Viscosity Solution of Optimal Stopping Problem
for Stochastic Systems with Bounded Memory
∗
Mou-Hsiung Chang
†Tao Pang
‡Moustapha Pemy
§April 5, 2012
Abstract
We consider a finite time horizon optimal stopping problem for a sys-tem of stochastic functional differential equations with a bounded memory. Under some sufficiently smooth conditions, a Hamilton-Jacobi-Bellman (HJB) variational inequality for the value function is derived via dynam-ical programming principle. It is shown that the value function is the unique viscosity solution of the HJB variational inequality.
KEYWORDS:optimal stopping, stochastic control, stochastic functional dif-ferential equations.
AMS 2000 subject classifications:primary 60G40, 60G20; secondary 60H30, 93E20
∗The research of this paper is partially supported by a grant W911NF-04-D-0003 from the U. S. Army Research Office
†Mathematical Sciences Division, U. S. Army Research Office, P. O. Box 12211, RTP, NC 27709, USA. Email: [email protected]
‡Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA. Email: [email protected] (Corresponding Author)
§Department of Mathematics, Towson University, Towson, MD 21252-0001, USA. Email: [email protected]
1
Introduction
Optimal control and optimal stopping problems over a finite or an infinite time horizon for Itˆo’s diffusion processes arise in many areas of science, engineering, and finance (see e.g. Fleming and Soner [14], Øksendal [35], Shiryaev [39], Karazas and Shreve [19] and references contained therein). The value function of these problems are normally expressed as a viscosity or a generalized solution of a Hamilton-Jacobi-Bellman equation (HJBE) or a Hamilton-Jacobi-Bellman variational inequality (HJBVI) that involves a second order parabolic or elliptic partial differential equation in a finite dimensional Euclidean space (see e.g. Lions [30] and [32]).
In an attempt to achieve better accuracy and to account for the delayed effect of the state variables in the modelling of real world stochastic control problems, the stochastic delay equations and controlled stochastic delay equations have been the subject of intensive studies in recent years by many researchers such as Elsanousi [11], Larrssen [25], Elsanousi et al [13], Øksendal and Sulem [37], Larrssen and Risebro [27], and Bauer and Rieder [5].
The controlled or uncontrolled stochastic delay equations considered by the aforementioned researchers are described by the following special classes of equa-tions that contain discrete and averaged delays:
dX(s) = αs, X(s), X(s−r), Z 0 −r eλθX(s+θ)dθ, u(s)ds (1) + βs, X(s), X(s−r), Z 0 −r eλθX(s+θ)dθ, u(s)dW(s), s∈[t, T], or dX(s) = αs, X(s), X(s−r), Z 0 −r eλθX(s+θ)dθds (2) + βs, X(s), X(s−r), Z 0 −r eλθX(s+θ)dθdW(s), s∈[t, T].
In the above equations, W(·) = {W(s), s ≥0} is an m-dimensional standard Brownian motion defined on a certain filtered probability space (Ω,F, P,F),
u(·) ={u(s), s∈[t, T]}is a control process taking values in the control setU in an Euclidean space,αandβ areRn andRn×m-valued functions defined on
[0, T]×Rn×Rn×Rn×U,
or
[0, T]×Rn×Rn×Rn,
andλ >0 is a given constant.
Based on the above two equations or their variants, Larssen [25] obtained an HJB equation for an optimal control problem, Elsanousi et al [13] consid-ered a singular control problem and obtained certain explicitly available solu-tions, Øksendal and Sulem [37] derived the maximum principle for the optimal
stochastic control. If the dynamics of the control problem with delay exhibit a special structure, Larssen and Risebro [27] and Bauer and Rieder [5] showed that the value function actually lives in a finite-dimensional space and the orig-inal problem can be reduced to a classical stochastic control problem without delay. Elsanousi and Larssen [12] treated an optimal control problem and its applications to consumption for (1) under partial observation. We also men-tion that optimal stopping problems were studied in Elsanousi’s unpublished dissertation [11] for such special type of equations.
This paper extends the results obtained for finite dimensional diffusion pro-cesses and stochastic delay equations described in (2) and investigates an opti-mal stopping problem over a finite time horizon for a general system of stochastic functional differential equations (SFDE) described below:
dX(s) =f(s, Xs)dt+g(s, Xs)dW(s), s∈[t, T], (3)
whereT >0 and t∈[0, T], respectively, denote the terminal time and an ini-tial time of the optimal stopping problem. Again,W(·) = {W(s), s≥0} is a standardm-dimensional Brownian motion, and the driftf(s, Xs) and the
diffu-sion coefficientg(s, Xs) (taking values in Rn and Rn×m, respectively) depend
explicitly on the segmentXs of the state processX(·) ={X(s), s∈[t−r, T]}
over the time interval [s−r, s], whereXs: [−r,0]→Rn is defined byXs(θ) =
X(s+θ), θ ∈[−r,0]. The consideration of such a system enables us to model many real world problems that have aftereffects (see e.g. Kolmanovsky and Shaikhet [24] and references contained therein for application examples). It is clear that this equation also includes (2) as a special case and many other equations that can not be modelled in the form of (2). Whenr= 0, it is also clear that the SFDE (3) reduces to the following Itˆo’s diffusion process (without delay):
dX(s) =f(s, X(s))ds+g(s, X(s))dW(s), s∈[t, T].
This paper treats a finite time horizon optimal stopping problem (see Section 2 for the problem statement). We derive an infinite dimensional HJB variational inequality (HJBVI) for the value function via a dynamic programming principle (see e.g. Larssen [25]). It is shown that the value function is the unique viscosity solution of the HJBVI. The proof of uniqueness involves embedding the function spaceW1,2((−r,0);Rn) into the Banach spaceC([−r,0];Rn) and extending the
concept of viscosity solution for controlled Itˆo’s diffusion process developed by Crandall et al [9], and Lions [30] and [32] to an infinite dimensional setting.
Although infinite dimensional HJBVIs for optimal stopping problems and their applications to pricing of American option have been studied very recently by a few researchers, they either considered stochastic delay equation of special form (2)(see e.g. Gapeev and Reiss [16] and [17]) or stochastic equations in Hilbert spaces (see e.g. Gatarek and ´Swiech [15] and Barbu and Marinelli [4]). This paper differs from the aforementioned papers in the following significant ways: i) The segmented solution process {Xs, s ∈ [t, T]} is a strong Markov
process in the Banach space C([−r,0];Rn) whose norm is not differentiable
[15] and [4]; ii) the infinite-dimensional HJBVI uniquely involves the extensions
DV(t, ψ) andD2V(t, ψ) of first and second order Fr´echet derivativesDV(t, ψ)
andD2V(t, ψ) fromC∗andC†to (C⊕B)∗and (C⊕B)†(see Subsection 3.1 for definitions of these spaces), respectively; and iii) the infinite-dimensional HJBVI also involves the infinitesimal generatorSV(t, ψ) of the semigroup of shift oper-ators value functions that does not appear in the special class of equations (2) in the aforementioned papers.
This paper is organized as follows. The notation and preliminary results that are needed for formulating the optimal stopping problem as well as the problem statement are contained in Section 2. In Section 3, the HJBVI for the value function is heuristically derived using Bellman’s type dynamic program-ming principle. The verification theorem is also proved there. In Section 4, the continuity of the value function is proved. Although continuous, the value func-tion is not known to be smooth enough to be a classical solufunc-tion of the HJBVI in general cases. It is shown in Section 4, however, that the value function is the unique viscosity solution of the HJBVI.
2
The Optimal Stopping Problem
Letr > 0 be a fixed constant, and let J = [−r,0] denote the duration of the bounded memory of the stochastic functional differential equations considered in this paper. For the sake of simplicity, denoteC(J;Rn), the space of continuous functions φ : J → Rn, by C. Note that C is a real separable Banach space under the sup-norm defined by
kφk= sup
θ∈J
|φ(θ)|, φ∈C
where| · |is the Euclidean norm inRn.
In addition to the space C, we also consider L2(
J;Rn), the Hilbert space of Lebesque squared-integrable functionsφ :J →Rn equipped with the inner
product (· | ·) and the Hilbertian normk · k2defined by
(φ|ψ) = Z 0 −r hφ(θ), ψ(θ)idθ and kφk2= (φ|φ) 1 2, ∀φ, ψ∈L2(J,Rn),
whereh ·,· ithe inner product inRn.
Note that the space C can be continuously embedded into L2(
J;Rn) (see e.g. Rudin [38]). In fact, it is easy to verify that
kφk2≤
√
rkφk, ∀φ∈C.
Convention 2.1 Throughout the rest of the paper, letT >0 denote the termi-nal time andt ∈[0, T]be an initial time of the optimal stopping problem. We shall use the following conventional notation for functional differential equations (see Hale [18]):
If ψ∈C([t−r, T];Rn)ands∈[t, T], letψ
s∈C be defined by
In addition, throughout the rest of the paper, we will useKand Λ to denote generic constants and their values may change from line to line.
Let {W(s), s ≥ 0} be an m-dimensional standard Brownian motion de-fined on a complete filtered probability space (Ω,F, P;F), where F denotes the smallest σ-field that contains all subsets of Ω, andF = {F(s), s ≥ 0} is theP-augmentation of the natural filtration{FW(s), s≥0} generated by the
Brownian motion{W(s), s≥0},i.e.,ifs≥0,
FW(s) =σ{W(t),0≤t≤s}
and
F(s) =FW(s)∨ {A⊂Ω|∃B∈ F such thatA⊂B andP(B) = 0},
where the operator ∨ denotes that F(s) is the smallest σ-algebra such that
FW(s)⊂ F(s) and
{A⊂Ω|∃B∈ F such thatA⊂B andP(B) = 0} ⊂ F(s).
Consider the following system of stochastic functional differential equations:
dX(s) =f(s, Xs)ds+g(s, Xs)dW(s), s∈[t, T]; (4)
with the initial functionXt=ψt, whereψtis a givenC-valued random variable
that isF(t)-measurable. Here,f : [0, T]×C→Rn andg: [0, T]×C→Rn×m
are given deterministic functionals.
Let L2(Ω,C) be the space of C-valued random variables Ξ : Ω → C such
that kΞkL2 = Z Ω kΞ(ω)k2dP(ω) 12 <∞.
LetL2(Ω,C;F(t)) be the space of those Ξ∈L2(Ω,C) which areF(t)-measurable.
Definition 2.2 A process {X(s;t, ψt), s ∈ [t−r, T]} is said to be a (strong)
solution of (4) on the interval [t−r, T]and through the initial datum (t, ψt)∈
[0, T]×L2(Ω,C;F(t))if it satisfies the following conditions:
1. Xt(·;t, ψt) =ψt;
2. X(s;t, ψt)isF(s)-measurable for eachs∈[t, T];
3. The process {X(s;t, ψt), s∈ [t, T]} is continuous and it satisfies the
fol-lowing stochastic integral equationP-a.s.
X(s) =ψt(0) + Z s t f(λ, Xλ)dλ+ Z s t g(λ, Xλ)dW(λ), s∈[t, T]. (5)
In addition, the (strong) solution process{X(s;t, ψt), s∈[t−r, T]}of (4) is said
to be (strongly) unique if{X˜(s;t, ψt), s∈[t−r, T]} is also a (strong) solution
of (4) on[t−r, T] and through the same initial datum(t, ψt), then
P{X(s;t, ψt) = ˜X(s;t, ψt),∀s∈[t, T]}= 1.
Throughout the rest of the paper, we assume that the functionalsf : [0, T]×
C→Rn, andg: [0, T]×C→Rn×mare continuous functionals and they satisfy the following conditions (Assumption 2.3 & 2.4) to ensure the existence and uniqueness of a (strong) solution{X(s;t, ψt), s∈ [t−r, T]} for each (t, ψt)∈
[0, T]×L2(Ω, C;F(t)). (See Mohammed [33, 34].)
Assumption 2.3 There exists a constant K >0such that
|f(t, ϕ)−f(s, φ)|+|g(t, ϕ)−g(s, φ)|
≤ K(|t−s|+kϕ−φk) ∀(t, ϕ),(s, φ)∈[0, T]×C.
Assumption 2.4 There exists a constant K >0such that
|f(t, φ)|+|g(t, φ)| ≤K(1 +kφk) ∀(t, φ)∈[0, T]×C.
Let {X(s;t, ψt), s ∈ [t, T]} be the solution of (4) through the initial
da-tum (t, ψt) ∈ [0, T] ×C. We consider the corresponding C-valued process
{Xs(t, ψt), s∈[t, T]}defined by
Xs(θ;t, ψt)≡X(s+θ;t, ψt), ∀θ∈J. (6) For each t∈[0, T], define G(t)≡S
s∈[t,T]G(t, s) whereG(t, s) is defined by
G(t, s) =σ(X(u;t, ψt), t≤u≤s).
Note that, it can be shown that for eachs∈[t, T],
G(t, s) =σ(Xu(t, ψt), t≤u≤s).
This is due to the sample path’s continuity of the process{X(s;t, ψt), s∈[t, T]}.
One can then establish the following Markov property (see Mohammed [33], [34]).
Theorem 2.5 Let Assumptions 2.3 and 2.4 hold. Then the correspondingC -valued solution process of (4) describes a C-valued Markov process in the fol-lowing sense:
For any (t, ψt)∈[0, T]×L2(Ω,C), the Markovian property
P{Xs(t, ψt)∈B|G(t, α)}=P{Xs(t, ψt)∈B|Xα(t, ψt)} ≡p(α, Xα(t, ψt), s, B)
holds a.s. fort ≤α≤s andB ∈ B(C), where B(C)is the Borel σ-algebra of subsets ofC.
In the above, the functionp: [t, T]×C×[t, T]× B(C)→[0,1] denotes the transition probabilities of theC-valued Markov process{Xs(t, ψt), s∈[t, T]}.
A random functionτ: Ω→[0,∞] is said to be aG(t)-stopping time if
{τ ≤s} ∈ G(t, s), ∀s≥t.
LetT be the collection of allG(t)-stopping times, and letTT
t be the collection
of allG(t)-stopping timesτ ∈ T such thatt≤τ ≤T a.s.. For eachτ∈ TT t , let
the sub-σ-algebraG(t, τ) ofF be defined by
G(t, τ) ={A∈ F |A∩ {t≤τ≤s} ∈ G(t, s)∀s∈[t, T]}.
With a little bit more effort, one can show that the correspondingC-valued solution process of (4) is also a strong Markov process inC. That is
P{Xs(t, ψt)∈B|G(t, τ)}=P{Xs(t, ψt)∈B|Xτ(t, ψt)} ≡p(τ, Xτ(t, ψt), s, B)
holds a.s. for allτ∈ TT
t , all deterministics∈[τ, T], and B∈ B(C).
If the drift f and the diffusion coefficient g are time-independent, i.e.,
f(s, φ)≡f(φ) andg(s, φ)≡g(φ), then (4) reduces to the following autonomous system:
dX(s) =f(Xs)ds+g(Xs)dW(s). (7)
In this case, we usually assume the initial datum (t, ψt) = (0, ψ) and
de-note the solution process of (7) through (0, ψ) and on the interval [−r, T] by {X(s;ψ), s ∈ [−r, T]}. Then the corresponding C-valued solution process
{Xs(ψ), s∈[−r, T]} of (7) is a strong Markov process with time-homogeneous
probability transition functionp(ψ, s, B)≡p(0, ψ, s, B) =p(t, ψ, t+s, B) for all
s, t≥0,ψ∈C, andB ∈ B(C).
Assume Land Ψ are twok · k2-Lipschitz continuous real-valued functionals
on [0, T]×C with at most polynomial growth in L2(
J;Rn). In other words, there exist positive constantsK, Λ, andk≥1 such that
|L(t, ψ)−L(s, φ)|+|Ψ(t, ψ)−Ψ(s, φ)| ≤ K(|t−s|+kψ−φk2)
≤ K(|t−s|+kψ−φk) ∀(t, ψ),(s, φ)∈[0, T]×C. (8) and
|L(t, φ)|+|Ψ(t, φ)| ≤Λ(1 +kφk2)k, ∀(t, φ)∈[0, T]×C. (9)
Given the initial datum (t, ψ)∈[0, T]×C, our objective is to find an optimal stopping time τ∗ ∈ TT
t that maximizes the following expected performance
index: J(τ;t, ψ)≡E Z τ t e−ρ(s−t)L(s, Xs)ds+e−ρ(τ−t)Ψ(τ, Xτ) , (10)
where ρ > 0 denotes a discount factor. In this case, the value function V : [0, T]×C→Ris defined to be
V(t, ψ)≡ sup
τ∈TT t
For the autonomous case,i.e.,
dX(s) =f(Xs)dt+σ(Xs)dW(s), s∈[t, T], (12)
the following optimal stopping problem is a special case of what will be treated in this paper: Find an optimal stopping time τ∗ ∈ TT
0 that maximizes the
following expected performance index:
J(τ;ψ)≡E Z τ 0 e−ρsL(Xs)ds+e−ρτΨ(Xτ) . (13)
In this case, the value functionV :C→Ris defined to be
V(ψ)≡ sup
τ∈TT
0
J(τ;ψ). (14)
3
HJB Variational Inequality
3.1
The Infinitesimal Generator
Let C∗ and C† be the space of bounded linear functionals Φ : C → R and bounded bilinear functionals ˜Φ : C×C → R, of the space C, respectively. They are equipped with the operator norms which will be, respectively, denoted byk · k∗ andk · k†.
LetB={v1{0}, v∈Rn}, where1{0}: [−r,0]→Ris defined by
1{0}(θ) =
0 forθ∈[−r,0),
1 forθ= 0. We form the direct sum
C⊕B={φ+v1{0}|φ∈C, v∈Rn}
and equip it with the norm, also denoted byk · k, defined by
kφ+v1{0}k= sup θ∈[−r,0]
|φ(θ)|+|v|, φ∈C, v∈Rn.
Again, let (C⊕B)∗ and (C⊕B)† be spaces of bounded linear and bilinear
functionals ofC⊕B, respectively.
The following two results can be found in Lemma 3.1 and Lemma 3.2 on pp 79-83 of Mohammed [33].
If Γ ∈ C∗, then Γ has a unique (continuous) bilinear extension Γ : (¯ C⊕
B)∗→Rsatisfying the following weak continuity property: (W1) If {ξ(k)}∞
k=1 is a bounded sequence in C such that ξ(k)(θ)→ξ(θ)as k→ ∞ for allθ∈J for someξ∈C⊕B, thenΓ(ξ(k))→Γ(¯ ξ)as k→ ∞. The
If Γ∈C†, then Γhas a unique (continuous) linear extensionΓ¯ ∈(C⊕B)†
satisfying the following weak continuity property: (W2) If {ξ(k)}∞
k=1 and {ζ (k)}∞
k=1 are bounded sequences in C such that ξ(k)(θ)→ξ(θ)andζ(k)(θ)→ζ(θ)ask→ ∞for allθ∈Jfor someξ, ζ ∈C⊕B,
thenΓ(ξ(k), ζ(k))→Γ(¯ ξ, ζ)ask→ ∞.
For a sufficiently smooth functional Φ :C → R, we can define its Fr´echet derivatives with respect toφ∈C. (For more details about Fr´echet derivatives, please refer to [2].) From the results stated above, we know that its first order Fr´echet derivative,DΦ(φ)∈C∗, has a unique and (continuous) linear extension
DΦ(φ)∈(C⊕B)∗. Similarly, its second order Fr´echet derivative,D2Φ(φ)∈C†,
has a unique and continuous linear extensionD2Φ(φ)∈(C⊕B)†.
For a Borel measurable function Φ :C→R, we also define
S(Φ)(φ)≡ lim h→0+ 1 h h Φ( ˜φh)−Φ(φ) i (15)
for all φ ∈ C, where ˜φ : [−r, T] → Rn is an extension of φ : [−r,0] → Rn
defined by ˜ φ(t) = φ(t) ift∈[−r,0) φ(0) ift≥0, and again ˜φt∈Cis defined by
˜
φt(θ) = ˜φ(t+θ), θ∈[−r,0].
Let ˆD(S), the domain of the operator S, be the set of Φ : C →R such that the above limit exists for eachφ∈C. Define D(S) as the set of all functionals Ψ : [0, T]×C→Rsuch that Ψ(t,·)∈Dˆ(S),∀t∈[0, T].
Throughout the end, letClip1,2([0, T]×C) be the space of functions Φ : [0, T]×
C→Rwhich satisfies the following conditions:
1. ∂∂tΦ : [0, T]×C →R, DΦ : [0, T]×C→C∗ andD2Φ : [0, T]×C →C†
exist and are continuous.
2. Its second order Fr´echet derivativeD2Φ satisfies the following global Lip-schitz condition:
kD2Φ(t, φ)−D2Φ(t, ϕ)k† ≤Kkφ−ϕk ∀t∈[0, T], φ, ϕ∈C.
The following example shows that ˆD(S) is not a subset ofC2
lip(C). Therefore,
it is not redundant to require that a function Φ∈Clip1,2([0, T]×C)∩ D(S) for deriving the infinite-dimensional HJB variational inequality.
Example. Assume n= 1. Let Φ :C→Rbe independent of the time variable and be defined by
Φ(φ) =φ−r
2
Then it is clear that Φ∈C2(C) but Φ∈/ Dˆ(S), since∀ ϕ∈C, DΦ(φ)(ϕ) =ϕ−r 2 , D2Φ(φ)(ϕ, ϕ) = 0,
andD2Φ is globally Lipschitz. However,
S(Φ)(φ) = lim t→0+ 1 t h Φ( ˜φt)−Φ(φ) i = lim t→0+ ˜ φ t−r 2 −φ −r 2 t
which exists only ifφhas a right derivative at−r
2.
Let Φ : [0, T]×C → R be a Borel measurable function and consider the following two smoothness conditions:
Smoothness Condition (i). Φ∈Clip1,2([0, T]×C).
Smoothness Condition (ii).Φ∈ D(S),i.e.,Φ(t,·)∈Dˆ(S) for eacht∈[0, T]. The following result will be used later on in this paper.
Theorem 3.1 (Mohammed [33], [34]) Suppose that Φ∈C([0, T]×C) sat-isfies Smoothness conditions (i) and (ii). Let{Xs, s∈ [t, T]} be theC-valued
Markov solution process of Equation (4) with the initial data(t, ϕt)∈[0, T]×C.
Then lim ↓0 E[Φ(t+, Xt+)]−Φ(t, ϕt) = ∂ ∂tΦ(t, ϕt) +S(Φ)(t, ϕt) +DΦ(t, ϕt)(f(t, ϕt)1{0}) +1 2 m X j=1 D2Φ(t, ϕ t)(g(t, ϕt)(ej)1{0}, g(t, ϕt)(ej)1{0}), (16)
whereej, j= 1,2,· · ·, m, is thejth vector of the standard basis in Rm.
3.2
Heuristic Derivation
We consider the following HJBVI: max Ψ−V,∂V ∂t +AV +L−ρV = 0 (17) on [0, T]×C, where AV(t, ψ) ≡ S(V)(t, ψ) +DV(t, ψ)(f(t, ψ)1{0}) +1 2 m X i=1 D2V(t, ψ)(g(t, ψ)e i1{0}, g(t, ψ)ei1{0}). (18)
The above inequality shall be interpreted as follows. ∂V ∂t +AV +L−ρV ≤0 and V ≥Ψ (19) and ∂V ∂t +AV +L−ρV (V −Ψ) = 0 (20) on [0, T]×C, where Ais defined by (18).
We heuristically derive the above variational inequality as follows. A rigorous derivation will be provided as a byproduct when we prove that the value function is a viscosity solution of the HJBVI in the next section. First, we prove that
V(t, ψ)≥Ψ(t, ψ) for all (t, ψ)∈[0, T]×C. Let ˜τ ∈ TT
t . If ˜τ=t, then by (10), we can get J(˜τ;t, ψ) = Ψ(t, ψ). Therefore, V(t, ψ) = sup τ∈TT t J(τ;t, ψ)≥J(˜τ;t, ψ) = Ψ(t, ψ). (21)
The following dynamic programming principle (see Larssen [25]) will be used to derive our HJBVI:
V(t, ψ)≥E " Z t+δ t e−ρ(s−t)L(s, Xs)ds+e−ρδV(t+δ, Xt+δ) # , ∀δ≥0. (22)
From this principle and Theorem 3.1, we have lim δ↓0 E[e−ρδV(t+δ, X t+δ)−V(t, ψ)] δ = lim δ↓0 E[e−ρδV(t+δ, Xt+δ)−V(t+δ, Xt+δ)] δ + lim δ↓0 E[V(t+δ, Xt+δ)−V(t, ψ)] δ = −ρV(t, ψ) +∂V ∂t(t, ψ) +AV(t, ψ) ≤ −L(t, ψ),
for all (t, ψ)∈[0, T]×Cprovided thatV ∈Clip1,2([0, T]×C)∩ D(S). That is,
∂V
∂t +AV +L−ρV ≤0. (23)
From (21) and (23), it follows that max{Ψ−V,∂V
on [0, T]×C. The derivation of the inequality
max{Ψ−V,∂V
∂t +AV +L−ρV} ≥0 (25)
can be found in the next section.
We therefore have the following result.
Theorem 3.2 SupposeV : [0, T]×C is the value function defined by (11) and satisfies Smoothness Conditions (i) and (ii). Then the value functionV satisfies the following HJB variational inequality:
max Ψ−V,∂V ∂t +AV +L−ρV = 0 (26) on[0, T]×C, andV(T, ψ) = Ψ(ψ), ∀ψ∈C.
Note that it is not known that the value functionV satisfies the Smoothness Conditions (i) and (ii). Therefore, we need to consider viscosity solutions instead of classical solutions for HJB variational inequality (26). In fact, it will be shown that the value function is a unique viscosity solution of the HJB variational inequality (26). These results shall be given in the next section.
4
Viscosity Solution
In this section, we shall show that the value functionV defined by (11) is actually the unique viscosity solution of the HJB variational inequality (26). First, let us define the viscosity solution of (26) as follows.
Definition 4.1 Letw∈C([0, T]×C). We say thatwis a viscosity subsolution of (26) if for everyΓ : [0, T]×C→Rsatisfying Smoothness Conditions (i)-(ii) on[0, T]×C, we have min Γ(t, ψ)−Ψ(t, ψ), ρΓ(t, ψ)−∂Γ ∂t(t, ψ)− AΓ(t, ψ)−L(t, ψ) ≤0
at every (t, ψ)∈[0, T]×C which is a local maximum of w−Γ, with Γ(t, ψ) =
w(t, ψ).
We say thatwis a viscosity supersolution of (26) if, for everyΓ : [0, T]×C→
Rsatisfying Smoothness Conditions (i)-(ii) on[0, T]×C, we have
min Γ(t, ψ)−Ψ(t, ψ), ρΓ(t, ψ)−∂Γ ∂t(t, ψ)− AΓ(t, ψ)−L(t, ψ) ≥0.
at every(t, ψ)∈[0, T]×C which is a local minimum ofw−Γ , withΓ(t, ψ) =
w(t, ψ).
We say thatwis a viscosity solution of (26) if it is a viscosity supersolution and a viscosity subsolution of (26).
As we can see in the definition, a viscosity solution must be continuous. So first we will show that the value functionV defined by (11) has this property. Actually, we have the following result:
Lemma 4.2 The value function V : [0, T]×C → R defined in (11) is con-tinuous and there exist constants K > 0 and k ≥ 1 such that, for every
(t, ψ)∈[0, T]×C, we have
|V(t, ψ)| ≤K(1 +kψk2)k. (27)
Proof. It is clear thatV has at most polynomial growth, sinceLand Φ have at most polynomial growth with the samek≥1 as in (9).
Let Ξ(s) = Xs(t, ψ), s∈[t, T], be the C-valued solution of (4) with initial
data (t, ψ)∈[0, T]×C. In view of Remark (v) following the proof of Theorem I.2 and in the proof of Theorem III.1 pp. 33 of Mohammed [34], the trajectory map (t, ψ) → Xs(t, ψ) from [0, T]×C to L2(Ω,C) is globally Lipschitz in ψ
uniformly with respect tot on compact sets, and continuous int for fixed ψ. Therefore, given twoC-valued solutions
Ξ1(s) =Xs(t, ψ1) and Ξ2(s) =Xs(t, ψ2), s∈[t, T]
of (4) with initial data (t, ψ1) and (t, ψ2), respectively, we have
EkΞ1(s)−Ξ2(s)k ≤Kkψ1−ψ2k2, (28)
where K is a positive constant that depends on the Lipschitz constant in As-sumptions 2.3 andT.
Using the Lipshitz continuity of L,Ψ : [0, T]×C → R, there exists yet another constant Λ>0 such that,
|J(τ;t, ψ1)−J(τ;t, ψ2)| ≤ΛEkΞ1(τ)−Ξ2(τ)k. (29)
Therefore, using (29) and (28), we have
|V(t, ψ1)−V(t, ψ2)| ≤ sup τ∈TT t |J(τ;t, ψ1)−J(τ;t, ψ2)| ≤ Λ sup τ∈TT t EkΞ1(τ)−Ξ2(τ)k ≤ Λkψ1−ψ2k2. (30)
This implies the (uniform) continuity ofV(t, ψ) with respect toψ.
We next show the continuity of V(t, ψ) with respect to t. Let Ξ1(s) = Xs(t1, ψ), s∈[t1, T] and Ξ2(s) =Xs(t2, ψ), s∈[t2, T], be twoC-valued
solu-tions of (4) with initial data (t1, ψ) and (t2, ψ), respectively.
Without loss of generality, we assume thatt1< t2. Then we can get
= E Z t2 t1 e−ρ(ξ−t1)[L(ξ,Ξ 1(ξ))]dξ + Z τ t2 e−ρ(ξ−t2)[L(ξ,Ξ 1(ξ))−L(ξ,Ξ2(ξ))]dξ +e−ρ(τ−t1)Ψ(τ,Ξ 1(τ))−e−ρ(τ−t2)Ψ(τ,Ξ2(τ)) . (31)
Therefore, there exists a constant Λ>0 such that
|J(τ;t1, ψ)−J(τ;t2, ψ)| ≤ Λ |t1−t2|EkΞ1(τ)k+EkΞ1(τ)−Ξ2(τ)k . (32)
Let ε > 0 be any give small constant. Using the compactness of [0, T] and the uniform continuity of the trajectory map in t, we know that there exists a constant η > 0 such that if|t1−t2| < η then EkΞ1(s)−Ξ2(s)k ≤ 2Λε. In
addition, there exists a constantK >0 such that
E " sup s∈[t1,T] kΞ1(s)k # ≤K ∀t1∈[0, T].
Then for|t1−t2|<min
η, ε 2ΛK , we have |J(τ;t1, ψ)−J(τ;t2, ψ)| ≤ ε 2 + ε 2 =ε. Consequently, |V(t1, ψ)−V(t2, ψ)| ≤ε.
This completes the proof. tu
Before we show that the value function is a viscosity solution of the HJB variational inequality (26), we need to prove some results related to the dynamic programming principle (22) which can also be found in [25]. The results are given next in Lemma 4.3 and Lemma 4.5.
Lemma 4.3 Letα, β∈ TT
t beG(t)-stopping times andt >0 such thatt≤α≤
β a.s.. Then we have
Ee−ρ(α−t)V(α, Xα) ≥ E Z β α e−ρ(ξ−t)L(ξ, Xξ)dξ +E e−ρ(β−t)V(β, Xβ). (33)
Proof. By virtue of (10) and (11), we can get
E e−ρ(β−t)V(β, Xβ) + Z β α e−ρ(ξ−t)L(ξ, Xξ)dξ
= sup τ∈TT β E Z τ β e−ρ(ξ−β)e−ρ(β−t)L(ξ, Xξ)dξ+e−ρ(τ−β)e−ρ(β−t)Ψ(τ, Xτ) +E Z β α e−ρ(ξ−t)L(ξ, Xξ)dξ = sup τ∈TT β E Z τ β e−ρ(ξ−t)L(ξ, Xξ)dξ+e−ρ(τ−t)Ψ(τ, Xτ) +E Z β α e−ρ(ξ−t)L(ξ, Xξ)dξ = sup τ∈TT β E Z τ α e−ρ(ξ−t)L(ξ, Xξ)dξ+e−ρ(τ−t)Ψ(τ, Xτ) ≤ sup τ∈TT α E Z τ α e−ρ(α−t)e−ρ(ξ−α)L(ξ, Xξ)dξ+e−ρ(α−t)e−ρ(τ−α)Ψ(τ, Xτ) = E[e−ρ(α−t)V(α, Xα)].
This completes the proof. tu
Now let us give the definition of-optimal stopping time which will be used in the next lemma.
Definition 4.4 For each > 0, a G(t)-stopping time τ ∈ TtT is said to be
-optimal if 0≤V(t, ψ)−E Z τ t e−ρ(ξ−t)L(ξ, Xξ)dξ+e−ρ(τ−t)V(τ, Xτ) ≤. (34)
Lemma 4.5 Let θ be a stopping time such that θ ≤ τ a.s., for any > 0,
whereτ∈ TtT is-optimal. Then,
V(t, ψ) =E Z θ t e−ρ(ξ−t)L(ξ, Xξ)dξ+e−ρ(θ−t)V(θ, Xθ) . (35)
Proof. Let θ be a stopping time such that θ ≤ τ a.s., for any -optimal
τ∈ TtT. Using Lemma 4.3, we have
E[e−ρ(θ−t)V(θ, Xθ)] ≥ E Z τ θ e−ρ(ξ−t)L(ξ, Xξ)dξ +E[e−ρ(τ−t)V(τ , Xτ)]. (36) This implies that
E[e−ρ(θ−t)V(θ, Xθ)] +E " Z θ t e−ρ(ξ−t)L(ξ, Xξ)dξ # ≥E Z τ t e−ρ(ξ−t)L(ξ, Xξ)dξ +E[e−ρ(τ−t)V(τ , Xτ)]. (37)
Note thatτis the-optimal stopping time, then we can get 0≤V(t, ψ)−E Z τ t e−ρ(ξ−t)L(ξ, Xξ)dξ+e−ρ(τ−t)V(τ, Xτ) ≤. (38)
On the other hand, by virtue of (37), we can get
V(t, ψ)−E e−ρ(θ−t)V(θ, Xθ) + Z θ t e−ρ(ξ−t)L(ξ, Xξ)dξ ≤ V(t, ψ)−E e−ρ(τ−t)V(τ , Xτ) + Z τ t e−ρ(ξ−t)L(ξ, Xξ)dξ . (39) Thus, we can get
0≤V(t, ψ)−E Z θ t e−ρ(ξ−t)L(ξ, Xξ)dξ+e−ρ(θ−t)V(θ, Xθ) ≤. (40)
Now we let→0 in the above inequality and we can get
V(t, ψ) =E Z θ t e−ρ(ξ−t)L(ξ, Xξ)dξ +E[e−ρ(θ−t)V(θ, Xθ)].
This completes the proof. tu
Now we are ready to show that the value function V defined by (11) is a viscosity solution of the HJBVI (26).
Theorem 4.6 The value function V is a viscosity solution of the HJB varia-tional inequality (26).
Proof. We need to prove that V is both a viscosity supersolution and a vis-cosity subsolution of (26).
First we prove that V is a viscosity supersolution. Let (t, ψ)∈ [0, T]×C
and Γ∈Clip1,2([0, T]×C)∩ D(S) satisfying Γ≤V on the neighborhood N(t, ψ) of (t, ψ) with Γ(t, ψ) =V(t, ψ), we want to prove the the viscosity supersolution inequality,i.e., min Γ(t, ψ)−Ψ(t, ψ), ρΓ(t, ψ)−∂Γ ∂t(t, ψ)− AΓ(t, ψ)−L(t, ψ) ≥0. (41)
We know thatV ≥Ψ onN(t, ψ) and Γ(t, ψ) =V(t, ψ), so we have Γ(t, ψ)−Ψ(t, ψ)≥0.
Therefore, we just need to prove that
ρΓ(t, ψ)−∂Γ
Since Γ∈Clip1,2([0, T]×C)∩D(S), by virtue of Theorem 3.1 pp. 78 of Mohammed [33], fort≤s≤T, we have E[e−ρ(s−t)Γ(s, Xs)−Γ(t, ψ)] = Eh Z s t e−ρ(ξ−t) ∂Γ(ξ, X ξ) ∂ξ +AΓ(ξ, Xξ)−ρΓ(ξ, Xξ) dξi. (42)
For anys∈[t, T] such that (s, Xs)∈N(t, ψ), from Lemma 4.3, we can get
V(t, ψ)≥E Z s t e−ρ(ξ−t)L(ξ, Xξ)dξ +Ehe−ρ(s−t)V(s, Xs) i . (43)
By virtue of (42), Γ≤V andV(t, ψ) = Γ(t, ψ), we can get
0 ≥ E Z s t e−ρ(ξ−t)L(ξ, Xξ)dξ +Ehe−ρ(s−t)V(s, Xs) i −V(t, ψ) ≥ E Z s t e−ρ(ξ−t)L(ξ, Xξ)dξ +Ehe−ρ(s−t)Γ(s, Xs) i −Γ(t, ψ) ≥ Eh Z s t e−ρ(ξ−t) L(ξ, Xξ) + ∂Γ(ξ, Xξ) ∂ξ +AΓ(ξ, Xξ) −ρΓ(ξ, Xξ) dξi. (44)
Dividing both sides of the above inequality by (s−t), we have
0 ≥ E 1 s−t Z s t e−ρ(ξ−t) L(ξ, Xξ) + ∂Γ(ξ, Xξ) ∂ξ +AΓ(ξ, Xξ) −ρΓ(ξ, Xξ) dξ . (45)
Now lets↓t in (45), and we obtain
∂Γ
∂t(t, ψ) +AΓ(t, ψ) +L(t, ψ)−ρΓ(t, ψ)≤0. (46)
which proves the inequality (41).
Next we want to prove that V is also a viscosity subsolution of (26). Let (t, ψ) ∈ [0, T]×C and Γ ∈ Clip1,2([0, T]×C)∩ D(S) satisfying Γ ≥ V on the neighborhoodN(t, ψ) of (t, ψ) with Γ(t, ψ) =V(t, ψ), we want to prove that
min Γ(t, ψ)−Ψ(t, ψ), ρΓ(t, ψ)−∂Γ ∂t(t, ψ)− AΓ(t, ψ)−L(t, ψ) ≤0. (47)
It is easy to get that
Therefore, we need to show that
ρΓ(t, ψ)−∂Γ
∂t(t, ψ)− AΓ(t, ψ)−L(t, ψ)≤0. (48)
Letθ∈ TT
t be a stopping time such thatθ≤τfor everyτ,-optimal stopping
time. Using Lemma 4.5, we can get
V(t, ψ) =E Z θ t e−ρ(ξ−t)L(ξ, Xξ)dξ +Ehe−ρ(θ−t)V(θ, Xθ) i . (49)
Using the Dynkin’s formula (see Mohammed [33]), we have
Ehe−ρ(θ−t)Γ(θ, Xθ) i −Γ(t, ψ) =E Z θ t e−ρ(ξ−t) ∂Γ(ξ, X ξ) ∂ξ +AΓ(ξ, Xξ)−ρΓ(ξ, Xξ) dξ .
Since Γ ≥ V on N(t, ψ) and Γ(t, ψ) = V(t, ψ), for all θ such that (θ, Xθ) ∈
N(t, ψ), we can get Ehe−ρ(θ−t)V(θ, Xθ) i −V(t, ψ) ≤ Ehe−ρ(θ−t)Γ(θ, Xθ) i −Γ(t, ψ) = Eh Z θ t e−ρ(ξ−t) ∂Γ(ξ, X ξ) ∂ξ +AΓ(ξ, Xξ)−ρΓ(ξ, Xξ) dξi
Combining this with (49), the above inequality implies 0≤E Z θ t e−ρ(ξ−t) L(ξ, Xξ) + ∂Γ(ξ, Xξ) ∂ξ +AΓ(ξ, Xξ)−ρΓ(ξ, Xξ) dξ . (50) Dividing (50) byE[(θ−t)] and sendingE[θ]→t, we deduce
∂Γ
∂t(t, ψ) +AΓ(t, ψ) +L(t, ψ)−ρΓ(t, ψ)≥0, (51)
which proves (48). Therefore,V is also a viscosity subsolution. This completes
the proof of the theorem. tu
In order to prove the uniqueness we need following results. We will use the next result proven in Ekeland and Lebourg [10] and also in a general form in Stegall [41] and Bourgain [6]. The reader is also referred to Crandall et al [9] and Lions [31] for an application example of this result in a setting similar to ours.
Lemma 4.7 Let Φ be a bounded above and upper semicontinuous real-valued function on a closed ballB of a Banach spaceX which has the Radon-Nikodym property. Then for any > 0 there exists an element u∗ ∈ X∗ with norm at
Note that every separable Hilbert space (X,k · kX) satisfies the
Radon-Nikodym property (see e.g. Ekeland and Lebourg [10]). In order to apply Lemma 4.7, we shall therefore restrict ourself to a subspaceXofC=C([−r,0];Rn) which is a separable Hilbert space and dense inC. One of the good candidates is the Sobolev spaceW1,2((−r,0);Rn), where
W1,2((−r,0);Rn) = {φ∈L2([−r,0];Rn); kφk1,2<∞},
wherekφk2
1,2 =kφk22+kφ0k22, with φ0 being the first derivative in the sense of
distribution ofφ.
From the Sobolev embedding theorems (see e.g. Adams [1]), it is known thatW1,2((−r,0);Rn)⊂Cand thatW1,2((−r,0);Rn) is dense inC. For more
about Sobolev spaces and corresponding results, one can refer to Adams [1].
Theorem 4.8 (Comparison Principle) Assume that V1(t, ψ) and V2(t, ψ)
are both continuous with respect to the argument (t, ψ) ∈ [0, T]×C and are respectively viscosity subsolution and supersolution of (26) with at most a poly-nomial growth, i.e., there exist constantsΛ>0 andk≥1 such that,
|Vi(t, ψ)| ≤Λ(1 +kψk2)k, for (t, ψ)∈[0, T]×C, i= 1,2.
Then, on every closed ballB ofW1,2((−r,0);Rn), we have
V1(t, ψ)≤V2(t, ψ) for all(t, ψ)∈[0, T]×B. (52) Before we give the proof of the above theorem, we first need to do some preparation works.
LetV1 andV2 be, respectively, a viscosity subsolution and supersolution of
(26). For any 0< δ, γ <1 and for all ψ, φ∈W1,2((−r,0);Rn) andt, s∈[0, T],
define Θδγ(t, s, ψ, φ) ≡ 1 δ kψ−φk2 2+kψ 0−φ0k2 2+|t−s| 2 +γ exp(1 +kψk2 2+kψ 0k2 2) + exp(1 +kφk 2 2+kφ 0k2 2) , (53) and Φδγ(t, s, ψ, φ)≡V1(t, ψ)−V2(s, φ)−Θδγ(t, s, ψ, φ), (54) whereψ0, φ0∈W1,2((−r,0);Rn) with ψ0(θ) = θ −rψ(−r−θ), φ 0(θ) = θ −rφ(−r−θ), ∀θ∈[−r,0].
Moreover, using the polynomial growth condition forV1 andV2, we have
lim
kψk2+kφk2→∞
The function Φδγ is a real-valued function that is bounded above and continuous
on [0, T]×[0, T]×W1,2((−r,0);Rn)×W1,2((−r,0);Rn), since the the embedding ofW1,2((−r,0);Rn) inC is continuous. Therefore, from Lemma 4.7 (which is applicable by virtue of (55)), for any 1> >0, there exits a continuous linear functionalT in the topological dual of W1,2((−r,0);Rn)×W1,2((−r,0);Rn),
with norm at most, such that the function Φδγ +T attains it maximum in
[0, T]×[0, T]×B×B, for any closed ballB ofW1,2((−r,0);Rn). (see Lemma
4.7).
Let B be a closed ball of W1,2((−r,0);Rn) centered at 0. Denote by
(tδγ, sδγ, ψδγ, φδγ) the maximum of Φδγ+T on [0, T]×[0, T]×B×B.
Without loss of generality, we assume that for any given δ, γ and , there exists a constantMδγ such that the maximum value Φδγ +T+Mδγ is zero.
In other words, we have
Φδγ(tδγ, sδγ, ψδγ, φδγ) +T(ψδγ, φδγ) +Mδγ= 0. (56)
We have the following lemmas.
Lemma 4.9 Let B be a closed ball ofW1,2((−r,0);Rn)centered at 0, and let
(tδγ, sδγ, ψδγ, φδγ)be the maximum ofΦδγ+T+Mδγon[0, T]×[0, T]×B×B for someδ, γ, ∈(0,1). Then, we have lim ↓0,δ↓0 kψδγ−φδγk22+kψ 0 δγ−φ 0 δγk 2 2+|tδγ−sδγ|2 = 0, (57)
Proof. Let rB be the radius of the ball B. Then, for all δ, γ, ∈(0,1), we
have kψδγk22≤ kψδγk21,2< r 2 B, and kφδγk22≤ kφδγk22,2< r 2 B.
Noting that (tδγ, sδγ, ψδγ, φδγ) is the maximum of Φδγ+T+Mδγ, we get
Φδγ(tδγ, tδγ, ψδγ, ψδγ) +T(ψδγ, ψδγ) + Φδγ(sδγ, sδγ, φδγ, φδγ) +T(φδγ, φδγ) ≤ 2Φδγ(tδγ, sδγ, ψδγ, φδγ) + 2T(ψδγ, φδγ). (58) It implies V1(tδγ, ψδγ)−V2(tδγ, ψδγ)−2γ(exp(1 +kψδγk22+kψ 0 δγk 2 2)) +T(ψδγ, ψδγ) +V1(sδγ, φδγ)−V2(sδγ, φδγ) −2γ(exp(1 +kφδγk22+kφ 0 δγk 2 2)) +T(φδγ, φδγ) ≤ 2V1(tδγ, ψδγ)−2V2(sδγ, φδγ)− 2 δ kψδγ−φδγk22+kψ 0 δγ−φ 0 δγk 2 2 +|tδγ−sδγ|2 −2γexp(1 +kψδγk22+kψ 0 δγk 2 2) + exp(1 +kφδγk22+kφ 0 δγk 2 2) + 2T(ψδγ, φδγ). (59)
From the above inequality, it is easy to get that 2 δ kψδγ−φδγk22+kψ 0 δγ−φ 0 δγk 2 2+|tδγ−sδγ|2 ≤ [V1(tδγ, ψδγ)−V1(sδγ, φδγ)] + [V2(tδγ, ψδγ)−V2(sδγ, φδγ)] + 2T(ψδγ, φδγ)−[T(ψδγ, ψδγ) +T(φδγ, φδγ)]. (60)
From the polynomial growth condition ofV1, V2, and the fact that the norm of
T is∈(0,1), we can find a constant Λ and a positive integer k≥2 such that
2 δ kψδγ−ψδγk22+kψ 0 δγ−φ 0 δγk 2 2+|tδγ−sδγ|2 ≤ Λ(1 +kψδγk2+kφδγk2)k ≤Λ(1 + 2r)k. (61) So kψδγ−φδγk22+kψ 0 δγ−φ 0 δγk 2 2+|tδγ−sδγ|2 ≤ δΛ(1 + 2r)k, (62)
In order to obtain (57), we sendδto zero in (62) using the above inequality. tu
Now let us introduce a functionalF :W1,2((−r,0);Rn)→Rdefined by
F(ψ)≡ kψk2
2. (63)
and the linear mapH :W1,2((−r,0);Rn)→Cdefined by
H(ψ)(θ)≡ θ
−rψ(−r−θ) =ψ
0(θ), θ∈[−r,0]. (64)
It is easy to verify that
H(ψ)(0) =ψ0(0) = 0, H(ψ)(−r) =ψ0(−r) =−ψ(0). (65)
H is continuous because kH(ψ)k ≤ kψk ≤Kkψk1,2 for some K >0, where the
last inequality comes from the Sobolev embedding inequality.
It is not hard to show that the map F is Fr´echet differentiable and its derivative is given byDF(u)h= 2(u|h). This comes from the fact that
kψ+hk2
2− kψk22= 2(ψ|h) +khk22,
and we can always find a constant Λ>0 such that
|kψ+hk2 2− kψk22−2(ψ|h)| khk1,2 = khk 2 2 khk1,2 ≤ Λkhk 2 1,2 khk1,2 = Λkhk1,2. (66) Moreover, we have 2(ψ+h| ·)−2(ψ| ·) = 2(h| · ).
We deduce thatF is twice differentiable and D2F(u)(h, k) = 2(h|k).
In addition, the map H is linear and twice continuously Fr´echet differen-tiable. Therefore,DH(ψ)(h) = H(h) andD2H(ψ)(h, k) = 0, for allψ, h, k ∈
W1,2((−r,0);Rn).
From the definition of Θδγ and the definition ofF, we can get that
Θδγ(t, s, ψ, φ) = 1 δ F(ψ−φ) +F(H(ψ)−H(φ)) +|t−s|2 + γ[e1+F(ψ)+F(H(ψ))+e1+F(φ)+F(H(φ))].
The following chain rule, quoted from [40] (Theorem 5.2.5 on page 208), is needed to get the Fr´echet derivatives of Θδγ:
Theorem 4.10 (Chain Rule) Let X,Y, andZ be real Banach spaces. If S:
X → Y and T : Y → Z are Fr´echet differentiable at x and y = S(x) ∈ Y, respectively, then U = T ◦S is Fr´echet differentiable at x and its Fr´echet derivative is given by
DxU(x) =DyT(S(x))DxS(x).
Given the above chain rule, we can say that Θδγ is Fr´echet differentiable.
Actually, forh, k∈W1,2((−r,0);Rn), we can get
DψΘδγ(t, s, ψ, φ)(h) = 2 δ h (ψ−φ|h) + (H(ψ−φ)|H(h))i + 2γe1+F(ψ)+F(H(ψ))[(ψ|h) + (H(ψ)|H(h))]. (67) Similarly, DφΘδγ(t, s, ψ, φ)(k) = 2 δ h (φ−ψ|k) + (H(φ−ψ)|H(k))i + 2γe1+F(φ)+F(H(φ))[(φ|k) + (H(φ)|H(k))]. (68) Furthermore, Dψ2Θδγ(t, s, ψ, φ)(h, k) = 2 δ h (h|k) + (H(h)|H(k))i + 2γe1+F(ψ)+F(H(ψ))h2((ψ|k) + (H(ψ)|H(k)))((ψ|h) + (H(ψ)|H(h))) + (k|h) + (H(k)|H(h))i. (69)
Similarly, D2φΘδγ(t, s, ψ, φ)(h, k) = 2 δ h (h|k) + (H(h)|H(k))i + 2γe1+F(φ)+F(H(φ))h2((φ|k) + (H(φ)|H(k)))((φ|h) + (H(φ)|H(h))) + (k|h) + (H(k)|H(h))i. (70)
Observe that we can extendDψΘδγ(t, s, ψ, φ) andD2ψΘδγ(t, s, ψ, φ), the first
and second order Fr´echet derivatives of Θδγ with respect to ψ, to the space
W1,2((−r,0);Rn)⊕B(see Section 3 of this paper or Lemma (3.1) and Lemma
(3.2) on pp 79-83 of Mohammed [33]) by setting DψΘδγ(t, s, ψ, φ)(h+v1{0}) = 2 δ h (ψ−φ|h+v1{0}) + (H(ψ−φ)|H(h+v1{0}) i + 2γe1+F(ψ)+F(H(ψ))[(ψ|h+v1{0}) + (H(ψ)|H(h+v1{0}))], (71) and D2 ψΘδγ(t, s, ψ, φ)(h+v1{0}, k+w1{0}) = 2 δ h (h+v1{0}|k+w1{0}) + (H(h+v1{0})|H(k+w1{0}) i + 2γe1+F(ψ)+F(H(ψ))h2((ψ|k+w1{0}) + (H(ψ)|H(k+w1{0}))) · ((ψ|h+v1{0}) + (H(ψ)|H(h+v1{0}))) + (k+w1{0}|h+v1{0}) + (H(k+w1{0})|H(h+v1{0})) i , (72) forv, w∈Rn andh, k∈W1,2((−r,0);Rn).
Moreover, it is easy to see that these extensions are continuous for that there exists a constant Λ>0 such that
|(ψ−φ|h+v1{0})| ≤ kψ−φk2· kh+v1{0}k2 ≤ Λkψ−φk2(khk+|v|); (73) |(ψ|h+v1{0})| ≤ kψk2· kh+v1{0}k2 ≤ Λkψk2(khk+|v|); (74) |(ψ|k+w1{0})| ≤ kψk2· kk+w1{0}k2 ≤ Λkψk2(kkk+|w|); (75) and |(k+w1{0}|h+v1{0})| ≤ kk+w1{0}k2kh+v1{0}k2 ≤ Λ(kkk+|w|)(khk+|v|). (76)
Similarly, we can extend the first and second order Fr´echet derivatives of Θδγ
with respect toφ, to the spaceW1,2((−r,0);Rn)⊕Band obtain similar expres-sions forDφΘδγ(t, s, ψ, φ)(k+w1{0}) andDφ2Θδγ(t, s, ψ, φ)(h+v1{0}, k+w1{0}).
The same is also true for the bounded linear functionalT whose extension
is still written asT.
In addition, it is easy to verify that for any φ ∈ W1,2((−r,0);Rn) and
v, w∈Rn we have (φ|v1{0}) = Z 0 −r hφ(s), v1{0}(s)ids= 0, (77) (w1{0}|v1{0}) = Z 0 −r hw1{0}(s), v1{0}(s)ids= 0, (78) H(v1{0}) = v1{−r}, (79) (H(ψ)|H(v1{0})) = 0, (H(w1{0})|H(v1{0})) = 0. (80)
These observations shall be used later on. Next we need several lemmas about the operatorS.
Lemma 4.11 Givenφ∈W1,2((−r,0);Rn), we have
S(F)(φ) =|φ(0)|2− |φ(−r)|2, (81)
S(F)(φ0) =−|φ(0)|2, (82)
whereF is the functional defined in (63) andS is the operator defined in (15). MoreoverS(F)is continuous from W1,2((−r,0);Rn) toR.
Proof. Recall that
S(F)(φ) = lim t↓0 1 t h F( ˜φt)−F(φ) i (83) for all φ ∈ W1,2((−r,0);Rn), where ˜φ : [−r, T] → Rn is an extension of φ
defined by ˜ φ(t) = φ(t) ift∈[−r,0) φ(0) ift≥0, (84)
and again ˜φt∈W1,2((−r,0);Rn) is defined by
˜ φt(θ) = ˜φ(t+θ), θ∈[−r,0]. Therefore, we have S(F)(φ) = lim t↓0 1 t h kφ˜tk22− kφk 2 2 i = lim t↓0 1 t Z 0 −r |φ˜t(θ)|2dθ− Z 0 −r |φ(θ)|2dθ
= lim t↓0 1 t Z 0 −r |φ˜(θ+t)|2dθ−Z 0 −r |φ(θ)|2dθ = lim t↓0 1 t Z t −r+t |φ˜(θ)|2dθ− Z 0 −r |φ(θ)|2dθ = lim t↓0 1 t Z 0 −r+t |φ˜(θ)|2dθ+ Z t 0 |φ˜(θ)|2dθ− Z 0 −r |φ(θ)|2dθ = lim t↓0 1 t Z 0 −r+t |φ(θ)|2dθ+ Z t 0 |φ(0)|2dθ− Z 0 −r |φ(θ)|2dθ = lim t↓0 1 t Z 0 −r+t |φ(θ)|2dθ−Z 0 −r |φ(θ)|2dθ + lim t↓0 1 t Z t 0 |φ(0)|2dθ = lim t↓0 1 t Z t 0 |φ(0)|2dθ−lim t↓0 1 t Z −r+t −r |φ(θ)|2dθ = |φ(0)|2− |φ(−r)|2. (85) Similarly, we have S(F)(φ0) =|φ0(0)|2− |φ0(−r)|2=−|φ(0)|2.
It is easy to verify that
|K(ψ)−K(φ)| ≤ |ψ(0)−φ(0)| ≤ kψ−φk ≤Λkψ−φk1,2 for some Λ>0.
Therefore, the map K : W1,2((−r,0);Rn) → R defined by K(ψ) = |ψ(0)| is
continuous.
Similarly the mapJ :W1,2((−r,0);Rn)→Rdefined byJ(ψ) =|ψ(−r)|is
continuous. ThereforeS(F) is continuous because S(F)(ψ) = K(ψ)2−J(ψ)2,
for allψ∈W1,2((−r,0);Rn). tu
LetSψ and Sφ denote the operatorS applied toψand φ, respectively. We
have the following lemma:
Lemma 4.12 Givenφ, ψ∈W1,2((−r,0);Rn),
Sψ(F)(φ−ψ) +Sφ(F)(φ−ψ) =|ψ(0)−φ(0)|2− |ψ(−r)−φ(−r)|2, (86)
and
Sψ(F)(φ0−ψ0) +Sφ(F)(φ0−ψ0) =−|ψ(0)−φ(0)|2, (87)
whereF is the functional defined in (63) andS is the operator defined in (15).
Proof. Define an operator ˜S:D( ˜S)→W1,2((−r,0);Rn) is defined by
˜ S(ψ) = lim t↓0 ˜ ψt−ψ t . (88)
where the domain D( ˜S) consists of thoseψ∈ W1,2((−r,0);Rn) for which the above limit exists.
It is easy to see that the domainD( ˜S) of ˜S contains differential functions of
W1,2((−r,0);Rn), thereforeD( ˜S) is dense inW1,2((−r,0);Rn).
We first assume thatψ∈ D( ˜S). By the definition ofS andF, we have
S(F)(ψ) = lim t↓0 1 t h F( ˜ψt)−F(ψ) i = lim t↓0 1 t h ( ˜ψt|ψ˜t)−(ψ|ψ) i = lim t↓0 1 t h ( ˜ψt|ψ˜t)−( ˜ψt|ψ) + ( ˜ψt|ψ)−(ψ|ψ) i = lim t↓0 1 t h ( ˜ψt|ψ˜t−ψ) + ( ˜ψt−ψ|ψ) i = lim t↓0 1 t h ( ˜ψt+ψ|ψ˜t−ψ) i = lim t↓0 h ˜ ψt+ψ i lim t↓0 1 t h ˜ ψt−ψ i = 2(ψ|S˜(ψ)). (89)
On the other hand, by virtue of Lemma 4.11, we have
S(F)(ψ) =|ψ(0)|2− |ψ(−r)|2. (90) Therefore, we have (ψ|S˜(ψ)) = 1 2 |ψ(0)|2− |ψ(−r)|2 . (91)
Since ˜S is a linear operator, we have (ψ−φ|S˜(ψ)−S˜(φ)) = (ψ−φ|S˜(ψ−φ)) = 1 2 |ψ(0)−φ(0)|2− |ψ(−r)−φ(−r)|2 . (92)
Given the above results, now we have
Sψ(F)(ψ−φ) +Sφ(F)(ψ−φ) = lim t↓0 1 t h kψ˜t−φk22− kψ−φk 2 2+kψ−φ˜tk22− kψ−φk 2 2 i = lim t↓0 1 t kψ˜tk22− kψk 2 2+kφ˜tk22− kφk 2 2 −2[( ˜ψt|φ)−(ψ|φ) + (ψ|φ˜t)−(ψ|φ)]
= S(F)(ψ) +S(F)(φ)−2[( ˜S(ψ)|φ) + (ψ|S˜(φ))] = 2(ψ|S˜(ψ)) + 2(φ|S˜(φ))−2[( ˜S(ψ)|φ) + (ψ|S˜(φ))] = 2(ψ−φ|S˜(ψ−φ))
= [|ψ(0)−φ(0)|2− |ψ(−r)−φ(−r)|2],
provided thatψ, φ∈ D( ˜S).
For anyψ, φ∈W1,2((−r,0);Rn), one can construct sequences{ψk}∞k=1and
{φk}∞k=1 in D( ˜S) such that
lim
k→∞kψk−ψk1,2= 0 and k→∞lim kφk−φk1,2= 0.
Consequently, by the continuity theS(F) operator, we have
Sψ(F)(ψ−φ) +Sφ(F)(ψ−φ) = lim k→∞ Sψ(F)(ψk−φk) +Sφ(F)(ψk−φk) = lim k→∞[|ψk(0)−φk(0)| 2− |ψ k(−r)−φk(−r)|2] = [|ψ(0)−φ(0)|2− |ψ(−r)−φ(−r)|2].
By the same argument, we can get
Sψ(F)(ψ0−φ0) +Sφ(F)(ψ0−φ0)
= |ψ0(0)−φ0(0)|2− |ψ0(−r)−φ0(−r)|2
= −|ψ(0)−φ(0)|2.
t u
Lemma 4.13 Given φ ∈ W1,2((−r,0);Rn), we define a new operator G as
follows
G(φ) =e1+F(φ)+F(φ0). (93)
Then we have,
S(G)(φ) = (−|φ(−r)|2)e1+F(φ)+F(φ0), (94)
whereF is the functional defined in (63) andS is the operator defined in (15). Proof. Recall that
S(G)(φ) = lim t↓0 1 t h G( ˜φt)−G(φ) i (95) for all φ ∈ W1,2((−r,0);Rn), where ˜φ : [−r, T] → Rn is an extension of φ
defined by ˜ φ(t) = φ(t) ift∈[−r,0) φ(0) ift≥0, (96)
and again ˜φt∈W1,2((−r,0);Rn) is defined by ˜ φt(θ) = ˜φ(t+θ), θ∈[−r,0]. Then we have S(G)(φ) = lim t↓0 1 t e1+R−0r|φ˜t(θ)|2dθ+R−0r|φ˜ 0 t(θ)|2dθ −e1+ R0 −r|φ(θ)| 2dθ+R0 −r|φ 0(θ)|2dθ = lim t↓0 1 t e1+ R0 −r|φ˜(θ+t)| 2dθ+R0 −r|φ˜ 0(θ+t)|2dθ −e1+R−0r|φ(θ)| 2dθ+R0 −r|φ 0(θ)|2dθ = lim t↓0 1 t e1+R−tr+t|φ˜(θ)| 2dθ+Rt −r+t|φ˜ 0(θ)|2dθ −e1+ R0 −r|φ(θ)| 2dθ+R0 −r|φ 0(θ)|2dθ = lim t↓0 1 t e1+ R0 −r+t|φ(θ)| 2dθ+Rt 0|φ(0)| 2dθ+R0 −r+t|φ 0(θ)|2dθ+Rt 0|φ 0(0)|2dθ −e1+R−0r|φ(θ)| 2dθ+R0 −r|φ 0(θ)|2dθ = lim t↓0 1 t e1+R−0r+t|φ(θ)| 2dθ+t|φ(0)|2+R0 −r+t|φ 0(θ)|2dθ+t|φ0(0)|2 −e1+R−0r|φ(θ)| 2dθ+R0 −r|φ 0(θ)|2dθ . (97) Using the L’Hospital rule on the last equality and (65), we obtain
S(G)(φ) = lim t↓0e 1+R0 −r+t|φ(θ)| 2dθ+t|φ(0)|2+R0 −r+t|φ 0(θ)|2dθ+t|φ0(0)|2 |φ(0)|2 −|φ(−r+t)|2+|φ0(0)|2− |φ0(−r+t)|2 = (|φ(0)|2− |φ(−r)|2− |φ0(−r)|2)e1+R0 −r|φ(θ)| 2dθ)+R0 −r|φ 0(θ)|2dθ = (|φ(0)|2− |φ(−r)|2− |φ(0)|2)e1+R−0r|φ(θ)| 2dθ+R0 −r|φ 0(θ)|2dθ = −|φ(−r)|2e1+F(φ)+F(φ0), (98)
which completes the proof. tu
Given all above results, now we are ready to prove Theorem 4.8.
Proof of Theorem 4.8. Let B a closed ball of W1,2((−r,0);Rn). For all
s, t∈[0, T] andψ, φ∈B, and for any given γ, δ, , we define
and
Γ2(s, φ)≡V1(tδγ, ψδγ)−Θδγ(tδγ, s, ψδγ, φ) +T(ψδγ, φδγ) +Mδγ (100)
where (tδγ, sδγ, ψδγ, φδγ) is maximum point of Φδγ+T+Mδγ over [0, T]×
[0, T]×B×B. Recall that
Φδγ(t, s, ψ, φ) =V1(t, ψ)−V2(s, φ)−Θδγ(t, s, ψ, φ),
and Φδγ+T+Mδγ reaches its maximum value zero at (tδγ, sδγ, ψδγ, φδγ)
over [0, T]×[0, T]×B×B.
By the definition of Γ1 and Γ2, it is easy to verify that
Γ1(t, ψ)≥V1(t, ψ), Γ2(s, φ)≤V2(s, φ), ∀t, s∈[0, T] andφ, ψ∈B,
and
V1(tδγ, ψδγ) = Γ1(tδγ, ψδγ) andV2(sδγ, φδγ) = Γ2(sδγ, φδγ).
Using the definitions of the viscosity subsolution,V1 and Γ1, we have
minnV1(tδγ, ψδγ)−Ψ(t, ψδγ), ρV1(tδγ, ψδγ)− ∂Γ1 ∂t (tδγ, ψδγ) − A(Γ1)(tδγ, ψδγ)−L(tδγ, ψδγ) o ≤0. (101) By the definitions of the operatorA and Γ1, and by virtue of (67),(68), (69),
(70), (71), (72), (77), (78), and (80), we can get
A(Γ1)(tδγ, ψδγ) = S(Γ1)(tδγ, ψδγ) +DψΘδγ(· · ·)(f(tδγ, ψδγ)1{0}) +1 2 m X j=1 D2 ψΘδγ(· · ·) g(tδγ, ψδγ)(ej)1{0}, g(tδγ, ψδγ)(ej)1{0} = S(Γ1)(tδγ, ψδγ).
Note that Θδγ(· · ·) is an abbreviation for Θδγ(tδγ, sδγ, ψδγ, φδγ) in the above
equation and the following.
It follows from (101) and the above inequality together that min V1(tδγ, ψδγ)−Ψ(tδγ, ψδγ), ρV1(tδγ, ψδγ)− S(Γ1)(tδγ, ψδγ)− ∂Γ1 ∂t (tδγ, ψδγ) −L(tδγ, ψδγ) ≤0. (102)
Similarly, using the definitions of the viscosity supersolution ofV2 and Γ2 and
by the virtue of the same techniques similar to (102) , we have min V2(sδγ, φδγ)−Ψ(sδγ, φδγ), ρV2(sδγ, φδγ)− S(Γ2)(sδγ, φδγ)− ∂Γ2 ∂s (sδγ, φδγ) −L(sδγ, φδγ) ≥0. (103)
By virtue of (99),(100) and (53), we can get
∂Γ1 ∂t (tδγ, ψδγ) = 2 δ(tδγ−sδγ), (104) ∂Γ2 ∂s (sδγ, φδγ) =− 2 δ(sδγ−tδγ) = 2 δ(tδγ−sδγ), (105)
Therefore, the inequality (102) is equivalent to
V1(tδγ, ψδγ)−Ψ(tδγ, ψδγ)≤0, (106)
or
ρV1(tδγ, ψδγ)− S(Γ1)(tδγ, ψδγ)−
2
δ(tδγ−sδγ)−L(tδγ, ψδγ)≤0. (107)
Similarly, by virtue of (105), the inequality (103) is equivalent to
V2(sδγ, φδγ)−Ψ(sδγ, φδγ)≥0, (108)
and
ρV2(sδγ, φδγ)− S(Γ2)(sδγ, φδγ)−
2
δ(tδγ−sδγ)−L(sδγ, φδγ)≥0. (109)
If (106) holds, using (108), we can get that there exists a constant Λ>0 such that V1(tδγ, ψδγ)−V2(sδγ, φδγ) ≤ Ψ(tδγ, ψδγ)−Ψ(sδγ, φδγ) ≤ Λ |tδγ−sδγ|+kψδγ−φδγk2 . (110)
Thus applying Lemma 4.9 to (110), we have lim sup
δ↓0,↓0
(V1(tδγ, ψδγ)−V2(sδγ, φδγ))≤0. (111)
On the other hand, if (107) holds, then by virtue of (107) and (109), we can obtain
ρ(V1(tδγ, ψδγ)−V2(sδγ, φδγ))
≤ S(Γ1)(tδγ, ψδγ)− S(Γ2)(sδγ, φδγ)
From the definition (15) of S, it is clear that S is linear and takes zero on constants. Recall that Γ1(t, ψ) =V2(sδγ, φδγ) + Θδγ(t, sδγ, ψ, φδγ)−T(ψδγ, φδγ)−Mδγ, and Γ2(s, φ) =V1(tδγ, ψδγ)−Θδγ(tδγ, s, ψδγ, φ) +T(ψδγ, φδγ) +Mδγ.
Then we can get
S(Γ1)(tδγ, ψ) =Sψ(Θδγ)(tδγ, sδγ, ψ, φδγ), and S(Γ2)(sδγ, φ) =−Sφ(Θδγ)(tδγ, sδγ, ψδγ, φ). Therefore, S(Γ1)(tδγ, ψδγ)− S(Γ2)(sδγ, φδγ) = Sψ(Θδγ)(tδγ, sδγ, ψδγ, φδγ) +Sφ(Θδγ)(tδγ, sδγ, ψδγ, φδγ).(113) Recall that Θδγ(t, s, ψ, φ) = 1 δ F(ψ−φ) +F(ψ0−φ0) +|t−s|2 +γ(G(ψ) +G(φ)).
Therefore, we can get
Sψ(Θδγ) +Sφ(Θδγ) (tδγ, sδγ, ψδγ, φδγ) ≡ Sψ(Θδγ)(tδγ, sδγ, ψδγ, φδγ) +Sφ(Θδγ)(tδγ, sδγ, ψδγ, φδγ) = 1 δ[Sψ(F)(ψδγ−φδγ) +Sφ(F)(ψδγ−φδγ) +Sψ(F)(ψδγ0 −φ 0 δγ) +Sφ(F)(ψδγ0 −φ 0 δγ)] +γ[Sψ(G)(ψδγ) +Sφ(G)(φδγ)]. (114)
Using Lemma 4.12 and Lemma 4.13, we deduce
Sψ(Θδγ) +Sφ(Θδγ) (tδγ, sδγ, ψδγ, φδγ) = 1 δ − |ψδγ(−r)−φδγ(−r)|2 −γ |ψδγ(−r)|2e1+F(ψδγ)+F(ψ 0 δγ) +|φδγ(−r)|2e1+F(φδγ)+F(φ 0 δγ) ≤ 0. (115)
Thus, by virtue of (113), we have lim sup δ↓0,↓0 h S(Γ1)(tδγ, ψδγ)− S(Γ2)(sδγ, φδγ) i ≤0. (116)
Combined with (112), the above inequality implies that lim sup ↓0,δ↓0 ρ(V1(tδγ, ψδγ)−V2(sδγ, φδγ)) ≤ lim sup ↓0,δ↓0 S(Γ1)(tδγ, ψδγ)− S(Γ2)(sδγ, φδγ) + [L(tδγ, ψδγ)−[L(sδγ, φδγ)] ≤ lim sup ↓0,δ↓0 [L(tδγ, ψδγ)−L(sδγ, φδγ)] . (117)
Using the Lipschitz continuity ofLand Lemma 4.9, we can get that lim sup δ↓0,↓0 |L(tδγ, ψδγ)−L(sδγ, φδγ)| ≤ lim sup δ↓0,↓0 Λ|tδγ−sδγ|+kψδγ−φδγk2 = 0, (118)
Moreover, by virtue of (118) and (117), we have lim sup
↓0,δ↓0
ρ(V1(tδγ, ψδγ)−V2(sδγ, φδγ))≤0. (119)
Since (tδγ, sδγ, ψδγ, φδγ) is the maximum of Φδγ+Tin [0, T]×[0, T]×B×B,
for all (t, ψ)∈[0, T]×B, we have
Φδγ(t, t, ψ, ψ) +T(ψ, ψ)≤Φδγ(tδγ, sδγ, ψδγ, φδγ) +T(ψδγ, φδγ). Then we get V1(t, ψ)−V2(t, ψ) ≤ V1(tδγ, ψδγ)−V2(sδγ, φδγ) −1 δ kψδγ−φδγk22+kψ 0 δγ−φ 0 δγk 2 2+|tδγ−sδγ|2 + 2γexp(1 +kψk22+kψ 0 k22) +γ exp(1 +kψδγk22+kψ 0 δγk 2 2) + exp(1 +kφδγk22+kφ 0 δγk 2 2) +T(ψδγ, φδγ)−T(ψ, ψ) ≤ V1(tδγ, ψδγ)−V2(sδγ, φδγ) + 2γexp(1 +kψk2 2+kψ 0k2 2) +T(ψδγ, φδγ)−T(ψ, ψ), (120)
where the last inequality comes from the fact thatδ >0 andγ >0. By virtue of (111) and (119), when we take the lim sup on (120) asδ, andγgo to zero, we can obtain V1(t, ψ)−V2(t, ψ) ≤ lim sup γ↓0,↓0,δ↓0 V1(tδγ, ψδγ)−V2(sδγ, φδγ) + 2γexp(1 +kψk22+kψ 0 k22) +T(ψδγ, φδγ)−T(ψ, ψ) ≤ 0. (121) Therefore, we have V1(t, ψ)≤V2(t, ψ), ∀(t, ψ)∈[0, T]×B. (122)
This completes the proof of Theorem 4.8. tu
Corollary 4.14 LetV1(t, ψ)andV2(t, ψ)be respectively a viscosity subsolution and a viscosity supersolution of (26) with at most a polynomial growth. Then we have
V1(t, ψ)≤V2(t, ψ), ∀(t, ψ)∈[0, T]×W1,2((−r,0);Rn). (123)
Proof. Let (t, ψ) ∈ [0, T]×W1,2((−r,0);Rn). Then we can always find a closed ball B of W1,2((−r,0);Rn) such that ψ ∈ B. Using Theorem 4.8, we know thatV1(t, φ)≤V2(t, φ), for allφ∈B. Therefore,V1(t, ψ)≤V2(t, ψ). This
implies that
V1(t, ψ)≤V2(t, ψ), ∀(t, ψ)∈[0, T]×W1,2((−r,0);Rn).
This completes the proof. tu
Corollary 4.15 LetV1(t, ψ)andV2(t, ψ)be respectively a viscosity subsolution
and a viscosity supersolution of (26) with at most a polynomial growth. Then, we have
V1(t, ψ)≤V2(t, ψ), ∀(t, ψ)∈[0, T]×C. (124)
Proof. Since W1,2((−r,0);Rn) is dense in C, for anyψ ∈C we can always
find a sequence (ψk)k withψk∈W1,2((−r,0);Rn) such that
lim
k→∞kψk−ψk →0.
Using Corollary 4.14 we know thatV1(t, ψk)≤V2(t, ψk) for allt∈[0, T].
There-fore, for allt∈[0, T], using the continuity ofV1 andV2, we can get V1(t, ψ) = lim
This completes the proof. tu
Now we give the main result of this paper:
Theorem 4.16 (Uniqueness) The value functionV : [0, T]×C→Rdefined by (11) is the unique viscosity solution of the HJBVI (26).
Proof. Since the value function V : [0, T]×C → R is a viscosity solution (and hence is both a subsolution and a supersolution) of the HJBVI (26), the uniqueness result of the viscosity solution follows immediately from the above
comparison principle. tu
Acknownledgement
The authors thank Professor Micheal O’Leary for his useful suggestions.
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