Solving Singular Control from Optimal Switching
Xin Guo
∗
Pascal Tomecek
†
May 13, 2008
Abstract
This report summarizes some of our recent work (Guo and Tomecek (2008b,a)) on
a new theoretical connection between singular control of finite variation and optimal
switching problems. This correspondence not only provides a novel method for analyz-
ing multi-dimensional singular control problems, but also builds links among singular
controls, Dynkin games, and sequential optimal stopping problems.
1 Introduction
In our recent work (Guo and Tomecek (2008b)), we established a generic theoretical con-
nection between singular control and optimal switching problems: we defined a consistency
property for collections of switching controls, and proved that there is an exact correspon-
dence between the set of finite variation c`agl`ad processes and the set of consistent collections
of switching controls.
This correspondence allows one to analyze multi-dimensional control problem under a
general setting for the regularity properties and the smooth fit principle directly: one can
obtain an integral representation for the value function of a general class of singular control
problem in terms of the values of corresponding optimal switching problems.
As a byproduct, we showed that the value of a Dynkin game can be represented as the
difference between the values of two related switching problems, thereby linking the general
reversible investment problem, the Dynkin game, and the optimal switching problem.
Continuing our analysis on singular control problems with possible non-smooth payoff
functions, we (Guo and Tomecek (2008a)) analyzed a class of singular control problems for
which value functions are not necessarily smooth. Necessary and sufficient conditions for
the well-known smooth fit principle, along with the regularity of the value functions, are
given. Explicit solutions for the optimal policy and for the value functions are provided. In
∗
Department of Industrial Engineering and Operations Research, UC Berkeley, CA 94720-1777, Phone
(510) 642 3615, Email (xinguo@ieor.berkeley.edu). Generous supp ort from OpenLink Fund at the Cole-
man Fung Risk Management Research Center at UC Berkeley is gratefully acknowledged.
†
J.P. Morgan, New York, email: pascal.i.tomecek@jpmorgan.com
1
particular, when payoff functions satisfy the usual Inada conditions, the boundaries between
action and no-action regions are smooth and strictly monotonic as postulated and exploited in
the existing literature (Dixit and Pindyck (1994); Davis, Dempster, Sethi, and Vermes (1987);
Kobila (1993); Abel and Eberly (1997); Øksendal (2000); Scheinkman and Zariphopoulou
(2001); Merhi and Zervos (2007); Alvarez (2006)). Illustrative examples for both smooth and
non-smooth cases are discussed to highlight the pitfall of solving singular control problems
with a priori smoothness assumptions.
Previous work Singular control problems have been studied extensively in both the math-
ematics and economics, starting from the well-known monotone fuel follower problem, for
which explicit solutions can be found in Bather and Chernoff (1967a,b); Beneˇs et al. (1980);
Karatzas (1983) and Harrison and Taksar (1983). In mathematical economics, a typical
(ir)reversible investment problem can be formulated as a singular control problem in which
a company, by adjusting its production capacity through expansion and contraction ac-
cording to market fluctuations, wishes to maximize its overall expected net profit over an
infinite horizon. This problem has been investigated by numerous authors (See for instance
Davis et al. (1987); Kobila (1993); Abel and Eberly (1997); Baldursson and Karatzas (1997);
Øksendal (2000); Scheinkman and Zariphopoulou (2001); Wang (2003); Chiarolla and Hauss-
mann (2005); Bank (2005); Guo and Pham (2005), and Merhi and Zervos (2007)). For a
standard reference on irreversible investment, see Dixit and Pindyck (1994).
Our approach of connecting singular control problems and related optimal stopping prob-
lems dates back to the seminal paper of Bather and Chernoff (1967a), and has since been
developed and applied to monotone singular control problems by Karatzas (1983, 1985);
Karatzas and Shreve (1984, 1985, 1986); El Karoui and Karatzas (1988, 1989, 1991), and
Baldursson and Karatzas (1997)
1
. Indeed, our integral representation theorem for the
reversible investment problem is in part inspired by the elegant integration arguments of
Baldursson and Karatzas (1997) for irreversible investment. Another closely related body
of work is Boetius and Kohlmann (1998); Boetius (2001, 2003, 2005). However, the connec-
tions between the singular control problem, the entry-exit problem, and Dynkin’s game in
their works are established within the framework of forward backward stochastic differential
equations and require a finite time horizon with the restrictive assumption that the control
has only an additive affect on the diffusion.
Our contribution Compared to all previous works and approaches, the correspondence
between singular controls and switching controls in our paper does not depend on the specific
form of the control problem. Thus, our methodology may be applied to cases for which the
underlying randomness is not necessarily captured by a diffusion and the payoff function is
not necessarily smooth.
1
Recently, Hamad`ene and Hassani (2006) observed that both the Dynkin game and the two regime optimal
switching problem lead to BSDE’s with two reflecting barriers.
2
2 Correspondence between Singular Controls and Switch-
ing Controls
The correspondence established in Guo and Tomecek (2008b) is analogous to the well-known
correspondence between a non-decreasing, F-adapted, c`agl`ad singular control (ξ
t
)
t≥0
and a
collection of stopping times (τ
ξ
(z))
z∈R
, given by
τ
ξ
(z) = inf{t ≥ 0 : ξ
t
> z}, and ξ
t
= sup{z ∈ R : τ
ξ
(z) < t}.
2.1 Definitions
Let (Ω, F, P ) be a complete probability space and F = {F
t
; 0 ≤ t < ∞} a filtration satisfying
the usual hypotheses. Let I ⊂ R be an open (possibly unbounded) interval, and
¯
I be its
closure.
Definition 2.1. Given y ∈
¯
I, an admissible singular control is a pair (ξ
+
t
, ξ
−
t
)
t≥0
of F-
adapted, non-decreasing c`agl`ad processes such that ξ
+
(0) = ξ
−
(0) = 0, Y
t
:= y + ξ
+
t
− ξ
−
t
∈
¯
I, ∀t ∈ [0, ∞), and dξ
+
, dξ
−
are supported on disjoint subsets.
We denote here A
y
to be the set of admissible strategies corresponding to an initial
capacity level of y.
Since dξ
+
, dξ
−
are supported on disjoint subsets, ξ
+
and ξ
−
are the positive and negative
variation of Y , respectively. By the uniqueness of the variation decomposition, there is a
one-to-one correspondence between strategies (ξ
+
, ξ
−
) ∈ A
y
and F-adapted c`agl`ad finite
variation processes Y with Y
0
= y and Y
t
∈
¯
I for all t.
Throughout the paper, (Y
t
)
t≥0
is a finite variation control process with Y
0
= y.
Definition 2.2. A switching control α = (τ
n
, κ
n
)
n≥0
consists of an increasing sequence
of stopping times (τ
n
)
n≥0
and a sequence of new regime values (κ
n
)
n≥0
that are assumed
immediately after each stopping time.
When there are only two distinct regimes, an optimal switching problem is often referred
to as the starting and stopping problem (Brekke and Øksendal (1994); Hamad`ene and Jean-
blanc (2004), etc.) or the entry and exit problem (Boetius (2003); Duckworth and Zervos
(2000), etc.). Following convention, we label the two regimes 0 and 1.
Definition 2.3. A switching control α = ( τ
n
, κ
n
)
n≥0
is admissible if the following hold
almost surely: τ
0
= 0, τ
n+1
> τ
n
for n ≥ 1, τ
n
→ ∞, and for all n ≥ 0, κ
n
∈ {0, 1} is F
τ
n
measurable, with κ
n
= κ
0
for even n and κ
n
= 1 − κ
0
for odd n.
Alternatively, an admissible switching control has a more mathematically convenient
representation given by its regime indicator function.
3
Proposition 2.4. There is a one-to-one correspondence between admissible switching con-
trols and the regime indicator function I
t
(ω), which is an F-adapted c`agl`ad process of finite
variation, so that I
t
(ω) : Ω × [0, ∞) → {0, 1}, with
I
t
:=
∞
X
n=0
κ
n
1
{τ
n
<t≤τ
n+1
}
, I
0
= κ
0
. (1)
Definition 2.5. Let y ∈
¯
I be given, and for each z ∈ I, let α(z) = (τ
n
(z), κ
n
(z))
n≥0
be a
switching control. The collection (α(z))
z∈I
is consistent if
α(z) is admissible for Lebesgue-almost every z ∈ I, (2)
I
0
(z) := κ
0
(z) = 1
{z≤y}
, for Lebesgue-almost every z ∈ I, (3)
and for all t < ∞,
Z
I
(I
+
t
(z) + I
−
t
(z))dz < ∞, almost surely, and (4)
I
t
(z) is decreasing in z for P ⊗ dz-almost every (ω, z). (5)
Here I
t
(z), I
+
t
(z) and I
−
t
(z) are I
t
= κ
0
+ I
+
t
− I
−
t
, and I
+
t
(I
−
t
) is the positive (negative)
variation of the corresponding regime indicator function such that
I
+
t
:=
∞
X
n>0,κ
n
=1
1
{τ
n
<t}
, I
+
0
= 0 and I
−
t
:=
∞
X
n>0,κ
n
=0
1
{τ
n
<t}
, I
−
0
= 0.
For I
t
(z) to be decreasing in z for P ⊗dz-almost every (ω, z), it means there exists a set
E ⊂ Ω ×
¯
I such that P ⊗ dz(E) = 0 and if (ω, z
0
), (ω, z
1
) ∈ (Ω ×
¯
I)\E with z
0
≤ z
1
, then
I
t
(ω, z
0
) ≥ I
t
(ω, z
1
).
2.2 Bijection
The bijection between the singular control and the switching control was established based
on a relatively old result in analysis (Evans and Gariepy, 1992, Theorem 5.5.1).
Proposition 2.6 (From Singular Controls to Switching Controls). Given (ξ
+
, ξ
−
) ∈ A
y
, de-
fine a switching control α(z) = (τ
n
(z), κ
n
(z))
n≥0
for each z ∈ I through the regime indicator
function I
t
(z) := lim
s↑t
1
{Y
s
>z}
. Then, the resulting collection (α(z))
z∈I
of switching controls
is consistent.
Proposition 2.7 (From Switching Controls to Singular Controls). Given y ∈
¯
I and a
consistent collection of switching controls (α(z))
z∈I
, define two processes ξ
+
and ξ
−
by setting
ξ
+
0
= 0, ξ
−
0
= 0, and for t > 0: ξ
+
t
:=
R
I
I
+
t
(z)dz, ξ
−
t
:=
R
I
I
−
t
(z)dz. Then
1. The pair (ξ
+
, ξ
−
) ∈ A
y
is an admissible singular control,
4
2. Up to indistinguishability,
Y
t
= y +
Z
∞
y
I
t
(z)1
{z∈I}
dz +
Z
y
−∞
(I
t
(z) − 1)1
{z∈I}
dz, and
3. For all t, we almost surely have
Y
t
= ess sup{z ∈ I : I
t
(z) = 1} = ess inf{z ∈ I : I
t
(z) = 0},
where ess sup ∅ := inf I and ess inf ∅ := sup I.
Proposition 2.8 (One-to-One Mapping). The mapping from consistent collections of switch-
ing controls to singular controls defined by Proposition 2.7 is one-to-one.
Theorem 2.9 (Bijection). The mappings in Propositions 2.6 and 2.7 define a bijection
between admissible singular controls (ξ
+
, ξ
−
) ∈ A
y
and consistent collections of switching
controls (up to equivalence).
Given this correspondence, we shall use the following terminology in the sequel. Given a
singular control (ξ
+
, ξ
−
) ∈ A
y
, the corresponding collection of switching controls (α(z))
z∈I
refers to the one defined in Proposition 2.6; given a consistent collection of switching controls,
the corresponding singular control refers to that in Proposition 2.7.
2.3 Change of Variable Formula
With the bijection established in Theorem 2.9, we established a change of variable formula
for integration with respect to the variation of a singular control.
Proposition 2.10. Let (ξ
+
, ξ
−
) ∈ A
y
be an admissible singular control and (α(z))
z∈I
be
the corresponding collection of switching controls. For every c`adl`ag process g : Ω ×[0, ∞] →
[0, ∞) with g(∞) ≡ 0,
Z
[0,∞)
g(t)dξ
+
t
=
Z
I
X
n>0
κ
n
=1
g(τ
n
(z))dz, a.s.,
and
Z
[0,∞)
g(t)dξ
−
t
=
Z
I
X
n>0
κ
n
=0
g(τ
n
(z))dz, a.s.
In particular, when Y is non-decreasing (i.e. ξ
−
≡ 0),
¯
I = [0, ∞) and y ≥ 0, we have
τ
n
(z) ≡ 0 for all n > 1, and for n = 1 when z ≤ y. In this case, our change of variable
formula reduces to the one for monotone controls in Baldursson and Karatzas (1997), after
adjusting for notational differences,
Z
[0,∞)
g(t)dξ
+
t
=
Z
∞
y
g(τ
1
(z))dz.
5
