On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
On solvability of a two-sided singular control
problem
Pekka Matomäki
Turku School of Economics
University of Turku
Finland
3.2.2011
On two-sided
singular
control
Table of contents
Pekka
Matomäki,
Turku School
of Economics
Introduction
1 Introduction
Solving the
problem
Sensitiveness
of the solution
2 Solving the problem
Conclusion
Connection
with the
Dynkin game
3 Sensitiveness of the solution
References
4 Conclusion
5 Connection with the Dynkin game
On two-sided
singular
control
Outline
Pekka
Matomäki,
Turku School
of Economics
Introduction
1 Introduction
Solving the
problem
Sensitiveness
of the solution
2 Solving the problem
Conclusion
Connection
with the
Dynkin game
3 Sensitiveness of the solution
References
4 Conclusion
5 Connection with the Dynkin game
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Example of a singular/reflected
control
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, applications
• The dividend payments problems (see e.g. Paulsen
(2008))
• A company pays dividends to the owners and in proportion
the owners have a duty to reinvest if the income process
falls too minimal.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, applications
• The dividend payments problems (see e.g. Paulsen
(2008))
• A company pays dividends to the owners and in proportion
the owners have a duty to reinvest if the income process
falls too minimal.
• Costly reversible investment problem (Guo and Pham
(2005))
• Invest in a project at upper boundary, and disinvest at the
lower boundary.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, applications
• The dividend payments problems (see e.g. Paulsen
(2008))
• A company pays dividends to the owners and in proportion
the owners have a duty to reinvest if the income process
falls too minimal.
• Costly reversible investment problem (Guo and Pham
(2005))
• Invest in a project at upper boundary, and disinvest at the
lower boundary.
• Monotone fuel follower problem (Jacka (2002))
• Controlling a vessel in a stormy sea.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, applications
• The dividend payments problems (see e.g. Paulsen
(2008))
• A company pays dividends to the owners and in proportion
the owners have a duty to reinvest if the income process
falls too minimal.
• Costly reversible investment problem (Guo and Pham
(2005))
• Invest in a project at upper boundary, and disinvest at the
lower boundary.
• Monotone fuel follower problem (Jacka (2002))
• Controlling a vessel in a stormy sea.
• Inventory theory (Harrison and Taksar (1983))
• Order extra amount of goods at the lower boundary.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, applications
• The dividend payments problems (see e.g. Paulsen
(2008))
• A company pays dividends to the owners and in proportion
the owners have a duty to reinvest if the income process
falls too minimal.
• Costly reversible investment problem (Guo and Pham
(2005))
• Invest in a project at upper boundary, and disinvest at the
lower boundary.
• Monotone fuel follower problem (Jacka (2002))
• Controlling a vessel in a stormy sea.
• Inventory theory (Harrison and Taksar (1983))
• Order extra amount of goods at the lower boundary.
• Controlling a dam (Faddy (1974)).
• Release optimally water from a dam.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, applications
• The dividend payments problems (see e.g. Paulsen
(2008))
• A company pays dividends to the owners and in proportion
the owners have a duty to reinvest if the income process
falls too minimal.
• Costly reversible investment problem (Guo and Pham
(2005))
• Invest in a project at upper boundary, and disinvest at the
lower boundary.
• Monotone fuel follower problem (Jacka (2002))
• Controlling a vessel in a stormy sea.
• Inventory theory (Harrison and Taksar (1983))
• Order extra amount of goods at the lower boundary.
• Controlling a dam (Faddy (1974)).
• Release optimally water from a dam.
• One of the first to study this problem was Bather and
Chernoff (1966): ”Sequential decisions in the control of a
spaceship”.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
The setup
• We assume that the state space is (0, ∞) with natural
boundaries and that the underlying diffusion Xt is a
general Itô diffusion:
dXt = µ(Xt )dt + σ(Xt )dWt ,
X0 = x,
where µ(x) is a drift term, σ(x) is a volatility term and
Wt is standard Brownian motion.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
The setup
• We assume that the state space is (0, ∞) with natural
boundaries and that the underlying diffusion Xt is a
general Itô diffusion:
dXt = µ(Xt )dt + σ(Xt )dWt ,
X0 = x,
where µ(x) is a drift term, σ(x) is a volatility term and
Wt is standard Brownian motion.
• There are control processes Ut and Dt , both
right-continuous and increasing.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
The setup
• We assume that the state space is (0, ∞) with natural
boundaries and that the underlying diffusion Xt is a
general Itô diffusion:
dXt = µ(Xt )dt + σ(Xt )dWt ,
X0 = x,
where µ(x) is a drift term, σ(x) is a volatility term and
Wt is standard Brownian motion.
• There are control processes Ut and Dt , both
right-continuous and increasing.
• The associated control process Zt = Xt + Ut − Dt , where
• Ut is the upward control
• Dt is the downward control.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
The setup
• We assume that the state space is (0, ∞) with natural
boundaries and that the underlying diffusion Xt is a
general Itô diffusion:
dXt = µ(Xt )dt + σ(Xt )dWt ,
X0 = x,
where µ(x) is a drift term, σ(x) is a volatility term and
Wt is standard Brownian motion.
• There are control processes Ut and Dt , both
right-continuous and increasing.
• The associated control process Zt = Xt + Ut − Dt , where
• Ut is the upward control
• Dt is the downward control.
• Using upward control costs q,
Using downward control gives gain p,
and q > p.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Problem
Our problem is
V (x) = sup Ex
(Ut ,Dt )
Z
∞
e
0
where π is a gain function.
−rt
(π(Zt )dt + pdDt − qdUt ) ,
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Problem
Our problem is
V (x) = sup Ex
(Ut ,Dt )
Z
∞
e
−rt
(π(Zt )dt + pdDt − qdUt ) ,
0
where π is a gain function.
• Our aim is to find the conditions, under which
Connection
with the
Dynkin game
References
∗
(a) there exists a unique reflecting control (Utz , Dty ).
(b) this reflecting control is optimal among all possible
controls.
∗
Conclusion
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Problem
Our problem is
V (x) = sup Ex
(Ut ,Dt )
Z
∞
e
−rt
(π(Zt )dt + pdDt − qdUt ) ,
0
where π is a gain function.
• Our aim is to find the conditions, under which
Connection
with the
Dynkin game
References
∗
(a) there exists a unique reflecting control (Utz , Dty ).
(b) this reflecting control is optimal among all possible
controls.
∗
Conclusion
• Reflecting controls Utz and Dty increase only when Zt = z
and Zt = y respectively (cf. local time).
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Problem
Our problem is
V (x) = sup Ex
(Ut ,Dt )
Z
∞
e
−rt
(π(Zt )dt + pdDt − qdUt ) ,
0
where π is a gain function.
• Our aim is to find the conditions, under which
Connection
with the
Dynkin game
References
∗
(a) there exists a unique reflecting control (Utz , Dty ).
(b) this reflecting control is optimal among all possible
controls.
∗
Conclusion
• Reflecting controls Utz and Dty increase only when Zt = z
and Zt = y respectively (cf. local time).
• Here is presented easily verifiable conditions under which
we can give a (quasi-)explicit form of the value function
and the reflected control is the optimal for two-sided
control problem
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Problem
Our problem is
V (x) = sup Ex
(Ut ,Dt )
Z
∞
e
−rt
(π(Zt )dt + pdDt − qdUt ) ,
0
where π is a gain function.
• Our aim is to find the conditions, under which
Connection
with the
Dynkin game
References
∗
(a) there exists a unique reflecting control (Utz , Dty ).
(b) this reflecting control is optimal among all possible
controls.
∗
Conclusion
• Reflecting controls Utz and Dty increase only when Zt = z
and Zt = y respectively (cf. local time).
• Here is presented easily verifiable conditions under which
we can give a (quasi-)explicit form of the value function
and the reflected control is the optimal for two-sided
control problem and also a sensitiveness analysis of the
solution is presented.
• This is Matomäki (2012).
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Example of a singular/reflected
control revisited
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, examples
• Rational harvesting
• Xt represents a forest, π ≡ 0
• In one-sided control problem only Dt , representing timber
harvesting, is present.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, examples
• Rational harvesting
• Xt represents a forest, π ≡ 0
• In one-sided control problem only Dt , representing timber
harvesting, is present.
• In two-sided control problem both Dt and Ut , representing
replanting of trees, are present.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, examples
• Rational harvesting
• Xt represents a forest, π ≡ 0
• In one-sided control problem only Dt , representing timber
harvesting, is present.
• In two-sided control problem both Dt and Ut , representing
replanting of trees, are present.
• Monotone fuel follower problem
• Xt represents a path of a spaceship
• π is the utility function of the desired trajectory
• Dt represents steering to the right and Ut steering to the
left.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Introduction, examples
• Rational harvesting
• Xt represents a forest, π ≡ 0
• In one-sided control problem only Dt , representing timber
harvesting, is present.
• In two-sided control problem both Dt and Ut , representing
replanting of trees, are present.
• Monotone fuel follower problem
• Xt represents a path of a spaceship
• π is the utility function of the desired trajectory
• Dt represents steering to the right and Ut steering to the
left.
• Heating a house
• Xt represents a temperature and π the temperature
dependent utility
• Dt represents cooling and Ut heating.
On two-sided
singular
control
Greenian approach
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• S ′ (x) = e
−
Rx
diffusion Xt .
2µ(t)
dt
σ 2 (t)
is the scale density function of the
On two-sided
singular
control
Greenian approach
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• S ′ (x) = e
−
Rx
2µ(t)
dt
σ 2 (t)
is the scale density function of the
diffusion Xt .
• m′ (x) =
2
σ 2 (x)S ′ (x)
the diffusion Xt .
is the density of the speed measure of
On two-sided
singular
control
Greenian approach
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• S ′ (x) = e
−
Rx
2µ(t)
dt
σ 2 (t)
is the scale density function of the
diffusion Xt .
• m′ (x) =
2
σ 2 (x)S ′ (x)
is the density of the speed measure of
the diffusion Xt .
• ψ and ϕ are, respectively, the increasing and decreasing
fundamental solution of the ordinary second-order
differential equation Au = ru, where
2
df
Af = 12 σ 2 (x) ddxf2 + µ(x) dx
is the differential operator
associated to the diffusion Xt .
On two-sided
singular
control
Greenian approach
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• S ′ (x) = e
−
Rx
2µ(t)
dt
σ 2 (t)
is the scale density function of the
diffusion Xt .
• m′ (x) =
2
σ 2 (x)S ′ (x)
is the density of the speed measure of
the diffusion Xt .
• ψ and ϕ are, respectively, the increasing and decreasing
fundamental solution of the ordinary second-order
differential equation Au = ru, where
2
df
Af = 12 σ 2 (x) ddxf2 + µ(x) dx
is the differential operator
associated to the diffusion Xt .
R∞
• (Rr π)(x) = Ex [ 0 e −rt π(Xt )dt] is the resolvent of π.
On two-sided
singular
control
Outline
Pekka
Matomäki,
Turku School
of Economics
Introduction
1 Introduction
Solving the
problem
Sensitiveness
of the solution
2 Solving the problem
Conclusion
Connection
with the
Dynkin game
3 Sensitiveness of the solution
References
4 Conclusion
5 Connection with the Dynkin game
On two-sided
singular
control
Necessary condition I
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
We search a necessary condition for a reflecting control to be
optimal and show it to have only one solution.
On two-sided
singular
control
Necessary condition I
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
We search a necessary condition for a reflecting control to be
optimal and show it to have only one solution.
With reflecting barriers z and y , the value function gets the
form


x ≥y
p(x − y ) + V (y )
V (x) = (Rr π)(x) + c1 ψ(x) + c2 ϕ(x)
z <x <y


q(x − z) + V (z)
x ≤ z.
This has four free parameters (z, y , c1 , c2 ). If we make an
educated guess and require V to be C 2 , we get a necessary
condition for these parameters.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Necessary condition II
In the line of Salminen (1985) and Lempa (2010) the optimal
reflecting barriers (z ∗ , y ∗ ) is the solution of the pair of
equations
(
Jq (z ∗ ) − Jp (y ∗ ) = 0
(N)
Iq (z ∗ ) − Ip (y ∗ ) = 0,
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Necessary condition II
In the line of Salminen (1985) and Lempa (2010) the optimal
reflecting barriers (z ∗ , y ∗ ) is the solution of the pair of
equations
(
Jq (z ∗ ) − Jp (y ∗ ) = 0
(N)
Iq (z ∗ ) − Ip (y ∗ ) = 0,
Connection
with the
Dynkin game
References
where

Z ∞

′
1


ϕt ρb (x) − ρb (t) mt dt
 Jb (x) = −
B
Z xx

′
1


ψt ρb (x) − ρb (t) mt dt ,
 Ib (x) =
B
0
where B is a coefficient and ρb (x) = π(x) + b(µ(x) − rx).
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Assumptions
Let ρb (x) := π(x) + b(µ(x) − rx), for b = p, q and assume
that
R∞
• µx , πx , σx ∈ C 1 (R+ ) and Ex [ 0 e −rt |ρb (t)|dt] < ∞ for
b = p, q,
• q > p,
• µ′ (x) < r ,
• there are x̃b ∈ R+ such that
d
dx ρb (x)
T 0 whenever
x S x̃b , for b ∈ [p, q],
Moreover, assume that
• Condition (N) holds.
These are sufficient for the solution to be two-sided reflected
control.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Assumptions
Let ρb (x) := π(x) + b(µ(x) − rx), for b = p, q and assume
that
R∞
• µx , πx , σx ∈ C 1 (R+ ) and Ex [ 0 e −rt |ρb (t)|dt] < ∞ for
b = p, q,
• q > p,
• µ′ (x) < r ,
• there are x̃b ∈ R+ such that
d
dx ρb (x)
T 0 whenever
x S x̃b , for b ∈ [p, q],
Moreover, assume that
• ρb (∞) = −∞ and that ρ′b (0+) > 0, for b = p, q,
• limx↓0
R∞
x
ϕ′ (t)/S ′ (t)dt = −∞.
These are sufficient for the solution to be two-sided reflected
control.
On two-sided
singular
control
The idea of the proof
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Let (z ∗ , y ∗ ) be solution to the necessary condition (N).
On two-sided
singular
control
The idea of the proof
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Let (z ∗ , y ∗ ) be solution to the necessary condition (N).
∗
Step 1 Prove that the associated reflecting control (Utz , Dty ) is
the optimal one among all possible controls.
∗
On two-sided
singular
control
The idea of the proof
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Let (z ∗ , y ∗ ) be solution to the necessary condition (N).
∗
Step 1 Prove that the associated reflecting control (Utz , Dty ) is
the optimal one among all possible controls.
∗
Step 2 Prove that there exists a unique optimal pair (z ∗ , y ∗ )
satisfying the necessary condition (N).
On two-sided
singular
control
The idea of the proof
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Let (z ∗ , y ∗ ) be solution to the necessary condition (N).
∗
Step 1 Prove that the associated reflecting control (Utz , Dty ) is
the optimal one among all possible controls.
∗
Step 2 Prove that there exists a unique optimal pair (z ∗ , y ∗ )
satisfying the necessary condition (N).
• The harder one.
On two-sided
singular
control
The idea of the proof
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Let (z ∗ , y ∗ ) be solution to the necessary condition (N).
∗
Step 1 Prove that the associated reflecting control (Utz , Dty ) is
the optimal one among all possible controls.
∗
• Not totally leisure, but an easier one.
Step 2 Prove that there exists a unique optimal pair (z ∗ , y ∗ )
satisfying the necessary condition (N).
• The harder one.
On two-sided
singular
control
The idea of the proof
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Let (z ∗ , y ∗ ) be solution to the necessary condition (N).
∗
Step 1 Prove that the associated reflecting control (Utz , Dty ) is
the optimal one among all possible controls.
∗
• Not totally leisure, but an easier one.
Step 2 Prove that there exists a unique optimal pair (z ∗ , y ∗ )
satisfying the necessary condition (N).
• The harder one.
In the next two slides we demonstrate how these can be proved.
On two-sided
singular
control
Properties of value function V
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
(i) V ∈ C 1
(ii) (A − r )V + π ≤ 0
(iii) p ≤ V ′ (x) ≤ q for all x > 0
Theorem 1
Let V ∗ be the optimal value function for studied control
problem. Then if V is admissible and satisfies the three
conditions above, V = V ∗ .
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Properties of value function V
Value function associated to (z ∗ , y ∗ ) is

∗
∗

p(x − y ) + V (y )
V (x) = (Rr π)(x) − Jq (z ∗ )ψ(x) + Iq (z ∗ )ϕ(x)


q(x − z ∗ ) + V (z ∗ )
x ≥ y∗
z∗ < x < y∗
x ≤ z ∗,
which satisfies the following conditions:
(i) V ∈ C 1
(ii) (A − r )V + π ≤ 0
(iii) p ≤ V ′ (x) ≤ q for all x > 0
Theorem 1
Let V ∗ be the optimal value function for studied control
problem. Then if V is admissible and satisfies the three
conditions above, V = V ∗ .
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Properties of value function V
Value function associated to (z ∗ , y ∗ ) is

∗
∗

p(x − y ) + V (y )
V (x) = (Rr π)(x) − Jq (z ∗ )ψ(x) + Iq (z ∗ )ϕ(x)


q(x − z ∗ ) + V (z ∗ )
x ≥ y∗
z∗ < x < y∗
x ≤ z ∗,
which satisfies the following conditions:
(i) V ∈ C 1 (quite clear)
(ii) (A − r )V + π ≤ 0
(iii) p ≤ V ′ (x) ≤ q for all x > 0
Theorem 1
Let V ∗ be the optimal value function for studied control
problem. Then if V is admissible and satisfies the three
conditions above, V = V ∗ .
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Properties of value function V
Value function associated to (z ∗ , y ∗ ) is

∗
∗

p(x − y ) + V (y )
V (x) = (Rr π)(x) − Jq (z ∗ )ψ(x) + Iq (z ∗ )ϕ(x)


q(x − z ∗ ) + V (z ∗ )
x ≥ y∗
z∗ < x < y∗
x ≤ z ∗,
which satisfies the following conditions:
(i) V ∈ C 1 (quite clear)
(ii) (A − r )V + π ≤ 0 (from straight calculation)
(iii) p ≤ V ′ (x) ≤ q for all x > 0
Theorem 1
Let V ∗ be the optimal value function for studied control
problem. Then if V is admissible and satisfies the three
conditions above, V = V ∗ .
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Properties of value function V
Value function associated to (z ∗ , y ∗ ) is

∗
∗

p(x − y ) + V (y )
V (x) = (Rr π)(x) − Jq (z ∗ )ψ(x) + Iq (z ∗ )ϕ(x)


q(x − z ∗ ) + V (z ∗ )
x ≥ y∗
z∗ < x < y∗
x ≤ z ∗,
which satisfies the following conditions:
(i) V ∈ C 1 (quite clear)
(ii) (A − r )V + π ≤ 0 (from straight calculation)
(iii) p ≤ V ′ (x) ≤ q for all x > 0 (Shreve et al. (1984),
µ′ (x) < r )
Theorem 1
Let V ∗ be the optimal value function for studied control
problem. Then if V is admissible and satisfies the three
conditions above, V = V ∗ .
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Unique existence of (z ∗ , y ∗ )
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Unique existence of (z ∗ , y ∗ )
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Unique existence of (z ∗ , y ∗ )
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Unique existence of (z ∗ , y ∗ )
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Unique existence of (z ∗ , y ∗ )
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Sample path
On two-sided
singular
control
Outline
Pekka
Matomäki,
Turku School
of Economics
Introduction
1 Introduction
Solving the
problem
Sensitiveness
of the solution
2 Solving the problem
Conclusion
Connection
with the
Dynkin game
3 Sensitiveness of the solution
References
4 Conclusion
5 Connection with the Dynkin game
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Sensitiveness theorem
Recall that
• dXt = µ(Xt )dt + σ(Xt )dWt ;
R ∞
e −rt (π(Zt )dt + pdDt − qdUt )
is the value function and (z ∗ , y ∗ ) are the optimal barriers.
• V (x) = sup(Ut ,Dt ) Ex
Theorem
0
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Sensitiveness theorem
Recall that
• dXt = µ(Xt )dt + σ(Xt )dWt ;
R ∞
e −rt (π(Zt )dt + pdDt − qdUt )
is the value function and (z ∗ , y ∗ ) are the optimal barriers.
• V (x) = sup(Ut ,Dt ) Ex
0
Theorem
(A1) V (x) is p-increasing and q-decreasing.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Sensitiveness theorem
Recall that
• dXt = µ(Xt )dt + σ(Xt )dWt ;
R ∞
e −rt (π(Zt )dt + pdDt − qdUt )
is the value function and (z ∗ , y ∗ ) are the optimal barriers.
• V (x) = sup(Ut ,Dt ) Ex
0
Theorem
(A1) V (x) is p-increasing and q-decreasing.
(A2) Inactive region (z ∗ , y ∗ ) shrinks as p increases and widens
as q increasing.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Sensitiveness theorem
Recall that
• dXt = µ(Xt )dt + σ(Xt )dWt ;
R ∞
e −rt (π(Zt )dt + pdDt − qdUt )
is the value function and (z ∗ , y ∗ ) are the optimal barriers.
• V (x) = sup(Ut ,Dt ) Ex
0
Theorem
(A1) V (x) is p-increasing and q-decreasing.
(A2) Inactive region (z ∗ , y ∗ ) shrinks as p increases and widens
as q increasing.
(B1) V (x) is σ-decreasing.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Sensitiveness theorem
Recall that
• dXt = µ(Xt )dt + σ(Xt )dWt ;
R ∞
e −rt (π(Zt )dt + pdDt − qdUt )
is the value function and (z ∗ , y ∗ ) are the optimal barriers.
• V (x) = sup(Ut ,Dt ) Ex
0
Theorem
(A1) V (x) is p-increasing and q-decreasing.
(A2) Inactive region (z ∗ , y ∗ ) shrinks as p increases and widens
as q increasing.
(B1) V (x) is σ-decreasing.
(B2) Inactive region (z ∗ , y ∗ ) widens as σ increases.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Sensitiveness illustrated
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Limiting theorem
Define xi , i = 1, 2 and x̃q as the unique points for which
Ip (x1 ) = 0, Jq (x2 ) = 0, ρ′q (x̃q ) = 0.
Theorem
(A) If q ր ∞ then (z ∗ , y ∗ ) → (0, x1 ).
• We approach a case Ut∗ ≡ 0, cf. Alvarez and Lempa
(2008).
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Limiting theorem
Define xi , i = 1, 2 and x̃q as the unique points for which
Ip (x1 ) = 0, Jq (x2 ) = 0, ρ′q (x̃q ) = 0.
Theorem
(A) If q ր ∞ then (z ∗ , y ∗ ) → (0, x1 ).
• We approach a case Ut∗ ≡ 0, cf. Alvarez and Lempa
Conclusion
Connection
with the
Dynkin game
References
(2008).
(B) If
π ′ (x)
≥ 0 and p ց 0, then (z ∗ , y ∗ ) → (x2 , ∞).
• We approach a case Dt∗ ≡ 0.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Limiting theorem
Define xi , i = 1, 2 and x̃q as the unique points for which
Ip (x1 ) = 0, Jq (x2 ) = 0, ρ′q (x̃q ) = 0.
Theorem
(A) If q ր ∞ then (z ∗ , y ∗ ) → (0, x1 ).
• We approach a case Ut∗ ≡ 0, cf. Alvarez and Lempa
Conclusion
Connection
with the
Dynkin game
References
(2008).
(B) If
π ′ (x)
≥ 0 and p ց 0, then (z ∗ , y ∗ ) → (x2 , ∞).
• We approach a case Dt∗ ≡ 0.
(C) If q − p → 0, then (z ∗ , y ∗ ) → (x̃q , x̃q ), and the value
function approaches, from below, a function
q(x − x̃q ) + 1r (qµ(x̃q ) + π(x̃q )) .
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Limiting theorem
Define xi , i = 1, 2 and x̃q as the unique points for which
Ip (x1 ) = 0, Jq (x2 ) = 0, ρ′q (x̃q ) = 0.
Theorem
(A) If q ր ∞ then (z ∗ , y ∗ ) → (0, x1 ).
• We approach a case Ut∗ ≡ 0, cf. Alvarez and Lempa
Conclusion
Connection
with the
Dynkin game
References
(2008).
(B) If
π ′ (x)
≥ 0 and p ց 0, then (z ∗ , y ∗ ) → (x2 , ∞).
• We approach a case Dt∗ ≡ 0.
(C) If q − p → 0, then (z ∗ , y ∗ ) → (x̃q , x̃q ), and the value
function approaches, from below, a function
q(x − x̃q ) + 1r (qµ(x̃q ) + π(x̃q )) .
(D) Assume that π ′ (x) ≥ 0. If q, p → ∞, then
(z ∗ , y ∗ ) ց (0, 0). If q, p → 0, then (z ∗ , y ∗ ) ր (∞, ∞).
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Sensitiveness illustrated revisited
On two-sided
singular
control
Notes on the limits
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
(A&B) We get also a direction for the convergent. I.e.
• y ∗ < x1 : using a heating example, in the absence of
heater, we cool the house later.
On two-sided
singular
control
Notes on the limits
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
(A&B) We get also a direction for the convergent. I.e.
• y ∗ < x1 : using a heating example, in the absence of
heater, we cool the house later.
• z ∗ > x2 : using a dividend payments problem with
obligative reinvestment example, in the absence of
dividend payments we reinvest later.
On two-sided
singular
control
Notes on the limits
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
(A&B) We get also a direction for the convergent. I.e.
• y ∗ < x1 : using a heating example, in the absence of
heater, we cool the house later.
• z ∗ > x2 : using a dividend payments problem with
obligative reinvestment example, in the absence of
dividend payments we reinvest later.
(C) The limiting control (z ∗ , y ∗ ) = (x̃q , x̃q ) is not an
admissible one.
On two-sided
singular
control
Notes on the limits
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
(A&B) We get also a direction for the convergent. I.e.
• y ∗ < x1 : using a heating example, in the absence of
heater, we cool the house later.
• z ∗ > x2 : using a dividend payments problem with
obligative reinvestment example, in the absence of
dividend payments we reinvest later.
(C) The limiting control (z ∗ , y ∗ ) = (x̃q , x̃q ) is not an
admissible one.
(D) At the limits, neither the control Dt0 nor Ut∞ are
admissible.
On two-sided
singular
control
Outline
Pekka
Matomäki,
Turku School
of Economics
Introduction
1 Introduction
Solving the
problem
Sensitiveness
of the solution
2 Solving the problem
Conclusion
Connection
with the
Dynkin game
3 Sensitiveness of the solution
References
4 Conclusion
5 Connection with the Dynkin game
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• We found conditions under which two-sided reflecting
control was optimal.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• We found conditions under which two-sided reflecting
control was optimal.
• Proof combined existing techniques from Harrison (1985),
Shreve et al. (1984), Alvarez (2008) and Lempa (2010).
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• We found conditions under which two-sided reflecting
control was optimal.
• Proof combined existing techniques from Harrison (1985),
Shreve et al. (1984), Alvarez (2008) and Lempa (2010).
• We found the sensitiveness with respect to volatility σ,
gain p and cost q.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• We found conditions under which two-sided reflecting
control was optimal.
• Proof combined existing techniques from Harrison (1985),
Shreve et al. (1984), Alvarez (2008) and Lempa (2010).
• We found the sensitiveness with respect to volatility σ,
gain p and cost q.
• We found a one-sided case as a limit from a two-sided
case.
On two-sided
singular
control
Some inconvenient issues
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• Assumption limx↓0
R∞
x
ϕ′ (t)/S ′ (t)dt = −∞.
On two-sided
singular
control
Some inconvenient issues
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
R∞
• Assumption limx↓0 x ϕ′ (t)/S ′ (t)dt = −∞.
• Somewhat idealisation, for example no fixed cost of using
control, the gain and cost of using controls are coefficients.
On two-sided
singular
control
Some inconvenient issues
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
R∞
• Assumption limx↓0 x ϕ′ (t)/S ′ (t)dt = −∞.
• Somewhat idealisation, for example no fixed cost of using
control, the gain and cost of using controls are coefficients.
• If we take for example fishing pool, Theorem says that
fishing a fish every now and then is the optimal harvesting
strategy.
On two-sided
singular
control
Some inconvenient issues
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
R∞
• Assumption limx↓0 x ϕ′ (t)/S ′ (t)dt = −∞.
• Somewhat idealisation, for example no fixed cost of using
control, the gain and cost of using controls are coefficients.
• If we take for example fishing pool, Theorem says that
fishing a fish every now and then is the optimal harvesting
strategy.
• =⇒ Impulse control problem.
On two-sided
singular
control
Outline
Pekka
Matomäki,
Turku School
of Economics
Introduction
1 Introduction
Solving the
problem
Sensitiveness
of the solution
2 Solving the problem
Conclusion
Connection
with the
Dynkin game
3 Sensitiveness of the solution
References
4 Conclusion
5 Connection with the Dynkin game
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Notation
Former our diffusion was
dXt = µ(Xt )dt + σ(Xt )dWt ,
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
with discount parameter r .
X0 = x
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Notation
Former our diffusion was
dXt = µ(Xt )dt + σ(Xt )dWt ,
X0 = x
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
with discount parameter r .Let us now introduce an associated
diffusion
d X̂t = µ(X̂t ) + σ(X̂t )σ ′ (X̂t ) dt + σ(X̂t )dWt , X̂0 = x
with discounting θ := r − µ′ .
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Notation
Former our diffusion was
dXt = µ(Xt )dt + σ(Xt )dWt ,
X0 = x
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
with discount parameter r .Let us now introduce an associated
diffusion
d X̂t = µ(X̂t ) + σ(X̂t )σ ′ (X̂t ) dt + σ(X̂t )dWt , X̂0 = x
with discounting θ := r − µ′ . (Its infinitesimal generator
d2
1 2
2 σ (x) dx 2
+ µ(x) + σ(x)σ ′ (x)
d
− r − µ′ (x)
dx
is got by derivating the infinitesimal generator of Xt :
2
d
d
A − r = 12 σ 2 (x) dx
2 + µ(x) dx − r .
So one could, informally, say that X̂t is the derivative diffusion
of Xt .)
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
The Dynkin game
• Two players: sup-player, inf-player.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
The Dynkin game
• Two players: sup-player, inf-player.
Introduction
• Both chooses own stopping time τ and γ.
Solving the
problem
• The game ends at the time τ ∧ γ := min{τ , γ}.
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
The Dynkin game
• Two players: sup-player, inf-player.
Introduction
• Both chooses own stopping time τ and γ.
Solving the
problem
• The game ends at the time τ ∧ γ := min{τ , γ}.
Sensitiveness
of the solution
• If inf-player stops first, she pays q to sup-player.
Conclusion
Connection
with the
Dynkin game
References
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
The Dynkin game
• Two players: sup-player, inf-player.
Introduction
• Both chooses own stopping time τ and γ.
Solving the
problem
• The game ends at the time τ ∧ γ := min{τ , γ}.
Sensitiveness
of the solution
• If inf-player stops first, she pays q to sup-player.
Conclusion
• If sup-player stops first, she receives p from inf-player.
Connection
with the
Dynkin game
References
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
The Dynkin game
• Two players: sup-player, inf-player.
Introduction
• Both chooses own stopping time τ and γ.
Solving the
problem
• The game ends at the time τ ∧ γ := min{τ , γ}.
Sensitiveness
of the solution
• If inf-player stops first, she pays q to sup-player.
Conclusion
• If sup-player stops first, she receives p from inf-player.
Connection
with the
Dynkin game
• As long as the game is in progress, inf-player keeps paying
References
sup-player at the rate π ′ (x) per time unit.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
The Dynkin game
• Two players: sup-player, inf-player.
Introduction
• Both chooses own stopping time τ and γ.
Solving the
problem
• The game ends at the time τ ∧ γ := min{τ , γ}.
Sensitiveness
of the solution
• If inf-player stops first, she pays q to sup-player.
Conclusion
• If sup-player stops first, she receives p from inf-player.
Connection
with the
Dynkin game
• As long as the game is in progress, inf-player keeps paying
References
sup-player at the rate π ′ (x) per time unit.
• The payoff is
Π(x; τ, γ) =
Z
τ ∧γ
0
+ e−
e−
Rt
R τ ∧γ
0
0
θ(X̂s )ds ′
θ(X̂s )ds
π (X̂t )dt
(q1τ >γ + p1τ <γ ) .
On two-sided
singular
control
The Dynkin game
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
• The aim is to solve minmax problem
Conclusion
Connection
with the
Dynkin game
References
u(x) = sup inf E [Π(x; τ, γ)] = inf sup E [Π(x; τ, γ)] .
τ
γ
γ
τ
On two-sided
singular
control
Connection between control
problem and Dynkin game
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Karatzas and Wang (2001), Theorem 3.1 & 3.2
∗
∗
Assume that (U z , D y ) is the optimal for the control problem.
On two-sided
singular
control
Connection between control
problem and Dynkin game
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Karatzas and Wang (2001), Theorem 3.1 & 3.2
∗
∗
Assume that (U z , D y ) is the optimal for the control problem.
Then the presented Dynkin game has a saddle point solution
(τ ∗ , γ ∗ ), where
τ ∗ := inf{t ≥ 0 | X̂t = y ∗ }
γ ∗ := inf{t ≥ 0 | X̂t = z ∗ }
On two-sided
singular
control
Connection between control
problem and Dynkin game
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Karatzas and Wang (2001), Theorem 3.1 & 3.2
∗
∗
Assume that (U z , D y ) is the optimal for the control problem.
Then the presented Dynkin game has a saddle point solution
(τ ∗ , γ ∗ ), where
τ ∗ := inf{t ≥ 0 | X̂t = y ∗ }
γ ∗ := inf{t ≥ 0 | X̂t = z ∗ }
and the value u of the Dynkin game satisfies V ′ (x) = u(x).
On two-sided
singular
control
A notice
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• Control problem has one actor, the controller, who tries to
maximise the payoff.
On two-sided
singular
control
A notice
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• Control problem has one actor, the controller, who tries to
maximise the payoff. Dynkin game has two actors, supand inf-players, one of them tries to maximise and the
other one minimise the payoff.
On two-sided
singular
control
A notice
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• Control problem has one actor, the controller, who tries to
maximise the payoff. Dynkin game has two actors, supand inf-players, one of them tries to maximise and the
other one minimise the payoff. Despite of this, the
solutions are connected.
On two-sided
singular
control
A notice
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
• Control problem has one actor, the controller, who tries to
maximise the payoff. Dynkin game has two actors, supand inf-players, one of them tries to maximise and the
other one minimise the payoff. Despite of this, the
solutions are connected.
• In control problem, at z the controller pays penalty −q
when he uses control U z .
On two-sided
singular
control
A notice
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
• Control problem has one actor, the controller, who tries to
maximise the payoff. Dynkin game has two actors, supand inf-players, one of them tries to maximise and the
other one minimise the payoff. Despite of this, the
solutions are connected.
Connection
with the
Dynkin game
• In control problem, at z the controller pays penalty −q
References
• At y the controller gains p when he uses control D y .
when he uses control U z .
On two-sided
singular
control
A notice
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
• Control problem has one actor, the controller, who tries to
maximise the payoff. Dynkin game has two actors, supand inf-players, one of them tries to maximise and the
other one minimise the payoff. Despite of this, the
solutions are connected.
Connection
with the
Dynkin game
• In control problem, at z the controller pays penalty −q
References
• At y the controller gains p when he uses control D y .
• Controller wants to minimise the usage of control U z and
when he uses control U z .
simultaneously maximise the usage of D y .
On two-sided
singular
control
A notice
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
• Control problem has one actor, the controller, who tries to
maximise the payoff. Dynkin game has two actors, supand inf-players, one of them tries to maximise and the
other one minimise the payoff. Despite of this, the
solutions are connected.
Connection
with the
Dynkin game
• In control problem, at z the controller pays penalty −q
References
• At y the controller gains p when he uses control D y .
• Controller wants to minimise the usage of control U z and
when he uses control U z .
simultaneously maximise the usage of D y . Sounds
somewhat like Dynkin game.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
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Systems. Wiley, New York.
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Brownian motion. Mathematics of Operations Research,
8(3):439–453.
Jacka, S. (2002). Avoiding the origin: a finite-fuel stochastic control
problem. The Annals of Applied Probability, 12:1378–1389.
On two-sided
singular
control
Pekka
Matomäki,
Turku School
of Economics
Introduction
Solving the
problem
Sensitiveness
of the solution
Conclusion
Connection
with the
Dynkin game
References
Karatzas, I. and Wang, H. (2001). Connections between
bounded-variation control and dynkin games. In Menaldi, J. L.,
Sulem, A., and Rofman, E., editors, Optimal Control and Partial
Differential Equations; Volume in Honor of Professor Alain
Bensoussan’s 60th Birthday, pages 353–362. IOS Press,
Amsterdam.
Lempa, J. (2010). A note on optimal stopping of diffusions with a
two-sided optimal rule. Operations Research Letters, 38:11–16.
Matomäki, P. (2012). On solvability of a two-sided singular control
problem. Preprint.
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diffusion processes with fixed and proportional costs. SIAM
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Salminen, P. (1985). Optimal stopping of one-dimensional diffusions.
Mathematische Nachrichten, 124:85–101.
Shreve, S., Lehoczky, J., and Gaver, D. (1984). Optimal
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