Download Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces

Non-standard Skorokhod convergence of
Lévy-driven convolution integrals in Hilbert
spaces
Ilya Pavlyukevich
Friedrich–Schiller–Universität Jena
Markus Riedle
King’s College London
Workshop
Stochastic Processes and Differential Equations
in Infinite Dimensional Spaces
London, 31.03–03.04.2014
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1
1. Convolution integrals w.r.t. a Lévy process. Example I
Mattingly, Suidan and Vanden–Eijnden (2007)[6]; P. and Sokolov (2008)[8]:
Invinitely dimensional transport system perturbed by a one-dimensional Lévy
process L “ pLtqtě0.
l
$
1
2
1
9t
’
9
a
“
´a
´
νa
`
L
’
t
t
t
’
’
&a9 2 “ a1 ´ a3 ´ νa2
t
t
t
t
’
¨¨¨
’
’
’
%a9 n “ an´1 ´ an`1 ´ νan,
t
t
t
t
¨
,
˛
ν
1 0 0 ¨¨¨
˚´1 ν 1 0 ¨ ¨ ¨‹
‹
A“˚
˝ 0 ´1 ν 1 ¨ ¨ ¨‚
¨¨¨
ně1
a9 t “ ´Aat ` L9 te1, a0 “ 0.
The explicit solution is a convolution integral
żt
ant
Hnpt ´ sq dL9 s,
“
0
Jnp2tq ´νt
Hnptq “ n
e ,
t
Jn, n ě 1, Bessel functions of the first kind.
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1
2
2. Convolution integrals w.r.t. a Lévy process. Example II
Chechkin, Gonchar, Szydłowski, Physics of Plasmas (2002)[1].
L “ pLtqtě0 is an isometric α-stable Lévy process in R3,
Ee
ixu,Lt y
“e
´t}u}α
,
u P R3 ,
α P p0, 2q.
Langevin equation for a particle in an external magnetic field B and Lévy
9
electric field L:
1:
X “ rX9 ˆ Bs ´ ν X9 ` L9 γ ą 0,
γ
or
$
˛
¨
9 “ γV,
’
X
’
ν
´B3 B2
&
´
¯
V9 “ γ rV ˆ Bs ´ νV `L9 , A “ ˝ B3
ν
´B1‚
’
looooooooomooooooooon
’
´B2 B1
ν
%
“:´AV
In other words, X is an integrated OU process:
żt
Xt “ γ
Vs ds,
0
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żt
Vt “ ´γ
AVs ds ` Lt
0
2
3
3. Explicit solution
Ornstein–Uhlenbeck process:
żt
Vt “ ´γ
żt
AVs ds ` Lt
e´γ pt´sqA dLs
Vt “
ñ
0
0
Integrated Ornstein–Uhlenbeck process (Fubini):
ż t”ż s
żt
AVs ds “ γ
AXt “ γ
0
0
ż t”ż t
0
“γ
0
´1
ı
Ae´γ ps´uqA dLu ds
ı
Ae´γ ps´uqA ds dLu
u
ż t´
“A A
1´e
´γ pt´uqA
¯
dLu
0
ż t´
¯
1 ´ e´γ pt´uqA dLu
“
0
Goal: limiting behaviour of X as γ Ñ 8, especially exit times.
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4
4. Convergence of f.d.d.
Theorem 1. For any t ě 0
P
AXt Ñ Lt,
!
EeiupAXt´Ltq “ E exp ´ iu lim
n
“ lim
n
“ lim
n
n
ź
γ Ñ 8.
n
ÿ
e´γ pt´sk qA∆Lsk
k“1
´γ pt´sk qA
Ee´iue
k“1
n
ź
e
)
∆Lsk
›
›α
´γ pt´sk qA ›
›
∆sk ´ue
k“1
!
“ exp lim
n
n
ÿ
›
› )
› ´γ pt´sk qA›α
∆sk ›ue
›
k“1
ż t›
›
!
)
› ´γ pt´sqA›α
α
“ exp }u}
›e
› ds Ñ 1, γ Ñ 8
0 loooooomoooooon
Ñ0, s‰t, γ Ñ8
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5
5. Behaviour of the paths
Convergence of in a path space to ensure, for instance, convergence of exit
times:
ż t´
¯
1 ´ e´γ pt´sqA dLs,
AXt “
Aą0
0
Dimension d “ 1 (see Hintze & P.[2]):
1
2
3
4
5
- 0.1
- 0.2
- 0.3
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6
6. Dimension d “ 2
ż t´
AXt “
¯
1 ´ e´γ pt´sqA dLs
0
- 30
- 20
30
60
20
40
10
20
- 10
10
30
- 60
- 40
- 20
20
- 10
- 20
- 20
- 40
- 30
- 60
ˆ
A“
νą0
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ν 0
0 ν
˙
ˆ
A“
40
60
˙
ν 0
,
0 µ
ν, µ ą 0,
퉵
6
7
7. Dimension d “ 2
ż t´
AXt “
¯
1 ´ e´γ pt´sqA dLs
0
60
40
40
20
20
- 40
- 20
20
40
- 60
- 40
- 20
20
40
60
- 20
- 20
- 40
- 40
- 60
ˆ
A“
1 ´1
1 1
λ1,2 “ 1 ˘ i
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˙
ˆ
A“
˙
1 ´3
3 1
λ1,2 “ 1 ˘ 3i
7
8
8. Dimension d “ 3
ż t´
AXt “
¯
1 ´ e´γ pt´sqA dLs
0
100
0
- 50
50
- 100
- 150
- 100
- 50
0
¨
0
˛
ν
´B3 B2
ν
´B1‚,
A “ ˝ B3
´B2 B1
ν
ν “ 1, B “ p2, ´3, 1q
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9
9. Setting and goals (R. & P., arXiv:1311.1342[7])
Let U, V be separable Hilbert spaces, LpU, V q the space of bounded linear
operators with the norm } ¨ }U ÑV .
L be a U -valued Lévy process and
żT
!
)
2
2
H pU, V q :“ F : r0, T s Ñ LpU, V q : F measurable and
}F psq}U ÑV ds ă 8
0
For càdlàg F, tFγ uγ ą0 Ă H2pU, V q consider convolution integrals
żt
F ˚ Lptq “
żt
F pt ´ sq dLpsq
and
Fγ ˚ Lptq “
0
Fγ pt ´ sq dLpsq.
0
Goal: convergence in probability in an appropriate path space
F ˚ Lp¨q Ñ Fγ ˚ Lp¨q
from the convergence F p¨q Ñ Fγ p¨q, γ Ñ 0.
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10. OU-process in dimension 1
´
¯
Fγ ptq “ 1 ´ e´γt Ir0,8qptq F ptq “ Ip0,8qptq
żt
Fγ ˚ Lt “ p1 ´ e´γ pt´sqq dLpsq
0
żt
żt
Ir0,tqpsq dLpsq “ F ˚ Lt “ Lt´
Ip0,8qpt ´ sq dLpsq “
Ñ
0
0
limPp|Lt´h ´ Lt| ą εq Ñ 0 (stochastic continuity)
hÓ0
1
2
3
4
5
- 0.1
- 0.2
- 0.3
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11
11. Skorohod M1-covergence in R (I)
For x P Dpr0, T s, Rq define a completed graph Γx:
Γx :“ tpx0, 0qu Y tpz, tq P R ˆ p0, T s : z “ cxt´ ` p1 ´ cqxt for some c, c P r0, 1su,
Γx Ă R2.
x
Γx
0
T
0
T
Natural order on Γx:
pz, tq ď pz 1, t1q
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if
t ă t1 or t “ t1 and |xt´ ´ z| ď |xt´ ´ z 1|.
11
12
12. Skorohod M1-convergence II
Parametric representation of Γx: continuous nondecreasing w.r.t. order
mapping
pzu, tuq : r0, 1s Ñ Γx.
Denote Πx the set of all parametric representations of Γx.
Skorohod M1-convergence on Dpr0, T s, Rq:
xn Ñ x
ô
for any pz, tq P Πx there is pz n, tnq Ă Πxn such that
!
)
max sup |zun ´ zu|, sup |tnu ´ tu| Ñ 0, n Ñ 8.
uPr0,1s
uPr0,1s
This topology in metrizable and the space DpR`, R; M1q is Polish (see Whitt,
Chapter 12.8).
Weak convergence in DpR`, Rd; M1q, see Whitt[10], Chapter 12.
Weak J1, J2, M1, M2 convergences in Dpr0, T s, Rq, see Skorohod[9].
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13. M1-convergence in probability in infinite dimensions.
Strong M1-convergence
Let V be a Hilbert (Banach) space, f1, f2 P Dpr0, T s, V q. Define a completed
graph and a parameterization of fi as in R1, and
!
)
dM pf1, f2q :“ inf |r1 ´ r2|8 _ }u1 ´ u2}V : pri, uiq P Πpfiq, i “ 1, 2 .
Strong M1-convergence in probability: for Xn, X càdlàg, V -valued stochastic
processes on a common probability space
Xn
P,M1 -strongly
Ñ
X
if
P
dM pXn, Xq Ñ 0
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14. Weak M1-convergence
Weak M1-convergence in probability: for Xn, X càdlàg, V -valued stochastic
processes on a common probability space
Xn
P,M1 -weakly
Ñ
X
if for any v P V
P
dM pxXn, vy, xX, vyq Ñ 0
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pin R1q
14
15
15. Product M1-convergence
Let e “ tek ukě1 be a orthonormal basis in V . Consider an e-dependent metric
8
ÿ
1 dM pxf, ek y, xg, ek yq
deM pf, gq “
k 1 ` d pxf, e y, xg, e yq
2
M
k
k
k“1
Pointwise convergence:
deM pfn, f q Ñ 0
ô
for any k ě 1
dM pxfn, ek y, xf, ek yq Ñ 0
For Xn, X càdlàg, V -valued stochastic processes on a common probability
space, a basis e,
Xn
P,e-product
Ñ
X
ô
P
deM pXn, Xq Ñ 0
Relations between the modes of convergence:
strong
M1 ñ weak M1
looooooooooooooooomooooooooooooooooon
ñ
product M1
coinside in finite-dimensional spaces
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16. A convenient convergence criterium
Recall:
weak convergence in R1
Xn ñ X
ô
#
a) convergence of f.d.d.
b) tightness
Tightness in terms of the oscillation function: For x, y P R (or P V) consider the
straight line segment
vx, yw :“ tz P R : z “ x ` cpy ´ xq, c P r0, 1su.
M1-oscillation function M : R3 Ñ r0, 8q, (V3 Ñ r0, 8q)
#
M px1, x, x2q :“
mint|x ´ x1|, |x2 ´ x|u,
0, x P vx1, x2w,
M pf, δq :“
sup
if x R vx1, x2w,
M pf pt1q, f ptq, f pt2qq.
0ďt1 ătăt2 ďT,t2 ´t1 ďδ
Tightness follows from
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lim lim sup PpM pXn, δq ě εq “ 0.
δ Ó0
nÑ8
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17
17. Strong M1-convergence in probability
Theorem. Let X, Xn, n ě 1, be stochastically continuous V-valued càdlàg
processes, t P r0, T s. Then the following conditions are equavalent:
1. Xn
P,M1 -strongly
Ñ
X in Dpr0, T s, Vq
2. the following two conditions are satisfied:
(a) for any t P r0, T s
P
Xnptq Ñ Xptq;
(b) for every ε ą 0 the oscillation function obeys
lim lim sup PpM pXn, δq ě εq “ 0,
δ Ó0
where
M pf, δq “
nÑ8
sup
M pf pt1q, f ptq, f pt2qq.
0ďt1 ătăt2 ďT,t2 ´t1 ďδ
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18. Product M1-convergence in probability (II)
Proof: check the definition, construct appropriate parameterizations...
Compare with the result by A. Jakubowski (Tightness criteria for random
measures with application to the principle of conditioning in Hilbert spaces,
1988)[3].
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19. Product M1-convergence in probability
Theorem. Let X, Xn, n ě 1, be stochastically continuous V-valued càdlàg
processes, t P r0, T s, e “ tek ukě1 a orthonormal basis in V . Then the following
conditions are equavalent:
1. Xn
P,e-product
Ñ
X in Dpr0, T s, V; deM q
2. the following two conditions are satisfied for every k ě 1:
(a) for any t P r0, T s
P
xXnptq, ek y Ñ xXptq, ek y;
(b) for every ε ą 0 the oscillation function obeys
lim lim sup PpM pxXn, ek y, δq ě εq “ 0,
δ Ó0
nÑ8
where
M pf, δq “
sup
M pf pt1q, f ptq, f pt2qq in R1.
0ďt1 ătăt2 ďT,t2 ´t1 ďδ
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20. Weak M1-convergence in probability
Theorem. Let X, Xn, n ě 1, be stochastically continuous V-valued càdlàg
processes, t P r0, T s. Then the following conditions are equavalent:
1. Xn
P,M1 -weakly
Ñ
X in Dpr0, T s, Vq
2. the following two conditions are satisfied for every v P V:
(a) for any t P r0, T s
P
xXnptq, vy Ñ xXptq, vy;
(b) for every ε ą 0 the oscillation function obeys
lim lim sup PpM pxXn, vy, δq ě εq “ 0,
δ Ó0
nÑ8
where
M pf, δq “
sup
M pf pt1q, f ptq, f pt2qq in R1.
0ďt1 ătăt2 ďT,t2 ´t1 ďδ
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21. Weak M1-convergence of convolution integrals
Theorem. Let L be a U -valued Lévy process. Let F, Fγ P H2pU, V q, γ ą 0, be
functions satisfying
piq F ˚p¨qv, Fγ˚ p¨qv P Dpr0, T s; U q
for all v P V and γ ą 0;
piiq sup }xFγ p¨qu, vy}8 ă 8 for all u P U, v P V ;
γ ą0
piiiq lim sup sup
αÑ0
pivq lim dM
γ Ñ8
8
ÿ
γ ą0 j “1
`
}xFγ˚ p¨qv, iαhα
j y}T V2 “ 0
˘
xFγ p¨qu, vy, xF p¨qu, vy “ 0
for all v P V ;
for all u P U, v P V.
Then
´ż t
0
¯
Fγ pt ´ sqLpsq
tPr0,T s
´ż t
Ñ
0
¯
F pt ´ sqdLpsq
tPr0,T s
M1-weakly in probability in Dpr0, T s, V q.
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22. Idea of the proof
0. General assumptions: F ˚ L and Fγ ˚ L have cadlag paths; F ˚ W has
continuous paths.
1. Fix v P V and consider a real-valued processes
ż
ż
@ t
D
@ t
D
Xγ ptq :“
Fγ pt ´ sq dLpsq, v , Xptq :“
F pt ´ sq dLpsq, v .
0
0
Xγ , X are stochastically continuous.
2. Convergence of marginals in probability follows directly from the estimates
of characteristic functions.
3. Decompose L “ W ` ξα ` ηα
W is a Q-Brownian motion
ξα is a Lévy martingale with }∆ξαptq}U ď α
ηα is a cPP with drift.
4. Xγ “ Cγ ` Aγ ` Bγ .
5. Condition (iv) guarantees the convergence of the oscillation function of Cγ
(use Borell’s inequality).
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6. Small jump part: For ξα define a covariance operator Rα by
xRαu, u1y :“ Exξαp1q, uyxξαp1q, u1y. R is positive semidefinite and symmetric;
there is another separable Hilbert space Hα and
embedding iα : Hα Ñ U such
ş
that Rα “ iαi˚α. and }i˚αu}Hα “ xRαu, uy ď }u}2 }r}ďα νpdrq Ñ 0, α Ó 0.
α
˚ α
α
Choose a basis thα
u
in
H
and
u
P
U
with
i
u
“
h
k
ě
1
α
α
k
k
k
k . Then
ξαptq “
8
ÿ
α
iαhα
k xξα ptq, uk y
k“1
where lkα “ xξαptq, uα
k y are uncorrelated real-valued LP.
7. Use a generalization of the estimates by Marcus and Rosiński (2003[4],
2005[5]): let f P H2pU, Rq, then
8
ˇż t
ˇ
ÿ
?
ˇ
ˇ
E sup ˇ f pt ´ sq dξ αpsqˇ ď κ 2T
}xf p¨q, iαhα
k y}T V2 ,
tPr0,T s
0
k“1
? ż 1a
κ “ 32 2
lnp1{sq ds.
0
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8. Compound Poisson part.
Aγ ptq “
N
ptq
ÿ
xFγ˚ pt ´ τj qv, ∆ηαpτj qyIrτj ,8qptq.
j “0
Due to the absolute continuity of τj , the summands
Rj pωq “ xFγ˚ pt ´ τj pωqqv, ∆ηαpτj pωqqyIrτj pωq,8qptq
have common points of discontinuity with probability 0 and the
M1-convergence follows and.
Nÿ
pt,ω q
xFγ˚ pt
M1 ,R1
´ τj pωqqv,∆ηαpτj pωqqyIrτj pωq,8qptq Ñ
j “0
Nÿ
pt,ω q
xF ˚pt ´ τj pωqqv, ∆ηαpτj pωqqyIrτj pωq,8qptq
j “0
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25. Integrated Ornstein–Uhlenbeck process
V separable Hilbert space, A a generator of a strongly continuous semigroup
S “ pStq in V , U another separable Hilbert space, L a Lévy process in U , G a
bounded linear operator G : U Ñ V . Consider
żt
żt
AY psq ds ` GLptq
Y ptq “ γ
ñ
Spt ´ sqG dLpsq,
Y ptq “
0
0
żt
Y psq ds
Xptq “ γ
0
Theorem. Assume that the semigroup is diagonalizable, i.e. there is a ONB
e “ tek ukě1 in V such that
Sptqek “ e´λk tek ,
t ě 0, k ě 1,
and inf k λk ą 0. Then
P
AX Ñ ´GL in the deM1 -product topology in Dpr0, T s, V q.
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References
[1] A. V. Chechkin, V. Yu. Gonchar, and M. Szydłowski. Fractional kinetics for relaxation and superdiffusion in a magnetic field.
Physics of Plasmas, 9(1):78–88, 2002.
[2] R. Hintze, and I. Pavlyukevich. Small noise asymptotics and first passage times of integrated Ornstein–Uhlenbeck processes
driven by α-stable Lévy processes. Bernoulli, 20(1), 265–281, 2014.
[3] A. Jakubowski, A. Tightness criteria for random measures with application to the principle of conditioning in Hilbert spaces.
Probability and Mathematical Statistics, 9(1), 95–114, 1988.
[4] M. B. Marcus and J. Rosiński. Sufficient conditions for boundedness of moving average processes. In Stochastic inequalities and
applications, vol. 56 of Progress in Probability, Birkhäuser, 113–128, 2003.
[5] M. B. Marcus and J. Rosiński. Continuity and boundedness of infinitely divisible processes: A Poisson point process approach.
Journal of Theoretical Probability, 18(1), 109–160, 2005.
[6] J. C. Mattingly, T. M. Suidan and E. Vanden–Eijnden Anomalous dissipation in a stochastically forced infinite-dimensional system
of coupled oscillators. The Journal of Statistical Physics, 128(5), 1145–1152, 2007.
[7] I. Pavlyukevich and M. Riedle. Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces,
http://arxiv.org/abs/1311.1342
[8] I. Pavlyukevich and I. M. Sokolov One-dimensional space-discrete transport subject to Lévy perturbations, The Journal of
Statistical Physics, 133(1), 205–215, 2008.
[9] A. V. Skorohod. Limit theorems for stochastic processes. Theory of Probability and its Applications, 1:261–290, 1956.
[10] W. Whitt. Stochastic-process limits: an introduction to stochastic-process limits and their application to queues. Springer, 2002.
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