The problem
Searching optimal stopping in Phase 1
Results
References
Optimal timing in a combined investment and exit
problem
Pekka Matomäki
Turku School of Economics
28.1.2011
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Table of contents
1
The problem
Introduction
How to approach?
2
Searching optimal stopping in Phase 1
The necessary condition
Existence and optimality
3
Results
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Outline
1
The problem
Introduction
How to approach?
2
Searching optimal stopping in Phase 1
The necessary condition
Existence and optimality
3
Results
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Introduction
Consider a company operating in the presence of uncertainty.
At any time it can decide to
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Introduction
Consider a company operating in the presence of uncertainty.
At any time it can decide to
invest irreversibly into an improved technology resulting in
higher profits.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Introduction
Consider a company operating in the presence of uncertainty.
At any time it can decide to
invest irreversibly into an improved technology resulting in
higher profits.
or exit the market.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Introduction
Consider a company operating in the presence of uncertainty.
At any time it can decide to
invest irreversibly into an improved technology resulting in
higher profits.
or exit the market.
Which option the company should use and when?
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Introduction
Consider a company operating in the presence of uncertainty.
At any time it can decide to
invest irreversibly into an improved technology resulting in
higher profits.
or exit the market.
Which option the company should use and when?
If the company decides to invest, when is the right time to
exit afterwards?
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Introduction
Consider a company operating in the presence of uncertainty.
At any time it can decide to
invest irreversibly into an improved technology resulting in
higher profits.
or exit the market.
Which option the company should use and when?
If the company decides to invest, when is the right time to
exit afterwards?
First study by Kwon (2010)
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Introduction
Consider a company operating in the presence of uncertainty.
At any time it can decide to
invest irreversibly into an improved technology resulting in
higher profits.
or exit the market.
Which option the company should use and when?
If the company decides to invest, when is the right time to
exit afterwards?
First study by Kwon (2010)
Speech is Matomäki (2010)
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
The setup
We assume that the profit flow Xt of the company is a general
Itô diffusion:
dXt = µ1 (Xt )dt + σ1 (Xt )dWt ,
X0 = x,
where µ1 (x) is a drift term, σ1 (x) is a volatility term and Wt
is standard Brownian motion.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
The setup
We assume that the profit flow Xt of the company is a general
Itô diffusion:
dXt = µ1 (Xt )dt + σ1 (Xt )dWt ,
X0 = x,
where µ1 (x) is a drift term, σ1 (x) is a volatility term and Wt
is standard Brownian motion.
The company has two options to use:
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
The setup
We assume that the profit flow Xt of the company is a general
Itô diffusion:
dXt = µ1 (Xt )dt + σ1 (Xt )dWt ,
X0 = x,
where µ1 (x) is a drift term, σ1 (x) is a volatility term and Wt
is standard Brownian motion.
The company has two options to use:
1
Invest irreversibly lump sum to ”shift” to a new higher profit
flow Yt :
dYt = µ2 (Yt )dt + σ2 (Yt )dWt
Y0 = y ,
with µ2 > µ1 .
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
The setup
We assume that the profit flow Xt of the company is a general
Itô diffusion:
dXt = µ1 (Xt )dt + σ1 (Xt )dWt ,
X0 = x,
where µ1 (x) is a drift term, σ1 (x) is a volatility term and Wt
is standard Brownian motion.
The company has two options to use:
1
Invest irreversibly lump sum to ”shift” to a new higher profit
flow Yt :
dYt = µ2 (Yt )dt + σ2 (Yt )dWt
2
Y0 = y ,
with µ2 > µ1 .
Exit the market.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
The setup
We assume that the profit flow Xt of the company is a general
Itô diffusion:
dXt = µ1 (Xt )dt + σ1 (Xt )dWt ,
X0 = x,
where µ1 (x) is a drift term, σ1 (x) is a volatility term and Wt
is standard Brownian motion.
The company has two options to use:
1
Invest irreversibly lump sum to ”shift” to a new higher profit
flow Yt :
dYt = µ2 (Yt )dt + σ2 (Yt )dWt
2
Y0 = y ,
with µ2 > µ1 .
Exit the market.
If the company invests, it still has the option to exit.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Problem
When is the optimal time to invest or exit?
cost Yt
sunk
Xt
Exit
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Problem
When is the optimal time to invest or exit?
If the investment is made, when is the optimal time to exit
afterwards?
cost Yt
sunk
Exit
Xt
Exit
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Why?
Big investments most likely affect the profit flow of a
company.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Why?
Big investments most likely affect the profit flow of a
company.
∗ Buying new equipments in the factory.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Why?
Big investments most likely affect the profit flow of a
company.
∗ Buying new equipments in the factory.
∗ Oil company opening a new oil field (or shutting an old one).
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Why?
Big investments most likely affect the profit flow of a
company.
∗ Buying new equipments in the factory.
∗ Oil company opening a new oil field (or shutting an old one).
The studies of investment problems overlook the exiting
option.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Why?
Big investments most likely affect the profit flow of a
company.
∗ Buying new equipments in the factory.
∗ Oil company opening a new oil field (or shutting an old one).
The studies of investment problems overlook the exiting
option.
∗ Often conceivable option.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Why?
Big investments most likely affect the profit flow of a
company.
∗ Buying new equipments in the factory.
∗ Oil company opening a new oil field (or shutting an old one).
The studies of investment problems overlook the exiting
option.
∗ Often conceivable option.
The studies of exit problems overlook investment option.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Why?
Big investments most likely affect the profit flow of a
company.
∗ Buying new equipments in the factory.
∗ Oil company opening a new oil field (or shutting an old one).
The studies of investment problems overlook the exiting
option.
∗ Often conceivable option.
The studies of exit problems overlook investment option.
∗ Possible mechanism to prolong the operation of a company.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
How to approach?
Following the principle of dynamic programming, the proof is split
into two cases:
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
How to approach?
Following the principle of dynamic programming, the proof is split
into two cases:
If we have invested, when is the right time to exit (phase 2).
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
How to approach?
Following the principle of dynamic programming, the proof is split
into two cases:
If we have invested, when is the right time to exit (phase 2).
When should we invest or exit in the first place (phase 1).
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
How to approach?
Following the principle of dynamic programming, the proof is split
into two cases:
If we have invested, when is the right time to exit (phase 2).
When should we invest or exit in the first place (phase 1).
In both phases we face an optimal stopping problem.
Phase 2 is a pure exit problem, and the solution is known.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
How to approach?
Following the principle of dynamic programming, the proof is split
into two cases:
If we have invested, when is the right time to exit (phase 2).
When should we invest or exit in the first place (phase 1).
In both phases we face an optimal stopping problem.
Phase 2 is a pure exit problem, and the solution is known.
Phase 1 is the problematic case.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Phase 2
Solve the optimal stopping problem
Z τ2
−rs
V2 (y ) = sup Ey
e π2 (Ys )ds ,
τ2
0
which arises after the investment.
π2 is a continuous and increasing function (revenue function).
r > 0 is the discount rate.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Phase 2
Solve the optimal stopping problem
Z τ2
−rs
V2 (y ) = sup Ey
e π2 (Ys )ds ,
τ2
0
which arises after the investment.
π2 is a continuous and increasing function (revenue function).
r > 0 is the discount rate.
Solution is known (Alvarez (2001)).
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Phase 1
Solve the optimal stopping problem
Z τ1
−rs
−r τ1
V1 (x) = sup Ex
e π1 (Xs )ds + e
max{V2 Xτ1 − k, 0} .
τ1
0
Here
k is the cost of the investment.
π1 is a continuous and increasing function (revenue function).
V2 is the solution of the second phase.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Phase 1
Solve the optimal stopping problem
Z τ1
−rs
−r τ1
V1 (x) = sup Ex
e π1 (Xs )ds + e
max{V2 Xτ1 − k, 0} .
τ1
0
Here
k is the cost of the investment.
π1 is a continuous and increasing function (revenue function).
V2 is the solution of the second phase.
This can be written as
V1 (x) = sup Ex e −r τ1 g (Xτ1 ) ,
τ1
where g is a gain function.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Assumptions
Our aim is to find the solution under least restricting assumptions.
We use the following assumptions
Assume that there exists x0 such that (A − r )g (x) < 0 for all
x > x0 .
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Introduction
How to approach?
Assumptions
Our aim is to find the solution under least restricting assumptions.
We use the following assumptions
Assume that there exists x0 such that (A − r )g (x) < 0 for all
x > x0 .
Assume that
Z ∞
Z
−rs
lim Ey
e π2 (Ys )ds −k > lim Ex
y →∞
0
Pekka Matomäki Turku School of Economics
x→∞
∞
e
−rs
0
Combined investment and exit problem
π1 (Xs )ds
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Outline
1
The problem
Introduction
How to approach?
2
Searching optimal stopping in Phase 1
The necessary condition
Existence and optimality
3
Results
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The idea of the proof
We proceed following way:
• Restrict to study two-sided threshold rules
τ(a,b) = inf{t ≥ 0 | Xt ∈
/ (a, b)}.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The idea of the proof
We proceed following way:
• Restrict to study two-sided threshold rules
τ(a,b) = inf{t ≥ 0 | Xt ∈
/ (a, b)}.
1 Prove that there is a unique optimal two-sided threshold rule
τ(a∗ ,b∗ ) .
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The idea of the proof
We proceed following way:
• Restrict to study two-sided threshold rules
τ(a,b) = inf{t ≥ 0 | Xt ∈
/ (a, b)}.
1 Prove that there is a unique optimal two-sided threshold rule
τ(a∗ ,b∗ ) .
2 Prove that there are no stopping strategies that give higher
payoff than this two-sided threshold rule.
These points completely solve the optimal stopping problem in
phase 1.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The idea of the proof
We proceed following way:
• Restrict to study two-sided threshold rules
τ(a,b) = inf{t ≥ 0 | Xt ∈
/ (a, b)}.
1 Prove that there is a unique optimal two-sided threshold rule
τ(a∗ ,b∗ ) .
2 Prove that there are no stopping strategies that give higher
payoff than this two-sided threshold rule.
These points completely solve the optimal stopping problem in
phase 1.
Item 2 is proved using the fact that the solution is the
minimal r -excessive majorant (quite easy).
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The idea of the proof
We proceed following way:
• Restrict to study two-sided threshold rules
τ(a,b) = inf{t ≥ 0 | Xt ∈
/ (a, b)}.
1 Prove that there is a unique optimal two-sided threshold rule
τ(a∗ ,b∗ ) .
2 Prove that there are no stopping strategies that give higher
payoff than this two-sided threshold rule.
These points completely solve the optimal stopping problem in
phase 1.
Item 2 is proved using the fact that the solution is the
minimal r -excessive majorant (quite easy).
For item 1 (not so easy) we
first derive a necessary condition for optimality;
second develop a fixed point method to show that there is a
unique solution to the necessary condition.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The road to necessary condition
For an arbitrary pair (a, b), for the Phase 1 we can write
n
o
V(a,b) (x) = Ex e −r τ(a,b) g (Xτ(a,b) )
ϕx ψ b − ϕb ψ x
ϕa ψ x − ϕx ψ a
g (a) +
g (b)
ϕa ψ b − ϕb ψ a
ϕa ψ b − ϕb ψ a
=: h1 (a, b)ψ(x) + h2 (a, b)ϕ(x)
=
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The road to necessary condition
For an arbitrary pair (a, b), for the Phase 1 we can write
n
o
V(a,b) (x) = Ex e −r τ(a,b) g (Xτ(a,b) )
ϕx ψ b − ϕb ψ x
ϕa ψ x − ϕx ψ a
g (a) +
g (b)
ϕa ψ b − ϕb ψ a
ϕa ψ b − ϕb ψ a
=: h1 (a, b)ψ(x) + h2 (a, b)ϕ(x)
=
We attain necessary optimal condition when requiring that
∂h1 /∂a = ∂h1 /∂b = 0 = ∂h2 /∂a = ∂h2 /∂b.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The necessary condition
The necessary condition (cf. Salminen (1985), Theorem 4.7)
The boundary points a∗ and b∗ satisfy

 I (b∗ ) − I (a∗ ) = 0
 J(b∗ ) − J(a∗ ) = 0,

 I (x) = ψ′ (x) g (x) −
S ′ (x)
where
 J(x) = g ′′ (x) ϕ(x) −
S (x)
Pekka Matomäki Turku School of Economics
g ′ (x)
S ′ (x) ψ(x)
ϕ′ (x)
S ′ (x) g (x),
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The necessary condition
The necessary condition (cf. Salminen (1985), Theorem 4.7)
The boundary points a∗ and b∗ satisfy

 I (b∗ ) − I (a∗ ) = 0
 J(b∗ ) − J(a∗ ) = 0,

 I (x) = ψ′ (x) g (x) −
S ′ (x)
where
 J(x) = g ′′ (x) ϕ(x) −
S (x)
g ′ (x)
S ′ (x) ψ(x)
ϕ′ (x)
S ′ (x) g (x),
Now our task is to find a unique pair (a∗ , b∗ ) that satisfies the
previous pair of equations.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
The necessary condition
The necessary condition (cf. Salminen (1985), Theorem 4.7)
The boundary points a∗ and b∗ satisfy

 I (b∗ ) − I (a∗ ) = 0
 J(b∗ ) − J(a∗ ) = 0,

 I (x) = ψ′ (x) g (x) −
S ′ (x)
where
 J(x) = g ′′ (x) ϕ(x) −
S (x)
g ′ (x)
S ′ (x) ψ(x)
ϕ′ (x)
S ′ (x) g (x),
Now our task is to find a unique pair (a∗ , b∗ ) that satisfies the
previous pair of equations. Next we demonstrate graphically how
this can be done.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Existence of the points a∗ , b ∗
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Existence of the points a∗ , b ∗
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Existence of the points a∗ , b ∗
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Existence of the points a∗ , b ∗
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Existence of the points a∗ , b ∗
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Existence of the points a∗ , b ∗
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Existence of the points a∗ , b ∗
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Existence of the points a∗ , b ∗
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Optimality of V(a∗ ,b∗ )
To prove that V(a∗ ,b∗ ) (x) is the optimal solution, we need to show
that
(i) V(a∗ ,b∗ ) (x) ≥ g (x) for all x ≥ 0
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Optimality of V(a∗ ,b∗ )
To prove that V(a∗ ,b∗ ) (x) is the optimal solution, we need to show
that
(i) V(a∗ ,b∗ ) (x) ≥ g (x) for all x ≥ 0
(ii) V(a∗ ,b∗ ) (x) is continuous for all x ≥ 0
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
The necessary condition
Existence and optimality
Optimality of V(a∗ ,b∗ )
To prove that V(a∗ ,b∗ ) (x) is the optimal solution, we need to show
that
(i) V(a∗ ,b∗ ) (x) ≥ g (x) for all x ≥ 0
(ii) V(a∗ ,b∗ ) (x) is continuous for all x ≥ 0
(iii) V(a∗ ,b∗ ) (x) satisfies
Ex {e −r τ V(a∗ ,b∗ ) (Xτ )} ≤ V(a∗ ,b∗ ) (x)
(⇐⇒ (A − r )V(a∗ ,b∗ ) (x) ≤ 0)
for all τ = inf{t ≥ 0 | Xt ∈
/ [c, d], 0 < c < d < ∞}.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Outline
1
The problem
Introduction
How to approach?
2
Searching optimal stopping in Phase 1
The necessary condition
Existence and optimality
3
Results
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Illustrating results
Solution is a two-sided threshold rule
Pekka Matomäki Turku School of Economics
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
Example of a sample path
b*
5
4
3
Xt
2
1
a*
5
10
15
20
25
30
35
150
Yt
100
50
20
40
60
Pekka Matomäki Turku School of Economics
80
100
120
140
Combined investment and exit problem
The problem
Searching optimal stopping in Phase 1
Results
References
References
Alvarez, L. H. R. (2001). Reward functionals, salvage values, and
optimal stopping. Mathematical Methods of Operations
Research, 54:315–337.
Kwon, H. D. (2010). Invest or exit: Optimal decisions in the face
of a declining profit stream. Operations Research, 58:638–649.
Matomäki, P. (2010). Optimal stopping in a combined exit and
invest situation. submitted.
Salminen, P. (1985). Optimal stopping of one-dimensional
diffusions. Mathematische Nachrichten, 124:85–101.
Pekka Matomäki Turku School of Economics
Combined investment and exit problem