Weak order in averaging principle for stochastic differential equations with jumps

0.2 MB·

by  Hongbo Fu

Weak order in averaging principle for stochastic
differential equations with jumps
Hongbo Fu, Li Wan
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1 Research Centerof Nonlinear Science, College of Mathematics and Computer Science,
0 Wuhan TextileUniversity,Wuhan, 430073, PR China
2
Jicheng Liu∗, Xianming Liu
n
a School of Mathematics and Statistics, Huazhong Universityof Science and Technology,
J
Wuhan, 430074, PR China

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2

]
R
P
Abstract
.
h
t This article deals with the weak error for averaging principle for two-time-
a
scale system of jump-diffusion stochastic differential equation. Under suitable
m
conditions, it is provedthat the rate of weak convergenceto the averagedeffec-
[
tive dynamics is of order 1 via an asymptotic expansion approach.

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Keywords: Jump-diffusion, averagingprinciple, invariant measure, weak
v
3 convergence,asymptotic expansion.
8 MSC 2010: primary 60H15, secondary 70K70
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7
0
1. Introduction
.
1
0
Weconsideratwo-time-scalesystemsofjump-diffusionstochasticdifferential
7
equation in form of
1
:
v dXǫ =a(Xǫ,Yǫ)dt+b(Xǫ)dB +c(Xǫ )dP , Xǫ =x (1.1)
t t t t t t− t 0
i
X 1 1
dYǫ = f(Xǫ,Yǫ)dt+ g(Xǫ,Yǫ)dW +h(Xǫ ,Yǫ )dNǫ, Yǫ =y,(1.2)
r t ǫ t t √ǫ t t t t− t− t 0
a
where Xǫ Rn,Yǫ Rm, the drift functions a(x,y) Rn,f(x,y) Rm, the
diffusiontfu∈nctionstb(∈x) Rn×d1,c(x) Rn,g(x,y) R∈m×d2 and h(x∈,y) Rm.
∈ ∈ ∈ ∈
B and W are the vector of d ,d -dimensional independent Brownian motions
t t 1 2
on a complete stochastic base (Ω, , ,P), respectively. P is a scalar simple
t t
F F
Poisson process with intensity λ , and Nǫ is a scalar simple Poisson process
1 t
∗Correspondingauthor at: School ofMathematics andStatistics, Huazhong Universityof
ScienceandTechnology, Wuhan, 430074, China.
Email addresses: [email protected] (HongboFu),[email protected] (LiWan),
[email protected] (JichengLiu),[email protected] (XianmingLiu)
Preprint submitted toArxiv January 30, 2017
with intensity λ2. The positive parameter ǫ is small and describes the ratio
ǫ
of time scales between Xǫ and Yǫ. Systems (1.1)-(1.2) with two time scale
t t
occur frequently in applications including chemical kinetics, signal processing,
complex fluids and financial engineering.
With the two separationof time scale, we can view the state variable of the
system as being divided into two parts: the “slow” variable Xǫ and the “fast”
t
variable Yǫ. It is often the case that we are interested only in the dynamics
t
of slow component. Then a simplified equation, which is independent of fast
variable and possesses the essential features of the system, is highly desirable.
Such a simplified equation is often constructed by averaging procedure as in
[2, 20] for deterministic ordinary differential equations, as well as the further
development[13,7,8,18,19,14,15,16]forstochasticdifferentialequationswith
continuous Gaussian processes. As far as averaging for stochastic dynamical
systemsininfinite dimensionalspaceis concerned,itis worthwhileto quote the
important work of [4, 5, 6, 26] and also the work of [9, 10].
In order to derive the averaging dynamics of the system (1.1)-(1.2), we in-
troduce the fast motionequationwith a frozen slow componentx Rn in form
∈
of
dYx =f(x,Yx)dt+g(x,Yx)dW +h(x,Yx )dN , Yx =y, (1.3)
t t t t t− t 0
whose solution is denoted by Yǫ(y). Under suitable conditions on f,g and h,
t
Yǫ(y) induces a unique invariant measure µx(dy) on Rm, which is ergodic and
t
ensures the averagedequation:
dX¯t =a¯(X¯t)dt+b(X¯t)dB˙t+c(X¯t−)dP˙t, X¯0 =x,
where the averagingnonlinear is defined by setting
a¯(x) = a(x,y)µx(dy)
ZRm
= lim Ea(x,Yx(y)).
t→+∞ t
In [12], it is shown that under the above conditions the slow motion Xǫ
t
convergesstronglytothesolutionX¯ oftheaboveaveragedequationwithjumps.
t
The order of convergence 1 in strong sense was provided in [17]. To the best
2
knowledge of the authors, the existing literature does not address the weak
order in averaging principle for jump diffusion stochastic differential systems.
Infact,itisfairtosaythattheweakconvergenceinstochasticaveragingtheory
of systems driven by jump noise is not yet fully developed, even though recent
years have witnessed some strong approximation results [1, 21, 22, 23, 24, 25].
In the present paper, we are interested in the rate of weak convergence of
theaveragingdynamicstothetruesolutionofslowmotionXǫ. Namely,wewill
t
determine the order, with respect to timescale parameter ǫ, of weak deviation
betweenoriginalsolutionofslowequationandthesolutionofthecorresponding
averaged equation. The main technique we adapted is to finding an expansion
with respect to ǫ of the solutions of the Kolmogorov equations associated with
2
the jump diffusion system. The solvability of the Poisson equation associated
with the generatoroffrozenequationprovidesanexpressionfor the coefficients
of the expansion. As a result, the boundedness for the coefficients of expansion
canbeprovedbysmoothingeffectofthecorrespondingtransitionsemigroupsin
the space of bounded and uniformly continuous functions, where some regular
conditionsis neededondriftanddiffusionterm. Ourresultshowthatthe weak
convergence rate to be 1 even when there are jump components in the system,
whichis the mainnoveltyofthis work. We wouldlike to stressthat asymptotic
method was first applied by Br´ehier [3] to an averaging result for stochastic
reaction-diffusion equations in the case of Gaussian noise of additive type was
includedonlyinthefastmotion. However,theextensionofthisargumentisnot
straightforward. The method used in the proof of weak order in [3] is strictly
relatedto the differentiability in time of averagingprocess. Therefore,once the
noiseisintroducedintheslowequation,difficulties willariseandthe procedure
becomes more complicated. Our result in this paper bridges such a gap, in
which the slow and the fast motions are both perturbed by noise with jumps.
The rest of the paper is structured as follows. Section 2 is devoted to no-
tation, assumptions and summarize preliminary results. The ergodicity of fast
process and the averagingdynamics of system with jumps is introduced in Sec-
tion3. Thenthe mainresultofthis article,whichis derivedvia the asymptotic
expansions and uniform error estimates, is presented in Section 4. In the fi-
nal section, we state and prove technical lemmas to which we appeal in the
preceding section.
Throughout the paper, the letter C below with or without subscripts will
denote generic positive constants independent of ǫ, whose value may change
from one line to another.
2. Assumptions and preliminary results
Foranyintegerd,thescalarproductandnormond dimensionalEuclidean
−
space Rd are denoted by , and Rd, respectively. For any integer k,
· · Rd k·k
wedenote byCk(Rd,R)th(cid:16)e sp(cid:17)aceofallk timesdifferentiable functions onRd,
b −
which have bounded uniformly continuous derivatives up to the k-th order.
In what follows, we shall assume that the drift and diffusion coefficients
arising in the system fulfill the following conditions.
(A1)Themappingsa(x,y),b(x),c(x),f(x,y),g(x,y) andh(x,y)areofclass
C2 and have bounded first and second derivatives. Moreover, we assume that
a(x,y),b(x) and c(x) are bounded.
(A2) There exists a constant α > 0 such that for any x Rn,y Rm it
∈ ∈
holds
yTg(x,y)gT(x,y)y α y Rm.
≥ k k
(A3) There exists a constant β > 0 such that for any y ,y Rm and
1 2
x Rn it holds ∈
∈
y y ,f(x,y ) f(x,y )+λ (h(x,y ) h(x,y ))
1 2 1 2 2 1 2
− − − Rm
(cid:16) (cid:17)
3
+ g(x,y ) g(x,y ) 2 +λ h(x,y ) h(x,y )2
1 2 Rm 2 1 2
k − k | − |
β y y 2 .
1 2 Rm
≤− k − k
Remark 2.1. Notice that from (A1) it immediately follows that the following
directional derivatives exist and are controlled:
Dxa(x,y) k1 Rn L k1 Rn,
k · k ≤ k k
Dya(x,y) l1 Rn L l1 Rm,
k · k ≤ k k
kDx2xa(x,y)·(k1,k2)kRn ≤Lkk1kRnkk2kRn,
kDy2ya(x,y)·(l1,l2)kRn ≤Lkl1kRmkl2kRm,
where L is a constant independent of x,y,k ,k ,l and l . For differentiability
1 2 1 2
of mappings b,c,f,g and h we possess the analogous results. For examples, we
have
kDx2xb(x)·(k1,k2)kRn ≤Lkk1kRnkk2kRn, k1,k2 ∈Rn,
kDy2yf(x,y)·(l1,l2)kRm ≤Lkl1kRmkl2kRmk, l1,l2 ∈Rm.
As far as the assumption (A2) is concerned, it is a sort of non-degeneracy
condition anditis assumedinorder tohavetheregularizingeffectoftheMarkov
transition semigroup associated with the fast dynamics. Assumption (A3) is
the dissipative condition which determine how the fast equation converges to its
equilibrium state.
As assumption (A1) holds, for any ǫ> 0 and any initial conditions x Rn
and y Rm, system (1.1)-(1.2) admits a unique solution, which, in ord∈er to
∈
emphasizethedependenceontheinitialdata,isdenotedby(Xǫ(x,y),Yǫ(x,y)).
t t
Moreover the following lemma holds (for a proof see e.g. [17]).
Lemma 2.1. Under the assumption (A1), (A2) and (A3), for any x Rn,
y Rm and ǫ>0 we have ∈
∈
EkXtǫ(x,y)k2Rn ≤CT(1+kxk2Rn +kyk2Rm), t∈[0,T] (2.1)
and
EkYtǫ(x,y)k2Rn ≤CT(1+kxk2Rn +kyk2Rm), t∈[0,T]. (2.2)
3. Frozen equation and averaging equation
Fixing ǫ = 1, we consider the fast equation with frozen slow component
x Rn,
∈
dYx(y)=f(x,Yx(y))dt+g(x,Yx(y))dW +h(x,Yx (y))dN ,
t t t t t− t (3.1)
(Y0x =y.
4
Under assumption (A1)-(A3), such a problem has a unique solution, which
satisfies [17]:
E Yx(y) 2 C(1+ x 2 +e−βt y 2 ), t 0. (3.2)
k t kRm ≤ k kRn k kRm ≥
Let Yx(y′) be the solution of problem (3.1) with initial value Yx = y′, the Itoˆ
t 0
formula implies that for any t 0,
≥
E Yx(y) Yx(y′) 2 y y′ 2 e−βt. (3.3)
k t − t kRm ≤k − kRm
Moreover,as discussionin [17] and [12], equation (3.1) admits a unique ergodic
invariant measure µx with finite 2 moments:
−
y 2 µx(dy) C(1+ x 2 ). (3.4)
Rm Rn
ZRmk k ≤ k k
Then, by averaging the coefficient a with respect to the invariant measure µx,
we can define an Rn-valued mapping
a¯(x):= a(x,y)µx(dy),x Rn.
ZRm ∈
Due to assumption (A1), it is easily to check that a¯(x) is 2-times differentiable
with bounded derivatives and hence it is Lipschitz-continuous such that
a¯(x1) a¯(x2) Rn C x1 x2 Rn, x1,x2 Rn.
k − k ≤ k − k ∈
According to the invariantproperty of µx, (3.4) and assumption(A1), we have
kEa(x,Ytx(y)−a¯(x)k2Rn = kZRmE a(x,Ytx(y)−a(x,Ytx(z) µx(dz)k2Rn
(cid:0) (cid:1)
E Yx(y) Yx(z) 2 µx(dz)
≤ ZRm k t − t kRm
e−βt y z 2Rmµx(dz)
≤ ZRmk − k
Ce−βt 1+ x 2 + y 2 . (3.5)
Rn Rm
≤ k k k k
(cid:0) (cid:1)
Now we can introduce the effective dynamical system
dX¯t(x)=a¯(X¯t(x))dt+b(X¯t(x))dB˙t+c(X¯t−(x))dP˙t,
(3.6)
(X¯0 =x.
As the coefficients a¯,b and c are Lipschitz-continuous, this equation admits a
unique solution such that
E X¯t(x) 2Rn CT(1+ x 2Rn), t [0,T]. (3.7)
k k ≤ k k ∈
5
4. Asymptotic expansion
WenowintroducetheKolmogorovoperatorassociatedwiththeslowmotion
equation, with frozen fast component y Rm, by setting
∈
1
Φ(x) = a(x,y),D Φ(x) + Tr D2 Φ(x) b(x)bT(x)
L1 x Rn 2 xx ·
+(cid:16)λ (Φ(x+c(x))(cid:17) Φ(x)), Φ(cid:2) C2(Rn,R). (cid:3)
1 − ∈ b
For any frozen slow component x Rm, the Kolmogorovoperator for equation
∈
(3.1) is given by
1
Ψ(y) = f(x,y),D Ψ(y) + Tr D2 Ψ(y) g(x,y)gT(x,y)
L2 y Rm 2 yy ·
(cid:16)+λ (Ψ(y+h(x,y(cid:17))) Ψ(y)),(cid:2)Ψ C2(Rm,R). (cid:3)
2 − ∈ b
Let φ C2(Rn,R) and define a function uǫ(t,x,y):[0,T] Rn Rm R by
∈ b × × →
uǫ(t,x,y)=Eφ(Xǫ(x,y)).
t
If we set
1
ǫ := + ,
L L1 ǫL2
we have that uǫ(t,x,y) is the solutio of the Kolmogorovequation:
∂ uǫ(t,x,y)= ǫuǫ(t,x,y),
∂t L (4.1)
(uǫ(0,x,y)=φ(x),
Also recall the Kolmogorov operator associated with the averaged equation is
defined as
1
¯Φ(x) = a¯(x),D Φ(x) + Tr D2 Φ(x) b(x)bT(x)
L x Rn 2 xx ·
+(cid:16)λ (Φ(x+c(x(cid:17),y)) Φ(x)(cid:2)), Φ C2(Rn,R). (cid:3)
1 − ∈ b
If we set
u¯(t,x)=Eφ(X¯ (x)),
t
we have
∂ u¯(t,x)= ¯u¯(t,x),
∂t L (4.2)
(u¯(0,x)=φ(x).
Then the weak difference at time T is equal to
Eφ(X¯ (x)) Eφ(Xǫ(x,y))=u¯(T,x) uǫ(T,x,y).
T − T −
With the above assumptions and notation we have the following result, which
is a direct consequence of Lemma 4.1, Lemma 4.2 and Lemma 4.5.
6
Theorem 4.1. Assume that x Rn and y Rm, Then, under assumptions
(A1), (A2) and (A3), for any ∈any T > 0 a∈nd φ C3(Rn,R), there exists a
∈ b
constant C such that
T,φ,x,y
Eφ(Xǫ(x,y)) Eφ(X¯ (x)) C ǫ.
T − T ≤ T,φ,x,y
(cid:12) (cid:12)
As a consequence(cid:12), it can be claimed that the(cid:12)weak order in averaging principle
for jump-diffusion stochastic systems is 1.
Inwhatfollows,forthesakeofsimplicity,weoftenomitthetemporalvariable
t and spatialvariables x and y. For example, for uǫ(t,x,y), we often write it as
uǫ. Our aimis to seek matchedasymptotic expansionsfor the uǫ(T,x,y) of the
form
uǫ =u +ǫu +rǫ, (4.3)
0 1
where u and u are smooth functions which will be constructed below, and rǫ
0 1
is the remainder term.
4.1. The leading term
Let us first determine the leading terms. Now, substituting (4.3) into (4.1)
yields
∂u ∂u ∂rǫ
0 +ǫ 1 + = u +ǫ u + rǫ
∂t ∂t ∂t L1 0 L1 1 L1
1 1
+ u + u + rǫ
ǫL2 0 L2 1 ǫL2
By comparing coefficients of powers of ǫ, we obtain
u =0, (4.4)
2 0
L
∂u
0 = u + u . (4.5)
∂t L1 0 L2 1
It follows from (4.4) that u is independent of y, which means
0
u (t,x,y)=u (t,x).
0 0
We also impose the initial condition u (0,x) = φ(x). Since µx is the invariant
0
measure of the Markov process with generator , we have
2
L
u (t,x,y)µx(dy)=0.
2 1
ZRmL
Thanks to (4.5), this yields
∂u ∂u
0(t,x) = 0(t,x)µx(dy)
∂t ZRm ∂t
7
= u (t,x)µx(dy)
1 0
ZRmL
1
= a(x,y),D u (t,x) µx(dy)+ Tr D2 u (t,x) b(x)bT(x)
ZRm(cid:16) x 0 (cid:17)Rn 2 xx 0 ·
+λ (Φ(x+c(x)) Φ(x)) (cid:2) (cid:3)
1
−
= ¯u (t,x),
0
L
so that u and u¯ are controlled by the same evolutionary equation. By using a
0
uniqueness argument, such u has to coincide with the solution u¯ and we have
0
the following lemma:
Lemma 4.1. Under assumption (A1), (A2) and (A3), for any x Rn, y
Rm and T >0, we have u (T,x,y)=u¯(T,x). ∈ ∈
0
4.2. Construction of u
1
Let us proceed to carry out the construction of u . Thanks to Lemma 4.1
1
and (4.2), the equation (4.5) can be rewritten as
¯u¯= u¯+ u ,
1 2 1
L L L
and hence we get an elliptic equation for u with form
1
u (t,x,y)= a¯(x) a(x,y),D u¯(t,x) := ρ(t,x,y), (4.6)
2 1 x
L − Rn −
(cid:16) (cid:17)
where ρ is of class C2 with respect to y, with uniformly bounded derivative.
Moreover,it satisfies for any t 0 and x Rn,
≥ ∈
ρ(t,x,y)µx(dy)=0.
ZRm
For any y Rm and s>0 we have
∈
∂
ρ(t,x,y) = f(x,y),D [ ρ(t,x,y)]
s y s
∂sP P Rm
(cid:16) 1 (cid:17)
+ Tr D2 [ ρ(t,x,y)] g(x,y)gT(x,y)
2 yy Ps ·
+λ2( (cid:2)s[ρ(t,x,y+h(x,y))] s[ρ(t,x,y(cid:3))]),
P −P
here
[ρ(t,x,y)]:=Eρ(t,x,Yx(y))
Ps s
satisfying
lim Eρ(t,x,Yx(y))= ρ(t,x,z)µx(dz)=0. (4.7)
s→+∞ s ZRm
Indeed, by the invariant property of µx and Lemma 5.1, we have
Eρ(t,x,Yx(y)) ρ(t,x,z)µx(dz)
s −
(cid:12) ZH (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8
= E[ρ(t,x,Yx(y)) ρ(t,x,Yx(z))µx(dz)]
(cid:12)ZRm s − s (cid:12)
(cid:12) (cid:12)
(cid:12) E a(x,Yx(z)) a(x,Yx(y)),D u¯(t,x)(cid:12) µx(dz)
≤Z(cid:12)Rm(cid:12) (cid:16) s − s x (cid:12)(cid:17)Rn(cid:12)
≤CZR(cid:12)(cid:12)mEkYsx(z)−Ysx(y)kRnµx(dz). (cid:12)(cid:12)
This, in view of (3.3) and (3.4), yields
Eρ(t,x,Yx(y)) ρ(t,x,z)µx(dz)
s −
(cid:12) ZH (cid:12)
(cid:12)(cid:12) C(1+ x Rn + y Rm)e−β2s, (cid:12)(cid:12)
≤(cid:12) k k k k (cid:12)
which implies the equality (4.7). Therefore, we get
+∞
f(x,y),D [ ρ(t,x,y)]ds
y s
(cid:16) 1 Z0+∞ P (cid:17)Rm
+ Tr D2 [ ρ(t,x,y)] g(x,y)gT(x,y)ds
2 yy Ps ·
(cid:20) Z0 (cid:21)
+∞ +∞
+λ [ρ(t,x,y+h(x,y))]ds [ρ(t,x,y)]ds
2 s s
P − P
(cid:18)Z0 Z0 (cid:19)
+∞ d
= [ρ(t,x,y)]ds
s
dsP
Z0
= lim Eρ(t,x,Yx(y)) ρ(t,x,y)
s→+∞ s −
= ρ(t,x,y)µx(dy) ρ(t,x,y)
−
ZH
= ρ(t,x,y),
−
which implies ( +∞ ρ(t,x,y)ds)= ρ(t,x,y). This means that the
L2 0 Ps −
R +∞
u (t,x,y):= Eρ(t,x,Yx(y))ds (4.8)
1 s
Z0
is the solution of Poissonequation (4.6).
Lemma 4.2. Under assumptions (A1), (A2) and (A3), for any x Rn,
y Rm and T >0, we have ∈
∈
u1(t,x,y) CT(1+ x Rn + y Rm), t [0,T]. (4.9)
| |≤ k k k k ∈
Proof. As known from (4.8), we have
+∞
u (t,x,y)= E a¯(x) a(x,Yx(y)),D u¯(t,x) ds.
1 Z0 (cid:16) − s x (cid:17)Rn
9
This implies that
+∞
|u1(t,x,y)| ≤ ka¯(x)−E[a(x,Ysx(y))]kRn ·kDxu¯(t,x)kRnds.
Z0
Then, due to Lemma 5.1 and (3.5), this yields:
+∞
u1(t,x,y) CT(1+ x Rn + y Rm) e−β2sds
| | ≤ k k k k
Z0
CT(1+ x Rn + y Rm).
≤ k k k k
4.3. Determination of remainder rǫ
Once u andu havebeen determined, we cancarryoutthe constructionof
0 1
the remainder rǫ. It is known that
(∂ ǫ)uǫ =0,
t
−L
which, together with (4.4) and (4.5), implies
(∂ ǫ)rǫ = (∂ ǫ)u ǫ(∂ ǫ)u
t t 0 t 1
−L − −L − −L
1 1
= (∂ )u ǫ(∂ )u
− t− ǫL2−L1 0− t− ǫL2−L1 1
= ǫ( u ∂ u ). (4.10)
1 1 t 1
L −
Inordertoestimatetheremaindertermrǫweneedthefollowingcruciallemmas.
Lemma 4.3. Under assumptions (A1), (A2) and (A3), for any x Rn,
y Rm and T >0, we have ∈
∈
∂u
1(t,x,y) CT(1+ x Rn + y Rm)
∂t ≤ k k k k
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. According(cid:12)to (4.8), we(cid:12)have
(cid:12) (cid:12)
∂u +∞ ∂
1(t,x,y)= E a¯(x) a(x,Yx(y)), D u¯(t,x) ds.
∂t − s ∂t x
Z0 (cid:18) (cid:19)Rn
By Lemma 5.6 introduced in Section 5, we have
∂u +∞ ∂
∂t1(t,x,y) ≤ E ka¯(x)−a(x,Ysx(y))kRn ·k∂tDxu¯(t,x)kRn ds
(cid:12) (cid:12) Z0 (cid:18) (cid:19)
(cid:12) (cid:12) +∞
(cid:12)(cid:12) (cid:12)(cid:12) ≤ CT Eka¯(x)−a(x,Ysx(y))kRnds
Z0
so that from (3.5) we have
∂u
1(t,x,y) CT(1+ x Rn + y Rm).
∂t ≤ k k k k
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
10