}\in S$.

    On the other hand, suppose $W({\bf x})= k_{\mathcal S}({\bf
    x}),\forall {\bf x}\in S$, then all the expectations in (\[kmW\]) must be zero. Assume $S={\mathbb{R}}^{n-1}\times[\beta,\gamma]$ and at least one of the inequalities is true: $\beta(\bar{\bf x})\neq
    g_1(\bar{\bf x}),\gamma(\bar{\bf x})\neq g_2(\bar{\bf x})$, then due to the continuity of these four functions we know that the sum of the first three integrals in (\[kmW\]) is positive by Theorem \[Wcond\]. And because the sum of the last two expectations in (\[kmW\]) is nonnegative, it can be seen that $W({\bf x})<
    k_{\mathcal S}({\bf x})$. Therefore in order to have $W({\bf x})=
    k_{\mathcal S}({\bf x})$, $S$ must be the region ${\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$.

    Again by Theorem \[Wcond\], we see that the processes ${\bf X}_t$ and $A_t^{(i)}$ ($i=1,2$) must all be continuous in order to eliminate the last two expectations in (\[kmW\]), which implies $A_t^{(i)}=A_t^{(i)c}$ ($i=1,2$) when $\beta(\bar{\bf
    x})=\tilde{a}(\bar{\bf x}),\gamma(\bar{\bf x})=\tilde{b}(\bar{\bf
    x})$. Therefore $({\bf X}_t,A_t^{(1)},A_t^{(2)})$ must be the reflecting diffusion on ${\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$.

The possible jumps, $\Delta A_t^{(i)},i=1,2,$ only happen at time zero. When the process ${\bf X}_t$ starts at a point outside the region ${\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$, the control brings it back to this region immediately, and after that, the process will be a continuous reflected diffusion. The confirmation of the last assertion is shown below.

If we let $\gamma=(0,0,...,0,1)^T$, then the reflected diffusion can be written as $$\label{refdif}
d{\bf X}_t=\mu({\bf X}_t) dt+\sigma({\bf X}_t) d{\bf B}_t+\gamma
dA_t^{(1)}-\gamma dA_t^{(2)},\ \ t>0,$$ where $A_t^{(1)}$ increases only at the boundary $\tilde{a}(\cdot)$ and $A_t^{(2)}$ increases only at the boundary $\tilde{b}(\cdot)$.

We notice that the reflection only happens to the last component of the process. Since the two curves $\tilde{a}(\cdot)$ and $\tilde{b}(\cdot)$ are smooth and uniformly Lipschitz, if we let $n({\bf x})$ be the inward normal for ${\bf x}$ at the boundary, then we can show that there exist positive constants $\nu_1,\nu_2$ such that $$\begin{aligned}
\forall {\bf x}&=&(\bar{\bf x},\tilde{a}({\bar{\bf x}})),\
(\gamma,n({\bf
x}))\geqslant \nu_1,\\
\forall {\bf x}&=&(\bar{\bf x},\tilde{b}({\bar{\bf x}})),\
(\gamma,n({\bf x}))\leqslant -\nu_2.\end{aligned}$$ By a localization technique and Theorem 4.3 in [@Lions84], it can be shown that there exists a solution $({\bf
X}_t,A_t^{(1)},A_t^{(2)})$ to the reflected diffusion (\[refdif\]). This problem is called the Skorohod problem.

Concluding Remarks {#concluding-remarks .unnumbered}
==================

In this paper, we studied a multi-dimensional stochastic singular control problem via Dynkin game and Dirichlet form. The value function of the Dynkin game satisfies a variational inequality problem, and the integrated form of this value function turns out to be the value function of the singular control problem. By characterizing the regularities of the value function of the Dynkin game and its integrated version, we showed the existence of a classical solution to the Hamilton-Jacobi-Bellman equation associated with this multi-dimensional singular control problem, and this kind of problems were traditionally solved through viscosity solutions. We also proved that, under some conditions, the optimal control policy is given by two curves and the controlled process is the reflected diffusion between these two curves. Unlike the one dimensional singular control problem, where under some conditions, the boundary of the optimal continuation region are given by two points [@Fuku02], it is much more difficult to characterize the boundaries of the continuation region in the multi-dimensional singular control problem. This paper investigates some conditions on the regularity of value function and the form of optimal singular control policies of multi-dimensional diffusion, and it provides a basis for the search of further conditions and further regularities in this realm.

Appendix {#appendix .unnumbered}
========

In this appendix we shall correct an error found in the paper [@Fuku02]. In the paper “Dynkin Games Via Dirichlet Forms and Singular Control of One-Dimensional Diffusion”[@Fuku02], the authors tried to show the existences of a smooth value function and an optimal policy to a one-dimensional stochastic singular control problem via Dynkin game and Dirichlet form. The value function $V(x)$ of a Dynkin game is known to exist [@Zab84], which is the solution of a variational inequality problem involving Dirichlet form. The integration of $V(x)$ turns out to be a smooth optimal return function $W(x)$ for a stochastic singular control problem. Thus the traditional technique of viscosity solution is avoided.

In their paper, the underlying process is a generalized one dimensional diffusion process given by $dX_t=\mu(X_t)dt+\sigma(X_t)dw_t$, in which $w_t$ is a Wiener process. It is found that a different diffusion process should be considered in the proofs, and as a result the main theorem of this paper should be amended.

In their paper, the infinitesimal generator is defined as (see page 693, Eq. 4.1 in [@Fuku02]) $$\label{gen}
Lu(x)=\frac{d}{dm}\frac{d}{ds}u(x)=\mu(x)u'(x)+\frac{1}{2}\sigma(x)^2
u''(x),$$ where $ds(x)=\dot{s}(x)dx,dm(x)=\dot{m}(x)dx$, and (see Eq. 4.2 in [@Fuku02]) $$\label{sam}
\dot{s}(x)
=\exp\left(-\int_0^x\frac{2\mu(y)}{\sigma(y)^2}dy\right),\quad
\dot{m}(x) =
\frac{2}{\sigma(x)^2}\exp\left(\int_0^x\frac{2\mu(y)}{\sigma(y)^2}dy\right).$$

The value function $W(x)$ of the stochastic singular control problem is assumed to satisfy the following PDE (see Eq. 3.23 on page 693 in [@Fuku02]) $$\alpha W(x) -\frac{d}{dm}\frac{d}{ds}W(x) = h(x),$$ or equivalently $$\alpha W(x) -\mu(x)W'(x)-\frac{1}{2}\sigma(x)^2 W''(x) = h(x),$$ where (see Eq. 3.21 and Eq. 3.22 in [@Fuku02]) $$\label{defh}
h(x) = \int_0^x H(y)\dot{s}(y)dy + C,$$ and $$\label{defW}
W(x) = \int_a^x
V(y)\dot{s}(y)dy+\frac{1}{\alpha}\left(-\frac{f_1'(a)}{\dot{m}(a)}+h(a)\right).$$

Then in the proof of Theorem 3.2 on page 693 in [@Fuku02], the authors constructed the function $$U(x)=\alpha W(x)-\frac{d}{dm}\frac{d}{ds}W(x)-h(x),$$ and claimed that $$\frac{1}{\dot{s}(x)}U'(x) = \alpha
V(x)-\frac{d}{ds}\frac{d}{dm}V(x)-H(x).$$ This is equivalent to $$\label{defU}
U(x)=\alpha W(x)-\mu(x)W'(x)-\frac{1}{2}\sigma(x)^2W''(x)-h(x),$$ and $$\label{hjbV}
\frac{1}{\dot{s}(x)}U'(x) = \alpha
V(x)-\mu(x)V'(x)-\frac{1}{2}\sigma(x)^2V''(x)-H(x).$$ However, by a careful examination, it can be seen that the above proposition is not true in general. The reason here is that $\mu(x)$ and $\sigma(x)$ are both functions of $x$, and when taking the derivative of $U(x)$, the product rule has to be applied. The details are shown below.

By the definition of $W(x)$ in (\[defW\]), $h(x)$ in (\[defh\]) and $\dot{s}(x)$ in (\[sam\]) in [@Fuku02], we get $$\begin{aligned}
W'(x) &&= V(x)\dot{s}(x),\nonumber\\
h'(x) &&= H(x)\dot{s}(x),\nonumber\\
\ddot{s}(x)&&=-\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2},\nonumber\end{aligned}$$ hence $$\begin{aligned}
W''(x) &&=
V'(x)\dot{s}(x)-V(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2},\nonumber\\
W'''(x)&&=V''(x)\dot{s}(x)-2V'(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2}+V(x)\dot{s}\frac{4\mu(x)^2}{\sigma(x)^4}\nonumber\\
&&\quad
-V(x)\dot{s}(x)\left(2\mu'(x)\sigma(x)^{-2}-4\mu(x)\sigma(x)^{-3}\sigma'(x)\right).\nonumber\end{aligned}$$ Now if we take the derivative of $U(x)$ in (\[defU\]) we get $$\begin{aligned}
U'(x)=&&\alpha
V(x)\dot{s}(x)-\mu'(x)V(x)\dot{s}(x)-\mu(x)V'(x)\dot{s}(x)+\mu(x)V(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2}\nonumber\\
&&-\sigma(x)\sigma'(x)V'(x)\dot{s}(x)+\sigma(x)\sigma'(x)V(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2}\nonumber\\
&&-\frac{1}{2}\sigma(x)^2\left(V''(x)\dot{s}(x)-2V'(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2}+V(x)\dot{s}\frac{4\mu(x)^2}{\sigma(x)^4}\right.\nonumber\\
&&\quad
\left.-V(x)\dot{s}(x)\left(2\mu'(x)\sigma(x)^{-2}-4\mu(x)\sigma(x)^{-3}\sigma'(x)\right)\right)-H(x)\dot{s}(x).\nonumber\end{aligned}$$ After simplifying this expression and comparing it with (\[hjbV\]) we should have the following $$0=-\sigma(x)\sigma'(x)V'(x)+2\mu(x)V'(x),$$ which does not hold in general. The following condition should be added to make it hold. $$\label{result}2\mu(x)=\sigma(x)\sigma'(x)$$

A second concern of this paper might be more profound. The Dirichlet form in this paper is defined as (see Eq. 3.3 on page 686 in [@Fuku02]) $$\label{dform}
\mathcal{E}(u,v)=\int_{-A}^Au'(x)v'(x)\frac{1}{\dot{m}(x)}dx,\quad
u,v\in\mathcal{F},$$ where $$\begin{aligned}
\mathcal{F}&&=H^1((-A,A);dx)\nonumber\\
&&=\{u\in L^2((-A,A);dx):\ u{\rm\ is\ absolutely\ continuous,\
}u'\in L^2((-A,A);dx)\}.\nonumber\end{aligned}$$

The authors claimed that this Dirichlet form $(\mathcal{E},\mathcal{F})$ is regular on $L^2([-A,A];ds)$ and the associated underlying process is a reflecting barrier diffusion on $[-A,A]$ with infinitesimal generator $\frac{d}{ds}\frac{d}{dm}$, i.e., the generator $L$ given in (\[gen\]). The correspondence is given by (see Corollary 1.3.1 on page 21 of [@Fuku11]) $$\label{da}
\mathcal{E}(u,v)=(-Lu,v),\quad u\in \mathcal{D}(L),v\in\mathcal{F},$$ where $\mathcal{D}(L)$ is the domain of $L$. Since the underlying process is a reflecting barrier diffusion on $[-A,A]$, $\mathcal{D}(L)$ is given by (see page 22 of [@Fuku11]) $$\begin{aligned}
\mathcal{D}(L)=\{&&u\in\mathcal{F}: u'{\rm\ is\ absolutely\
continuous,\ }\nonumber\\
&&u''\in L^2((-A,A);dx), u'(-A)=u'(A)=0\}.\nonumber\end{aligned}$$

Now we try the integration by parts on (\[dform\]) and get $$\begin{aligned}
\mathcal{E}(u,v)&&=\int_{-A}^A
u'(x)v'(x)\frac{\sigma(x)^2}{2}\exp\left(-\int_0^x\frac{2\mu(y)}{\sigma(y)^2}dy\right)dx\nonumber\\
&&=-\int_{-A}^A\left(\frac{\sigma(x)^2}{2}u''(x)+\sigma(x)\sigma'(x)u'(x)-\mu(x)u'(x)\right)v(x)\exp\left(-\int_0^x\frac{2\mu(y)}{\sigma(y)^2}dy\right)dx.\nonumber\end{aligned}$$

Once again, when the condition (\[result\]) holds, we get $$\frac{\sigma(x)^2}{2}u''(x)+\sigma(x)\sigma'(x)u'(x)-\mu(x)u'(x)=\frac{\sigma(x)^2}{2}u''(x)+\mu(x)u'(x)=Lu(x),$$ and (\[da\]) holds.

As a conclusion, if the condition (\[result\]) is added, then all the results in that paper still hold, but for a very particular Ito diffusion.

In the following we give another way to amend the results of that paper which makes the theorems more general. If we just simply consider the diffusion $$\label{ndiff}
dX_t = \gamma(X_t)dt + \sigma(X_t)dw_t,$$ where $$\gamma(x)=\sigma(x)\sigma'(x)-\mu(x),$$ and define the infinitesimal generator $$L_\gamma u(x) = \gamma(x)u'(x)+\frac{1}{2}\sigma(x)^2 u''(x),$$ while the Dirichlet form is still defined as in (\[dform\]) and $\dot{s}(x),\dot{m}(x)$ are still given in (\[sam\]), then we get $$\mathcal{E}(u,v)=(-L_\gamma u,v).$$ That means the underlying process associated with the Dirichlet form (\[dform\]) should be (\[ndiff\]). With this in mind, we can examine again the results of that paper [@Fuku02]. Results in Section 2 are classical on variational inequalities and optimal stopping. In Section 3, the part $\frac{d}{ds}\frac{d}{dm}$, whenever it appears before Theorem 3.2, should be replace by $L_\gamma$. Let the functions $h(x),W(x)$ still be defined as in Eqn. (3.21) (3.22) in that paper, respectively, then Theorem 3.2 holds intact. But in the proof of this theorem, after setting $$\begin{aligned}
U(x)&=&\alpha W(x)-\frac{d}{dm}\frac{d}{ds}W(x)-h(x)\nonumber\\
&=&\alpha
W(x)-\mu(x)W'(x)-\frac{1}{2}\sigma(x)^2W''(x)-h(x)\nonumber,\end{aligned}$$ and taking the derivative of both sides, we should get $$\begin{aligned}
\frac{1}{\dot{s}(x)}U'(x) &=& \alpha
V(x)-(\sigma(x)\sigma(x)'-\mu(x))V'(x)-\frac{1}{2}\sigma(x)^2V''(x)-H(x)\nonumber\\
&=&\alpha V(x)-L_\gamma V(x)-H(x).\nonumber\end{aligned}$$ Since Theorem 3.1 has been amended, the rest of the proof of Theorem 3.2 just follows. Section 4 of that paper is about a verification theorem, and the results there still hold.

It might be interesting to notice that when $\sigma$ is a constant, we get $\gamma(x)=-\mu(x)$, and $$\mathcal{E}(u,v)=\int_{-A}^A
u'(x)v'(x)\frac{\sigma^2}{2}\exp\left(\int_0^x\frac{2\gamma(y)}{\sigma^2}dy\right)dx.$$

Acknowledgments {#acknowledgments .unnumbered}
===============

In memory of Dr. Michael Taksar.

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[^1]: The idea of this paper was proposed by Dr. Michael Taksar. Dr. Taksar passed away in February, 2012, however, his contributions should always be remembered.

[^2]: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri, 65211 (yangyip@missouri.edu)
