Multi-dimensional Stochastic Singular Control Via Dynkin Game and - - PowerPoint PPT Presentation

Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form ∗

Yipeng Yang

* Under the supervision of Dr. Michael Taksar

Department of Mathematics University of Missouri-Columbia

Oct 11, 2012

Y.Yang (MU) Oct 11, 2012 1 / 35

Outline

Problem Formulation Related Literature Dirichlet Form and Dynkin Game Dynkin Game and Free Boundary Problem The Multi-dimensional Stochastic Singular Control Problem Concluding Remarks and Future Research References

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Problem Formulation

Problem Formulation

Given a probability space (Ω, F, Ft, X, θt, Px), we are concerned with a multi-dimensional diffusion on Rn: dXt = µ(Xt)dt + σ(Xt)dBt, X0 = x, (1) where Xt =     X1t . . . Xnt     , µ(Xt) =     µ1 . . . µn     , σ(Xt) =     σ11 · · · σ1m . . . . . . σn1 · · · σnm     , Bt =     B1t . . . Bmt     , (2) in which µi, σi,j (1 i n, 1 j m) are functions of X1t, ..., X(n−1)t satisfying the usual conditions, and Bt is m-dimensional Brownian motion with m n.

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Problem Formulation

There is a cost function associated with this process: kS(x) = Ex ∞ e−αth(Xt)dt + ∞ e−αt f1(Xt)dA(1)

t

+ f2(Xt)dA(2)

t

• ,

f1(x), f2(x) > 0, ∀x ∈ Rn. (3) And there is control on the underlying process: dX1t = µ1dt + σ11dB1t + · · · + σ1mdBmt, . . . . . . . . . dXnt = µndt + σn1dB1t + · · · + σnmdBmt + dA(1)

t

− dA(2)

t ,

X0 = x.

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Problem Formulation

◮ A control policy is defined as a pair (A(1) t , A(2) t ) = S of Ft adapted

processes which are right continuous and nondecreasing in t.

◮ A(1) t

− A(2)

t

is the minimal decomposition of a bounded variation process into the difference of two nondecreasing processes. Problem Formulation: one looks for a control policy S that minimizes the cost function kS(x), W (x) = min

S∈S kS(x),

(4) where S is the set of admissible policies.

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Problem Formulation

Applications?

◮ A decision maker observes the expenses of a company under a

multi-factor situation and wants to minimize the total expense by adjusting one factor.

◮ An investor observes the prices of several assets in a portfolio and

wants to maximize the total wealth by adjusting the investment on

• ne asset.

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Related Literature

Related Literature

◮ This is a free boundary multi-dimensional singular control problem. ◮ The classical approach is to use the dynamic programming principle

to derive the Hamilton-Jacobi-Bellman (HJB) equation and solve the PDE, if one can, e.g., [Pham (2009), Ma and Yong(1999)].

◮ Viscosity solution techniques, e.g.,

[Fleming and Soner(2006), Crandall, Ishii and Lions (1992)].

◮ Existence, uniqueness and regularity of the solution to the HJB

equation are hard to analyze [Soner and Shreve(1989)].

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Related Literature

◮ The value function of the stochastic singular control problem is

closely related to the value of a zero-sum game, called Dynkin game, e.g., [Fukushima and Taksar(2002), Taksar(1985), Guo and Tomecek(2008)].

◮ The value of the Dynkin game coincides with the solution of a

variational inequality problem involving Dirichlet forms, e.g., [Nagai(1978), Zabczyk (1984), Karatzas(2005)].

◮ Using an approach via Dynkin game and Dirichlet form,

[Fukushima and Taksar(2002)] proved the existence of a classical solution to a one-dimensional stochastic singular control problem.

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Dirichlet Form and Dynkin Game

Variational Inequality Problem Involving Dirichlet Form Let f1 ∈ F. Nagai [Nagai(1978)] showed that there exist a quasi continuous function w ∈ F which solves the variational inequality problem w −f1, Eα(w, u − w) 0, ∀u ∈ F with u −f1, and a properly exceptional set N such that for all x ∈ Rn/N, w(x) = sup

σ Ex

• e−ασ[−f1(Xσ)]
• = Ex
• e−αˆ

σ[−f1(Xˆ σ)]

• ,

where ˆ σ = inf{t 0; w(Xt) = −f1(Xt)}. Moreover, w is the smallest α-potential dominating the function −f1 m-a.e.

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Dirichlet Form and Dynkin Game

[Zabczyk (1984)] then extended this result to the solution of the zero-sum game ( Dynkin game) by showing that, there exist a quasi continuous function V (x) ∈ K which solves the variational inequality Eα(V , u − V ) 0, ∀u ∈ K, (5) where K = {u ∈ F : −f1 u f2 m-a.e.}, f1, f2 ∈ F, and a properly exceptional set N such that for all x ∈ Rn/N, V (x) = sup

σ inf τ Jx(τ, σ) = inf τ sup σ Jx(τ, σ)

(6) for any stopping times τ and σ, where Jx(τ, σ) = Ex

• e−α(τ∧σ) (−Iστf1(Xσ) + Iτ<σf2(Xτ))
• .

(7)

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Dirichlet Form and Dynkin Game

What is more, if define E1 = {x ∈ Rn/N : V (x) = −f1(x)}, E2 = {x ∈ Rn/N : V (x) = f2(x)}, then the hitting times ˆ τ = τE2, ˆ σ = τE1 is the saddle point of the game Jx(ˆ τ, σ) Jx(ˆ τ, ˆ σ) Jx(τ, ˆ σ) (8) for any x ∈ Rn/N and any stopping times τ, σ, where Jx is given in (7). In particular V (x) = Jx(ˆ τ, ˆ σ), ∀x ∈ Rn/N. (9)

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Dirichlet Form and Dynkin Game

[Fukushima and Menda (2006)] further showed that if the transition probability function of the underlying process satisfies the absolute continuity condition: pt(x, ·) ≪ m(·), (10) and f1, f2 are finite finely continuous functions satisfying the following separability condition:

Assumption

There exist finite α-excessive functions v1, v2 ∈ F such that, for all x ∈ Rn, − f1(x) v1(x) − v2(x) f2(x), (11) then there does not exist the exceptional set N, and the solution V (x) is finite finely continuous.

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Dynkin Game and Free Boundary Problem

Dynkin Game and Free Boundary Problem

We are concerned with a multi-dimensional Dynkin game over a region D = Rn−1 × (A(¯ x), B(¯ x)), where A, B are two bounded, smooth and uniformly Lipschitiz functions. The associated cost function is Jx(τ, σ) =Ex τ∧σ e−αtH(Xt)dt

• + Ex
• e−α(τ∧σ) (−Iστf1(Xσ) + Iτ<σf2(Xτ))
• ,

(12) where H is the holding cost, and f1, f2 are boundary penalty costs.

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Dynkin Game and Free Boundary Problem

Two players P1 and P2 observe the underlying process Xt in (1) with accumulated income, discounted at present time, equalling σ

0 e−αtH(Xt)dt, for any stopping time σ. ◮ If P1 stops the game at time σ, he pays P2 the amount of the

accumulated income plus the amount f2(Xσ), which after been discounted equals e−ασf2(Xσ).

◮ If the process is stopped by P2 at time σ, he receives from P1 the

accumulated income less the amount f1(Xσ), which after been discounted equals e−ασf1(Xσ). P1 tries to minimize his payment while P2 tries to maximize his income. Thus the value of this Dynkin game is given by V (x) = inf

τ sup σ Jx(τ, σ),

∀x ∈ Rn. (13)

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Dynkin Game and Free Boundary Problem

◮ Define the following Dirichlet form on D:

E(u, v) =

• D

∇u(x) · A∇v(x)m(dx), u, v ∈ F, (14) where F = {u ∈ L2(D) : u is continuous,

• D

∇u(x)T∇u(x)m(dx) < ∞}, A(x) = 1

2σσT is assumed to be uniformly elliptic, and

m(dx) = eb·xdx, in which b = A−1µ.

◮ Consider the solution V ∈ F, −f1 V f2 of

Eα(V , u − V ) (H, u − V ), ∀u ∈ F, −f1 u f2, (15) where Eα(u, v) = E(u, v) + α

• Rn u(x)v(x)m(dx).

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Dynkin Game and Free Boundary Problem

Theorem

Assume some usual conditions on H, f1, f2 ∈ F and the separability condition, we put Jx(τ, σ) =Ex τ∧σ e−αtH(Xt)dt

• + Ex
• e−α(τ∧σ) (−Iστf1(Xσ) + Iτ<σf2(Xτ))
• (16)

for finite stopping times τ, σ. Then the solution of (15) admits a finite and continuous value function of the game V (x) = inf

τ sup σ Jx(τ, σ) = sup σ inf τ Jx(τ, σ),

∀x ∈ Rn. (17)

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Dynkin Game and Free Boundary Problem

Theorem

Furthermore if we let E1 = {x ∈ Rn : V (x) = −f1(x)}, E2 = {x ∈ Rn : V (x) = f2(x)}, (18) then the hitting times ˆ τ = τE2, ˆ σ = τE1 is the saddle point of the game Jx(ˆ τ, σ) Jx(ˆ τ, ˆ σ) Jx(τ, ˆ σ) (19) for any x ∈ Rn and any stopping times τ, σ. In particular, ˆ τ, ˆ σ are finite a.s. and V (x) = Jx(ˆ τ, ˆ σ), ∀x ∈ Rn. (20) Regularities? Optimal control policies?

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Dynkin Game and Free Boundary Problem

Assumption

There exist smooth and uniformly Lipschitz continuous functions a(¯ x), b(¯ x), ¯ x ∈ Rn−1, such that, (α − L)f1(¯ x, a(¯ x)) + H(¯ x, a(¯ x)) = 0, (α − L)f2(¯ x, b(¯ x)) − H(¯ x, b(¯ x)) = 0, and A(¯ x) < a(¯ x) < 0 < b(¯ x) < B(¯ x), ∀¯ x ∈ Rn−1.

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Dynkin Game and Free Boundary Problem

Main Result

Theorem

Let V (x) be the solution of the multi-dimensional Dynkin game, then there exist unique smooth functions a(¯ x), b(¯ x) such that A(¯ x) < a(¯ x) < 0 < b(¯ x) < B(¯ x) and −f1(x) < V (x) < f2(x), ∀x ∈ Rn−1 × (a, b), (21) V (x) = −f1(x), ∀x ∈ Rn−1 × (−∞, a], V (x) = f2(x), ∀x ∈ Rn−1 × [b, ∞), (22) ∂Vu(¯ x, a(¯ x)) = −∂f1u(¯ x, a(¯ x)), ∂Vu(¯ x, b(¯ x)) = ∂f2u(¯ x, b(¯ x)), ∀¯ x ∈ Rn−1, (23) where ∂Vu represents the directional derivative along the u direction.

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Dynkin Game and Free Boundary Problem

Theorem (Continued)

Furthermore V is C 1,··· ,1,1 on Rn, C 2,··· ,2 on Rn−1 × (a, b) ∪ Rn−1 × (−∞, a) ∪ Rn−1 × (b, ∞) and αV (x) − LV (x) = H(x), ∀x ∈ Rn−1 × (a, b), αV (x) − LV (x) > H(x), ∀x ∈ Rn−1 × (−∞, a), αV (x) − LV (x) < H(x), ∀x ∈ Rn−1 × (b, ∞), (24) where L is the infinitesimal generator.

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Dynkin Game and Free Boundary Problem

Sketch of Proof

Proposition

For any (¯ x, xn) with xn < a(¯ x), (α − L)f1(¯ x, a(¯ x)) + H(¯ x, a(¯ x)) < 0, and for any (¯ x, xn) with xn > a(¯ x), (α − L)f1(¯ x, a(¯ x)) + H(¯ x, a(¯ x)) > 0. Similarly, for any (¯ x, xn) with xn < b(¯ x), (α − L)f2(¯ x, b(¯ x)) − H(¯ x, b(¯ x)) > 0, and for any (¯ x, xn) with xn > b(¯ x), (α − L)f2(¯ x, b(¯ x)) − H(¯ x, b(¯ x)) < 0.

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Dynkin Game and Free Boundary Problem

Proposition

There exist A(¯ x) < 0 < B(¯ x), ∀¯ x ∈ Rn−1 such that the diffusion M = (Xt, Px) on D associated with the Dirichlet form (14) satisfies Eξ1 τ0∧τA e−αtH(Xt)dt

• < −2M,

Eξ2 τ0∧τB e−αtH(Xt)dt

• > 2M,

(25) for some ξ1 ∈ Rn−1 × (A(¯ x), 0), and ξ2 ∈ Rn−1 × (0, B(¯ x)), where τ0, τA, τB denote the hitting times to the graphs of xn = 0, A(¯ x), B(¯ x) respectively.

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Dynkin Game and Free Boundary Problem

Proposition

◮ It is not optimal for P1 to stop the game when Xt < 0 (or

equivalently H(Xt) < 0), and it is not optimal for P2 to stop the game when Xt > 0 (or equivalently H(Xt) > 0).

◮ For any starting point x0 = (¯

x0, xn) ∈ Rn of the game, if xn A(¯ x0), it is optimal for P2 to stop the game immediately; and if xn B(¯ x0), it is optimal for P1 to stop the game immediately.

◮ Let (ˆ

τ, ˆ σ) be the saddle point in (16), then ˆ τ, ˆ σ are finite a.s., hence V (x) = Jx(ˆ τ, ˆ σ).

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Dynkin Game and Free Boundary Problem

Proposition

◮ −f1(¯

x, 0) < V (¯ x, 0) < f2(¯ x, 0), ∀¯ x ∈ Rn−1.

◮ V (x) > −f1(x) for x ∈ Rn−1 × (0, B) and V (x) < f2(x) for

x ∈ Rn−1 × (A, 0).

◮ For each x ∈ E1,

(α − L)f1(x) + H(x) 0, and for each x ∈ E2, (α − L)f2(x) − H(x) 0, where E1, E2 were given in (18).

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Dynkin Game and Free Boundary Problem

Proposition

◮ If x = (¯

x, xn) ∈ E1, then for any point (¯ x, y) with y < xn, (α − L)f1(¯ x, y) + H(¯ x, y) < 0. If x = (¯ x, xn) ∈ E2, then for any point (¯ x, y) with y > xn, (α − L)f2(¯ x, y) − H(¯ x, y) < 0.

◮ If

(α − L)f1(¯ x, xn) + H(¯ x, xn) < 0, ∀¯ x ∈ Rn−1, then it is optimal for P2 to stop the game immediately. If (α − L)f2(¯ x, xn) − H(¯ x, xn) > 0, ∀¯ x ∈ Rn−1, then it is optimal for P1 to stop the game immediately.

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Dynkin Game and Free Boundary Problem

Conclusion: Rn−1 × (−∞, a] = E1, Rn−1 × [b, ∞) = E2, Rn−1 × (a, b) = E. Remarks: The optimal control is given by two curves a(¯ x) and b(¯ x), ¯ x ∈ Rn−1.

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The Multi-dimensional Stochastic Singular Control Problem

The Multi-dimensional Stochastic Singular Control Problem

Define h(x), W (x), x ∈ Rn, as h(¯ x, y) = y H(¯ x, u)du + C(¯ x), (26) W (¯ x, y) = y

a(¯ x)

V (¯ x, u)du, ¯ x ∈ Rn−1, y ∈ R, (27) where C(¯ x) is a function of ¯ x such that lim

y→a(¯ x)+ αW (¯

x, y) − LW (¯ x, y) − h(¯ x, y) = 0, then h(¯ x, y) and W (¯ x, y) satisfy the following:

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The Multi-dimensional Stochastic Singular Control Problem

Theorem

W is C 2,··· ,2 on Rn and there exist unique smooth and uniformly Lipschitz functions a(¯ x), b(¯ x) such that A(¯ x) < a(¯ x) < 0 < b(¯ x) < B(¯ x) and αW (x) − LW (x) = h(x), ∀x ∈ Rn−1 × (a, b), αW (x) − LW (x) < h(x), ∀x ∈ Rn−1 × (−∞, a) ∪ Rn−1 × (b, ∞), −f1(x) < ∂ ∂xn W (x) < f2(x), ∀x ∈ Rn−1 × (a, b), ∂ ∂xn W (x) = −f1(x), ∀x ∈ Rn−1 × (−∞, a], ∂ ∂xn W (x) = f2(x), ∀x ∈ Rn−1 × [b, ∞), and ∀¯ x ∈ Rn−1, 1 k n, ∂2 ∂xn∂xk W (¯ x, a(¯ x)) = − ∂f1 ∂xk (¯ x, a(¯ x)), ∂2 ∂xn∂xk W (¯ x, b(¯ x)) = ∂f2 ∂xk (¯ x, b(¯ x)).

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The Multi-dimensional Stochastic Singular Control Problem

Sketch of Proof

◮ The function

αW (x) − LW (x), is continuous, so is C(x).

◮ For fixed ¯

x, consider the function U(y) = αW (¯ x, y) − LW (¯ x, y) − h(¯ x, y) with U′(y) = αV (¯ x, y) − LV (¯ x, y) − H(¯ x, y), and we know U(a(¯ x)) = 0. Notice that U′(y) = 0 for a(¯ x) < y < b(¯ x); U′(y) > 0 for y < a(¯ x); U′(y) < 0 for y > b(¯ x), the function U(y) is continuous, it can be seen that αW (¯ x, y) − LW (¯ x, y) < h(¯ x, y), for y < a(¯ x) or y > b(¯ x).

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The Multi-dimensional Stochastic Singular Control Problem

Define the following notations: ∆A(i)

t

= A(i)

t

− A(i)

t−,

t 0, i = 1, 2, ∆Xt = Xt − Xt−, t 0, ∆W (Xt) = W (Xt) − W (Xt−), t 0. Let γ = (0, 0, ..., 0, 1)T, then the reflected diffusion can be written as dXt = µ(Xt)dt + σ(Xt)dBt + γdA(1)

t

− γdA(2)

t ,

(28) where A(1)

t

increases only at the boundary a and A(2)

t

increases only at the boundary b. We call a quadruplet S = (S, Xt, A(1)

t , A(2) t ) (S = (A(1) t , A(2) t ) for

simplicity) admissible policy if it satisfies some usual conditions.

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The Multi-dimensional Stochastic Singular Control Problem

Let kS(x) be the cost function given by the following kS(x) = Ex ∞ e−αth(Xt)dt

• + Ex

∞ e−αt f1(Xt)dA(1),c

t

+ f2(Xt)dA(2),c

t

• + Ex

 

0t<∞

e−αt Xnt−+∆A(1)

t

Xnt−

f1(Xt)dy + Xnt−

Xnt−−∆A(2)

t

f2(Xt)dy   , then:

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The Multi-dimensional Stochastic Singular Control Problem

Theorem

• 1. For any admissible policy S, W (x) kS(x), ∀x ∈ Rn.
• 2. W (x) = kS(x), ∀x ∈ Rn, if and only if S = Rn−1 × [a, b], and the

process Xt is the reflecting diffusion on S, i.e., the optimal policy is such that A(1)

t

increases only when Xt is on the boundary (¯ x, a(¯ x)) and A(2)

t

increases only when Xt is on the boundary (¯ x, b(¯ x)), ∀¯ x ∈ Rn−1. Proved using verification theorem and the conditions of W in Theorem 12.

Corollary

Under the given conditions on H, f1, f2 and the definition (26) for function h, the solution W ∈ C 2,··· ,2(Rn) and the functions a(¯ x), b(¯ x) in Theorem 12 are uniquely determined. The function W (x) (x ∈ Rn) coincides with the optimal value function.

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The Multi-dimensional Stochastic Singular Control Problem

Skorohod Problem

◮ The two curves a and b are smooth and uniformly Lipschitz. ◮ Let n(x) be the inward normal for x at the boundary, then it can be

shown that there exist positive constants ν1, ν2 such that ∀x = (¯ x, a(¯ x)), (γ, n(x)) ν1, ∀x = (¯ x, b(¯ x)), (γ, n(x)) −ν2.

◮ Using a localization technique and Theorem 4.3 in

[Lions and Sznitman (1984)] we can show that there exists a solution (Xt, A(1)

t , A(2) t ) to the reflected diffusion (28).

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Concluding Remarks and Future Research

Concluding Remarks

• 1. The value function V of the multi-dimensional Dynkin game is

characterized as the solution of a variational inequality problem involving Dirichlet form.

• 2. The integrated form of V is shown to be the optimal value function

W of the multi-dimensional singular control problem.

• 3. Regularities of V imply the smoothness of W , hence the existence of

a classical solution to the HJB equation.

• 4. The optimal control policy is shown to be given by two curves and the

controlled process is the reflected diffusion between these two curves.

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Concluding Remarks and Future Research

Future Research

• 1. Time inhomogeneous stochastic singular control via game theory and

Dirichlet form.

• 2. Finite horizon stochastic singular control problems.
• 3. Option pricing via game theoretical models.

Thank you!

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References

M.G. Crandall, H. Ishii and P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations, American Mathematical Society. Bulletin. New Series 27 (1) pp. 1–67, 1992. W.H. Fleming and H.M Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 2nd edition, 2006.

• M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and

Symmetric Markov Processes, Walter de Gruyter, Berlin, New York, 1994.

• M. Fukushima and M. Taksar, Dynkin Games Via Dirichlet Froms and

Singular Control of One-Dimensional Diffusion, SIAM J. Control Optim., 41(3) pp. 682–699, 2002.

• M. Fukushima and K. Menda, Refined Solutions of Optimal Stopping

Games for Symmetric Markov Processes, Technology Reports of Kansai University, 48 pp. 101–110, 2006.

• X. Guo and P. Tomecek, Solving singular control from optimal

switching, Special issue for Asian Pacific Financial Market, 2008.

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References

• I. Karatzas and I.M. Zamfirescu, Game approach to the optimal

stopping problem, Stochastics, 77(5) pp. 401–435, 2005. P.L. Lions and A.S. Sznitman, Stochastic Differential Equations with Reflecting Boundary Conditions, Communications on Pure and Applied Mathematics, Vol. XXXVII pp. 511–537, 1984.

• J. Ma and J. Yong, Dynamic Programming for Multidimensional

Stochastic Control Problems, ACTA MATHEMATICA SINICA, 15(4)

• pp. 485–506, 1999.
• H. Nagai, On An Optimal Stopping Problem And A Variational

Inequality, J. Math. Soc. Japan, 30 pp. 303–312, 1978.

• H. Pham, Continuous-time Stochastic Control and Optimization with

Financial Applications, Springer-Verlag Berlin Heidelberg 2009. H.M. Soner and S.E. Shreve, Regularity of the Value Function for a Two-Dimensional Singular Stochastic Control Problem, SIAM J. Control and Optimization, 27(4) pp. 876–907, 1989.

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References

• M. Taksar, Average Optimal Singular Control and a Related Stopping

Problem, Math. Oper. Res., 10 pp. 63–81, 1985.

• J. Zabczyk, Stopping Games for Symmetric Markov Processes,
• Probab. Math. Statist., 4(2) pp. 185–196, 1984.

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