FCLTs for the Quadratic Variation of a CTRW and for ...€¦ · process to accumulate into a large...
Transcript of FCLTs for the Quadratic Variation of a CTRW and for ...€¦ · process to accumulate into a large...
FCLTs for the Quadratic Variation of a CTRWand for certain stochastic integrals
Noelia Viles Cuadros(joint work with Enrico Scalas)
Universitat de Barcelona
FCPNLO-BCAMBilbao, November 7 2013
Damped harmonic oscillator subject to a random force
The equation of motion is informally given by
x(t) + γx(t) + kx(t) = ξ(t), (1)
where x(t) is the position of the oscillating particle with unit mass attime t, γ > 0 is the damping coefficient, k > 0 is the spring constantand ξ(t) represents white Levy noise.
I. M. Sokolov,Harmonic oscillator under Levy noise: Unexpected properties inthe phase space.Phys. Rev. E. Stat. Nonlin Soft Matter Phys 83, 041118 (2011).
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The formal solution is
x(t) = F (t) +
∫ t
−∞G (t − t ′)ξ(t ′)dt ′, (2)
where G (t) is the Green function for the homogeneous equation.The solution for the velocity component can be written as
v(t) = Fv (t) +
∫ t
−∞Gv (t − t ′)ξ(t ′)dt ′, (3)
where Fv (t) = ddtF (t) and Gv (t) = d
dtG (t).
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• Replace the white noise with a sequence of instantaneous shots ofrandom amplitude at random times.
• They can be expressed in terms of the formal derivative ofcompound renewal process, a random walk subordinated to acounting process called continuous-time random walk.
A continuous time random walk (CTRW) is a pure jump processgiven by a sum of i.i.d. random jumps {Yi}i∈N separated by i.i.d.random waiting times (positive random variables) {Ji}i∈N.
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Fractional Poisson Process
Let {Ji}i∈N be a sequence of i.i.d. and positive β-stable rv’s with themeaning of waiting times between jumps and β ∈ (0, 1).Let
Tn =n∑
i=1
Ji
represent the epoch of the n-th jump and the counting processassociated is the fractional Poisson process
Nβ(t) = max{n : Tn 6 t}.
which counts the number of jumps up to time t > 0.
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Compound Fractional Poisson Process
Let {Yi}i∈N be a sequence of i.i.d. and α-stable rv’s (independent ofJi ) with the meaning of jumps taking place at every epoch andα ∈ (0, 2].
If we subordinate a CTRW to the fractional Poisson process, weobtain the compound fractional Poisson process, which is not Markov
XNβ(t) =
Nβ(t)∑i=1
Yi . (4)
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Our goal
Under the following distributional assumptions,
• the jumps {Yi}i∈N are rv’s in the DOA of an α-stable process withindex α ∈ (0, 2];
• the waiting times {Ji}i∈N are rv’s in the DOA of a β-stable processwith index β ∈ (0, 1);
We want to study the limits of the following stochastic process 1
nβ/α
Nβ(nt)∑i=1
G
(t − Ti
n
)Yi
t>0
and
1
nβ/α
Nβ(nt)∑i=1
Gv
(t − Ti
n
)Yi
t>0
.
But limits in which sense?
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Our goal
Under the following distributional assumptions,
• the jumps {Yi}i∈N are rv’s in the DOA of an α-stable process withindex α ∈ (0, 2];
• the waiting times {Ji}i∈N are rv’s in the DOA of a β-stable processwith index β ∈ (0, 1);
We want to study the limits of the following stochastic process 1
nβ/α
Nβ(nt)∑i=1
G
(t − Ti
n
)Yi
t>0
and
1
nβ/α
Nβ(nt)∑i=1
Gv
(t − Ti
n
)Yi
t>0
.
But limits in which sense?
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α-stable Levy processes
A continuous-time process L = {Lt}t>0 with values in R is called aLevy process if its sample paths are cadlag at every time point t, andit has stationary, independent increments.
An α-stable process is a real-valued Levy process Lα = {Lα(t)}t>0
with initial value Lα(0) that satisfies the self-similarity property
1
t1/αLα(t)
L= Lα(0), ∀t > 0.
If α = 2 then the α-stable Levy process is the Wiener process.
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The Skorokhod space
The Skorokhod space, denoted by D = D([0,T ],R) (with T > 0), isthe space of real functions x : [0,T ]→ R that are right-continuouswith left limits:
1. For t ∈ [0,T ), x(t+) = lims↓t x(s) exists and x(t+) = x(t).
2. For t ∈ (0,T ], x(t−) = lims↑t x(s) exists.
Functions satisfying these properties are called cadlag functions.
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Skorokhod topologies
The Skorokhod space provides a natural and convenient formalismfor describing the trajectories of stochastic processes with jumps:
• Poisson processes;
• Levy processes;
• martingales and semimartingales;
• empirical distribution functions;
• discretizations of stochastic processes.
Topology: It can be assigned a topology that intuitively allows us towiggle space and time a bit (whereas the traditional topology ofuniform convergence only allows us to wiggle space a bit).
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J1 and M1 topologies
Skorokhod (1956) proposed four metric separable topologies on D,denoted by J1, J2, M1 and M2.
The difference between J1 and M1 topologies is:
• M1-topology allows numerous small jumps for the approximatingprocess to accumulate into a large jump for the limit process
• J1-topology requires a large jump for the limit process to beapproximated by a single large jump for the approximating one.
A. Skorokhod.Limit Theorems for Stochastic Processes.Theor. Probability Appl. 1, 261–290, 1956.
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Our problem
Under the following distributional assumptions,
• the jumps {Yi}i∈N are rv’s in the DOA of an α-stable process withindex α ∈ (0, 2];
• the waiting times {Ji}i∈N are rv’s in the DOA of a β-stable processwith index β ∈ (0, 1);
We want to study the Functional Central Limit Theorem of 1
nβ/α
Nβ(nt)∑i=1
f
(Ti
n
)Yi
t>0
,
for all f continuous and bounded function.
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Convergence to the β-stable subordinator
For t > 0, we define
Tt :=
btc∑i=1
Ji .
We have
{c−1/βTct}t>0J1−top⇒ {Dβ(t)}t>0, as c → +∞.
A β-stable subordinator {Dβ(t)}t>0 is a real-valued β-stable Levyprocess with nondecreasing sample paths.
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Convergence to the inverse β-stable subordinator
For any integer n > 0 and any t > 0:
{Tn 6 t} = {Nβ(t) 6 n}.
Theorem (Meerschaert & Scheffler (2001))
{c−1/βNβ(ct)}t>0J1−top⇒ {D−1β (t)}t>0, as c → +∞.
The functional inverse of {Dβ(t)}t>0 can be defined as
D−1β (t) := inf{x > 0 : Dβ(x) > t}.
It has a.s. continuous non-decreasing sample paths and withoutstationary and independent increments.14 of 27
Convergence to the symmetric α-stable Levy process
Assume the jumps Yi belong to the strict generalized DOA of somestable law with α ∈ (0, 2], then
Theorem (Meerschaert & Scheffler (2004))c−1/α[ct]∑i=1
Yi
t>0
J1−top⇒ {Lα(t)}t>0, when c → +∞.
M. Meerschaert, H. P. Scheffler.Limit theorems for continuous-time random walks with infinitemean waiting times.J. Appl. Probab., 41 (3), 623–638, 2004.
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Functional Central Limit Theorem
Theorem (Meerschaert & Scheffler (2004))
Under the distributional assumptions considered above for the waitingtimes Ji and the jumps Yi , we havec−β/α
Nβ(ct)∑i=1
Yi
t>0
M1−top⇒ {Lα(D−1β (t))}t>0, when c → +∞,
(5)in the Skorokhod space D([0,+∞),R).
M. Meerschaert, H. P. Scheffler.Limit theorems for continuous-time random walks with infinitemean waiting times.J. Appl. Probab., 41 (3), 623–638, 2004.
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Main ingredients of the proof
• Triangular array approach
∆ = {(Y (c)i , J
(c)i ) : i > 1, c > 0}.
It is assumed that ∆ is given so that btc∑
i=1
Y(c)i ,
btc∑i=1
J(c)i
t>0
J1−top⇒ {(Lα(t),Dβ(t))}t>0 , c →∞.
• Continuous-Mapping Theorem (CMT)
W. Whitt,Stochastic-Process Limits: An Introduction toStochastic-Process Limits and Their Application to Queues.Springer, New York (2002).
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Example: Quadratic Variation
The quadratic variation of
X (t) =
Nβ(t)∑i=1
Yi
is
[X ](t) = [X ,X ](t) =
Nβ(t)∑i=1
[X (Ti )− X (Ti−1)]2 =
Nβ(t)∑i=1
Y 2i .
E. Scalas, N. Viles,On the Convergence of Quadratic variation for CompoundFractional Poisson Processes.Fractional Calculus and Applied Analysis, 15, 314–331 (2012).
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FCLT for the Quadratic Variation
Theorem (Scalas & V. (2012))
Under the distributional assumptions considered above, we have that 1
n2/α
[nt]∑i=1
Y 2i ,
1
n1/βTnt
t>0
J1−top⇒n→+∞
{(L+α/2(t),Dβ(t))}t>0,
in D([0,+∞),R+ × R+) . Moreover, we have also
1
n2β/α
Nβ(nt)∑i=1
Y 2i
M1−top⇒ L+α/2(D−1β (t)), as n→ +∞,
in D([0,+∞),R+), where L+α/2(t) denotes an α2 -stable positive Levy
process.19 of 27
FCLT for certain stochastic integrals
Distributional assumptions
• Jumps {Yi}i∈N: i.i.d. symmetric α-stable random variables suchthat Y1 belongs to DOA of an α-stable random variable withα ∈ (0, 2].
• Wating times {Ji}i∈N: i.i.d. random variables such that J1 belongsto DOA of some β-stable random variables with β ∈ (0, 1).
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Our preliminary results
Proposition (Scalas & V.)
Let f ∈ Cb(R). Under the distributional assumptions and a suitablescaling we have that 1
nβ/α
bnβtc∑i=1
f
(Ti
n
)Yi
t>0
J1−top⇒{∫ t
0f (Dβ(s))dLα(s)
}t>0
,
as n→ +∞.
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Our preliminary results
Proposition (Scalas & V.)
Let f ∈ Cb(R). Under the distributional assumptions and a suitablescaling,
Nβ(nt)∑i=1
f
(Ti
n
)Yi
t>0
J1−top⇒
{∫ D−1β (t)
0f (Dβ(s))dLα(s)
}t>0
as n→ +∞, where∫ D−1β (t)
0f (Dβ(s))dLα(s)
a.s.=
∫ t
0f (s)dLα(D−1β (s)).
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FCLT for stochastic integrals
Theorem (Scalas & V.)Let f ∈ Cb(R). Under the distributional assumptions and the scaling, 1
nβ/α
Nβ(nt)∑i=1
f
(Ti
n
)Yi
t>0
M1−top⇒n→+∞
{∫ t
0
f (s)dLα(D−1β (s))
}t>0
,
in D([0,+∞),R) with M1-topology.
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Theorem (Kurtz & Protter) for stochastic integrals
For each n, let (Hn,X n) be an Fnt -adapted process in
DMkm×Rm([0,∞)) and let X n be an Fnt -semimartingale. Fix δ > 0
and define X n,δ = X n − Kδ(Xn). Let X n,δ = Mn,δ + An,δ be a
decomposition of X n,δ into an Fnt -local martingale and a process with
finite variation. Suppose for each θ > 0, there exist stopping times τ θnsuch that P(τ θn 6 θ) 6 1
θ and furthermore
supn
E[[Mn,δ,Mn,δ]t∧τθn + TV (An,δ, t ∧ τ θn )
]< +∞.
If (Hn,X n)J1−top⇒ (H,X ) on DMkm×Rm([0,∞)), then X is a
semimartingale w.r.t a filtration to which H and X are adapted and(Hn,X n,
∫HndX n
)J1−top⇒
(H,X ,
∫HdX
), as n→ +∞,
on DMkm×Rm×Rk ([0,∞)).24 of 27
Application (Scalas & V.)
Let G (t) be the Green function of the equation (1) and Gv (t) itsderivative. Under the distributional assumptions and the suitablescaling, it follows that 1
nβ/α
Nβ(nt)∑i=1
G
(t − Ti
n
)Yi
t>0
M1−top⇒{∫ t
0
G (t − s)dLα(D−1β (s))
}t>0
,
and 1
nβ/α
Nβ(nt)∑i=1
Gv
(t − Ti
n
)Yi
t>0
M1−top⇒n→+∞
{∫ t
0
Gv (t − s)dLα(D−1β (s))
}t>0
,
in D([0,+∞),R) with M1-topology.
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Conclusions:
• We have studied the convergence of a class of stochastic integralswith respect to the Compound Fractional Poisson Process.
• Under proper scaling hypotheses, these integrals converge to theintegrals w.r.t a symmetric α-stable process subordinated to theinverse β-stable subordinator.
Extensions:
• It is possible to approximate some of the integrals discussed inKobayashi (2010) by means of simple Monte Carlo simulations.This will be the subject of a forthcoming applied paper.
• To extend this result to the integration of stochastic processes.
• To study the coupled case using the limit theorems provided inBecker-Kern, Meerschaert and Schefller (2002).
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Thank you for your attention!
Eskerrik Asko
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