Invariant measures of reflected stochastic delay differential equations with jumps

0.24 MB·English

by  Lijun Bo

Invariant measures of reflected stochastic delay differential
equations with jumps
∗
Lijun Bo Chenggui Yuan
† ‡
3
1 This version: December 11, 2013
0
2

n
Abstract In this paper, we consider a class of multi-dimensional reflected stochastic delay differ-
a
J ential equations with jumps. Based on the existence and uniqueness of the strong solution to the

3 equation, we prove that the Markov semigroup generated by the segment process corresponding

to the solution admits a unique invariant measure on the Skorohod space when the coefficients of
]
R equation satisfy a class of monotone conditions. Finally, we establish a relationship between the
P regulator and the local time of the solution and discuss a local time property at large time under
.
h the stationary setting .
t
a
m Keywords: Stochastic delay differential equations; jump reflection; invariant measure; local time.
[ MSC: 60J60; 60K10.

1
v 1 Introduction and framework
2
4
4 We take a complete filtered probability space (Ω, ,( ; t 0),P) carrying an ( ; t 0)-adapt
t t
0 n N-dimensional Brownian motion W = (W (tF); tF 0)≥ and an ( ; t 0F)-ada≥pt Poisson
1. ran∈dom measure (N((0,t] A); A ( )) wijth inten≥sityn(×ν1(A)t; A Ft ( )≥), where t > 0 and
0 × ∈ B E ∈ B E
= Rd/ 0 . Here the filtration ( ; t 0) is assumed to satisfy the usual conditions. In this
3 t
E { } F ≥
1 paper, we consider the following d N-dimensional reflected stochastic delay differential equation
: ∈
v with jumps (RSDDEJ for short):
i
X
dX(t) = b(t,X(t),X(t τ))dt+σ(t,X(t),X(t τ))dW(t)
r − −
a (1) + g(t,X(t ),X((t τ) ),ρ)N˜(dρ,dt)+dK(t), on t 0,

− − − ≥
X(t) = ξ(t)E Rd, on t [ τ,0],
R ∈ + ∈ −
where τ >0 is a deterministic delay level, the initial data ξ() D([ τ];Rd), the space of all
· ∈ − +
right-continuous functions with left limits from [ τ,0] to Rd and N˜(dρ,dt):= N(dρ,dt) ν(dρ)dt
− + −
defines the compensated version of the Poisson measure N. Here the characteristic measure ν of
N is a σ-finite measure on ( , ( )).
E B E
WewritethesolutionprocessX = (X(t); t τ),thecoefficientfunctions(b,g)inRSDDEJ(1)
≥ −
asd 1columnvectors andthediffusioncoefficient σ asad n-matrix. Namely X(t) = (Xi(t)) ,
d 1
× × ×
∗ThisworkispartiallysupportedbyNSFofChina(No. 11001213). Theauthorswouldliketothankparticipants
at Beijing-Swansea Workshop on Stochastic Processes organized by Profs. M.F. Chen, N.Jacob and F.Y. Wang.
†Department of Mathematics, Xidian University,Xi’an 710071, PR China. E-mail: [email protected]
‡Department of Mathematics, Swansea University,Swansea SA28PP, UK. E-mail: [email protected]
1
b(, , ) = (b (, , )) , σ(, , ) = (σ (, , )) , g(, , , ) = (g (, , , )) . In addition, K =
i d 1 ij d n i d 1
(K·i·(t·); t 0·)· · is×a d-d·im·e·nsional no·n·n·ega×tive pr·o·ce·s·s, which·is· c·a·lled×the regulator for the
d 1
≥ ×
d-dimensional solution process X at the orthant. Moreover, the regulator K can be uniquely
determined by the following properties up to a positive constant factor (see Harrison [H86]):
(a) For i = 1,2,...,d, thepathsoft Ki(t)arenon-decreasing, right-continuous withleftlimits
→
(r.c.l.l. for short) and Ki(0) = Ki(0 ) = 0;
−
(b) For all t 0, it holds that
≥
t
(2) X(s),dK(s) = 0,
h i
Z0
where x,y = d x y for x = (x ) and y = (y ) Rd.
h i i=1 i i i d×1 i d×1 ∈
For any d n-matrix a = (a ) , define a = Tr[aaT], where aT is the transpose of a
ij d n
and Tr[aaT] de×Pnotes the trace of the×matrix aaTk. kDefine x = x,x for any x = (x ) Rd.
p| | h i i d×1 ∈
We work in the following assumptions concerning coefficient functions (b,σ,g) in RSDDEJ (1)
p
throughout the paper:
(A1) there exists a constant α > 0,α > α > 0 such that
1 2
2 x,b(t,x,y) + σ(t,x,y) 2 + g(t,x,y,ρ)2ν(dρ) α α x 2+α y 2,
1 2
h i k k | | ≤ − | | | |
ZE
for all t R and x,y Rd.
∈ + ∈ +
(A2) there exist two constants β > β > 0 such that
1 2
2 x xˆ,b(t,x,y) b(t,xˆ,yˆ) + σ(t,x,y) σ(t,xˆ,yˆ) 2
h − − i k − k
+ g(t,x,y,ρ) g(t,xˆ,yˆ,ρ)2ν(dρ) β x xˆ 2+β y yˆ2,
1 2
| − | ≤ − | − | | − |
ZE
for all t R and x,xˆ,y,yˆ Rd.
∈ + ∈ +
An illustrative example for the condition (A1) and (A2) is to take the drift coefficient b(t,x,y) =
γ(t)x+θ(t)y with γ(t),θ(t) > 0. We assume that γ = inf γ(t) and θ = sup θ(t) are finite.
−For (t,x,xˆ,y,yˆ) R Rd Rd Rd Rd, it hold∗s thatt≥0 ∗ t≥0
∈ +× +× +× +× +
θ 2
2 x xˆ,b(t,x,y) b(t,xˆ,yˆ) (2γ ε2)x xˆ 2+ | ∗| y yˆ2,
h − − i ≤ − ∗− | − | ε2 | − |
for any ε > 0. Moreover the coefficient functions (σ,g) are assumed to satisfy σ(t,0,0) = 0 and
g(t,0,0,ρ) = 0 for all (t,ρ) R and the following Lipschitzian-type conditions:
+
∈ ×E
σ(t,x,y) σ(xˆ,yˆ,ρ) 2 ℓ (t)(x xˆ 2+ y yˆ2),
σ
k − k ≤ | − | | − |
g(t,x,y,ρ) g(t,xˆ,yˆ,ρ)2 ℓ (t,ρ)(x xˆ 2+ y yˆ2),
g
| − | ≤ | − | | − |
where ℓ (t) and ℓ (t,ρ) are positive functions satisfying ℓ := sup (ℓ (t)+ ℓ (t,ρ)ν(dρ)) <
σ g ∗σ,g t 0 σ g
+ . If γ ℓ > θ , we can always find a constant ε> 0 such that≥α := 2γ Eε2 ℓ > α :=
θ∗∞2 ∗− ∗σ,g ∗ 1 ∗−R − ∗σ,g 2
|ε2| +ℓ∗σ,g. Thus, the assumptions (A1) and (A2) are satisfied.
2
Given the Brownian motion W and Poisson random measure N, we call the ( ; t 0)-pair of
t
F ≥
r.c.l.l. processes (X,K) = ((X(t); t τ),(K(t); t 0)) is a strong solution to (1), if they solve
≥ − ≥
the following stochastic integral equation:
t t
X(t) = ξ(0)+ b(s,X(s),X(s τ))ds+ σ(s,X(s),X(s τ))dW(s)
0 − 0 −
(3) + t g(s,X(s ),X((s τ) ),ρ)N˜(dρ,ds)+K(t) Rd, on t 0,
 0 R − − − R ∈ + ≥
X(t) = ξ(t) ERd, on t [ τ,0],
R ∈R + ∈ −
and theRd-valued regulator K satisfies the properties (a) and (b).
+
Wefirsthavethefollowingremarkconcerningtheexistenceanduniquenessofthestrongsolution
to RSDDEJ (1).
Remark 1.1. Under the assumptions (A1) and (A2), RSDDEJ (1) admits a unique strong so-
lution defined as above. As a matter of fact, the existence of the unique strong solution to (1) can
be guaranteed by the following weaker conditions than (A1) and (A2), namely
(A1’) there exists a positive constant α¯ such that
2 x,b(t,x,y) + σ(t,x,y) 2 + g(t,x,y,ρ)2ν(dρ) α¯(1+ x 2+ y 2),
h i k k | | ≤ | | | |
ZE
for all t R and x,y Rd.
∈ + ∈ +
(A2’) there exist two positive constants β¯ ,β¯ such that
1 2
2 x xˆ,b(t,x,y) b(t,xˆ,yˆ) + σ(t,x,y) σ(t,xˆ,yˆ) 2
h − − i k − k
+ g(t,x,y,ρ) g(t,xˆ,yˆ,ρ)2ν(dρ) β¯ x xˆ 2+β¯ y yˆ2,
1 2
| − | ≤ | − | | − |
ZE
for all t R and x,xˆ,y,yˆ Rd.
∈ + ∈ +
The proof is similar to that of [K99] and [vRS], we omit it here. Our aim is to use the conditions
(A1) and (A2) to study invariant measures for RSDDEJ (1).
For i= 1,2,...,d, letKi,c(t)= Ki(t) ∆Ki(s)bethecontinuouspartoftheith-regulator
− s t
Ki(t), wherethet-timejump’ssize∆Ki(t) = K≤i(t) Ki(t )withleftlimitKi(t ):= lim Ki(s).
It will be seen that the continuous counterPpart Ki,−c = (K−i,c; t 0) behaves lik−e the locasl↑ttime of
≥
the ith-element Xi of the solution process X when Xi is treated as a r.c.l.l. semimartingale (see
Section 4inthecurrentpaper). However, thejumpofKi happenswhenXi Ki jumpsdownbelow
−
barrierzeroduetotheappearanceofsomenegativejump. Thisphenomenonisusuallycalled“jump
reflection” in the literature (e.g., Slomin´ski and Wojciechowski [SW10] and Nam [N10]). Moreover,
the corresponding jump’s size of the ith-regulator is given by
−
(4) ∆Ki(t) = g (s,X(s ),X((s τ) ),ρ)N(dρ,ds)+Xi(t ) ,
i
− − − −
"Z{t}ZE #
where [x ] = max x ,0 for x R and [x] = (x ) for x Rd. Write Kc = (Ki,c(t); t
i − {− i } i ∈ − −i d×1 ∈ ≥
0) . From the “jump reflection”, we also have that, for all t 0, it holds that
d 1
× ≥
t
(5) X(s),dKc(s) = 0.
h i
Z0
3
The similar one-sided Lipschitzian condition and monotone condition as (A1) and (A2) has
been discussed in Bao, et al. [BTY09] for stochastic delay equation without jump reflection, in
Marin-RubioandReal[MRR]andZhang[Z94]forreflectedstochasticdifferentialequationswithout
jumps. In particular, the Picard’s successive approximation used in Xu and Zhang [XZ09] can deal
with the existence and uniqueness of strong solutions to RSDDEJ (1) when the regulator K is
described as a local time. However, due to the existing of negative jumps in (1), the regulator
K has jumps whose sizes can be identified by (4). Let (X ; n = 0 N) be the corresponding
n
{ }∪
Picard’s approximating sequence to the d-dimensional solution processes. Then, the successive
approximation to the jump ∆K(t) can be established through (4), namely
−
∆K (t) = g(s,X (s ),X ((s τ) ),ρ)N(dρ,ds)+X (t ) , n N.
n n 1 n 1 n 1
"Z{t}ZE − − − − − − − # ∈
See Proposition 2.4 in Slomin´ski and Wojciechowski [SW10] for more details. The literature con-
cerning stochastic delay differential equations with (or without) jumps is extensive (see e.g., [A10],
[F 11], [KW10], [ S10], [ S07], [HMS11], [RRG06], [W11], and the references therein). Recently
∅ ∅ ∅
Kinnally and Williams [KW10] discussed the existence and uniqueness of stationary solutions to
a class of reflected SDDE driven by Brownian motions. The stability property in distribution of
Brownian-driven reflected Markov-modulated SDDE was considered in Bo and Yuan [BY11]. The
stability in distribution implies the existence and uniqueness of invariant measures for the corre-
sponding segment processes. To the best of our knowledge, it seems that there exists not much
literature to investigate SDDE with jump reflection.
An outline of the paper is as follows. In Section 2, we establish an estimate for the second-order
moment associated to the segment process of the solution to RSDDEJ (1) and consider an expo-
nential integrability of the solution when the drift and diffusion coefficients are uniformly bounded.
The existence and uniqueness of invariant measures associated with d-dimensional segment process
is proved in Section 3. In Section 4, we discuss the relationship between the regulator K and the
local time related to the strong solution process and a local time property in the stationary setting.
Additional Notation. For any f() D(I;Rd), f := sup f(t), where I R. For the
· ∈ + k kI t∈I | | ⊂
d-dimensional solution process X = (X(t); t τ), the corresponding d-dimensional segment
≥ −
process (X ; t 0) is defined as X (θ) = X(t +θ) with τ θ 0, correspondingly X :=
t t t
≥ − ≤ ≤ k k
sup X(t+θ). Throughout the paper, we use the conventions:
τ θ 0
− ≤ ≤ | |
d
∞
:= , and := ,
Zc Z(c,d] Zc Z(c,∞)
for any real numbers c< d.
2 Moment estimates of the solution
Thissection concentrates ontheestimates ofthesecond-ordermomentrelated tothed-dimensional
segmentprocessandtheexponentialmomentforthed-dimensionalsolutionprocessX = (X(t); t
≥
τ) to RSDDEJ (1).
−
Before discussing these moment estimates, we first present the following auxiliary results which
will serve to establish final moment estimates.
4
Lemma 2.1. Let X = (X(t); t τ) be the strong solution to RSDDEJ (1). Then, for any
≥ −
F() C2(Rd), it holds that
· ∈ +
t
F(X(t)) = F(ξ(0))+ F(X(s)),b(s,X(s),X(s τ)) ds
h∇ − i
Z0
t t
+ F(X(s)),σ(s,X(s),X(s τ))dW(s) + F(X(s)),dKc(s)
h∇ − i h∇ i
Z0 Z0
t
+ F(X(s)),g(s,X(s),X(s τ),ρ) N˜(dρ,ds)
h∇ − i
Z0 Z
1 tE
+ Tr (σσT)(s,X(s),X(s τ))D2F(X(s)) ds
2 −
Z0
t (cid:2) (cid:3)
+ [F([X(s )+g(s,X(s ),X((s τ) ),ρ)]+) F(X(s ))
− − − − − −
Z0 ZE
(6) F(X(s )),g(s,X(s ),X((s τ) ),ρ) ]N(dρ,ds),
−h∇ − − − − i
where [x]+ = (x+) and x+ = max x ,0 with x = (x ) Rd, F(x) denotes the gradient
of F(x), D2F(x)i ids×t1he the di d-mat{rixi of}second-orderipad×rt1ia∈l deriv∇atives of F(x) and Kc(t) =
×
(Ki,c(t)) corresponds tothe continuous component of the regulator K(t)= (Ki(t)) with t 0.
d 1 d 1
× × ≥
Proof. By virtue of Itˆo formula with jumps (see e.g., Protter [P04]), we have
t
F(X(t)) = F(ξ(0))+ F(X(s )),dX(s)
h∇ − i
Z0
1 t
(7) + Tr (σσT)(s,X(s),X(s τ))D2F(X(s)) ds
2 −
Z0
(cid:2) (cid:3)
+ [F(X(s )+∆X(s)) F(X(s )) F(X(s )),∆X(s) ].
− − − −h∇ − i
0<s t
X≤
For i = 1,2,...,d, define the process with pure jumps:
t
(8) Yi(t) = g (s,X(s ),X((s τ) ),ρ)N(dρ,ds), t 0.
i
− − − ≥
Z0 ZE
In terms of (1), the random jump amplitude of the ith-element Xi (i = 1,2,...,d) is clearly given
by ∆Xi(t) = ∆Yi(t)+∆Ki(t) for t > 0. Using the following key representation of the jump’s size
of the ith-regulator Ki with i = 1,2,...,d (see (4)):
(9) ∆Ki(t) = [∆Yi(t)+Xi(t )] t > 0,
−
−
we arrive at
(10) ∆Xi(t) = ∆Yi(t)+[∆Yi(t)+Xi(t )] =:ϕ(Xi(t ),∆Yi(t)), t > 0.
−
− −
The function ϕ(x,y) = (ϕ(x ,y )) with x = (x ) Rd and y = (y ) Rd, where
i i d×1 i d×1 ∈ + i d×1 ∈
ϕ(x ,y ) = x 1l + y 1l for i = 1,2,...,d. Using the equality x + ϕ(x ,y ) =
[x +i yi]+ an−d siub{xsti+ityui≤te0}the foill{oxwi+inygi>e0q}uality into the equality (7), i i i
i i
F(X(s )+∆X(s)) F(X(s )) F(X(s )),∆X(s)
− − − −h∇ − i
5
= F([X(s )+∆Y(s)]+) F(X(s )) F(X(s )),∆Y(s) F(X(s )),∆K(s) ,
− − − −h∇ − i−h∇ − i
we obtain the equality (6), where we have used the finite variation property of the regulator
K = (K(t); t 0) and the following equality:
≥
t t
F(X(s )),dK(s) F(X(s )),∆K(s) = F(X(s)),dKc(s) , t > 0.
h∇ − i− h∇ − i h∇ i
Z0 0<s t Z0
X≤
Thus, we complete the proof of the lemma. (cid:3)
Corollary 2.1. Let λ R. Then, the d-dimensional strong solution process X = (X(t); t τ)
∈ ≥ −
to RSDDEJ (1) satisfies the following inequality:
t t
eλt X(t)2 ξ(0)2 +2 eλs X(s),b(s,X(s),X(s τ)) ds+λ eλs X(s)2ds
| | ≤ | | h − i | |
Z0 Z0
t t
+ eλsdM(s)+ eλs σ(s,X(s),X(s τ)) 2ds
k − k
Z0 Z0
t
(11) + eλs g(s,X(s),X(s τ),ρ)2ν(dρ)ds,
| − |
Z0 ZE
where the process M = (M(t); t 0) is defined by
≥
t
M(t) = 2 X(s),σ(s,X(s),X(s τ))dW(s)
h − i
Z0
t
+ [ [X(s )+g(s,X(s ),X((s τ) ),ρ)]+ 2 X(s )2]N˜(dρ,ds).
− − − − −| − |
Z0 ZE (cid:12) (cid:12)
Proof. For x Rd, we tak(cid:12)e thesmooth function F(x) = x 2 in Lem(cid:12) ma2.1. Thenthe correspond-
∈ + | |
ing equality (6) reads
t t
X(t)2 = ξ(0)2 +2 X(s),b(s,X(s),X(s τ)) ds+ σ(s,X(s),X(s τ)) 2ds
| | | | h − i k − k
Z0 Z0
t
+M(t)+ [ [X(s )+g(s,X(s ),X((s τ) ),ρ)]+ 2 X(s )2
− − − − −| − |
Z0 ZE (cid:12) 2 X(s ),g(s,X(s ),X((s τ) ),ρ) ](cid:12)ν(dρ)ds,
−(cid:12) h − − − − i(cid:12)
where we used the support property (5), namely
t
X(s),dKc(s) = 0, t 0.
h i ∀ ≥
Z0
For (t,x,y,ρ) R Rd Rd , applying the following inequality
∈ +× +× +×E
(12) [x+g(t,x,y,ρ)]+ 2 x 2 2 x,g(t,x,y,ρ) g(t,x,y,ρ)2,
−| | − h i ≤ | |
and conclude the(cid:12)(cid:12) validity of the ine(cid:12)(cid:12)quality (11) by using integration by parts. (cid:3)
6
Corollary 2.2. Let H() C2(Rd). Define the following function as
· ∈ +
1
QH(t,x,y) = H(x),b(t,x,y) + Tr (σσT)(t,x,y)(D2H(x)+ H(x) H(x))
h∇ i 2 ∇ ⊗∇
(cid:2) (cid:3)
(13) + exp H([x+g(t,x,y,ρ)]+) H(x) 1 H(x),g(t,x,y,ρ) ν(dρ),
− − −h∇ i
ZE
(cid:2) (cid:8) (cid:9) (cid:3)
on (t,x,y) R Rd Rd. We further define the positive process by
∈ +× +× +
t
(14) YE(t) := exp H(X(t)) H(ξ(0)) QH(s,X(s),X(s τ))ds , t 0,
− − − ≥
(cid:18) Z0 (cid:19)
where X = (X(t); t τ) is the d-dimensional strong solution process to RSDDEJ (1). Then the
≥ −
positive process YE = (YE(t); t 0) satisfies the following equality:
≥
t
(15) YE(t) = 1+ME(t)+ YE(s) H(X(s)),dKc(s) , t 0,
h∇ i ≥
Z0
where the process ME = (ME(t); t 0) is an ( ; t 0)-local martingale taken values on R.
t
≥ F ≥
Proof. For x Rd, take the smooth function F(x) = exp(H(x)) in Lemma 2.1. Using the rep-
∈ +
resentation of derivatives F(x) = F(x) H(x) and D2F(x) = F(x)[D2H(x)+ H(x) H(x)],
∇ ∇ ∇ ⊗∇
we arrive at
dexp H(X(t)) H(ξ(0))
{ − } = QH(t,X(t),X(t τ))dt+dMˆ(t)+ H(X(t)),dKc(t) ,
exp H(X(t)) H(ξ(0)) − h∇ i
{ − }
where Mˆ = (Mˆ(t); t 0) is a real-valued ( ; t 0)-local martingale. Then the equality (15)
t
follows fromapplyingi≥ntegration bypartstoeFxp H≥(X(t)) H(ξ(0)) exp( tQH(s,X(s),X(s
{ − } − 0 −
τ))ds). (cid:3)
R
The Corollary 2.1 can be used to establish the estimate of the second-order moment for the
d-dimensional segment process (X ; t 0) corresponding to the d-dimensional solution process
t
≥
X = (X(t); t τ).
≥ −
Proposition 2.1. Under the condition (A1), it holds that
(16) supE X 2 < + .
t
k k ∞
t 0
≥
(cid:2) (cid:3)
Proof. Let λ > 0. By virtue of the inequality (11) in Corollary 2.1, we arrive at, under the
condition (A1),
E eλt X(t)2
| |
h i t
E ξ(0)2 +2E eλs X(s),b(s,X(s),X(s τ)) ds
≤ | | h − i
(cid:20)Z0 (cid:21)
(cid:2) t(cid:3) t
+E λ eλs X(s)2ds +E eλs σ(s,X(s),X(s τ)) 2ds
| | k − k
(cid:20) Z0 (cid:21) (cid:20)Z0 (cid:21)
t
+E eλs g(s,X(s),X(s τ),ρ)2ν(dρ)ds
| − |
(cid:20)Z0 Z (cid:21)
E
7
t t t
E ξ(0)2 +α eλsds+(λ α )E eλs X(s)2ds +α E eλs X(s τ)2ds
1 2
≤ | | − | | | − |
Z0 (cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21)
(cid:2) (cid:3) α 0 t
= E ξ(0)2 + (eλt 1)+eλτE eλv ξ(v)2dv +(λ α +α eλτ)E eλs X(s)2ds .
1 2
| | λ − | | − | |
(cid:20)Z τ (cid:21) (cid:20)Z0 (cid:21)
(cid:2) (cid:3) −
Now we can choose a constant λ > 0 such that λ α +α eλ∗τ = 0, since α > α . Then, for all
∗ ∗ 1 2 1 2
−
t 0,
≥
α 0
E X(t)2 e λ∗t (eλ∗t 1)+E ξ(0)2 +eλ∗τE eλ∗v ξ(v)2dv
−
| | ≤ λ − | | | |
(cid:26) ∗ (cid:20)Z−τ (cid:21)(cid:27)
(cid:2) (cid:3) α (cid:2) 0 (cid:3)
(17) +E ξ(0)2 +eλ∗τE eλ∗v ξ(v)2dv .
≤ λ | | | |
∗ (cid:20)Z−τ (cid:21)
(cid:2) (cid:3)
Let θ [ τ,0]. For any t > τ, it follows from Lemma 2.1 that
∈ −
t+θ
X(t+θ)2 = X(t τ)2 +2 X(s),b(s,X(s),X(s τ)) ds
| | | − | h − i
Zt−τ
t+θ
+ σ(s,X(s),X(s τ)) 2ds
k − k
Zt−τ
t+θ
+2 X(s),σ(s,X(s),X(s τ))dW(s)
h − i
Zt τ
−
t+θ
+2 X(s),g(s,X(s),X(s τ),ρ) N˜(dρ,ds)
h − i
Zt−τ ZE
t+θ
+ [ [X(s )+g(s,X(s ),X((s τ) ),ρ)]+ 2 X(s )2
− − − − −| − |
Zt2−τXZ(Es(cid:12)),g(s,X(s ),X((s τ) ),ρ) ]N(dρ,ds(cid:12)).
(cid:12) (cid:12)
− h − − − − i
Using the Burkh¨older-Davis-Gundy inequality, we obtain
t+θ
E sup X(s),σ(s,X(s),X(s τ))dW(s)
h − i
"−τ≤θ≤0(cid:12)(cid:12)Zt−τ (cid:12)(cid:12)#
1 (cid:12) t (cid:12)
E sup(cid:12) X(t+θ)2 +CE σ(s,X(s),X(s(cid:12) τ)) 2ds ,
≤ 2 | | k − k
"−τ≤θ≤0 # (cid:20)Zt−τ (cid:21)
and similarly
t+θ
E sup X(s),g(s,X(s ),X((s τ) ),ρ) N˜(dρ,ds)
h − − − i
"−τ≤θ≤0(cid:12)(cid:12)Zt−τ ZE (cid:12)(cid:12)#
1 (cid:12) t (cid:12)
E sup(cid:12) X(t+θ)2 +CE g(s,X(s),X(s τ),ρ)2ν(cid:12)(dρ)ds ,
≤ 4 | | | − |
"−τ≤θ≤0 # (cid:20)Zt−τ ZE (cid:21)
where C > 0 is some positive constant. On the other hand, by using the inequality (12), it follows
that
t+θ
E sup [[X(s )+g(s,X(s ),X((s τ) ),ρ)]+ 2 X(s )2
| − − − − | −| − |
"−τ≤θ≤0Zt−τ ZE
8
2 X(s ),g(s,X(s ),X((s τ) ),ρ) ]N(dρ,ds)
− h − − − − i
#
t+θ
E sup g(s,X(s ),X((s τ) ),ρ)2N(dρ,ds)
≤ | − − − |
"−τ≤θ≤0Zt−τ ZE #
t
E g(s,X(s),X(s τ),ρ)2ν(dρ)ds .
≤ | − |
(cid:20)Zt τ Z (cid:21)
− E
Hence, it holds that
t
(18) E X 2 4E X(t τ)2 +C E X(s τ)2 ds,
t
k k ≤ | − | | − |
Zt−τ
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
for some positive constant C which is independent of time t. The required assertion follows from
(17). (cid:3)
Thefollowing resultrelates tothe exponential momentof thed-dimensional solution process X,
which is a reflected delay version of Ro¨chner and Zhang [RZ07]’s exponential integrability of the
solution without reflection and delay when the drift and diffusion coefficients (b,σ) are uniformly
bounded on (t,x,y) [0,T] Rd Rd with T > 0. In particular, we can establish the exponential
∈ × +× +
moment estimate of the following RSDDEJ without drift and diffusive parts:
t
X(t) = ξ(0)+ g(s,X(s ),X((s τ) ),ρ)N˜(dρ,ds)+K(t)
− − −
Z0 ZE
(19) =: ξ(0)+Y˜(t)+K(t), t 0,
≥
where the d-dimensional process Y˜ = (Y˜i(t); t 0) is the compensated version of the pure
d 1
jump process Y = (Yi(t); t 0) defined as (8≥). ×
d 1
≥ ×
Lemma 2.2. For the characteristic measure and the jump coefficient (ν,g), suppose that there
exists a constant ℓ > 0 such that
g
(20) g(t,x,y,ρ) ℓ h(t,ρ), (t,x,y,ρ) [0,T] Rd Rd ,
| | ≤ g ∀ ∈ × +× +×E
where T > 0 and h(t,ρ) is a nonnegative measurable function satisfying
(21) sup h2(t,ρ)e̺h(t,ρ)ν(dρ)< + ,
∞
0 t T
≤ ≤ ZE
for any finite ̺ > 0. If the drift and diffusion coefficients (b,σ) are uniformly bounded on (t,x,y)
∈
[0,T] Rd Rd, then
× +× +
(22) E exp a sup X(t) < + ,
| | ∞
" 0 t T !#
≤ ≤
for any finite a > 0.
Proof. We adopt the test function used in Ro¨chner and Zhang [RZ07] to discuss our reflected
delay case. Consider the following smooth function on Rd,
+
(23) H (x) = 1+λ x 2, λ >0, and x Rd.
λ | | ∈ +
p
9
Then, the gradient H (x) = λH 1(x)x with x Rd. For any i,j = 1,2,...,d and x Rd, the
∇ λ λ− ∈ + ∈ +
partial derivatives
∂H (x) ∂2H (x) ∂H (x)∂H (x)
(24) λ √λ, and λ + λ λ 2λ.
∂x ≤ ∂x ∂x ∂x ∂x ≤
i i j i j
Recall the real-valued process YE = (YE(t); t 0) given by (15). Note that, for all t 0,
≥ ≥
t t
0 H (X(s)),dKc(s) = λH 1(X(s)) X(s),dKc(s)
≤ h∇ λ i λ− h i
Z0 Z0
t
λ X(s),dKc(s) = 0,
≤ h i
Z0
by employing the support property (5). This implies that
t
H (X(s)),dKc(s) = 0, t 0.
λ
h∇ i ∀ ≥
Z0
Then by Corollary 2.2, we have that
t
YE(t) := exp H (X(t)) H (ξ(0)) QHλ(s,X(s),X(s τ))ds , t 0
λ λ λ
− − − ≥
(cid:18) Z0 (cid:19)
is a positive ( ; t 0)-local martingale and hence it is a supermartingale, where the function
t
F ≥
QH(t,x,y) is given by (13) with the function H(x) replaced by H (x). Using the inequality
λ λ
H ([x+g(t,x,y,ρ)]+) H (x+g(t,x,y,ρ)) and the estimates of derivatives (24), it is not difficult
λ λ
≤
to prove that, for any (t,x,y) [0,T] Rd Rd,
∈ × +× +
QHλ(t,x,y) C 1+ sup h2(t,ρ)e√λh(t,ρ)ν(dρ) := C < + ,
1 2
≤ ∞
" 0 t T #
≤ ≤ ZE
under the conditions (20) and (21), where positive constants C = C (d,λ) and C = C (d,λ,T)
1 1 2 2
dependon thedimension numberd, theparameter λ and thetime level T only. Based on theabove
estimate of the function QHλ(t,x,y), the desired result follows from Proposition 4.2 and Corollary
4.3 in Ro¨chner and Zhang [RZ07]. (cid:3)
3 Invariant measures
In this section, we will establish the existence and uniqueness of invariant measures of the d-
dimensional segment process X (θ) = X(t+θ) with τ θ 0 under the condition (A1) and
t
− ≤ ≤
(A2).
For the initial data ξ D([ τ,0];Rd), we use (Xξ; t 0) to represent the corresponding
∈ − + t ≥
segment process to the d-dimensional solution process X = (X(t); t τ) with X(t) = ξ(t) on
≥ −
[ τ,0]. Define the Markov semigroup associated with the segment process by
−
(25) f(ξ)= E f Xξ ,
Pt t
h (cid:16) (cid:17)i
10