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 Y.Yang (MU) Oct 11, 2012 2 / 35
Problem Formulation Problem Formulation Given a probability space (Ω , F , F t , X , θ t , P x ), we are concerned with a multi-dimensional diffusion on R n : d X t = µ ( X t ) dt + σ ( X t ) d B t , X 0 = x , (1) where X 1 t µ 1 σ 11 · · · σ 1 m B 1 t . . . . . X t = . , µ ( X t ) = . , σ ( X t ) = . . , B t = . , . . . . . X nt µ n σ n 1 · · · σ nm B mt (2) in which µ i , σ i , j (1 � i � n , 1 � j � m ) are functions of X 1 t , ..., X ( n − 1) t satisfying the usual conditions, and B t is m -dimensional Brownian motion with m � n . Y.Yang (MU) Oct 11, 2012 3 / 35
Problem Formulation There is a cost function associated with this process: �� ∞ � ∞ �� e − α t � f 1 ( X t ) dA (1) + f 2 ( X t ) dA (2) e − α t h ( X t ) dt + k S ( x ) = E x , t t 0 0 f 1 ( x ) , f 2 ( x ) > 0 , ∀ x ∈ R n . (3) And there is control on the underlying process: dX 1 t = µ 1 dt + σ 11 dB 1 t + · · · + σ 1 m dB mt , . . . . . . . . . dX nt = µ n dt + σ n 1 dB 1 t + · · · + σ nm dB mt + dA (1) − dA (2) t , t X 0 = x . Y.Yang (MU) Oct 11, 2012 4 / 35
Problem Formulation ◮ A control policy is defined as a pair ( A (1) t , A (2) t ) = S of F t adapted processes which are right continuous and nondecreasing in t . ◮ A (1) − A (2) is the minimal decomposition of a bounded variation t t process into the difference of two nondecreasing processes. Problem Formulation: one looks for a control policy S that minimizes the cost function k S ( x ), W ( x ) = min S∈ S k S ( x ) , (4) where S is the set of admissible policies. Y.Yang (MU) Oct 11, 2012 5 / 35
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 one asset. Y.Yang (MU) Oct 11, 2012 6 / 35
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)]. Y.Yang (MU) Oct 11, 2012 7 / 35
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. Y.Yang (MU) Oct 11, 2012 8 / 35
Dirichlet Form and Dynkin Game Variational Inequality Problem Involving Dirichlet Form Let f 1 ∈ F . Nagai [Nagai(1978)] showed that there exist a quasi continuous function w ∈ F which solves the variational inequality problem w � − f 1 , E α ( w , u − w ) � 0 , ∀ u ∈ F with u � − f 1 , and a properly exceptional set N such that for all x ∈ R n / N , � � e − ασ [ − f 1 ( X σ )] e − α ˆ σ [ − f 1 ( X ˆ � � w ( x ) = sup σ E x = E x σ )] , where σ = inf { t � 0; w ( X t ) = − f 1 ( X t ) } . ˆ Moreover, w is the smallest α -potential dominating the function − f 1 m-a.e. Y.Yang (MU) Oct 11, 2012 9 / 35
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 E α ( V , u − V ) � 0 , ∀ u ∈ K , (5) where K = { u ∈ F : − f 1 � u � f 2 m -a.e. } , f 1 , f 2 ∈ F , and a properly exceptional set N such that for all x ∈ R n / N , V ( x ) = sup σ inf τ J x ( τ, σ ) = inf τ sup σ J x ( τ, σ ) (6) for any stopping times τ and σ , where � e − α ( τ ∧ σ ) ( − I σ � τ f 1 ( X σ ) + I τ<σ f 2 ( X τ )) � J x ( τ, σ ) = E x . (7) Y.Yang (MU) Oct 11, 2012 10 / 35
Dirichlet Form and Dynkin Game What is more, if define E 1 = { x ∈ R n / N : V ( x ) = − f 1 ( x ) } , E 2 = { x ∈ R n / N : V ( x ) = f 2 ( x ) } , then the hitting times ˆ τ = τ E 2 , ˆ σ = τ E 1 is the saddle point of the game J x (ˆ τ, σ ) � J x (ˆ τ, ˆ σ ) � J x ( τ, ˆ σ ) (8) for any x ∈ R n / N and any stopping times τ, σ , where J x is given in (7). In particular ∀ x ∈ R n / N . V ( x ) = J x (ˆ τ, ˆ σ ) , (9) Y.Yang (MU) Oct 11, 2012 11 / 35
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: p t ( x , · ) ≪ m ( · ) , (10) and f 1 , f 2 are finite finely continuous functions satisfying the following separability condition: Assumption There exist finite α -excessive functions v 1 , v 2 ∈ F such that, for all x ∈ R n , − f 1 ( x ) � v 1 ( x ) − v 2 ( x ) � f 2 ( x ) , (11) then there does not exist the exceptional set N , and the solution V ( x ) is finite finely continuous. Y.Yang (MU) Oct 11, 2012 12 / 35
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 = R n − 1 × ( A (¯ x ) , B (¯ x )), where A , B are two bounded, smooth and uniformly Lipschitiz functions. The associated cost function is �� τ ∧ σ � e − α t H ( X t ) dt J x ( τ, σ ) = E x 0 (12) � e − α ( τ ∧ σ ) ( − I σ � τ f 1 ( X σ ) + I τ<σ f 2 ( X τ )) � + E x , where H is the holding cost, and f 1 , f 2 are boundary penalty costs. Y.Yang (MU) Oct 11, 2012 13 / 35
Dynkin Game and Free Boundary Problem Two players P 1 and P 2 observe the underlying process X t in (1) with accumulated income, discounted at present time, equalling � σ 0 e − α t H ( X t ) dt , for any stopping time σ . ◮ If P 1 stops the game at time σ , he pays P 2 the amount of the accumulated income plus the amount f 2 ( X σ ), which after been discounted equals e − ασ f 2 ( X σ ). ◮ If the process is stopped by P 2 at time σ , he receives from P 1 the accumulated income less the amount f 1 ( X σ ), which after been discounted equals e − ασ f 1 ( X σ ). P 1 tries to minimize his payment while P 2 tries to maximize his income. Thus the value of this Dynkin game is given by ∀ x ∈ R n . V ( x ) = inf τ sup σ J x ( τ, σ ) , (13) Y.Yang (MU) Oct 11, 2012 14 / 35
Dynkin Game and Free Boundary Problem ◮ Define the following Dirichlet form on D : � E ( u , v ) = ∇ u ( x ) · A ∇ v ( x ) m ( d x ) , u , v ∈ F , (14) D where � F = { u ∈ L 2 ( D ) : u is continuous , ∇ u ( x ) T ∇ u ( x ) m ( d x ) < ∞} , D 2 σσ T is assumed to be uniformly elliptic, and A ( x ) = 1 m ( d x ) = e b · x d x , in which b = A − 1 µ . ◮ Consider the solution V ∈ F , − f 1 � V � f 2 of E α ( V , u − V ) � ( H , u − V ) , ∀ u ∈ F , − f 1 � u � f 2 , (15) where � E α ( u , v ) = E ( u , v ) + α R n u ( x ) v ( x ) m ( d x ) . Y.Yang (MU) Oct 11, 2012 15 / 35
Dynkin Game and Free Boundary Problem Theorem Assume some usual conditions on H, f 1 , f 2 ∈ F and the separability condition, we put �� τ ∧ σ � e − α t H ( X t ) dt J x ( τ, σ ) = E x 0 (16) � e − α ( τ ∧ σ ) ( − I σ � τ f 1 ( X σ ) + I τ<σ f 2 ( X τ )) � + E x for finite stopping times τ, σ . Then the solution of (15) admits a finite and continuous value function of the game ∀ x ∈ R n . V ( x ) = inf τ sup σ J x ( τ, σ ) = sup σ inf τ J x ( τ, σ ) , (17) Y.Yang (MU) Oct 11, 2012 16 / 35
Dynkin Game and Free Boundary Problem Theorem Furthermore if we let E 1 = { x ∈ R n : V ( x ) = − f 1 ( x ) } , E 2 = { x ∈ R n : V ( x ) = f 2 ( x ) } , (18) then the hitting times ˆ τ = τ E 2 , ˆ σ = τ E 1 is the saddle point of the game J x (ˆ τ, σ ) � J x (ˆ τ, ˆ σ ) � J x ( τ, ˆ σ ) (19) for any x ∈ R n and any stopping times τ, σ . In particular, ˆ τ, ˆ σ are finite a.s. and ∀ x ∈ R n . V ( x ) = J x (ˆ τ, ˆ σ ) , (20) Regularities? Optimal control policies? Y.Yang (MU) Oct 11, 2012 17 / 35
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