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arXiv:2207.01126v1 [math.PR] 3 Jul 2022
ON THE BAILOUT DIVIDEND PROBLEM WITH PERIODIC DIVIDEND PAYMENTS
FOR SPECTRALLY NEGATIVE MARKOV ADDITIVE PROCESSES
DANTE MATA, HAROLD A. MORENO-FRANCO, KEI NOBA, AND JOS ´
E-LUIS P ´
EREZ
ABS TR ACT. This paper studies the bailout optimal dividend problem with regime switching under the
constraint that dividend payments can be made only at the arrival times of an independent Poisson pro-
cess while capital can be injected continuously in time. We show the optimality of the regime-modulated
Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative
Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a
single spectrally negative L´evy process with a final payoff at an exponential terminal time and characterise
the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global
regime-switching problem into an equivalent local optimization problem with a final payoff up to the first
regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be
proven by using the results from the auxiliary problem and approximations via recursive iterations.
Keywords: regime switching; spectrally one-sided L´evy processes; scale functions; periodic and
singular control strategies.
Mathematics Subject Classification: 60G51, 93E20, 91G80
1. INTRO DUCTI ON
In the bailout model of de Finetti’s dividend problem, the goal is to find a joint optimal dividend
and capital injection strategy in order to maximise the expected net present value (NPV) of dividend
payments minus the capital injections. A spectrally negative L´evy process, namely a L´evy process with
no positive jumps, has been used to model the surplus for an insurance company that has a diffusive
behavior because of the premiums and jumps downwards by claim payments. In the seminal paper [4],
Avram et al. showed that it is optimal to inject capital by reflecting the surplus process from below at
zero and pay dividends from above at a suitable chosen threshold.
As in [4], most of the existing continuous-time models assume that the dividends can be paid at all
times and instantaneously (see, e.g., [4,5,6,15]); but in reality, dividend-payout decisions can only
be made at discrete times, for that reason, the modeling of optimal dividend-payout in discrete random
times has recently drawn much attention; see, e.g., [1,2,3,23,24,25].
With this in mind, in this work, we impose the constraint that dividend payments can only be made
at discrete times given by the arrival times of a Poisson process, independent of the surplus process. In
1
2 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
addition, the classical bailout restriction requires that capital must be injected continuously in time so
that the controlled process remains non-negative uniformly in time.
In this paper, we consider the bailout dividend problem in a more general framework, where the un-
derlying surplus is driven by a spectrally negative Markov additive process. This process can be seen
as a family of spectrally negative L´evy processes switching via an independent Markov chain. The
regime-switching model is often used to capture the changes in market behavior due to macroeconomic
transitions or macroeconomic readjustments, such as technological development, epidemics, and geopo-
litical issues. The continuous-time Markov chain is commonly used to approximate some stochastic
factors that affect the underlying state processes. In addition, a negative jump is introduced each time
there is a change in the current regime. This jump is independent of the family of L´evy processes and
the Markov chain and can be interpreted as the cost for the company to adapt to the new regime. The
regime-switching model turns out to be attractive in financial applications as it provides tractable and
explicit structures, and it has become a vibrant research topic in the past decades. Some recent work
motivated by different financial applications can be found in [7,11,16,19].
Optimal dividend problems in the context with regime-switching have been studied mostly in the
framework of jump diffusion models, see e.g. [5,13,14,15]; however, recently Noba et al. [24] studied
the more general spectrally negative L´evy framework. Similar to the single regime work, these works
have shown that optimal dividend strategies fit in the type of barrier strategies as well. These previous
studies, assume that dividends can be paid continuously in time, so it becomes an open question whether
a barrier dividend policy is optimal in the regime switching case under the constraint that dividend
payments can only be made at the jump times of an independent Poisson process.
This paper aims to provide a positive answer to the optimality of the periodic-classical barrier divi-
dend strategy, namely the periodic dividend payment and classical capital injection but modulated by the
regime states. Our approach on showing the optimality of barrier strategies relies on purely probabilistic
methods and is based on fluctuation identities for spectrally negative L´evy processes reflected at Pois-
sonian times. The motivation behind periodic-classical barrier strategies arises from the works by Noba
et al. [23] and P´erez and Yamazaki [25], where optimality was shown in the single regime context for
spectrally negative and spectrally positive L´evy processes, respectively. However, our analysis differs
from [23] due to the complexity caused by different regimes. The verification of optimality of barrier
strategies is expected to be much more involved than in [23] as the barrier in each regime is coupled
with other regime modulated barriers through the definition of the value function. The HJB variational
inequalities that arise for the global control problem become a system of coupled variational inequalities
based on the regime states. In order to reduce complexity and deal with the switch in regimes, we borrow
the idea of stochastic control to use the dynamic programming principle and localize the problem up to
the period of the first regime switch, see [24] and [28] for similar optimal dividend problems.
Our verification of optimality can be summarised as follows:
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 3
(1) First, we study an auxiliary bailout dividend problem with a terminal payoff until an independent
exponential time driven by a single spectrally negative L´evy process. In this part, we compute the
expected NPV of dividends minus capital injections under a periodic-classical reflection strategy
explicitly in terms of the scale function, and perform the “guess and verify” procedure common
in the literature. It is noteworthy that we present a novel result, where we compute the resolvent
density for the spectrally negative L´evy process with periodic reflection above up to the first
downcrossing time below 0, as well as the resolvent density for the spectrally negative L´evy
process with periodic reflection above and classical reflection below. The candidate optimal
barrier is chosen using the conjecture that the slope of the value function at the barrier becomes
one, then we proceed to verify the optimality of the selected barrier strategy by showing that the
candidate value function solves the proper variational inequalities.
(2) After studying the single spectrally negative L´evy model, we define an iteration operator by
proving the dynamic programming principle similar to [24] and [28]. We can show the existence
of the candidate optimal barriers modulated by the regime states using the results from step (1).
Then we proceed to prove that both the value function and the expected NPV under the regime-
modulated periodic-classical reflection strategy are solutions of a functional equation, and then
we prove via iterative methods that the expected NPV of the candidate barrier strategy agrees
with the value function. This completes the second step of the verification and the optimality of
the barrier type control is successfully retained in the general framework as conjectured.
The rest of this paper is structured as follows. Section 2introduces some mathematical preliminaries
regarding spectrally negative L´evy processes. In Section 3we formulate the bailout dividend problem
with regime switching in the spectrally negative Markov additive model, with periodic dividend decision
times. The main result in this section confirms the optimality of the regime modulated periodic-classical
reflection strategy. Section 4then formulates the auxiliary bailout dividend problem with Poissonian
dividend decision times and with a final payoff at an independent exponential time. Our main result in
this section gives the optimality of the periodic-classical reflection strategy. In Section 5we compute the
expected NPV of the periodic-classical reflection strategy in terms of the scale function; in addition, we
present new results on fluctuation theory for spectrally negative L ´evy processes, namely the computation
of resolvents for the process with periodic reflection above and for the process with periodic reflection
above and classical reflection below. Sections 6and 7give the construction and existence of the candidate
optimal strategy and the rigorous verification of the optimality for the auxiliary problem, respectively.
Finally, in Section 8we define an auxiliary iteration operator and provide the verification of optimality
of the regime modulated barrier strategy via iterative arguments. Throughout the paper, the right hand
derivative of a real function fis denoted by f′
+(x), whenever it exists.
4 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
2. PRELIM INAR IES ON S PE CT RAL LY NEGATIVE L´
EVY PRO CE SSES
Let us consider a spectrally negative L´evy process X= (X(t); t≥0) defined on a probability space
(Ω,F,P)where F:= {Ft:t≥0}denotes the right-continuous filtration generated by X. For x∈R,
we denote by Pxthe law of Xwhen it starts at xand write for convenience Pin place of P0. Accordingly,
we shall write Exand Efor the associated expectation operators.
We denote by ψX: [0,∞)→Rto the Laplace exponent of the process X, i.e.
EeθX(t)=: eψX(θ)t, t, θ ≥0,
given by the L´
evy-Khintchine formula
(2.1) ψX(θ) := γθ +η2
2θ2+Z(−∞,0) eθz −1−θz1{z>−1}Π(dz), θ ≥0.
Here, γ∈R,η≥0, and Πis the L´evy measure of Xdefined on (−∞,0) which satisfies
Z(−∞,0)
(1 ∧z2)Π(dz)<∞.
It is well known that Xhas paths of bounded variation if and only if η= 0 and R(−1,0) |z|Π(dz)is finite.
In this case Xcan be written as
X(t) = ct −S(t), t ≥0,
where
c:= γ−Z(−1,0)
zΠ(dz)
and (S(t); t≥0) is a driftless subordinator. We assume that the process Xdoes not have monotone
paths, and therefore we must have c > 0and we can write
ψX(θ) = cθ +Z(−∞,0) eθz −1Π(dz), θ ≥0.
2.1. Scale functions. For fixed q≥0, let W(q):R→[0,∞)be the scale function of the spectrally
negative L´evy process X. This takes the value zero on the negative half-line, and on the positive half-line
it is a continuous and strictly increasing function defined by its Laplace transform:
Z∞
0
e−θxW(q)(x)dx=1
ψX(θ)−q, θ > Φ(q),(2.2)
where ψXis as in (2.1) and
Φ(q) := sup{λ≥0 : ψX(λ) = q}.
(2.3)
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 5
We also define, for all x∈R,
W(q)(x) := Zx
0
W(q)(y)dy, W(q)(x) := Zx
0Zz
0
W(q)(y)dydz,
Z(q)(x) := 1 + qW(q)(x), Z(q)(x) := Zx
0
Z(q)(z)dz=x+qW(q)(x).
Because W(q)(x) = 0 for −∞ < x < 0, we have
W(q)(x) = 0, W (q)(x) = 0, Z(q)(x) = 1,and Z(q)(x) = x, x ≤0.
Remark 2.1. (1) W(q)is differentiable a.e.. In particular, if Xis of unbounded variation or the L´
evy
measure is atomless, it is known that W(q)is C1(R\{0}); see, e.g., [8, Theorem 3].
(2) As in Lemma 3.1 of [17],
W(q)(0) =
0if Xis of unbounded variation,
1
cif Xis of bounded variation,
From the identity (6) in [21],
W(q+r)(x)−W(q)(x) = rZx
0
W(q+r)(u)W(q)(x−u)du,
Z(q+r)(x)−Z(q)(x) = rZx
0
W(q+r)(u)Z(q)(x−u)du,
x∈R.(2.4)
We also define, for q, r ∈(0,∞)and x∈R,
Z(q)(x, Φ(q+r)) := eΦ(q+r)x1−rZx
0
e−Φ(q+r)zW(q)(z)dz
(2.5)
=rZ∞
0
e−Φ(q+r)zW(q)(z+x)dz > 0,
where the second equality holds due to (2.2).
By differentiating (2.5) with respect to the first argument,
Z(q)′(x, Φ(q+r)) := ∂
∂x Z(q)(x, Φ(q+r)) = Φ(q+r)Z(q)(x, Φ(q+r)) −rW (q)(x), x > 0.(2.6)
Finally, for b≥0and x∈R, we define
W(q,r)
b(x) := W(q)(x) + rZx
b
W(q+r)(x−y)W(q)(y)dy,
Z(q,r)
b(x) := Z(q)(x) + rZx
b
W(q+r)(x−y)Z(q)(y)dy,
Z(q,r)
b(x) := Z(q)(x) + rZx
b
W(q+r)(x−y)Z(q)(y)dy.
(2.7)
6 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
Notice that the identities in (2.7) reduce to
(2.8) W(q,r)
b(x) = W(q)(x), Z(q,r)
b(x) = Z(q)(x),Z(q,r)
b(x) = Z(q)(x),when x∈[0, b].
In addition, for x∈Rwe have
(2.9) W(q,r)
0(x) = W(q+r)(x), Z(q,r)
0(x) = Z(q+r)(x).
For a comprehensive study on the scale functions and their application, see [17,18].
Finally, let us introduce the following notation that will be used throughout this paper. For x, b ∈[0,∞),
and a measurable function h:R7→ R, we define
ρ(q)
b(x;h) := Zb
0
W(q)(x−y)h(y)dy,(2.10)
ρ(q,r)
b(x;h) := ρ(q)
b(x;h) + rZx
b
W(q+r)(x−y)ρ(q)
b(y;h)dy, ,(2.11)
Ξ(q,r)(b;h) := Z∞
0
h(y+b)e−Φ(q+r)ydy+rZ∞
b
e−Φ(q+r)(y−b)ρ(q)
b(y;h)dy.(2.12)
3. THE BAI LO UT OPTI MA L DIV IDEND PROB LE M WIT H PE RI ODIC DIVIDEN D PAYME NT S AND
REGI ME S WI TCHING
We formulate the dividend problem when the surplus is driven by a Markov additive process (MAP)
with negative jumps, and present our main result that states the optimality of barrier strategies.
3.1. Spectrally negative Markov additive processes. Let us consider a bivariate process (X, H ) =
{(X(t), H(t)); t≥0}, where the component His a continuous-time Markov chain with finite state
space E={1,··· , N}and generator matrix Q= (λij )i,j ∈E. When the chain His in state i,Xbehaves
as a spectrally negative L´evy process Xi. In addition, when then process Hchanges to a state j6=i,
the process Xjumps according to a non-positive random variable Jij with i, j ∈E. The components
(Xi)i∈E, H, and (Jij)i,j∈Eare assumed to be independent and are defined on some filtered probability
space (Ω,F,F,P), where F:= (Ft)t≥0is the right-continuous complete filtration generated by the
processes (X, H)and the family of random variables (Jij )i,j ∈E. We denote by P(x,i)the law of the
process conditioned on the event {X(0) = x, H (0) = i}; likewise we denote by E(x,i)the associated
expectation operator.
Throughout this work we assume that for each i∈E, the Laplace exponent of the L´evy process Xi,
ψXi: [0,∞)→R, is given by the L´
evy -Khintchine formula
ψXi(θ) = γiθ+η2
i
2θ2+Z(−∞,0) eθz −1−θz1{z>−1}Π(i, dz), θ ≥0,
where γi∈R, ηi≥0, and Π(i, ·)is the L´evy measure of Xion (−∞,0) that satisfies R(−∞,0) (1 ∧
x2)Π(i, dx)<∞. In addition, as in Section 2, if Xihas paths of bounded variation its Laplace exponent
is given by ψXi(θ) = ciθ+R(−∞,0)(eθz −1)Π(i, dz), θ ≥0, where ci:= γi−R(−1,0) zΠ(i, dz).
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 7
Throughout this work we denote by Pi
xthe law of the L´evy process Xicontidioned on the event
{Xi(0) = x}and by Ei
xits associated expectation operator.
3.2. Bailout optimal dividend problem with Poissonian decision times and regime switching. A
strategy is a pair of non-decreasing, right-continuous, and adapted processes π:= (Lπ
r(t), Rπ
r(t)) con-
sisting of the cumulative amount of dividends Lπ
rand those of capital injection Rπ
r.
Throughout this paper we will consider that the dividend payments can only be made at the arrival
times Tr:= (T(i); i≥1) of a Poisson process Nr= (Nr(t); t≥0) with intensity r > 0, which is
defined on (Ω,F,F,P), where F={Ht}t≥0is the right-continuous complete filtration generated by
(X, N r). We assume that Xand Nrare independent on the previous probability space. In other words,
we consider that Lπ
radmits the form
Lπ
r(t) = Z[0,t]
νπ(s)dNr(s), t ≥0,(3.1)
for some c`agl`ad process νπadapted to the filtration generated by X,Hand Nr.
The process Rπ
ris non-decreasing, right-continuous, and F-adapted, with Rπ
r(0−) = 0. Contrary to
the dividend payments, capital injection can be made continuously in time. In addition, the process Rπ
r
must satisfy
(3.2) E(x,i)Z[0,∞)
e−I(t)dRπ
r(t)<∞, x ≥0, i ∈E,
where I(t) := Rt
0q(H(s))ds, and q:E→R+represents the Markov-modulated rate of discounting.
The corresponding controlled process associated to the strategy πis given by Uπ
r(0−) = X(0) and
Uπ
r(t) := X(t)−Lπ
r(t) + Rπ
r(t), t ≥0.
We denote by Athe set of strategies satisfying the constraints mentioned above and that Uπ
r(t)≥0for
all t≥0a.s.. We call a strategy πadmissible if π∈ A.
We consider that β > 1is the constant cost per unit of capital injected in all regimes. Our aim is to
maximize the expected net present value (NPV)
(3.3) Vπ(x, i) := E(x,i)Z[0,∞)
e−I(t)dLπ
r(t)−βZ[0,∞)
e−I(t)dRπ
r(t), x ≥0, i ∈E,
over all π∈ A. Hence, our goal is to find the value function of the problem
(3.4) V(x, i) := sup
π∈A
Vπ(x, i), x ≥0,
and obtain an optimal strategy π∗∈ A whose expected NPV, Vπ∗, agrees with Vif such a strategy exists.
Throughout this paper we assume the following.
Assumption 3.1. We assume that E[Xi(1)] = ψ′
Xi(0+) >−∞ for i∈E.
Assumption 3.2. For all i, j ∈Ewith i6=j, we assume that maxi,j∈EE[|Jij |]<∞.
8 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
We claim that the dynamic programming principle for the value function of the control problem holds
valid, which will play a key role in the verification via iteration operators later on (see Section 8). We
defer its proof to the Appendix (see Subsection C.1).
Proposition 3.1. For x∈Rand i∈E, we have
(3.5) V(x, i) = sup
π∈A
E(x,i)Z[0,ζ)
e−I(t)dLπ
r(t)−βZ[0,ζ)
e−I(t)dRπ
r(t) + e−I(ζ)Vˆ
V?(Uπ
r(ζ), H(ζ)),
where ζdenotes the epoch of the first regime switch.
3.3. Markov-modulated periodic-classical barrier strategies. For our candidate optimal control, we
will consider the Markov-modulated reflection strategy, say π0,b= (L0,b
r(t), R0,b
r(t); t≥0), at a suitable
reflection threshold b= (b(i))i∈E. Namely, dividends are paid as a lump sum whenever the surplus
process is above at b(H(T(i))), where T(i)∈ Tris the i-th arrival time of the Poisson process Nr,
while it is pushed upward by capital injection whenever it attempts to down cross zero. The resulting
surplus process becomes the spectrally negative MAP with periodic and classical reflection, denoted by
U0,b
r(t) := X(t)−L0,b
r(t)+R0,b
r(t), t ≥0. We can describe explicitly the cumulative dividend payments
associated to the Markov-modulated barrier strategy as
L0,b
r(t) = X
T(i)∈Tr
T(i)≤t
(U0,b
r(T(i)−)−b(H(T(i)))) ∨0, t ≥0.
By a modification of Remark 3.5 in [24], it follows that the Markov-modulated barrier strategy π0,bis
indeed admissible.
We state the main result of our paper, and its proof will be provided by an iterative construction of the
value function Vin Section 8.
Theorem 3.1. Under Assumptions 3.1 and 3.2, there exists b∗= (b∗(i))i∈Esuch that the Markov-
modulated reflection strategy with Poissonian decision times, π0,b∗, is optimal and the value function of
the problem (3.4)is given by
V(x, i) = Vπ0,b∗(x, i),for x≥0, i ∈E.
4. OPTIMA L S TR ATEG IES FO R AN AUX ILIARY PO ISSONIA N BAIL-OU T D IV IDEND PROB LE M W IT H
AN EXPONE NTIAL TERMI NAL TIME
In this section we introduce a Poissonian bail-out dividend problem with an exponential terminal time,
in a model with a single spectrally negative L´evy process, which is closely related with the problem
mentioned in Section 3, due to Proposition 3.1. To introduce the problem, let us first assume that the
uncontrolled process is given by a spectrally negative L´evy process Xwith Laplace exponent, denoted
by ψX, as in Section 2.
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 9
As in Section 3.2, we consider that the dividend payments can only be made at the arrival times
Tr:= (T(i); i≥1) of a Poisson process Nr= (Nr(t); t≥0) with intensity r > 0, which is defined on
(Ω,F,F,P), where F={Ht}t≥0is the right-continuous complete filtration generated by (X, Nr). We
assume that the processes Xand Nrare independent.
We consider strategies π:= (Lπ
r(t), Rπ
r(t); t≥0) ∈ A where Lπ
radmits the form R[0,t]vπ(s)dNr(s),
t≥0, and vπis a c`agl`ad process adapted to the filtration F. On the other hand, the process Rπ
ris
nondecreasing, right-continuous and F-adapted, with Rπ
r(0−) = 0 satisfying
(4.1) ExZ[0,∞)
e−qtdRπ
r(t)<∞, x ≥0,
and Uπ
r(t)≥0a.s.. The rate of discounting qis a positive constant.
Let ζbe an exponential random variable with parameter λ > 0, independent of X, representing a
random terminal time. We consider that a payoff is made upon termination, given by a function w:
[0,∞)→R. Then, assuming that β > 1is the cost per unit of injected capital, the objective is to
maximize the expected NPV
vπ(x) := ExZ[0,ζ)
e−qtdLπ
r(t)−βZ[0,ζ)
e−qtdRπ
r(t) + e−qζ w(Uπ
r(ζ)), x ≥0,(4.2)
over the set of all admissible strategies A. Hence the problem is to compute the value function
(4.3) v(x) := sup
π∈A
vπ(x), x ≥0,
and obtain an optimal strategy π∗such that vπ∗=v, if such a strategy exists.
We make the following assumptions:
Assumption 4.1. We assume that E[X(1)] = ψ′
X(0+) >−∞.
Assumption 4.2. We assume that wis a concave function with
w′
+(0+) ≤βand w′
+(∞) := lim
x→∞ w′
+(x)∈[0,1].
4.1. Spectrally negative processes with Parisian reflection above. Let Tr={T(i) : i≥1}be the set
of jump times of an independent Poisson process with rate r > 0. We construct the L´evy process with
Parisian reflection above at the level b≥0, denoted by Ub
r:= {Ub
r(t) : t≥0}, as follows: the process
is observed only at times belonging to the set Trand is pushed down to the level bif and only if it is
observed above b. Formally, we have:
Ub
r(t) = X(t), t ∈[0, T +
0(1)),
where
(4.4) T+
b(1) := inf{T(i)∈ Tr:X(T(i)) > b}.
10 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
The process then jumps downward by X(T+
b(1))−bso that Ub
r(T+
b(1)) = b. For T+
b(1) ≤t < T +
b(2) :=
inf{T(i)> T +
b(1) : Ub
r(T(i)−)> b}, we have Ub
r(t) = X(t)−(X(T+
b(1)) −b). The process Ub
rcan
be constructed by repeating this procedure.
Suppose Lb
r(t)is the cumulative amount of (Parisian) reflection until time t≥0. Then we have
Ub
r(t) = X(t)−Lb
r(t), t ≥0,
with
Lb
r(t) := X
T+
b(i)≤tUb
r(T+
b(i)−)−b, t ≥0,(4.5)
where (T+
b(n); n≥1) can be constructed inductively by (4.4) and
T+
b(n+ 1) := inf{T(i)> T +
b(n) : Ub
r(T(i)−)> b}, n ≥1.
4.2. Periodic-classical barrier strategies. The objective of this section is to show the optimality of the
periodic-classical barrier strategy
¯π0,b := {(L0,b
r(t), R0,b
r(t)); t≥0}.
The controlled process U0,b
rbecomes the L´
evy process with Parisian reflection above and classical re-
flection below, which can be constructed as follows.
Let R0,b
r(t) := (−inf0≤s≤tX(s)) ∨0for t≥0, then we have
U0,b
r(t) = X(t) + R0,b
r(t),0≤t < b
T+
b(1)
where b
T+
b(1) := inf{T(i) : X(T(i)−) + Rr(T(i)−)> b}. The process then jumps down by
X(b
T+
b(1)) + Rr(b
T+
b(1)) −bso that U0,b
r(b
T+
b(1)) = b. For b
T+
b(1) ≤t < b
T+
b(2) := inf{T(i)>b
T+
b(1) :
U0,b
r(T(i)−)> b},U0,b
r(t)is the process reflected at 0of the process (X(t)−X(b
T+
b(1))+ b;t≥b
T+
b(1)).
The process U0,b
rcan be constructed by repeating this procedure. It is clear that it admits a decomposition
U0,b
r(t) = X(t)−L0,b
r(t) + R0,b
r(t), t ≥0,
where L0,b
r(t)and R0,b
r(t)are, respectively, the cumulative amounts of Parisian and classical reflection
until time t.
Notice that for b≥0, the strategy ¯π0,b := {(L0,b
r(t), R0,b
r(t)); t≥0}is admissible for the problem
described at the beginning of this section, because (4.1) holds by Proposition 5.2 and Assumption 4.1.
Its expected NPV of dividends minus the costs of capital injection and payoff at an exponential time is
denoted by
vb(x) : = ExZ[0,ζ)
e−qtdL0,b
r(t)−βZ[0,ζ)
e−qtdR0,b
r(t) + e−qζ w(U0,b
r(ζ))
(4.6)
=ExZ[0,∞)
e−θtdL0,b
r(t)−βZ[0,∞)
e−θtdR0,b
r(t) + λZ∞
0
e−θtw(U0,b
r(t))dt, x ≥0,
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 11
where θ:= q+λ.
The main result for this section confirms the optimality of the periodic-classical barrier strategy for
the auxiliary control problem.
Theorem 4.1. Under Assumptions 4.1 and 4.2, there exists a constant barrier 0≤b∗<∞such that the
periodic-classical reflection strategy at the threshold b∗is optimal, i.e., π0,b∗is an optimal strategy for
the problem (4.3)and the value function is given by
v(x) = vπ0,b∗(x) = vb∗(x),for x≥0.
5. EXPRESSI ON O F vbUSING T HE S CA LE FU NC TI ON.
In this section we will write an expression for the expected NPV of total costs vbas in (4.6). For
convenience, let us denote
vLR
b(x) = ExZ[0,∞)
e−θtdL0,b
r(t)−βZ[0,∞)
e−θtdR0,b
r(t), vw
b(x) = ExZ[0,∞)
e−θtw(U0,b
r(t))dt.
It is clear that vb(x) = vLR
b(x) + λvw
b(x)for x≥0. We also have that the expected NPV of dividend
payments and capital injection vLR
bhas already been computed in Lemma 3.1 in [23] , which is given by
vLR
b(x) = −C1
bZ(θ,r)
b(x)−rZ(θ)(b)W(r+θ)(x−b)−rW (r+θ)(x−b)(5.1)
+βZ(θ,r)
b(x) + ψ′
X(0+)
θ−rZ(θ)(b)W(θ+r)(x−b),
with
(5.2) C1
b=r(βZ (θ)(b)−1)
θΦ(r+θ)Z(θ)(b, Φ(r+θ)) +β
Φ(r+θ).
Therefore, it only remains to compute the expected NPV of running costs vw
b. To this end, we provide
the following result and the proof is deferred to Appendix A(see Section A.2).
Proposition 5.1. For x, b ∈R,q > 0, and a positive measurable function hon Rwith compact support,
g(q)(x;h) := ExZ∞
0
e−qth(U0,b
r(t))dt
(5.3)
=(C(q,r)(b;h) + ρ(q)
b(b;h))
Z(q)(b)Z(q,r)
b(x)−ρ(q,r)
b(x;h)
−rC(q,r)(b;h)W(q+r)(x−b)−Zx−b
0
h(z+b)W(q+r)(x−b−z)dz,
12 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
where
C(q,r)(b;h) =
Ξ(q,r)(b;h)−rρ(q)
b(b;h)
Z(q)(b)Z∞
b
e−Φ(q+r)(y−b)Z(q)(y)dy
r
Z(q)(b)Z∞
b
e−Φ(q+r)(y−b)Z(q)(y)dy−r
Φ(q+r)
.
(5.4)
Now we provide an expression of the expected NPV of the periodic-classical barrier strategy with
additional running costs vb, given in (4.6), in terms of scale functions. We omit the proof as it is a direct
consequence of (5.1) and Proposition 5.1, due to the fact that
vw
b(x) = g(θ)(x;w) = (C(θ,r)(b;w) + ρ(θ)
b(b;w))
Z(θ)(b)Z(θ,r)
b(x)−ρ(θ,r)
b(x;w)(5.5)
−rC(θ,r)(b;w)W(θ+r)(x−b)−Zx−b
0
w(z+b)W(θ+r)(x−b−z)dz.
Proposition 5.2. For b≥0and x∈R,
vb(x) = −C1
bZ(θ,r)
b(x)−rZ(θ)(b)W(r+θ)(x−b)−rW (θ+r)(x−b)(5.6)
+βZ(θ,r)
b(x) + ψ′(0+)
θ−rZ(θ)(b)W(r+θ)(x−b)
+λ(C(θ,r)(b;w) + ρ(θ)
b(b;w))
Z(θ)(b)Z(θ,r)
b(x)−ρ(θ,r)
b(x;w)
−rC(θ,r)(b;w)W(r+θ)(x−b)−Zx−b
0
w(z+b)W(r+θ)(x−b−z)dz,
where C1
band C(θ,r)(b;w)are given in (5.2)and (5.4)respectively.
Additionally, in the next result we provide an expression for the resolvent of L ´evy process with Parisian
reflection above at the threshold b,Ub
r. The proof is deferred to the Appendix (see Section A.3).
Proposition 5.3. For x, b ∈R,q > 0, and a positive measurable function hon Rwith compact support,
˜g(q)(x;h) := Ex"Zτ−
0(r)
0
e−qth(Ub
r(t))dt#
(5.7)
=(˜
C(q,r)(b;h) + ρ(q)
b(b;h))
W(q)(b)W(q,r)
b(x)−ρ(q,r)
b(x;h)
−r˜
C(q,r)(b;h)W(q+r)(x−b)−Zx−b
0
h(y+b)W(q+r)(x−b−y)dy,
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 13
where
(5.8) ˜
C(q,r)(b;h) =
Ξ(q,r)(b;h)−ρ(q)
b(b;h)
W(q)(b)Z(q)(b; Φ(q+r))
Z(q)(b; Φ(q+r))
W(q)(b)−r
Φ(q+r)
.
6. SELECT IO N OF A CAND IDATE OPTIMA L BARRIER b∗
We focus on the periodic barrier strategy defined at the beginning of Section 4.2 and choose the
candidate optimal barrier b∗, which satisfies that v′
b∗(b∗) = 1 if such b∗>0exists, and set it to be 0
otherwise.
Recall that the expected NPV of the periodic-classical barrier strategy is given by expression (5.6).
We first analyse the smoothness of the function vb.
Lemma 6.1. For all b≥0, and x∈R\ {0, b},
v′
b(x) = −θC1
bW(θ,r)
b(x)−rW(θ+r)(x−b) + βZ(θ,r)
b(x)(6.1)
+λ" θC(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)−w(0)!W(θ,r)
b(x)
−Zx−b
0
w′
+(z+b)W(r+θ)(x−b−z)dz+ρ(θ,r)
b(x;w′
+),
and, if Xhas paths of unbounded variation,
v′′
b(x) = −θC1
bW(θ)′(x) + rW (θ+r)(x−b)W(θ)(b) + rZx
b
W(θ+r)(x−y)W(θ)′(y)dy
(6.2)
−rW (θ+r)(x−b) + βθW (θ,r)
b(x) + rW (θ+r)(x−b)Z(θ)(b)
+λ" θC(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)−w(0)!
×W(θ)′(x) + rW (θ+r)(x−b)W(θ)(b) + Zx
b
W(θ+r)(x−y)W(θ)′(y)dy
−Zb
0
W(θ)′(x−y)w′
+(y)dy+rZx
b
W(θ+r)′(x−y)ρ(θ)
b(y;w′
+)dy
−Zx−b
0
w′
+(z+b)W(θ+r)′(x−b−z)dz.
Proof. The first and second (if Xhas paths of unbounded variation) derivatives of vLR ′
bare computed in
Lemma 3.2 in [23]. Hence, it remains to compute the first and second derivatives of vw
b.
14 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
Differentiating Z(θ,r)
b(x)and ρ(θ,r)
b(x, w)(given in (2.7) and (2.11) respectively), using (2.4) and inte-
gration by parts, it can be checked that for x > 0we have
Z(θ,r)′
b(x) = θW (θ,r)
b(x) + rZ(θ)(b)W(θ+r)(x−b),
ρ(θ,r)′
b(x;w) = W(θ)(x)w(0) −W(θ)(x−b)w(b) + ρ(θ)
b(x;w′
+)
+rW(θ+r)(0)ρ(θ)
b(x;w) + Zx
b
W(θ+r)′(x−y)ρ(θ)
b(y;w)dy
=ρ(θ,r)
b(x, w′
+) + rW (θ+r)(x−b)ρ(θ)
b(b;w) + w(0)W(θ,r)
b(x)−W(θ+r)(x−b)w(b).
From here, we obtain by differentiating (5.5)
vw′
b(x) = C(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)θW (θ,r)
b(x) + rZ(θ)(b)W(θ+r)(x−b)−rC(θ,r)(b;w)W(θ+r)(x−b)
(6.3)
−ρ(θ,r)
b(x, w′
+) + rW (θ+r)(x−b)ρ(θ)
b(b;w) + w(0)W(θ,r)
b(x)−W(θ+r)(x−b)w(b)
−w(b)W(r+θ)(x−b) + Zx−b
0
w′
+(z+b)W(θ+r)(x−b−z)dz
= θC(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)−w(0)!W(θ,r)
b(x)−ρ(θ,r)
b(x;w′
+)
−Zx−b
0
w′
+(z+b)W(θ+r)(x−b−z)dz,
where we used a change of variable to compute the derivative of Rx−b
0w(z+b)W(θ+r)(x−b−z)dz
given in the last term of the first equality.
In addition, when Xhas paths of unbounded variation, we have using (2.7), (2.11) and Remark 2.1(2)
W(θ,r)′
b(x) = W(θ)′(x) + rW (θ+r)(x−b)W(θ)(b) + Zx
b
W(θ+r)(x−y)W(θ)′(y)dy,
ρ(θ,r)′
b(x;w′
+) = ρ(θ)′
b(x;w′
+) + rZx
b
W(θ)′(x−y)ρ(θ)
b(y;w′
+)dy.(6.4)
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 15
Then, by differentiating (6.3) and using (6.4) together with Remark 2.1(2)
vw′′
b(x) = θC(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)−w(0)!
(6.5)
×W(θ)′(x) + rW (θ+r)(x−b)W(θ)(b) + Zx
b
W(θ+r)(x−y)W(θ)′(y)dy
−Zb
0
W(θ)′(x−y)w′
+(y)dy+rZx
b
W(θ+r)′(x−y)ρ(θ)
b(y;w′
+)dy
−Zx−b
0
w′
+(z+b)W(θ+r)′(x−b−z)dz.
The proof is completed.
Remark 6.1. Using (6.1)and (6.2)together with Remark 2.1(1) we have that the mapping x7→ vw
b(x)is
continuously differentiable (resp. twice continuously differentiable) on R\{0, b}when Xis of bounded
variation (resp. unbounded variation). In addition, we have by (6.3)that
vw′
b(b+) = θC(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)−w(0)!W(θ)(b)−ρ(θ)
b(b;w′) = vw′
b(b−).
On the other hand, if Xis of unbounded variation, by (6.5)and Remark 2.1(2)
vw′′
b(b+) = θC(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)−w(0)!W(θ)′(b)−Zb
0
W(θ)′(b−y)w′
+(y)dy=vw′′
b(b−).
By the smoothness of the scale function, together with Lemma 3.3 from [23] and Remark 6.1 we
obtain the following result.
Lemma 6.2 (Smoothness of vb).For all b≥0we have:
(i) When Xhas paths of bounded variation, vbis continuously differentiable on R\ {0};
(ii) When Xhas paths of unbounded variation, vbis twice continuously differentiable on R\ {0}.
6.1. Selection and existence of the optimal barrier. In this section, we will define and prove the
existence of the threshold b∗≥0under which the strategy ¯πb∗= (L0,b∗
r(t), R0,b∗
r(t)) is optimal. For that
purpose, we provide some preliminary results.
Remark 6.2. (1) Fix b≥0. Let Ub
rbe the Parisian reflected process of Xfrom above at b(without
classical reflection) as given in Section 4.1, and
τ−
0(r) := inf{t > 0 : Ub
r(t)<0}.
16 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
Then, by Corollary 3 in [26], for any x∈Rand q > 0,
Exhe−qτ −
0(r)i=Z(q,r)
b(x)−rZ(q)(b)W(q+r)(x−b)(6.6)
−qZ(q)(b, Φ(q+r))
Z(q)′(b, Φ(q+r)) W(q,r)
b(x)−rW (q)(b)W(q+r)(x−b),
where in particular
Ebhe−qτ −
0(r)i=Z(q)(b)−qZ(q)(b, Φ(q+r))
Z(q)′(b, Φ(q+r))W(q)(b).(6.7)
(2) By (4.1) and Remark 4.1 from [23],
(6.8) vLR ′
b(b) =
θβEbhe−θτ −
0(r)i−1
Φ(θ+r)Z(θ)(b)−Eb[e−θτ−
0(r)]W(θ)(b) + 1.
We now provide the following auxiliary result and we defer the proof to Appendix B.
Lemma 6.3. For b≥0we have
vw′
b(b) = θW (θ)(b) C(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)!−w(0)W(θ)(b)−ρ(θ)
b(b;w′
+)(6.9)
=
θEb"Zτ−
0(r)
0
e−θtw′
+(Ub
r(t))dt#
Φ(θ+r)Z(θ)(b)−Eb[e−θτ−
0(r)]W(θ)(b).
Using that v′
b(b) = vLR ′
b(b) + λvw′
b(b)together with (6.8) and (6.9), gives
(6.10) v′
b(b) = θ
Φ(θ+r)
G(b)
Z(θ)(b)−Ebhe−θτ −
0(r)iW(θ)(b) + 1,
where
G(b) := β−1−Eb"Zτ−
0(r)
0
e−θt βθ −λw′
+(Ub
r(t))dt#
(6.11)
=βZ (θ)(b)−1 + λ˜
C(θ,r)(b;w′
+)−βθ W (θ)(b)Z(θ)(b, Φ(θ+r))
Z(θ)′(b, Φ(θ+r)) b≥0,
where the last equality is true because of (5.7) (taking w′
+instead of h) and (6.7). We propose as
candidate for the optimal barrier
b∗:= inf{b≥0 : G(b)≤0}.(6.12)
In the next result we provide a necessary and sufficient condition for the optimal barrier b∗to be 0.
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 17
Proposition 6.1. We have that 0≤b∗<∞. Moreover, we have that b∗= 0 if and only if Xhas paths of
bounded variation and
(6.13) β−1≤1
c
1
Φ(θ+r)−r
c
θβ −λΦ(θ+r)Z∞
0
e−Φ(θ+r)zw′
+(z)dz.
Proof. Due to Assumption 4.2 we have that b7→ βθ −λw′
+(b)is non-decreasing, hence
b7→ Eb"Zτ−
0(r)
0
e−θt βθ −λw′
+(Ub
r(t))dt#
is non-decreasing as well. It follows that the mapping b7→ G(b)is non-increasing.
On the other hand, due to spatial homogeneity of L ´evy processes, we have that the {Ub
r(t); t≤τ−
0(r)}
started at Ub
r(0) = bis equal in law to {b+U0
r;t≤˜τ−
−b(r)}started at U0
r(0) = 0, where ˜τ−
−b(r) :=
inf{t≥0 : U0
r(t)<−b}. Then, by dominated convergence we have
lim
b↑∞ G(b) = β−1−lim
b↑∞
E0Z∞
0
e−θt1{t≤˜τ−
−b(r)}βθ −λw′
+(b+U0
r(t))dt=λ
θw′
+(+∞)−1<0.
(6.14)
Now, using (6.11), we obtain
lim
b↓0G(b) = β1−θW (θ)(0+) 1
Φ(θ+r)−rW (θ)(0+) −1
+λΦ(θ+r)W(θ)(0+)
Φ(θ+r)−rW (θ)(0+) Z∞
0
e−Φ(θ+r)zw′
+(z)dz,
where we have used that limb↓0ρ(θ)
b(b;w′
+) = 0 and limb↓0Ξ(θ,r)(b;w′
+) = R∞
0e−Φ(q+r)yw′
+(z)dz. Hence,
we get the following cases:
(1) If Xhas paths of unbounded variation, then limb↓0G(b) = β−1>0. Thus, there exists a unique
b∗>0such that G(b∗) = 0.
(2) If Xhas paths of bounded variation we have
lim
b↓0G(b) = β−1−1
c
1
Φ(θ+r)−r
c
θβ −λΦ(θ+r)Z∞
0
e−Φ(θ+r)zw′
+(z)dz.
Thus, if (6.13) does not hold, then there exists a unique b∗>0such that G(b∗) = 0; otherwise, if
(6.13) holds, we set b∗= 0.
18 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
7. VERI FIC ATION OF OPTIMALIT Y
We shall show the optimality of the periodic-classical barrier strategy ¯π0,b∗, where the barrier b∗is
defined by (6.12).
Theorem 7.1. The strategy ¯π0,b∗is optimal and the value function of the problem (4.3)is given by
v=vb∗.
Let Lbe the infinitesimal generator associated with the process Xapplied to a C1(0,∞)(resp.,
C2(0,∞)) function ffor the case Xis of bounded (resp., unbounded) variation:
Lf(x):=γf ′(x) + 1
2η2f′′(x) + Z(−∞,0)
[f(x+z)−f(x)−f′(x)z1{−1<z<0}]Π(dz)f or x > 0.
(7.1)
In the next result we provide a verification lemma. The proof is essentially the same as Proposition
5.1 in [23] (which deals with the case in which the payoff function wis equal to zero), and thus we omit
it.
Throughout the rest of this section we extend the domain of vπto Rby setting vπ(x) = βx +vπ(0+)
for x < 0.
Lemma 7.1 (Verification lemma).Suppose that ˆπ∈ A is such that vˆπ∈ C(R)∩C1((0,∞)) (respectively,
C1(R)∩ C2((0,∞))) for the case that Xhas paths of bounded (respectively, unbounded) variation. In
addition, suppose that
(L − θ)vˆπ(x) + rmax
0≤l≤x{l+vˆπ(x−l)−vˆπ(x)}+λw(x)≤0, x ≥0,
v′
ˆπ(x)≤β, x ≥0,
inf
x≥0vˆπ(x)>−mfor some m > 0.
(7.2)
Then ˆπis an optimal strategy and vˆπ(x) = v(x)for all x≥0.
Notice that if vb∗satisfies the variational inequalities (7.2), then the strategy ¯π0,b∗is optimal, due to
the previous lemma. To show this, we shall provide some preliminary results.
Lemma 7.2. For b≥0, we have
(7.3) (L − θ)vb(x) =
−λw(x),if x∈(0, b),
−r{(x−b) + vb(b)−vb(x)} − λw(x),if x∈[b, ∞).
Proof. First, from Lemma 5.1 in [23] we have
(7.4) (L − θ)vLR
b(x) =
0,if x∈(0, b),
−r{(x−b) + vLR
b(b)−vLR
b(x)},if x∈[b, ∞).
BAILOUT PERIODIC DIVIDEND PROBLEM FOR MARKOV ADDITIVE PROCESSES 19
It remains to analyse the term (L − θ)vw
b(x)for x∈(0,∞).
(i) Suppose 0< x < b. By the proof of Theorem 2.1 in [6] it follows that
(7.5) (L − θ)Z(θ)(x) = 0, x > 0.
In addition, the proof of Lemma 4.5 of [10] implies
(7.6) (L − θ)ρ(q)
b(x;w) = w(x),0< x < b.
By combining (5.5), (7.5) and (7.6), we obtain
(7.7) (L − θ)vw
b(x) = −w(x),0< x < b.
(ii) Now, assume that x > b. From (5.4) and (5.6) in [23] we have
(L − θ)W(θ+r)(x−b) = 1 + rW (θ+r)(x−b),
(L − θ)Z(θ,r)
b(x) = rZ(θ,r)
b(x).
(7.8)
On the other hand, from the proof of Lemma 4.5 of [10] we have
(L − (θ+r)) Zx−b
0
W(θ+r)(x−b−y)w(y+b)dy=w(x).
Hence
(7.9) (L− θ)Zx−b
0
W(θ+r)(x−b−y)w(y+b)dy=w(x)+ rZx−b
0
W(θ+r)(x−b−y)w(y+b)dy.
In a similar way, we obtain by (2.11)
(L − θ)Zx
b
W(θ+r)(x−y)ρ(θ)
b(y;w)dy=ρ(θ)
b(x;w) + rZx
b
W(θ+r)(x−y)ρ(θ)
b(y;w)dy
=ρ(θ,r)
b(x;w).
By the proof of Lemma 4 in [4] we have that (L − θ)W(θ)(x−y) = 0 for x > y > 0, then by dominated
convergence we get
(L − θ)ρ(θ)
b(x;w) = (L − θ)Zb
0
W(θ)(x−y)w(y)dy= 0, x > b.
Therefore, using (2.11)
(7.10) (L − θ)ρ(θ,r)
b(x;w) = rρ(θ,r)
b(x;w).
20 D. MATA L ´
OPEZ, H. A. MORENO-FRANCO, K. NOBA, AND J. L. P ´
EREZ
Finally, by (5.5) together with (7.8)–(7.10) we obtain
(L − θ)vw
b(x) = r C(θ,r)(b;w) + ρ(θ)
b(b;w)
Z(θ)(b)!Z(θ,r)
b(x)−rC(θ,r)(b;w)(1 + rW(θ+r)(x−b))(7.11)
−rρ(θ,r)
b(x;w)−w(x) + rZx−b
0
W(θ+r)(x−b−y)w(y+