A Maximal Inequality for $p$th Power of Stochastic Convolution Integrals

0.23 MB·English

by  Erfan Salavati

A Maximal Inequality for pth Power of Stochastic
Convolution Integrals
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1
0
2 Erfan Salavati and Bijan Z. Zangeneh

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a Department of Mathematical Sciences
J
Sharif University of Technology
2
Tehran, Iran

]
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P
Abstract
.
h
Aninequalityforthepthpowerofthenormofastochasticconvolution
t
a integral in a Hilbert space is proved. The inequality is stronger than
m
analoguesinequalitiesintheliteratureinthesensethatitispathwiseand
[ not in expectation.
Anapplicationofthisinequalityisprovidedforthesemilinearstochas-
1
ticevolutionequationswithL´evynoiseandmonotonenonlineardrift. The
v
existence and uniqueness of the mild solutions in Lp for theses equations
2
isprovedandasufficientconditionforexponentialasymptoticstabilityof
0
the solutions is derived.
4
0
0 Mathematics Subject Classification: 60H10, 60H15, 60G51, 47H05, 47J35.
.
1 Keywords: StochasticConvolutionIntegral,Itoˆtypeinequality,StochasticEvo-
0
lution Equation, Monotone Operator, L´evy Noise.
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1
:
v 1 Introduction
i
X
Stochastic convolution integrals appear in many fields of stochastic analysis.
r
a They are integrals of the form
t
X = S dM
t t−s s
Z0
whereM isamartingalewithvaluesinaHilbertspace. Althoughtheyaregen-
t
eralizationofstochasticintegralsbuttheyaredifferentinmanyways. Forexam-
pletheyarenotsemimartingalesingeneralandhencetheusualresultsonsemi-
martingales, such as maximal inequalities (i.e. inequalities for sup kX k)
0≤s≤t s
and existence of ca`dla`g versions could not be applied directly to them.
Among first studies in this field one can note the works of Kotelenez [7] and
Ichikawa [5] where they consider stochastic convolution integrals with respect
1
to general martingales. They prove a maximal inequality in L2 for stochastic
convolution integrals (Theorem 1).
Stochasticconvolutionintegralsarisenaturallyinprovingexistence,uniqueness
and regularity of the solutions of semilinear stochastic evolution equations,
dX =AX dt+f(t,X )dt+g(t,X )dM
t t t t t
whereAisthegeneratorofaC semigroupoflinearoperatorsonaHilbertspace
0
and M is a martingale. The case that the coefficients are Lipschitz operators
t
is studied well and the theorems of existence, uniqueness and continuity with
respecttoinitialdataforthesolutionsinL2isproved,seee.gKotelenez[8]. The
proofsarebasedonthe maximalinequality forstochastic convolutionintegrals,
that is Theorem 1.
These results have been generalized in several directions. One is the maximal
inequalityforpthpowerofthenormofstochasticconvolutionintegrals. Tubaro
hasprovedanupperestimateforE[sup |x(s)|p]withp≥2inthecasethat
0≤s≤t
M isarealWienerprocess. Ichikawa[5]hasprovedmaximalinequalityforpth
t
power, p ≥ 2 in the special case that M is a Hilbert space valued continuous
t
martingale. ThecaseofgeneralmartingaleisprovedbyZangeneh[18]forp≥2
(see Theorem 5). Hamedani and Zangeneh [4] have generalized the maximal
inequality to 0<p<∞.
Brzezniak, Hausenblas and Zu [2] have derived a maximal inequality for pth
power of the norm of stochastic convolutions driven by Poisson random mea-
sures.
As far as we know, the maximal inequalities proved for stochastic convolution
integralsintheliteratureallinvolveexpectations. Theonlyresultthatprovides
a pathwise (almost sure) bound is Zangeneh [?] in which is proved Theorem 2
called Ito¨ type inequality. This inequality provides a pathwise estimate for the
square of the norm of stochastic convolution integrals and is the generalization
of the Ito¨ formula to stochastic convolution integrals.
In Section 2 we define and state some results about stochastic convolution in-
tegrals that will be used in the sequel.
In Section 3 we state and prove the main result of this article, i.e. Theorem 6,
which provides a pathwise bound for the pth power of stochastic convolution
integrals with respect to general martingales. The special case that the mar-
tingale is an Ito¨ integral with respect to a Wiener process has been proved by
Jahanipour and Zangeneh [6].
ThepathwisenatureofTheorem6enablesonetoapplyittosemilinearstochas-
tic evolution equations with non Lipschitz coefficients. We consider the drift
term to be a monotone nonlinear operator and the noise term to be a com-
pensated Poisson random measure and prove the exitence of the mild solution
in Lp in Theorem 15. The precise assumptions on coefficients wll be stated in
Section 4. An auxiliary result is a Bichteler-Jacod inequality in Hilbert spaces
proved in Theorem 9. This result has been stated and proved before in the
literature, for example in [10], but we give a new proof for it. We also show
the exponential stability of the mild solutions under certain conditions in The-
orem 19.
2
2 Stochastic Convolution Integrals
Let H be a separable Hilbert space with inner product h, i. Let S be a C
t 0
semigroup on H with infinitesimal generator A : D(A) → H. Furthermore we
assume the exponential growth condition on S , i.e. there exists a constant α
t
such that kS k≤eαt. If α=0, S is called a contraction semigroup.
t t
Inthissectionwereviewsomepropertiesandresultsaboutconvolutionintegrals
t
oftype X = S dM where M is a martingale. These are calledstochastic
t 0 t−s s t
convolution integrals. Kotelenez [8] gives a maximal inequality for stochastic
R
convolution integrals.
Theorem 1 (Kotelenez, [8]). Assume α ≥ 0. There exists a constant C such
that for any H-valued c`adla`g locally square integrable martingale M we have
t
t
E sup k S dM k2 ≤Ce4αTE[M] .
t−s s T
0≤t≤T Z0
Remark. Hamedani and Zangeneh[4] generalizedthis inequality to a stopped
maximal inequality for p-th moment (0 < p < ∞) of stochastic convolution
integrals.
Because of the presence of monotone nonlinearity in our equation, we need a
pathwise bound for stochastic convolutionintegrals. For this reasonthe follow-
ingpathwiseinequalityforthenormofstochasticconvolutionintegralshasbeen
proved in Zangeneh [18].
Theorem 2 (It¨otype inequality,Zangeneh[18]). Let Z be an H-valued c`adla`g
t
locally square integrable semimartingale. If
t
X =S X + S dZ ,
t t 0 t−s s
Z0
then
t t
kX k2 ≤e2αtkX k2+2 e2α(t−s)hX ,dZ i+ e2α(t−s)d[Z] ,
t 0 s− s s
Z0 Z0
where [Z] is the quadratic variation process of Z .
t t
We state here the Burkholder-Davis-Gundy (BDG) inequality and a corollary
to it, for future reference.
Theorem 3 (Burkholder-Davis-Gundy (BDG) inequality). For every p ≥ 1
there exists a constant C > 0 such that for any real valued square integrable
p
cadlag martingale M with M =0 and for any T ≥0,
t 0
p
E sup |M |p ≤C E[M]2.
t p T
0≤t≤T
Proof. See [14], page 37, and the reference there.
3
Corollary 4. Let p≥1 and C be the constant in the BDG inequality and M
p t
bean H-valuedsquareintegrablecadlag martingale andX an H-valuedadapted
t
process and T ≥0. Then for K >0,
t p
p
E sup hX ,dM i ≤ C E (X∗)p[M]2
s s p t t
0≤t≤T(cid:12)(cid:12)Z0 (cid:12)(cid:12) C (cid:16) C(cid:17)K
(cid:12)(cid:12) (cid:12)(cid:12) ≤ 2Kp E(Xt∗)2p+ p2 E[M]pt.
where X∗ =E sup kX k.
t t
0≤t≤T
Proof. See [18], Lemma 4, page 147.
We will needalso the followinginequality whichis ananalogousof Burkholder-
Davies-Gundy inequality for stochastic convolution integrals.
Theorem 5 (Burkholder Type Inequality, Zangeneh [18], Theorem 2, page
147). Let p≥2 and T >0. Let S be a contraction semigroup on H and M be
t t
an H-valued square integrable c`adla`g martingale for t∈[0,T]. Then
t
p
E sup k S dM kp ≤K E([M]2)
t−s s p T
0≤t≤T Z0
where K is a constant depending only on p.
p
3 Ito¨ Type Inequality for pth Power
WeusethenotionofsemimartingaleandIto¨’sformulaasisdescribedinMetivier[?].
Theorem 6 (It¨o type Inequality for pth power). Let p ≥ 2. Assume Z(t) =
V(t)+ M(t) is a semimartingale where V(t) is an H-valued process with fi-
nite variation |V|(t) and M(t) is an H-valued square integrable martingale with
quadratic variation [M](t). Assume that
E[M](T)2p <∞ E|V|(T)p <∞
Let X (ω) be F measurable and square integrable. Define X(t) = S(t)X +
0 0 0
t
S(t−s)dZ(s). Then we have
0
R t
kX(t)kp ≤ epαtkX kp+p epα(t−s)kX(s−)kp−2hX(s−),dZ(s)i
0
Z0
1 t
+ p(p−1) epα(t−s)kX(s−)kp−2d[M]c(s)
2
Z0
+ epα(t−s) kX(s)kp−kX(s−)kp−pkX(s−)kp−2hX(s−),∆X(s)i
0≤s≤t
X (cid:0) (cid:1)
Remark. 1. Forp=2thetheoremimpliestheItotypeinequality(Theorem2).
4
2. IfM isacontinuousmartingalethentheinequalitytakesthesimplerform
t
kX(t)kp ≤ epαtkX kp+p epα(t−s)kX(s−)kp−2hX(s−),dZ(s)i
0
Z0
1 t
+ p(p−1) epα(t−s)kX(s−)kp−2d[M](s)
2
Z0
Before proceeding to the proof of theorem we state and prove some lemmas.
Lemma 7. It suffices to prove theorem 2 for the case that α=0.
Proof. Define
S˜(t)=e−αtS(t), X˜(t)=e−αtX(t),
dZ˜(t)=e−αtdZ(t)
NowwehavedX˜(t)=S˜(t)X + t˜(S)(t−s)dZ˜(s). NotethatS˜ isacontraction
0 0 t
semigroup. It is easyto see that the statementfor X˜ implies the statement for
R t
X(t).
Hence from now on we assume α=0.
Lemma 8 (Ordinary Ito¨’s formula for pth power). Let p ≥2 and assume that
Z(t) is an H-valued semimartingale. Then
kZ(t)kp ≤kZ(0)kp+p tkZ(s−)kp−2hZ(s−),dZ(s)i+ p(p−1) tkZ(s−)kp−2d[M]c(s)
0 2 0
+ kZ(s)kp−kZ(s−)kp−pkZ(s−)kp−2hZ(s−),∆Z(s)i
0≤s≤t R R
ProPof. UseI(cid:0)to¨’sformula(Metivier[?],Theorem27.2,Page190)f(cid:1)orϕ(x)=kxkp
and note that
ϕ′(x)(h)=pkxkp−2hx,hi,
1 1 1
ϕ′′(x)(h⊗h)= p(p−2)kxkp−4hx,hihx,hi+ pkxkp−2hh,hi≤ p(p−1)kxkp−2khk2
2 2 2
Lemma 9. Assume v :[0,T] →D(A) is a function with finite variation (with
respect to the norm of D(A)) denoted by |v|(t). Assume that u ∈ D(A). Let
0
t
u(t)=S(t)u + S(t−s)dv(s). Then u(t) is D(A)-valued and satisfies
0 0
R t
u(t)=u + Au(s)ds+v(t)
0
Z0
Proof. (seealsoCurtainandPritchardpage30Theorem2.22forthespecialcase
dv(t)=f(t)dt.) Let q(t) be the Radon-Nikodym derivative of v(t) with respect
to |v|(t), i.e, q(t) is a D(A)-valued function which is Bochner measurable with
t
respect to d|v|(t) and v(t) = q(s)d|v|(s). We know that for every t ∈ [0,T],
0
kq(t)k≤1.
R
5
Recall from semigrop theory that one can equip D(A) with an inner product
by defining hx,yi := hx,yi+hAx,Ayi. By closedness of A it follows that
D(A)
under this inner product D(A) is a Hilbert space and A : D(A) → H is a
bounded linear map. Note that S(t) is also a semigroup on D(A). Hence u(t)
is a convolution integral in D(A) and hence has it’s value in D(A). We use the
following two simple identities that hold in D(A):
t t
S(t)x=x+ AS(r)xdr, S(t−s)x=x+ S(r−s)Axdr
Z0 Zs
We have
t
u(t) =S(t)u + S(t−s)dv(s)
0 0
t
=S(t)u + S(t−s)q(s)d|v|(s)
0 R0
t t t
=u + AS(r)u dr+ q(s)+A S(r−s)q(s)dr d|v|(s)
0 0 R 0 0 s
R R (cid:16) R (cid:17)
Now using Fubini’s theorem we find
t t t r
=u + q(s)d|v|(s)+ AS(r)u dr+ A S(r−s)q(s)d|v|(s)dr
0 0 0 0 0 0
t r
=u +v(t)+ A S(r)u dr+ S(r−s)dv(s)dr
0 R 0 R 0 0 R R
t
=u +v(t)+ Au(r)dr.
0 R0 (cid:0) R (cid:1)
R
Lemma 10. Assume V(t) is a D(A)-valued process with finite variation in
D(A) and M(t) is a D(A)-valued square integrable martingale and V(0) =
M(0) = 0. Let Z(t) = V(t) + M(t) and let X be D(A)-valued and F -
0 0
t
measurable and define X(t)=S(t)X + S(t−s)dZ(s). Then X(t) is D(A)-
0 0
valued and satisfies the following stochastic integral equation in H:
R
t
X(t)=S(t)X + AX(s)ds+Z(t)
0
Z0
Proof of Lemma. Note that S(t) is also a semigroup on D(A). Hence X(t) is
a stochastic convolution integral in D(A) and hence has it’s value in D(A).
t t
Write Y(t) = S(t)X + S(t−s)dV(s) and Y(t) = S(t−s)dM(s). Hence
0 0 0
X(t) = Y(t)+Y(t). We can apply lemma 9 to term Y(t) and deduce Y(t) =
R R
t t
X + AY(s)ds+V(t). Hence it suffices to prove Y(t)= AY(s)ds+M(t).
0 0 0
j
Let{e ,e ,e ,...}beabasisforHilbertspaceD(A). DefineM (t)=hM(t),e i
R1 2 3 R j
andMk(t)= k Mj(t). LetYk(t)= tS(t−s)dMk(s). Weusethefollowing
j=1 0
two simple identities that hold in D(A):
P R
t t
S(t)x=x+ AS(r)xdr, S(t−s)x=x+A S(r−s)xdr
Z0 Zs
6
We have
Yk(t) = tS(t−s)dMk(s)
0
= k tS(t−s)e dMj(s)
R 1 0 j
= k t e + tS(r−s)Ae dr dMj(s)
P1R0 j s j
=PMk(Rt)(cid:16)+ 0t sRtS(r−s)Aejdrd(cid:17)Mj(s)
Now using stochastic Fubini theorRemR(see [14] Theorem 8.14 page 119) we find
=Mk(t)+ t rS(r−s)Ae dMj(s)dr
0 0 j
=Mk(t)+ tA rS(r−s)dMj(s) dr
R0 R 0
=Mk(t)+RtAY(cid:16)Rk(s)ds. (cid:17)
0
Hence we find R
t
Yk(t)=Mk(t)+ AYk(s)ds (1)
Z0
We have EkM(T)−Mk(T)k2 →0 and by Theorem 1
D(A)
E sup kY(t)−Yk(t)k2 ≤CEkM(T)−Mk(T)k2 →0
D(A) D(A)
0≤t≤T
andsinceA:D(A)→H iscontinuousEsup kAY(t)−AYk(t)k2 →0and
0≤t≤T H
hence Ek tAY(s)ds− tAYk(s)dsk → 0. Hence by taking limits from both
0 0
sides of (1) we get
R R t
Y(t)=M(t)+ AY(s)ds.
Z0
Proof of Theorem 6. By using Lemma 7, we need only to prove for the case
α=0. In this case we have to prove
kX(t)kp ≤ kX kp+p tkX(s−)kp−2hX(s−),dZ(s)i+ 1p(p−1) tkX(s−)kp−2d[M]c(s)
0 0 2 0
+ kX(s)kp−kX(s−)kp−pkX(s−)kp−2hX(s−),∆X(s)i .
0≤s≤t R R
(2)
P (cid:0) (cid:1)
The main idea is that we approximate M(t) and V(t) by some D(A) valued
processes,and for D(A) valuedprocesseswe use ordinaryIto¨’s formula. This is
doneby Yosidaapproximations. We recallsomefacts fromsemigrouptheoryin
the following lemma. For proofs see Pazy [12].
Lemma 11. For λ > 0, λI −A is invertible. Let R(λ) = λ(λI −A)−1 and
A(λ)=AR(λ). We have:
(a) R(λ):H →D(A) and A(λ):H →H are bounded linear maps.
(b) for every x∈H, kR(λ)xk ≤kxk and hx,A(λ)xi≤0.
H H
(c) R(λ)S(t)=S(t)R(λ) and for x∈D(A), R(λ)Ax=AR(λ)x.
7
(d) for every x∈H, lim R(λ)x=x in H.
λ→∞
(e) for every x∈D(A), lim A(λ)x=Ax.
λ→∞
Now for n=1,2,3,... Define:
Vn(t)=R(n)V(t), Mn(t)=R(n)M(t), Zn(t)=Vn(t)+Mn(t)=R(n)Z(t)
t
Xn =R(n)X , Xn(t)=S(t)Xn+ S(t−s)dZn(s)
0 0 0
Z0
AccordingtoLemma11,Vn(t)isaD(A)-valuedfinite variationprocess,Mn(t)
isaD(A)-valuedmartingaleandZn(t)isaD(A)-valuedsemimartingale. Hence
by lemma 10, Xn(t) is an ordinary stochastic integral and hence we can apply
Lemma 8 to it and find
kXn(t)kp ≤kXnkp+p tkXn(s−)kp−2hXn(s−),AXn(s)ds+dVn(s)+dMn(s)i
0 0
+p(p−1) tkXn(s−)kp−2d[Mn]c(s)+Fn
2 0 R
where R
Fn = kXn(s)kp−Xn(s−)p−pkXn(s−)kp−2hXn(s−),∆Zn(s)i .
0≤s≤t
X (cid:0) (cid:1)
SinceAisthe generatorofacontractionsemigroup,wehavehAx,xi≤0,hence
we find
t
kXn(t)kp ≤ kXnkp+p kXn(s−)kp−2hXn(s−),dVn(s)i
0
Z0
An Bn
Cn
t
| {z } | {z }
+p kXn|(s−)kp−2hXn(s−{z),dMn(s)i }
(3)
Z0
Dn
t
+p(|p−1) kXn(s−){kzp−2d[Mn]c(s)+F}n.
2
Z0
En
We claim that the inequality (3)| (after choo{szing a suitabl}e subsequence) con-
verges term by term in to the following inequality and hence the following will
be proved:
t
kX(t)kp ≤ kX kp+p kX(s−)kp−2hX(s−),dV(s)i
0
Z0
A B
C
t
| {z } | {z }
+p kX|(s−)kp−2hX(s−{z),dM(s)i }
Z0
D
t
+p(|p−1) kX(s−{)zkp−2d[M]c(s)+}F
2
Z0
E
| {z }
8
where
F = kX(s)kp−X(s−)p−pkX(s−)kp−2hX(s−),∆Z(s)i .
0≤s≤t
X (cid:0) (cid:1)
We prove this claim in several steps.
(Step 1) We claim that E|Vn−V|(t)p → 0. Let q(t) be the Radon-Nykodim
derivative of V(t) with respect to |V|(t). We know that for every t,
kq(t)k≤1. We have
t p
E|Vn−V|(t)p =E k(R(n)−I)q(s)kd|V|(s)
(cid:18)Z0 (cid:19)
Note that for every s and ω, k(R(n)−I)q(s)k ≤2 and tends to zero and
since |V|(t) < ∞, a.s. by the Lebesgue’s dominated convergence theo-
t
rem, k(R(n)−I)q(s)kd|V|(s)→0, a.s. and is dominated by 2|V|(t).
0
NowsinceE|V|(t)p <∞andusingtheLebesgue’sdominatedconvergence
R p
theorem we find that E tk(R(n)−I)q(s)kd|V|(s) →0 and the claim
0
is proved. (cid:16)R (cid:17)
(Step 2) We claim that E[Mn − M](t)p2 → 0. Note that [Mn − M](t) ≤
2[Mn](t)+2[M](t) ≤ 4[M](t) and hence [Mn−M](t)p2 is dominated by
4p2[M](t)p2. On the other hand E[Mn −M](t) = EkMn(t)−M(t)k2 →
0. Hence [Mn − M](t) and consequently [Mn − M](t)p2 tend to 0 in
probability and therefore by Lebesgue’s dominated convergence theorem
it’s expectation also tends to 0.
(Step 3) We claim that
E sup kXn(s)−X(s)kp →0. (4)
0≤s≤t
We have
kXn(s)−X(s)kp ≤3pkS(s)(Xn−X )kp
0 0
A1
s
+3pk| S(s{−zr)d(Vn}(r)−V(r))kp
Z0
A2
s
+3p|k S(s−r)d({Mz n(r)−M(r))}kp.
Z0
A3
For A1 we have | {z }
E sup A1 ≤EkX0n−X0kp →0.
0≤s≤t
9
For A2 we have
E sup A2 ≤E|Vn−V|(t)p →0,
0≤s≤t
where we have used Step 1. For A3, we use Burkholder type inequality
(Theorem 5) for α=0 and find
E sup A3 ≤KpE [Mn−M](t)p2 →0,
0≤s≤t
(cid:16) (cid:17)
where we have used Step 2. Hence (4) is proved.
(Step 4) We claim that
E sup kXn(s)kp →E sup kX(s)kp (5)
0≤s≤t 0≤s≤t
By triangle inequality,
1 1
Esup kXn(s)kp p − Esup kX(s)kp p ≤
0≤s≤t 0≤s≤t
(cid:12)(cid:12)(cid:0)E sup0≤s≤tkXn(s)k−(cid:1)sup0(cid:0)≤s≤tkX(s)k p p1 ≤ (cid:1) (cid:12)(cid:12)
(cid:12) 1 (cid:12)
(cid:0)E(cid:12)sup0≤s≤t|kXn(s)k−kX(s)k|p p ≤ (cid:12) (cid:1)
(cid:12) 1 (cid:12)
Esup kXn(s)−X(s)kp p →0
(cid:0) 0≤s≤t (cid:1)
where in (cid:0)the last line we have used St(cid:1)ep 3. Hence (5) is proved and in
particular the sequence Esup kXn(s)kp is bounded for each t.
0≤s≤t
(Step 5) We claim that E|Cn−C|→0. We have
t
E|Cn−C|≤E| (kXn(s−)kp−2−kX(s−)kp−2)hXn(s−),dVn(s)i|
Z0
Cn
1
t
+E| kX(s−|)kp−2hXn(s−)−X(s−),d{Vzn(s)i|+ }
Z0
Cn
2
t
E|| kX(s−)kp−2hX(s−{)z,d(Vn(s)−V(s))i|. }
Z0
Cn
3
| n {z }
For the term C we have,
1
Cn ≤E (sup|kXn(s−)kp−2−kX(s−)kp−2|)(supkXn(s−)k)|Vn|(t)
1
Nowusing(cid:0)thesimpleinequality|a−b|r ≤|ar−br|forr ≥1anda,b∈(cid:1)R+
we have |kXn(s−)kp−2 − kX(s−)kp−2| ≤ |kXn(s−)kp − kX(s−)kp|p−p2.
Substituting and using the Holder inequality we find
≤ Esup|kXn(s−)kp−kX(s−)kp| p−p2 EsupkXn(s−)kp p1 (E|Vn|(t)p)p1
(cid:0) (cid:1) (cid:0) (cid:1)
10