Systems of variational inequalities for non-local operators related to optimal switching problems:...

manuscripta math. © Springer-Verlag Berlin Heidelberg 2014

Niklas L. P. Lundström · Kaj Nyström · Marcus Olofsson

Systems of variational inequalities for non-localoperators related to optimal switching problems:existence and uniqueness

Received: 8 July 2013 / Revised: 7 March 2014

Abstract. In this paper we study viscosity solutions to the system

min

{−Hui (x, t)− ψi (x, t), ui (x, t)− max

j �=i(−ci, j (x, t)+ u j (x, t))

}= 0,

ui (x, T ) = gi (x), i ∈ {1, . . . , d},where (x, t) ∈ R

N ×[0, T ]. Concerning H, we assume that H = L+I where L is a linear,possibly degenerate, parabolic operator of second order and I is a non-local integro-partialdifferential operator. A special case of this type of system of variational inequalities withterminal data occurs in the context of optimal switching problems when the dynamics ofthe underlying state variables is described by an N -dimensional Levy process. We establisha general comparison principle for viscosity sub- and supersolutions to the system undermild regularity, growth and structural assumptions on the data, i.e., on the operator H andon the continuous functions ψi , ci, j , and gi . Using the comparison principle we prove theexistence of a unique viscosity solution (u1, . . . , ud ) to the system by Perron’s method. Ourmain contribution is that we establish existence and uniqueness of viscosity solutions, in thesetting of Levy processes and non-local operators, with no sign assumption on the switchingcosts {ci, j } and allowing ci, j to depend on x as well as t .

1. Introduction and statement of main results

In this paper we consider the problem

min

{−Hui (x, t)− ψi (x, t), ui (x, t)− max

j �=i(−ci, j (x, t)+ u j (x, t))

}= 0,

ui (x, T ) = gi (x), i ∈ {1, . . . , d}, (1.1)

in RN ×[0, T ] ((x, t) ∈ R

N ×[0, T ]), T > 0, whereψi , ci, j , and gi are continuousfunctions and H is a non-local integro-partial differential operator. Concerning H,

N. L. P. Lundström and M. Olofsson were financed by Jan Wallanders och Tom HedeliusStiftelse samt Tore Browaldhs Stiftelse through the project Optimal switching problems andtheir applications in economics and finance, P2010-0033:1.

N. L. P. Lundström · K. Nyström · M. Olofsson(B): Department of Mathematics, UppsalaUniversity, 751 06, Uppsala, Sweden. e-mail: [email protected];[email protected]; [email protected]

Mathematics Subject Classification (2000): 35R09, 49L25, 45G15

DOI: 10.1007/s00229-014-0683-9

N. L. P. Lundström et al.

we assume that H = L + I, where

L =N∑

i, j=1

ai, j (x, t)∂xi x j +N∑

i=1

ai (x, t)∂xi + ∂t , (1.2)

for continuous functions ai, j and ai , and I is a non-local integro-partial differentialoperator which, for smooth functions φ, is defined as

I(x, t, φ) =∫

Rl\{0}K (x, t, z, φ, Dφ)ν (dz) ,

K (x, t, z, φ, p) = φ (x + η (x, t, z) , t)− φ (x, t)

−χ{|z|≤1}(z)N∑

k=1

ηk (x, t, z) pk, (1.3)

for (x, t) ∈ RN × R, Dφ = (∂x1φ, . . . , ∂xNφ), p ∈ R

N . Here ν is a positiveRadon measure and, for each i ∈ {1, . . . , N }, ηi is a function taking values in R

N .χ{|z|≤1} is the indicator function for the closed unit ball in R

l . Operators H = L+Ioccur, for instance, in the context of financial markets where the state variables aredescribed by an N -dimensional Levy process X = (

X x,ts

)solving the system of

stochastic differential equations

d X x,ts = a(X x,t

s , s)ds + σ(X x,ts , s)dWs

+∫

z∈Rl

η(X x,ts− , s, z)d N (ds, dz), (1.4)

for t ≤ s ≤ T , with initial condition X x,ts = x , 0 ≤ s ≤ t , (x, t) ∈ R×[0, T ]. Here

W = {Wt } denotes a standard N -dimensional Brownian motion and

N (ds, dz) ={

N (ds, dz) if |z| ≥ 1

N (ds, dz)− ν(dz)ds if |z| < 1,

where N is a Poisson random measure on [0,∞) × Rl with intensity measure

ν(dz)ds. H can, in the context of (1.4), be seen as the infinitesimal generatorassociated to X = {

X x,ts

}and we note that the diffusion part of the system is then

described by the N × N -dimensional matrix σ .In the Markovian setting, when the randomness stems from the Levy process

X in (1.4), the problem in (1.1) is a system of variational inequalities with inter-connected obstacles related to multi-mode optimal switching problems. Our mainresults concern existence and uniqueness of viscosity solutions to the system in(1.1) under mild regularity, growth and structural assumptions on the data, i.e., onthe operator H and on continuous functions ψi , ci, j , and gi .

In multi-mode optimal switching problems the system in (1.1) occurs withg = (g1, . . . , gd) ≡ (0, . . . , 0). To outline the setting for this class of problems,consider a production facility which can run the production in d, d ≥ 2, production

Variational inequalities for non-local operators

modes. Let X = {X x,t

s}

denote the vector of stochastic processes in (1.4) which,as discussed above, represents the market price of the underlying commodities andother financial assets that influence the production. Let the payoff rate in productionmode i , at time t , be ψi (Xt , t) and let ci, j (Xt , t) denote the switching cost forswitching from mode i to mode j at time t . A management strategy is a combinationof a non-decreasing sequence of stopping times {τk}k≥0, where, at time τk , themanager decides to switch production from its current mode to another one, anda sequence of indicators {ξk}k≥0, taking values in {1, . . . , d}, indicating the modeto which the production is switched. At τk the production is switched from modeξk−1 (current mode) to ξk . A strategy ({τk}k≥0, {ξk}k≥0) can be represented by asimple function μ defined as

μs =∑i≥1

ξiχ(τi ,τi+1](s)+ ξ0χ[τ0,τ1](s),

indicating the current mode of the production. Here, χB(s) is the indicator functionfor the measurable set B ⊂ R, i.e., χB(s) = 1 if s ∈ B and 0 otherwise. Whenthe production is run using a strategy μ, defined by ({τk}k≥0, {ξk}k≥0), over a finitehorizon [0, T ], the total expected profit up to time T is

J (μ) = E

⎡⎣

⎛⎝

T∫0

ψμs (Xs, s)ds −∑k≥1

cξk−1,ξk (Xτk , τk)

⎞⎠

⎤⎦ .

The optimal switching problem consists of finding an optimal management strategyμ∗, defined by ({τ ∗

k }k≥0, {ξ∗k }k≥0), such that

J (μ∗) = supμ

J (μ).

Let (Y 1t , . . . ,Y d

t ) be the value function associated with the optimal switching prob-lem, on the time interval [t, T ], where Y i

t stands for the optimal expected profit if,at time t , the production is in mode i . Under sufficient assumptions, see for ex-ample [2], it can then be proved that (Y 1

t , . . . ,Y dt ) = (u1(Xt , t), . . . , ud(Xt , t)),

where the vector of deterministic functions (u1(x, t), . . . , ud(x, t)) satisfies (1.1)with g = (g1, . . . , gd) ≡ (0, . . . , 0).

1.1. Assumptions

We here outline the assumptions we impose on H and on the functionsψi , ci, j , andgi . Firstly, focusing on H = L + I we impose additional structural assumptionson the matrix {ai, j }N

i, j=1. In particular, we assume that

ai, j (x, t) = (σ (x, t)σ ∗(x, t))i, j , i, j ∈ {1, . . . , N }, (A.1)

where σ = σ(x, t) is an N × N matrix and σ ∗ is the transpose of σ . Concerningregularity and growth conditions on {ai, j }N

i, j=1 and {ai }Ni=1, we assume that

(i) |ai (x, t)− ai (y, s)| + |σi, j (x, t)− σi, j (y, s)| ≤ A|x − y|,(i i) |ai (x, t)| + |σi, j (x, t)| ≤ A(1 + |x |), (A.2)


for some A, 1 ≤ A < ∞, for all i, j ∈ {1, . . . , N }, and whenever (x, t), (y, s) ∈R

N × [0, T ]. Here, |x | is the standard Euclidean norm of x ∈ RN . Note that (i)

implies (i i) in (A.2). Note also that {ai, j }Ni, j=1 is only assumed non-negative definite

and hence large sets of entries in the matrices {ai, j }Ni, j=1, {σi, j }N

i, j=1, may be zeroat points, resulting in degeneracies.

The above restrictions define the local part, L, of H. Focusing on the non-localpart, I, we assume that ν is a positive Radon measure defined on R

l \ {0} such that

∫0<|z|≤1

|z|2ν (dz)+∫

1<|z|e�|z|ν (dz) ≤ A (A.3)

for some constant A, 1 ≤ A < ∞, and for some� > 0. In addition we assume thatηk is, for k ∈ {1, . . . , N }, continuous in x and t , Borel measurable in z, and that

|ηk (x, t, z) | ≤ B min{|z|, 1},|ηk (x, t, z)− ηk (y, t, z) | ≤ B min{|z|, 1}|x − y|, (A.4)

for some constant B, 1 ≤ B < ∞, and for all x, y ∈ RN , t ∈ R, z ∈ R

l .This completes our definition of H. Secondly, concerning regularity and growthconditions on ψi , ci, j and gi we assume that

ψi , ci, j and gi are continuous functions,

|ψi (x, t)| + |ci, j (x, t)| + |gi (x)| ≤ B(1 + |x |γ ) for some

B, γ ∈ [1,∞),whenever (x, t) ∈ RN × [0, T ]. (A.5)

Thirdly, the structural assumptions on the functions ci, j that we impose are thefollowing:

(i) ci,i (x, t) = 0 for each i ∈ {1, . . . ., d},(i i) For any sequence i1, . . . , ik, i j ∈ {1, . . . , d} for each j ∈ {1, . . . , k},

we have ci1,i2(x, t)+ ci2,i3(x, t)+ · · · + cik−1,ik (x, t)+ cik ,i1(x, t) > 0,

for all (x, t) ∈ RN × [0, T ]. (A.6)

We note that (i i) above is the no-loop condition introduced in Ishii and Koike [9].Finally, concerning the interplay between the terminal data {gi } and the switchingcosts {ci, j }, at t = T , we assume that

gi (x) ≥ maxj �=i(−ci, j (x, T )+ g j (x)) (A.7)

for all i ∈ {1, . . . , d}, and for all x ∈ RN . Note that in the special case of the

optimal switching problem discussed above, i.e., g = (g1, . . . , gd) ≡ (0, . . . , 0),(A.7) implies that ci, j (x, T ) ≥ 0 for all i, j ∈ {1, . . . , d}, and for all x ∈ R

N .


1.2. Statement of main results

We here formulate our main results. For the definition of LSCp(RN × [0, T ]),

USCp(RN ×[0, T ]) and Cp(R

N ×[0, T ]) as well as for the definition of viscositysub- and supersolutions we refer to the bulk of the paper. We first prove the followingcomparison principle.

Theorem 1.1. Assume (A.1) – (A.7) and suppose that (u+1 , . . . , u+

d ) and (u−1 , . . . ,

u−d ) are viscosity super- and subsolutions, respectively, to the problem in (1.1).

Then u−i ≤ u+

i in RN × (0, T ] for all i ∈ {1, . . . , d}.

Before stating our existence theorems, we make the following definition.

Definition 1. A barrier from above for the system in (1.1), component i ∈ {1, . . . , d}and the point y ∈ R

N , u+,i,y , is a family of continuous supersolutions, {u+,i,y,ε}ε>0

= {(u+,i,y,ε1 , . . . , u+,i,y,ε

d )}ε>0, u+,i,y,εj ∈ Cp(R

N × [0, T ]) for j ∈ {1, . . . , d}, tosystem (1.1) such that

limε→0

u+,i,y,εi (y, T ) = gi (y).

A barrier from below for the system in (1.1), component i ∈ {1, . . . , d} andthe point y ∈ R

N , u−,i,y , is a family of continuous subsolutions, {u−,i,y,ε}ε>0 ={(u−,i,y,ε

1 , . . . , u−,i,y,εd )}ε>0, u−,i,y,ε

j ∈ Cp(RN × [0, T ]) for j ∈ {1, . . . , d}, to

system (1.1) such that

limε→0

u−,i,y,εi (y, T ) = gi (y).

To stress generality, we first prove the following theorem.

Theorem 1.2. Assume (A.1)–(A.7) and suppose that

(1) there exists, for each i ∈ {1, . . . , d} and y ∈ RN , a barrier from above u+ =

u+,i,y to the system in (1.1) in the sense of Definition 1,(2) there exists, for each i ∈ {1, . . . , d} and y ∈ R

N , a barrier from below u− =u−,i,y to the system in (1.1) in the sense of Definition 1.

Then there exists a viscosity solution (u1, . . . , ud), ui ∈ Cp(RN × [0, T ]) for

i ∈ {1, . . . , d}, to the problem in (1.1), u−i ≤ ui ≤ u+

i on RN × [0, T ], for

i ∈ {1, . . . , d}, and this solution is unique in the class Cp(RN × [0, T ]).

To establish existence of a viscosity solution (u1, . . . , ud) to the problem in (1.1),it hence remains to construct barriers in the sense of Definition 1. To constructbarriers from above and below we need to impose additional assumptions on theswitching costs ci, j and the terminal data gi . In particular, to construct barriers weneed the following structural assumptions,

ci1,i2(x, t)+ ci2,i3(x, t) ≥ ci1,i3(x, t) for any sequence of indices i1, i2, i3,

il ∈ {1, . . . , d}, for each l ∈ {1, 2, 3}, and for all (x, t) ∈ RN × [0, T ]. (A.8)

and

gi = g, for all i ∈ {1, . . . , d}, for some Lipschitz continuous function g, (A.9)

together with the additional regularity


ci, j ∈ C1,2(RN × [0, T ]), ∂xk ci, j ∈ L∞(RN × [0, T ]),|ψi (x, t)|, |g(x, t)|, |∂t ci, j (x, t)|, |∂xk xl ci, j (x, t)|(1 + |x |2) ≤ D(1 + |x |),for any i, j, k, l ∈ {1, . . . , d}, for all (x, t) ∈ R

N × [0, T ] and for some D,

1 ≤ D < ∞. (A.10)

The idea is then to use that, by (A.8), given any real valued function w, w+j =

w + ci, j lies above the obstacle at all points (x, t) ∈ RN × [0, T ]. Indeed,

w+j − max

k �= j(−c j,k(x, t)+ w+

k ) > − maxk �= j

(−ci, j (x, t)− c j,k(x, t)+ ci,k(x, t)) ≥ 0.

In particular, we prove that

u+,i,y,εj (x, t) = g(y)+ K

ε2 (T − t)

+L(e�(T −t) + 1)(|x − y|2 + ε)12 + ci, j (x, t),

u−,i,y,εj (x, t) = g(y)− K

ε2 (T − t)

−L(e�(T −t) + 1)(|x − y|2 + ε)12 ,

where L is the Lipschitz constant of g and K and � are large enough constants,constitute barriers from above and below to the problem in (1.1). From Theorem1.2 we can subsequently deduce that the following theorem holds.

Theorem 1.3. Assume (A.1) – (A.10). Then there exists a unique viscosity solution(u1, . . . , ud) to the problem in (1.1), unique in the class Cp(R

N × [0, T ]).Remark 1.1. Through H = L + I we see that H is written as the sum of the localoperator L and the non-local operator I. Note that by the assumptions in Theorem1.1-Theorem 1.3 the matrix {ai, j (x, t)} is only assumed to be non-negative definiteand as such it can vanish at points. Similarly the jump vector η = η(x, t, z) isallowed to vanish. A consequence of this is that there are no regularization effectsin the problem coming either from L or I. Therefore the system in (1.1) can not beexpected to have classical solutions and hence an appropriate notion of viscositysolutions is needed.

Remark 1.2. H can be seen as the infinitesimal generator associated with a Levyprocess described by (1.4). In this context, the assumptions stated in (A.3) excludesome Levy processes as a Levy measure ν in general only satisfies

∫min{|z|2, 1}ν(dz) < ∞.

However, for many, if not most, applications the class of Levy processes consid-ered in this paper is sufficiently rich since, e.g., any Levy process with compactlysupported Levy measure satisfies (A.3).


Remark 1.3. Our proofs of Theorems 1.1–1.3 are influenced by the correspondingarguments in Biswas et al. [2], El-Asri and Fakhouri [5] and Lundström et al. [11].In Lundström et al. [11], versions of Theorems 1.1–1.3 are proved in the casewhen H ≡ L, i.e., in the case of local operators. In the latter paper the problemof regularity of viscosity solutions was also treated in the context of operators ofKolmogorov type.

Remark 1.4. Theorems 1.1–1.3 remain true for the more general non-linear prob-lems

min

{− supα∈Ai

infβ∈Bi

[Hiα,βui (x, t)− ψ

α,βi (x, t)

],

ui (x, t)− maxj �=i(−ci, j (x, t)+ u j (x, t))

}= 0,

ui (x, T ) = gi (x), i ∈ {1, . . . , d},where Ai , Bi are compact metric spaces and Hi

α,β = Liα,β + I i

α,β , for each

i ∈ {1, . . . , d}, is as in (1.2), (1.3), for some σ ik,l,α,β , ai

k,l,α,β , aik,α,β , ν, ηi

α,β sat-isfying (A.1), (A.2), (A.3), and (A.4) uniformly in α, β. Equations of this typeare considered in Biswas et al. [2] and arise for example as the Bellman-Isaacsequations for zero-sum stochastic games.

1.3. Our contribution in relation to the current literature

In this paper we consider optimal switching problems for operators H = L + I,where L is a local operator and I is a non-local operator, allowing for generalswitching costs. As a general comment we note that there is currently a substan-tial literature devoted to the the case of local operators, i.e., H ≡ L, and thatconsiderably less is known in the case when non-local effects are allowed.

In the local setting there is a well-established connection between the theoryof reflected backward stochastic differential equations driven by Brownian motionand multi-mode optimal switching problems, see [3–5,7] and [8]. In general, the re-search on optimal switching problems focuses on two approaches: the probabilisticor stochastic approach and the deterministic approach. We note that the stochasticapproach heavily uses probabilistic tools such as the Snell envelope and backwardstochastic differential equations. For more on the stochastic approach we refer tothe references above concerning reflected backward stochastic differential equa-tions. The latter, deterministic approach focuses more on variational inequalitiesand partial differential equations and we refer, in the local setting, to El-Asri andHamadene [6] and Lundström et al. [11]. In general, in the local setting the twoapproaches are used in combination due to the connection between reflected back-ward stochastic differential equations and systems of variational inequalities. Theresults in the literature concerning the optimal switching problem, i.e., system (1.1)with g = (g1, . . . , gd) ≡ (0, . . . , 0), are derived under different assumptions onthe switching costs {ci, j }. We note that the switching costs in the references listedabove are essentially always assumed to be non-negative, i.e., ci, j (x, t) ≥ α for all


(x, t) ∈ RN × [0, T ], i, j ∈ {1, . . . , d}, i �= j , and for some α ≥ 0. Often, even

α > 0 is assumed, see El-Asri and Hamadene [6] and Djehiche and Hamadene [3]for instance. Furthermore, often additional restrictions are imposed on {ci, j }, likeci, j is independent of x , see [4] for instance, or that ci, j is even constant, see [2,3].The only papers we are aware of where the switching costs are allowed to changesign are El-Asri and Fakhouri [5] and Lundström et al. [11]. However, in El-Asriand Fakhouri [5] there are two additional conditions concerning the non-negativityof the switching costs: ci, j (x, T ) = 0 and the number of negative switching costsis limited in a certain sense, see condition (v) in El-Asri and Fakhouri [5]. In Lund-ström et al. [11] the assumptions imposed on the switching costs are in line withthe assumptions imposed in this paper.

In the non-local setting the connections between a theory of reflected backwardstochastic differential equation driven by Levy processes and multi-modes optimalswitching problem seems to be considerably less developed. In general the theoryfor non-local operators is currently an active area of research. We here refer to [1,2]and [12] for an account of this and for references, but to our knowledge [2] is the onlypaper that makes contribution to multi-mode optimal switching problems involvingnon-local operators, i.e., involving non-local effects through Levy processes thatare allowed to jump. Building on Biswas et al. [2] our main contribution is that whilein Biswas et al. [2], the switching costs are assumed to be non-negative constants,our baseline assumption on the functions {ci, j } is that we make no sign assumptionon the switching costs {ci, j } and that ci, j is allowed to depend on x as well as t . Inparticular, in Biswas et al. [2] the authors basically prove Theorems 1.1–1.3 for acombined optimal switching and control problem essentially assuming H = L+I,with L, I, as in (1.2), (1.3), respectively, (A.1)–(A.4), and (A.5) with γ = 2. Ourcontribution is that we prove Theorem 1.1–1.3 allowing for any γ ∈ [1,∞) in(A.5) and with more general switching costs than [2].

Naturally one may ask whether the allowance for possibly negative switchingcosts is of importance beyond the mathematical issues. In fact, it is very natural toallow for negative switching costs as these allow one to model the situation when, forexample, a government through environmental policies provides subsidies, grantsor other financial support to energy production facilities in case they switch to more‘green energy’ production or to more clean methods for production. In this case itis not a cost for the facility to switch, it is a gain. However, the final decision toswitch or not to switch is naturally also influenced by the running pay-offs {ψi }.1.4. Outline of the paper

The rest of this paper is organized as follows. Section 2 is of preliminary nature andwe here introduce some notation and state the definition of viscosity subsolutions,supersolutions and solutions. Section 3 is devoted to the proof of Theorem 1.1 andin Sect. 4 we prove Theorems 1.2 and 1.3.

2. Preliminaries

In this section we introduce some notation used throughout the paper and we defineviscosity sub- and supersolutions to the problem in (1.1). Throughout the paper we


let Q = {1, . . . , d} denote the (finite) set of available states and we let Q−i = Q\{i}for i ∈ {1, . . . , d}.

2.1. Notation

We denote by LSC(RN × [0, T ]) the set of lower semi-continuous functions, i.e.,all functions f : R

N × [0, T ] → R such that for all points (x0, t0) and for anysequence {(xn, tn)}n , limn→∞(xn, tn) → (x0, t0) in R

N × [0, T ], we have

lim infn→∞ f (xn, tn) ≥ f (x0, t0).

Likewise, we denote by USC(RN ×[0, T ]) the set of upper semi-continuous func-tions, i.e., all functions f : R

N × [0, T ] → R such that for all points (x0, t0) andfor any sequence {(xn, tn)}n , limn→∞(xn, tn) → (x0, t0) in R

N × [0, T ], we have

lim supn→∞

f (xn, tn) ≤ f (x0, t0).

Note that a function f is upper semi-continuous if and only if − f is lower semi-continuous. Also, a real function g is continuous if and only if it is both upper andlower semi-continuous. Given γ ∈ [1,∞), the function space LSCp,γ (R

N ×[0, T ])is defined to consist of functions h ∈ LSC(RN × [0, T ]) which satisfy the growthcondition

|h(x, t)| ≤ c(1 + |x |γ )for some c ∈ [1,∞), whenever (x, t) ∈ R

N × [0, T ]. USCp,γ (RN × [0, T ]) is

defined by analogy. Furthermore, Cp,γ (RN × [0, T ]) = USCp,γ (R

N × [0, T ]) ∩LSCp,γ (R

N ×[0, T ]). Cp(RN ×[0, T ]), USCp(R

N ×[0, T ]), LSCp(RN ×[0, T ])

are the unions, respectively, of the sets Cp,γ (RN ×[0, T ]), USCp,γ (R

N ×[0, T ]),LSCp,γ (R

N ×[0, T ]), with respect to γ ≥ 1. By C1,2(RN ×[0, T ))we denote theset of functions which are twice continuously differentiable in the spatial variablesand once continuously differentiable in the time variable, on R

N × [0, T ). We willby c denote a generic constant, 1 ≤ c < ∞, that may change value from line to line.

2.2. Viscosity solutions

We here define the notion of viscosity solutions to the problem in (1.1). Let H =L + I with L, I, as in (1.2), (1.3), respectively. Given (x, t) ∈ R

N × [0, T ],φ ∈ C1,2(RN × [0, T )), p ∈ R

N , κ ∈ (0, 1), we let

Iκ(x, t, φ, p) =∫

{z:|z|<κ}K (x, t, z, φ, p)ν (dz) ,

Iκ(x, t, φ, p) =∫

{z:|z|≥κ}K (x, t, z, φ, p)ν (dz) ,

where


K (x, t, z, φ, p) = φ (x + η (x, t, z) , t)− φ (x, t)

−χ{|z|≤1}(z)N∑

k=1

ηk (x, t, z) pk .

Note that I = Iκ + Iκ and that Iκ and Iκ give, respectively, the contributionto I from the ‘small’ and ‘large’ jumps. Using this notation we let Hκ(φ, u) =Hκ(x, t, φ, u) where

Hκ(x, t, φ, u) := Lφ(x, t)+ Iκ(x, t, φ, Dφ)+ Iκ(x, t, u, Dφ)

whenever (x, t) ∈ RN × [0, T ], φ ∈ C1,2(RN × [0, T )) and u ∈ LSCp(R

N ×[0, T ]) ∪ USCp(R

N × [0, T ]).Definition 2. (i)A vector (u+

1 , . . . , u+d ), ui ∈LSCp(R

N ×[0, T ]) for i ∈{1, . . . , d},is a viscosity supersolution to the problem in (1.1) if u+

i (x, T ) ≥ gi (x) wheneverx ∈ R

N , i ∈ {1, . . . , d}, and if the following holds. If (x0, t0) ∈ RN × (0, T ) and

if, for some i ∈ {1, . . . , d}, we have φi ∈ C1,2(RN × [0, T )) such that

(i) φi (x0, t0) = u+i (x0, t0),

(i i) (x0, t0) is a global maximum of φi − u+i ,

then

min

{−Hκ(φi , u+

i )− ψi , u+i − max

j �=i(−ci, j + u+

j )

}≥ 0 for all κ ∈ (0, 1).

(i i) A vector (u−1 , . . . , u−

d ), u−i ∈ USCp(R

N × [0, T ]) for i ∈ {1, . . . , d}, isa viscosity subsolution to the problem in (1.1) if u−

i (x, T ) ≤ gi (x) wheneverx ∈ R

N , i ∈ {1, . . . , d}, and if the following holds. If (x0, t0) ∈ RN × (0, T ) and

if, for some i ∈ {1, . . . , d}, we have φi ∈ C1,2(RN × [0, T )) such that

(i) φi (x0, t0) = u−i (x0, t0),

(i i) (x0, t0) is a global minimum of φi − u−i ,

then

min

{−Hκ(φi , u−

i )− ψi , u−i − max

j �=i(−ci, j + u−

j )

}≤ 0 for all κ ∈ (0, 1).

(i i i) If (u1, . . . , ud) is both a viscosity supersolution and subsolution to the problemin (1.1), then (u1, . . . , ud) is a viscosity solution to the problem in (1.1).

Remark 2.1. Note that it is natural, since the Levy measure is singular at 0 and hassome decay properties at infinity, to break the non-local operator into the pieces Iκand Iκ which give, respectively, the contribution from the ‘small’ and ‘large’ jumpsto I. To have Iκ (x, t, vi , Dφi )well defined the reasonable thing is to choose vi = φi

in Definition 2. However, for Iκ (x, t, vi , Dφi ) to be well defined the important thingis not regularity of vi but decay properties at infinity of vi , decay properties whichmust be compatible with the decay properties of the Levy measure ν. Hence it is


reasonable to use Iκ(x, t, vi , Dφi ) with vi = u±i in Definition 2 as long as u±

iis assumed to have moderate growth at infinity. In our case polynomial growth issufficient. We refer to Jakobsen and Karlsen [10] for an elaboration on this definitionand to Barles and Imbert [1] for alternative and equivalent definitions.

Remark 2.2. In the following we simply writeIκ (φ, p) = Iκ(x, t, φ, p),Iκ (φ, p) =Iκ(x, t, φ, p). Note that

K (x, t, z, φ1 + φ2, p1 + p2) = K (x, t, z, φ1, p1)+ K (x, t, z, φ2, p2)

and hence that

Iκ(φ1 + φ2, p1 + p2) = Iκ(φ

1, p1)+ Iκ(φ2, p2),

I κ(φ1 + φ2, p1 + p2) = I κ(φ1, p1)+ I κ(φ2, p2).

3. The comparison principle: proof of Theorem 1.1

The purpose of this section is to prove Theorem 1.1 and hence throughout thesection we adopt the assumption stated in Theorem 1.1. In particular, let H be asin (1.2) and assume (A.1) and (A.2). Assume that ψi , ci, j , and gi are as stated inTheorem 1.1 and assume that (u−

1 , . . . , u−d ) and (u+

1 , . . . , u+d ) are viscosity sub- and

supersolutions, respectively, to the problem in (1.1). We first prove the followinglemma.

Lemma 3.1. The following is true for any γ > 0. Let θ ≥ 0. Then there existsη > 0, independent of θ , such that if λ ≥ η, then (u+

1 , . . . , u+d ),

u+i := u+

i + θe−λt (|x |2γ+2 + 1) for i ∈ {1, . . . , d},is a viscosity supersolution of (1.1).

Proof. Since u+i ∈ LSCp(R

N × [0, T ]) we have u+i ∈ LSCp(R

N × [0, T ]). Let(x0, t0) ∈ R

N × [0, T ] and assume, for some i ∈ {1, . . . , d}, that φi ∈ C1,2(RN ×[0, T )), satisfies

(i) φi (x0, t0) = u+i (x0, t0),

(i i) (x0, t0) is a global maximum of φi − u+i .

To prove the lemma it is enough to prove that there exists η > 0, independent of θ ,such that if λ ≥ η, then

min

{−Hκ(φi , u+


j �=i(−ci, j + u+

j )

}≥ 0 for all κ ∈ (0, 1).

Let �i = φi − θe−λt (|x |2γ+2 + 1) and note that by construction �i − u+i has

a global maximum at (x0, t0). Using that u+i is a supersolution we have

min

{−Hκ(�i , u+


j �=i(−ci, j + u+

j )

}≥ 0 (3.1)


for all κ ∈ (0, 1). Using (3.1) we see that

u+i − max

j �=i(−ci, j + u+

j ) = u+i − max

j �=i(−ci, j + u+

j ) ≥ 0

since u+i − u+

i is independent of i . To conclude the proof we hence only have toensure that

− Hκ(φi , u+i )− ψi ≥ 0 at (x0, t0) and for all κ ∈ (0, 1). (3.2)

Let φ(x, t) = θe−λt (|x |2γ+2 + 1). Then �i = φi − φ and u+i = u+

i + φ. Toestablish (3.2) we first note, using (3.1), that, at (x0, t0),

0≤−Hκ(�i , u+i )− ψi =−L�i − Iκ(�i , D�i )− Iκ(u+

i , D�i )− ψi . (3.3)

However, using Remark 2.2 we see that

Iκ(�i , D�i ) = Iκ(φi , Dφi )+ Iκ(−φ,−Dφ),

Iκ(u+i , D�i ) = Iκ(u+

i , Dφi )+ Iκ(−φ,−Dφ). (3.4)

Hence, combining (3.3) and (3.4) we can conclude that

0 ≤ −Lφi − Iκ(φi , Dφi )− Iκ(u+i , Dφi )− ψi

+Lφ − Iκ(−φ,−Dφ)− Iκ(−φ,−Dφ)

= −Hκ(φi , u+i )− ψi + Lφ − I(−φ,−Dφ).

In particular, we have

−Hκ(φi , u+i )− ψi ≥ −Lφ + I(−φ,−Dφ).

Using this we first note that

−Lφ ≥ λθe−λt (|x |2γ+2 + 1)− F(θe−λt (|x |2γ+2 + 1))

where

F :=N∑

i, j=1

ai, j (x, t)∂xi x j +N∑

i=1

ai (x, t)∂xi . (3.5)

Using the assumption on the operator H stated in Theorem 1.1 we see that

|F(θe−λt (|x |2γ+2 + 1))| ≤ cθe−λt (|x |2γ+2 + 1)

and hence

−Lφ ≥ (λ− c)φ.


To estimate I(−φ,−Dφ) we first note, by definition, that

I(x, t,−φ,−Dφ) = −∫

Rl\{0}K (x, t, z, φ, Dφ)ν (dz) ,

K (x, t, z, φ, Dφ) = φ (x + η (x, t, z) , t)− φ (x, t)

−χ{|z|≤1}(z)N∑

k=1

ηk (x, t, z) ∂xk φ.

Hence,

|I(x, t,−φ,−Dφ)| ≤∫

B(0,1)\{0}|K (x, t, z, φ, Dφ)|ν (dz) ,

+∫

Rl\B(0,1)

|K (x, t, z, φ, Dφ)|ν (dz)

=: T1 + T2.

Now, using (A.4), and Taylor’s formula, we first see that

|K (x, t, z, φ, Dφ)| ≤ c|z|2φ(x, t) whenever z ∈ B(0, 1) \ {0},|K (x, t, z, φ, Dφ)| ≤ cφ(x, t) whenever z ∈ R

l \ B(0, 1).

Hence, using this, (A.3), and (A.4) we can conclude that

T1 ≤ cφ(x, t) and T2 ≤ cφ(x, t).

Putting the estimates together we can conclude that, at (x0, t0) and for all κ ∈ (0, 1),

− Hκ(φi , u+i )− ψi ≥ −Lφ + I(−φ,−Dφ) ≥ (λ− c)φ(x0, t0). (3.6)

In view of (3.2) we see that (3.6) completes the proof of the lemma. �

3.1. Proof of Theorem 1.1

Assume that (u+1 , . . . , u+

d ) is a viscosity supersolution and that (u−1 , . . . , u−

d ) is aviscosity subsolution, respectively, to the problem in (1.1). We want to prove that

u−i (x, t) ≤ u+

i (x, t), for all i ∈ {1, . . . , d}, (3.7)

whenever (x, t) ∈ RN × (0, T ]. In fact, we will prove a slightly modified version

of (3.7). We let u−i (x, t) = et u−

i (x, t) and u+i (x, t) = et u+

i (x, t) for all i ∈{1, . . . , d}. One can then easily verify that (u−

1 , . . . , u−d ) is a viscosity subsolution

to the problem

min

{u−

i −Hu−i (x, t)−ψi (x, t), u−

i (x, t)− maxj �=i(−ci, j (x, t)+u−

j (x, t))

}=0,

u−i (x, T ) = gi (x), (3.8)


where

ψi (x, t) = etψi (x, t), ci, j (x, t) = et ci, j (x, t), gi (x) = eT gi (x),

if and only if (u−1 , . . . , u−

d ) is a subsolution to (1.1). Similarly, (u+1 , . . . , u+

d ) isa viscosity supersolution to the problem in (3.8) if and only if (u+

1 , . . . , u+d ) is a

supersolution to (1.1). Proving (3.7) is now equivalent to proving

u−i (x, t) ≤ u+

i (x, t), for all i ∈ {1, . . . , d}, (3.9)

whenever (x, t) ∈ RN × (0, T ]. However, according to Lemma 3.1 it suffices to

prove

u−i (x, t) ≤ u+

i (x, t), for all i ∈ {1, . . . , d}, (3.10)

whenever (x, t) ∈ RN × (0, T ], where

u+i := u+

i + θe−(λ−1)t (|x |2γ+2 + 1), γ > 0, (3.11)

since we easily recover (3.9) by letting θ → 0 in (3.10). In fact, it is enough toprove that

u−i (x, t) ≤ u+

i (x, t)+ θ

t, for all i ∈ Q, (3.12)

whenever (x, t) ∈ RN × (0, T ] and for any θ > 0, since the desired result is still

retrieved in the limit as θ → 0. In particular, we will prove (3.12) for θ fixed andthen let θ → 0. We divide the proof of (3.12) into a number of steps as outlinedbelow.

Step 1. (Setting up an argument by contradiction) Let in the following B(0, R),R > 0, be the standard Euclidean ball of radius R centered at 0. Using thatu−

i ∈ USCp(RN × [0, T ]) for i ∈ Q, we have that there exists γ0 ≥ 1

2 suchthat |u−

i (x, t)| ≤ c(1 + |x |2γ0). We now fix this γ0 and in the following we will,with slight abuse of notation, drop the subscript from γ0 and simply write γ . Basedon γ = γ0 fixed we let u+

i (x, t) be defined as in (3.11). Then, using u+i we see that

there exists R > 0 such that

u−i (x, t)− u+

i (x, t) < 0 whenever (x, t) ∈ (RN \ B(0, R))× [0, T ]. (3.13)

From (3.13) we see that

sup(x,t)∈RN ×[0,T ]

maxi∈{1,...,d}

(u−

i (x, t)− u+i (x, t)− θ

t

)

= sup(x,t)∈B(0,R)×[0,T ]

maxi∈{1,...,d}

(u−

i (x, t)− u+i (x, t)− θ

t

)

= maxi∈{1,...,d}

(u−

i (x, t)− u+i (x, t)− θ

t

)


for some (x, t) ∈ B(0, R) × (0, T ]. We will now prove (3.12) by contradiction.Indeed, assume that

maxi∈{1,...,d}

(u−

i (x, t)− u+i (x, t)− θ

t

)> 0. (3.14)

Using that u−i (x, T ) ≤ u+

i (x, T ), for all x ∈ RN , by the definition of sub- and

supersolution, we see that (x, t) ∈ B(0, R)× (0, T ). For (x, t) ∈ B(0, R)× (0, T )fixed, we let A be the non-empty set of all j ∈ {1, . . . , d} such that(

u−j (x, t)− u+

j (x, t)− θ

t

)= max

i∈{1,...,d}

(u−

i (x, t)− u+i (x, t)− θ

t

).

Step 2. (Doubling of the variables) For given degrees of freedom β > 0, � > 0,and ε, 0 < ε � 1, we introduce the function ϕε : R

N × RN × [0, T ] → R,

ϕε(x, y, t) = 1

2ε|x − y|2 +�(|x − x |2γ+2 + |y − x |2γ+2)+ β(t − t)2.

Note that ϕε is non-negative. Furthermore, for j ∈ A fixed, and ε, 0 < ε � 1, weconsider the function

φ jε (x, y, t) = u−

j (x, t)− u+j (y, t)− θ

t− ϕε(x, y, t),

where (x, y, t) ∈ B(0, R) × B(0, R) × [0, T ]. Using that u−j is upper semi-

continuous and that u+j is lower semi-continuous we can conclude that φ j

ε is uppersemi-continuous and hence that there exist (xε, yε, tε) such that

φ jε (xε, yε, tε) = sup

(x,y,t)∈B(0,R)×B(0,R)×[0,T ]φ jε (x, y, t). (3.15)

Note that the points (xε, yε, tε) depend also on β and �. However, in this part ofthe argument β and � are kept fixed and hence this dependence is harmless in thefollowing. Using that 2φ j (xε, yε, tε) ≥ φ j (xε, xε, tε)+ φ j (yε, yε, tε) we see that

1

ε|xε − yε|2 ≤ u−

j (xε, tε)− u−j (yε, tε)+ u+

j (xε, tε)− u+j (yε, tε), (3.16)

and as the right hand side of (3.16) is bounded we have that |xε − yε| → 0 asε → 0. Using that (x, t) ∈ B(0, R) × (0, T ), and the construction of ϕε, we seethat

u−j (x, t)− u+

j (x, t)− θ

t= φ j

ε (x, x, t)

≤ φ jε (xε, yε, tε) ≤ u−

j (xε, tε)− u+j (yε, tε)− θ

tε.

(3.17)

Furthermore, we see that we must have, using the definition of (x, t), (xε, yε, tε),and the upper semi-continuity of u−

j − u+j , that

(xε, yε, tε) → (x, x, t) as ε → 0. (3.18)


The above display also shows that, for ε small enough, we have tε ∈ (0, T ) sincetε → t and t ∈ (0, T ). Note also that

u−j (xε, tε) → u−

j (x, t) and u+j (yε, tε) → u+

j (x, t) as ε → 0. (3.19)

Indeed, recall that u−j is upper semi-continuous and assume, taking (3.18) into

account, that lim supε→0 u−j (xε, tε) < u−

j (x, t). Then, using (3.17) we have that

lim infε→0 u+j (yε, tε) < u+

j (x, t) but this contradicts the lower semi-continuity for

u+j . Similarly, assuming that lim infε→0 u+

j (yε, tε) > u+j (x, t) we see that

lim supε→0

u−j (xε, tε) > u−

j (x, t),

which contradicts the upper semi-continuity for u−j . Repeating (3.17) we also have

that

u−j (x, t)− u+

j (x, t)− θ

t= φ j

ε (x, x, t)

≤ φ jε (xε, yε, tε) = u−

j (xε, tε)− u+j (yε, tε)− ϕε(xε, yε, tε)− θ

tε.

In particular,

ϕε(xε, yε, tε) ≤ u−j (xε, tε)− u−

j (x, t)

+ u+j (x, t)− u+

j (yε, tε)+ θ

t− θ

tε

and using (3.19) we see that

limε→0

ϕε(xε, yε, tε) = 0.

In particular,

1

ε|xε − yε|2 → 0 as ε → 0. (3.20)

Step 3. (Using the no-loop condition) To proceed we will now argue as in Ishiiand Koike [9], using the no-loop condition (A.6) (i i), to conclude that there existsk ∈ A such that

u−k (x, t) > max

i∈Q−k(u−

i (x, t)− ck,i (x, t)). (3.21)

Indeed, assume, on the contrary, that

u−k (x, t) ≤ max

i∈Q−k(u−

i (x, t)− ck,i (x, t))

for all k ∈ A and hence, in particular, that

u−k (x, t)+ ck, j (x, t) ≤ u−

j (x, t)


for some j ∈ Q−k . Since (u+1 , . . . , u+

d ) is a supersolution to (3.8) we also have that

u+k (x, t) ≥ u+

j (x, t)− ck, j (x, t).

Combining the two inequalities above yields

u−k (x, t)− u−

j (x, t) ≤ −ck, j (x, t) ≤ u+k (x, t)− u+

j (x, t),

and hence

u−k (x, t)− u+

k (x, t)− θ

t≤ u−

j (x, t)− u+j (x, t)− θ

t. (3.22)

But k ∈ A so (3.22) is actually an equality and hence j ∈ A. Repeating thisargument as many times as necessary we get the existence of a loop of indices{i1, i2, . . . , i p, i p+1} such that i1 = i p+1 and

ci1,i2 + ci2,i3 + · · · + ci p,i p+1 = 0.

This contradicts our assumptions on the switching costs and hence (3.21) musthold.Step 4. (Completing the argument by contradiction by applying a maximum princi-ple for semi-continuous functions) We now consider k ∈ A such that (3.21) holdsand we intend to derive a contradiction to the assumption in (3.14). First, using(3.19) and (3.21) we see that there exists ε, 0 < ε � 1, such that

u−k (xε, tε) > max

i∈Q−k(u−

i (xε, tε)− ck,i (xε, tε)) when 0 < ε < ε. (3.23)

In particular, (3.23) ensures that u−k is above the obstacle at the points {(xε, tε)}ε<ε.

We next intend to apply the so called maximum principle for semi-continuous func-tions, in our case adapted to the non-local system, in a neighborhood of (xε, yε, tε)and for u−

k , and along the lines of Barles and Imbert [1]. To do this we first have tocalculate ∂tϕε, ∂xiϕε, ∂yiϕε and ∂xi y jϕε. Doing this we see that

∂tϕε(x, y, t) = 2β(t − t),

Dxϕε(x, y, t) = 1

ε(x − y)+�(2γ + 2)(x − x)|x − x |2γ ,

Dyϕε(x, y, t) = −1

ε(x − y)+�(2γ + 2)(y − x)|y − x |2γ .

Furthermore,

D2x,yϕε(x, y, t) = 1

ε

(I −I

−I I

)+

(l(x) 0

0 l(y)

)

where

l(x) = �(2γ + 2)|x − x |2γ I + 2γ�(2γ + 2)〈x − x, x − x〉|x − x |2γ−2,

and I is the identity matrix. Let SN be the set of all N × N -dimensional symmetricmatrices and let ϕαε be the sup-convolution of ϕε as defined in Sect. 4 of Barles and


Imbert [1]. We may now use Lemma 1 in Barles and Imbert [1] (more preciselyCorollary 2) to conclude that there exists, for any κ > 0, α(κ) such that for 0 <α ≤ α there exist C, D ∈ R and X,Y ∈ SN , such that

C + D = 2β(tε − t)− θ

t2ε

,

(X 00 −Y

)≤ D2

x,yϕε(xε, yε, tε)+ oα(1)

(I 00 I

), (3.24)

and such that

−C + u−k (xε, tε)−

N∑i, j=1

ai, j (xε, tε)Xi, j −N∑

i=1

ai (xε, tε)∂xiϕε(xε, yε, tε)

−Iκ(xε, tε, ϕαε (·, yε, tε), Dxϕε)

−Iκ(xε, tε, u−k (·, tε), Dxϕε)− ψk(xε, tε) ≤ 0, (3.25)

and

D + u+k (yε, tε)−

N∑i, j=1

ai, j (yε, tε)Yi, j +N∑

i=1

ai (yε, tε)∂yiϕε(xε, yε, tε)

−Iκ(yε, tε,−ϕαε (xε, ·, tε),−Dyϕε)

−Iκ(yε, tε, u+k (·, tε),−Dyϕε)− ψk(yε, tε) ≥ 0. (3.26)

In the above, we have used the little o notation, i.e., oα(1) → 0 as α → 0. Adding(3.25) and (3.26) we see that

−C − D + u−k (xε, tε)− u+

k (yε, tε)−N∑

i, j=1

(ai, j (xε, tε)Xi, j − ai, j (yε, tε)Yi, j )

−N∑

i=1

(ai (xε, tε)∂xiϕε(xε, yε, tε)+ ai (yε, tε)∂yiϕε(xε, yε, tε))

−(ψk(xε, tε)− ψk(yε, tε))

− (Iκ(xε, tε, ϕαε (·, yε, tε), Dxϕε)− Iκ(xε, tε,−ϕαε (xε, ·, tε),−Dyϕε)

)− (Iκ(yε, tε, u−

k (·, tε), Dxϕε)− Iκ(yε, tε, u+k (·, tε),−Dyϕε)

) ≤ 0.

Now, using standard arguments based on assumptions (A.1) and (A.2) it followsthat

∣∣∣∣∣N∑

i=1

(ai (xε, tε)∂xiϕε(xε, yε, tε)+ ai (yε, tε)∂yiϕε(xε, yε, tε)

)∣∣∣∣∣≤ c

(1

ε|xε − yε|2 +�

(1 + |xε||xε − x |2γ+1 + |yε||yε − x |2γ+1

) )


and ∣∣∣∣∣∣N∑

i, j=1

(ai, j (xε, tε)Xi, j − ai, j (yε, tε)Yi, j

)∣∣∣∣∣∣

≤ c

(1

ε|xε − yε|2 +�(1 + |xε|2|xε − x |2γ + |yε|2|yε − x |2γ )

).

Putting these estimates together we find that

−C − D + u−k (xε, tε)− u+

k (yε, tε)

−(Iκ(xε, tε, ϕαε (·, yε, tε), Dxϕε)− Iκ(xε, tε,−ϕαε (xε, ·, tε),−Dyϕε))

−(Iκ(yε, tε, u−k (·, tε), Dxϕε)− Iκ(yε, tε, u+

k (·, tε),−Dyϕε))

≤ c

ε|xε − yε|2 + c�h(xε, yε, x, y)+ ψk(xε, tε)− ψk(yε, tε)

where 0 ≤ h(xε, yε, x, y) ≤ c for all ε, 0 < ε ≤ ε. Hence, using the relation forC + D, see (3.24), we see that

u−k (xε, tε)− u+

k (yε, tε)− 2β(tε − t)+ θ

t2ε

−(Iκ(xε, tε, ϕαε (·, yε, tε), Dxϕε)− Iκ(xε, tε,−ϕαε (xε, ·, tε),−Dyϕε))



≤ c

ε|xε − yε|2 + c�h(xε, yε, x, y)+ ψk(xε, tε)− ψk(yε, tε).

To complete Step 4, and hence the proof of Theorem 1.1, we now need to estimatethe expressions involving the non-local operators. We state the appropriate estimatein the following lemma and postpone its proof to the next subsection.

Lemma 3.2. Using the assumptions and notation introduced above, we have∣∣∣∣−(Iκ(xε, tε, ϕαε (·, yε, tε), Dxϕε)− Iκ(xε, tε,−ϕαε (xε, ·, tε),−Dyϕε))



∣∣∣∣≤ c

ε|xε − yε|2 + c�h(xε, yε, x, y)+ f (κ)

ε+ oα(1), (3.27)

where 0 ≤ h(xε, yε, x, y) ≤ c for all ε, 0 < ε ≤ ε, and for some non-negativefunction f (κ) such that f (κ) → 0 as κ → 0.

Note that the points {(xε, yε, tε)}ε<ε are independent of κ and that the functionf (κ) is independent of ε. Assuming Lemma 3.2 we have that (3.27) holds andcombining the estimates in the last displays we see that


k (yε, tε)+ θ

t2ε

≤ 2β(tε − t)+ c

ε|xε − yε|2 + c�h(xε, yε, x, y)+ c�h(xε, yε, x, y)

+ψk(xε, tε)− ψk(yε, tε)+ f (κ)

ε+ oα(1).


Now, letting first α → 0 and then κ → 0 we get


k (yε, tε)+ θ

t2ε

≤ 2β(tε − t)+ c

ε|xε − yε|2 + c�h(xε, yε, x, y)+ c�h(xε, yε, x, y)

+ψk(xε, tε)− ψk(yε, tε).

Then, letting ε → 0, using (3.18), (3.20) and the continuity of ψk , we can concludethat

u−k (x, t)− u+

k (x, t)+ θ

t2 ≤ c�.

Finally, letting � → 0 in the last display we conclude that

u−k (x, t)− u+

k (x, t)+ θ

t2 ≤ 0,

which contradicts (3.14). This completes Step 4 and the proof of Theorem 1.1modulo Lemma 3.2.

3.2. Proof of Lemma 3.2

To prove Lemma 3.2 we have to prove the estimate in (3.27). However, by followingSection 5 of Barles and Imbert [1], we see that we may replace Iκ(xε, tε, ϕαε (·, yε,tε), Dxϕε) by Iκ(xε, tε, ϕε(·, yε, tε), Dxϕε)+ oα(1) and hence it suffices to prove(3.27) with ϕαε replaced by ϕε. To do this we first see, using Taylor’s formula, thedefinition of ϕε, (A.3) and (A.4), that

|Iκ(xε, tε, ϕε(·, yε, tε), Dxϕε)

− Iκ(yε, tε,−ϕε(xε, ·, tε),−Dyϕε)| ≤ f (κ)

ε

for all ε, 0 < ε ≤ ε, and for some non-negative function f (κ) as above. Bydominated convergence we have f (κ) → 0 as κ → 0 since

limκ→0

∫B(0,κ)

|z|2ν(dz) = limκ→0

∫B(0,1)

|z|2χ{|z|≤κ}ν(dz)

=∫

B(0,1)

limκ→0

(|z|2χ{|z|≤κ}

)ν(dz) = 0.

Hence, only the term involving Iκ remains. To conduct estimates we first decom-pose

−(Iκ(xε, tε, u−k (·, tε), Dxϕε)− Iκ(yε, tε, u+


= −(Iκ−(xε, tε, u−k (·, tε), Dxϕε)− Iκ−(yε, tε, u+


−(Iκ+(xε, tε, u−k (·, tε), Dxϕε)− Iκ+(yε, tε, u+



whereIκ− indicates integration with respect to z ∈ B(0, 1)\B(0, κ) andIκ+ indicatesintegration with respect to z ∈ R

l \ B(0, 1). Note that by construction, see (3.15),we can, working with index k, assume that the sequence {(xε, yε, tε)}ε<ε is suchthat

φkε (xε, yε, tε) = sup

(x,y,t)∈B(0,R)×B(0,R)×(0,T )φkε (x, y, t), where

φkε (x, y, t) = u−

k (x, t)− u+k (y, t)− θ

t− ϕε(x, y, t). (3.28)

To proceed we exploit that the maximizing property of the sequence {(xε, yε, tε)}ε<εin (3.28) implies, for z ∈ R

l \ B(0, κ), that

u−k (xε + η(xε, tε, z), tε)− u−

k (xε, tε)−∑

l

ηl(xε, tε, z)∂xlϕε(xε, yε, tε)

−u+k (yε + η(yε, tε, z), tε)+ u+

k (yε, tε)−∑

l

ηl(yε, tε, z)∂ylϕε(xε, yε, tε)

≤ ϕε(xε + η(xε, tε, z), yε + η(yε, tε, z), tε)− ϕε(xε, yε, tε)

−∑

l

ηl(xε, tε, z)∂xlϕε(xε, yε, tε)−∑

l

ηl(yε, tε, z)∂ylϕε(xε, yε, tε). (3.29)

To ease the notation in the calculations to follow we let ηεx (z) = η(xε, tε, z),ηεy(z) = η(yε, tε, z), and we let 〈·, ·〉 denote that standard scalar product in R

N . Toproceed we first note here that the left hand side of (3.29) is precisely the integrandof

Iκ(xε, tε, u−k (·, tε), Dxϕε)− Iκ(yε, tε, u+

k (·, tε),−Dyϕε), (3.30)

and hence, using (3.29), we see that we want to develop appropriate estimates forthe function

ϕε(xε + ηεx (z), yε + ηεy(z), tε)− ϕε(xε, yε, tε)

−〈(ηεx (z), ηεy(z)), (Dxϕε(xε, yε, tε), Dyϕε(xε, yε, tε))〉, (3.31)

where, as we recall,

ϕε(x, y, t) = 1

2ε|x − y|2 +�(|x − x |2γ+2 + |y − x |2γ+2)+ β(t − t)2.

A straight forward calculation shows that (3.31) can be simplified to

1

2ε|ηεx − ηεy |2

+�[

Aε−(2 + 2γ )(〈ηεx , (xε − x)〉|xε − x |2γ +〈ηεy, (yε − x)〉|yε − x |2γ

)],

(3.32)

where

Aε :=[|xε+ηεx − x |2+2γ − |xε − x |2+2γ + |yε + ηεy − x |2+2γ − |yε− x |2+2γ

].


We now first estimate the contribution to the expression in (3.30) coming fromintegration over z ∈ B(0, 1) \ B(0, κ), i.e., from the corresponding expressioninvolving Iκ−. Given z ∈ B(0, 1) \ B(0, κ) and using (A.4), we first see that

1

2ε|ηεx − ηεy |2 ≤ c

ε|z|2|xε − yε|2.

Concerning the second term of (3.32) we see, using the fundamental theorem ofcalculus and (A.4), that[

Aε − (2 + 2γ )(〈ηεx , (xε − x)〉|xε − x |2γ + 〈ηεy, (yε − x)〉|yε − x |2γ

)]

≤ c|z|2(1 + |xε − x |2γ+2 + |yε − x |2γ+2)

whenever z ∈ B(0, 1) \ B(0, κ). In particular, using the above estimates and (A.3)we can conclude that

|Iκ−(xε, tε, u−k (·, tε), Dxϕε)− Iκ−(yε, tε, u+

k (·, tε),−Dyϕε)|≤ c

(1

ε|xε − yε|2 +�(1 + |xε − x |2γ+2 + |yε − x |2γ+2)

). (3.33)

Finally, focusing on the contribution from the term Iκ+, through similar calculationsas above now using (A.3) on R

l \ B(0, 1), we find that (3.33) also holds with Iκ−replaced by Iκ+ and hence the proof of (3.27), and Theorem 1.1, is complete. �

4. Existence: proof of Theorem 1.2 and Theorem 1.3


To prove Theorem 1.2 we use Perron’s method. To construct a viscosity solutionto problem (1.1) using Perron’s method we define u = (u1, . . . , ud) through

ui (x, t) := inf{u+i (x, t) : (u+

1 , . . . , u+d ) is a supersolution of problem (1.1)}.

Note that u = (u1, . . . , ud) is well-defined based on the assumption of the existenceof a barrier from above. Let u∗ = (u∗

1, . . . , u∗d) and u∗ = (u∗,1, . . . , ud,∗) denote,

respectively, the upper and lower semi-continuous envelopes of u, i.e.,

u∗i (x, t)= lim sup

r→0{ui (y, s) : (y, s)∈(B (x, r)×(t − r, t + r))∩(RN ×(0, T ])},

ui,∗ (x, t) = − (−ui (x, t))∗ ,

for i ∈ {1, . . . , d}. Then, by definition and by Theorem 1.1 we have

u−i ≤ u∗

i , ui,∗ ≤ u+i and ui,∗ ≤ u∗

i ,

for all i ∈ {1, . . . , d}, and where u− = (u−1 , . . . , u−

d ) and u+ = (u+1 , . . . , u+

d ) aresub- and supersolutions to (1.1), respectively. The essence of Perron’s method isnow to prove that u∗ and u∗ are, respectively, a subsolution and a supersolutionto problem (1.1). It then follows by the comparison principle that u∗ ≤ u∗ and


hence u = u∗ = u∗ is a viscosity solution to the system in (1.1). We first provethat u∗ satisfies its corresponding terminal condition. To do this, fix a componenti ∈ {1, . . . , d} and a point y ∈ R

N . By assumption, there exists a barrier fromabove, u+,i,y = {u+,i,y,ε}ε>0, to the system in (1.1), in the sense of Definition 1.In particular, by definition ui (x, t) ≤ u+,i,y,ε

i (x, t) whenever (x, t) ∈ RN × [0, T ]

and for all ε > 0. Furthermore, since u+,i,y,ε is continuous we have that u∗i (x, t) ≤

(u+,i,y,εi )∗i (x, t) = u+,i,y,ε

i (x, t) for every ε > 0 and (x, t) ∈ RN × [0, T ]. In

particular, for component i and the point y we have that

u∗i (y, T ) ≤ lim

ε→0u+,i,y,ε

i (y, T ) = gi (y).

Since i and y are arbitrary in this argument we conclude that u∗ satisfies the correctterminal condition. That u∗ also satisfies the correct terminal condition is provedsimilarly using the assumption of barriers from below. Hence it only remains toprove that the remaining conditions for the property of being sub- and supersolutionsto the system hold for u∗ and u∗, respectively. However, after noticing that ourswitching costs are assumed to be continuous, see (A.5), this follows in the sameway, for the subsolution, as outlined on p. 70 in Biswas et al. [2], and for thesupersolution, as outlined on p. 71–72 in Biswas et al. [2]. We omit further detailsand conclude that the proof of Theorem 1.2 is complete.


In the light of Theorem 1.2 we see that Theorem 1.3 will follow if we find barriersfrom above and below, in the sense of Definition 1, for each i ∈ {1, . . . , d} andy ∈ R

N . To construct an appropriate barrier from above, for fixed i ∈ {1, . . . , d}and y ∈ R

N , we let, for all j ∈ {1, . . . , d},

u+,i,y,εj (x, t) = g(y)+ K

ε2 (T − t)

+L(e�(T −t) + 1)(|x − y|2 + ε)12 + ci, j (x, t), (4.1)

where K and� are non-negative degrees of freedom and L is the Lipschitz-constantof g(x). We first note that (A.7) together with (A.9) imply that ci, j (x, T ) ≥ 0 forall i, j ∈ {1, . . . , d}. Hence

u+,i,y,εj (x, T ) = g(y)+ 2L(|x − y|2 + ε)

12 + ci, j (x, T ) ≥ g(x),

whenever x ∈ RN and, in particular,

u+,i,y,εi (y, T ) = g(y)+ 2Lε

12 + ci,i (x, T ).

Hence, using (A.6) (i) we see that u+,i,y,εi (y, T ) = g(y)+ 2Lε

12 and

limε→0

u+,i,y,εi (y, T ) = g(y).


We next verify that u+,i,y,ε is a supersolution to (1.1). To do this we start byobserving that assumption (A.8) implies that u+,i,y,ε is above the obstacle. Indeed,

u+,i,y,εj (x, t)− max

k �= j(−c j,k(x, t)+ u+,i,y,ε

k (x, t))

> − maxk �= j

(−ci, j (x, t)− c j,k(x, t)+ ci,k(x, t)) ≥ 0.

Next, recall the notation L = ∂t + F with F as defined in (3.5). To complete theproof we now only have to prove that if K and � are large enough, then,

−Hu+,i,y,εj (x, t)− ψ j (x, t)

= −∂t u+,i,y,εj (x, t)− Fu+,i,y,ε

j (x, t)− I(x, t, u+,i,y,εj )− ψ j (x, t) ≥ 0.

(4.2)

In the below estimates, c will denote a constant independent of K , � and ε. Byassumption (A.10) we have

∂t u+,i,y,εj (x, t) ≤ − K

ε2 − �e�(T −t)(|x − y|2 + ε)12 + c(1 + |x |). (4.3)

Let in the following ||X || = max1≤i, j≤N |Xi, j | and note that by assumption (A.10)it follows that

supx∈RN

||∇2x ci, j (x, t)|| ≤ c

1 + |x | and

supx∈RN

||∇2x (|x − y|2 + ε)

12 || ≤ c

(1

ε32

+ 1

ε12

)≤ c

ε2 . (4.4)

Using the above display together with assumptions (A.1) and (A.2) we deduce, bystandard manipulations, that

|Fu+,i,y,εj | ≤ c(e�(T −t) + 1)(1 + 1

ε2 + |x |). (4.5)

Concerning the non-local term I(x, t, u+,i,y,εj ), using Taylor’s formula, assumption

(A.4) and (4.4) we deduce that if |z| ≤ 1 then∣∣∣∣u+,i,y,ε

j (x + η(x, t, z), t)− u+,i,y,εj (x, t)−

∑l

ηl(x, t, z)∂xl u+,i,y,εj (x, t)

∣∣∣∣≤ c|z|2 sup

x ′∈RN||∇2

x u+,i,y,εj (x ′, t)|| ≤ c

ε2 (e�(T −t) + 1)|z|2.

By assumption (A.10) we conclude Lipschitz continuity of u+,i,y,εj and hence, using

(A.4) we obtain, for |z| > 1, that∣∣∣∣u+,i,y,ε

j (x + η(x, t, z), t)− u+,i,y,εj (x, t)

∣∣∣∣ ≤ c|η(x, t, z)| ≤ c min{|z|, 1} ≤ c.


Using the two displays above and taking assumption (A.3) into account we see that

| − I(x, t, u+,i,y,εj )| ≤ c

ε2 (e�(T −t) + 1). (4.6)

Putting (4.3), (4.5) and (4.6) together we find that for K and � large enough

Hu+,i,y,εj (x, t)− ψ j (x, t)

≥ K

ε2 + �e�(T −t)(|x − y|2 + ε)12 − ce�(T −t)

(1 + 1

ε2 + |x |)

≥ 0.

Hence (4.2) holds and thus u+,i,y,ε is a supersolution to (1.1).To construct an appropriate barrier from below, for fixed i ∈ {1, . . . , d} and

y ∈ RN , we let, for all j ∈ {1, . . . , d},

u−,i,y,εj (x, t) = g(y)− K

ε2 (T − t)

−L(e�(T −t) + 1)(|x − y|2 + ε)12 , (4.7)

where again K and � are non-negative degrees of freedom and L is the Lipschitz-constant of g(x). The above argument can now be repeated to conclude that u−,i,y,εis a subsolution to (1.1) and hence the proof of Theorem 1.3 is complete. We omitfurther details. �Remark 4.1. Note that a local version of Theorem 1.3 is proved as Theorem 2.4in Lundström et al. [11]. In the proof of Theorem 2.4 in Lundström et al. [11]the barriers in (4.1) and (4.7) are also used. However, there are two typo errors inthe statements of these barriers in Lundström et al. [11]. Indeed, the factors e−λt

and K stated in the corresponding construction in Lundström et al. [11] should becorrected and replaced by eλ(T −t) and K

ε2 respectively, as above. The subsequentcalculation/argument in Lundström et al. [11] should also be modestly adjustedaccordingly.

References

[1] Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscositysolutions theory revisited. Annales de l’Institut Henri Poincaré 25, 567–585 (2008)

[2] Biswas, I.H., Jakobsen, E.R., Karlsen, K.H.: Viscosity solutions for a system ofintegro-PDEs and connections to optimal switching and control of jump-diffusionprocesses. Appl. Math. Optimiz. 62, 47–80 (2010)

[3] Djehiche, B., Hamadene, S.: On a finite horizon starting and stopping problem withrisk of abandonment. Int. J. Theoret. Appl. Finance 12, 523–543 (2009)

[4] Djehiche, B., Hamadene, S., Popier, A.: A finite horizon optimal multiple switchingproblem. SIAM J. Control Optimiz. 48, 2751–2770 (2010)

[5] El-Asri, B., Fakhouri, I.: Optimal multi-modes switching with the switching cost notnecessarily positive, arXiv:1204.1683v1, 2012

[6] El-Asri, B., Hamadene, S.: The finite horizon optimal multi-modes switching problem:the viscosity solution approach. Appl. Math. Optimiz. 60, 213–235 (2009)

http://arxiv.org/abs/1204.1683v1


[7] Hu, Y., Tang, S.: Multi-dimensional BSDE with oblique reflection and optimal switch-ing. Prob. Theory Relat. Fields 147, 89–121 (2010)

[8] Hamadéne, S., Zhang, J.: Switching problem and related system of reflected backwardSDEs. Stoch. Process. Appl. 120, 403–426 (2010)

[9] Ishii, H., Koike, S.: Viscosity solutions of a system of nonlinear second-order ellipticPDEs arising in switching games. Funkcialaj Ekvacioj 34, 143–155 (1991)

[10] Jakobsen, E.R., Karlsen, K.H.: A maximum principle for semicontinuous functionsapplicable to integro-partial differential equations. Nonlin. Differ. Equ. Appl. 13, 137–165 (2006)

[11] Lundström, N., Nyström, K., Olofsson, M.: Systems of variational inequalities in thecontext of optimal switching problems and operators of Kolmogorov type. Annali. di.Mathematica (2013). doi:10.1007/s10231-013-0325-y

[12] Pham, H.: Optimal stopping, free boundary, and American option pricing in a jump-diffusion model. Appl. Math. Optimiz. 35, 145–164 (1997)

http://dx.doi.org/10.1007/s10231-013-0325-y

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