FCLTs for the Quadratic Variation of a CTRW
and for certain stochastic integrals
Noèlia Viles Cuadros
(joint work with Enrico Scalas)
2nd Barcelona Summer School on Stochastic
Analysis,
CRM, Bellaterra, July 8 2014
Damped harmonic oscillator subject to a random force
The equation of motion is informally given by
ẍ(t) + γ ẋ(t) + kx(t) = ξ(t),
(1)
where x(t) is the position of the oscillating particle with unit mass at
time t, γ > 0 is the damping coefficient, k > 0 is the spring constant
and ξ(t) represents white Lévy noise.
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The formal solution is
Z
t
x(t) = F (t) +
G (t − t 0 )ξ(t 0 )dt 0 ,
(2)
−∞
where G (t) is the Green function and F (t) is a particular solution of
the homogeneous equation.
The solution for the velocity component can be written as
Z t
v (t) = Fv (t) +
Gv (t − t 0 )ξ(t 0 )dt 0 ,
(3)
−∞
where Fv (t) =
d
dt F (t)
and Gv (t) =
d
dt G (t).
I. M. Sokolov,
Harmonic oscillator under Lévy noise: Unexpected properties in
the phase space.
Phys. Rev. E. Stat. Nonlin Soft Matter Phys 83, 041118 (2011).
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• We want to replace the white noise with a sequence of
instantaneous shots of random amplitude at random times.
• The sequence of instantaneous shots of random amplitude at
random times can be expressed in terms of the formal derivative of
compound renewal process.
• A compound renewal process (or CTRW) is a random walk
subordinated to a counting process.
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Fractional Poisson Process
Let {Ji }i∈N be a sequence of positive i.i.d. rv’s with complementary
cumulative distribution function FJβ (t) given by Eβ (−t β ) the
one-parameter Mittag-Lefler function with β ∈ (0, 1).
Tn :=
n
X
Ji
i=1
represents the epoch of the n-th jump and
Nβ (t) = max{n : Tn 6 t}.
is the counting process associated.
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Compound Fractional Poisson Process
Let {Yi }i∈N be a sequence of i.i.d. random variables that represent
the jumps taking place at every epoch Ti .
If we subordinate a CTRW to the counting process Nβ (t), we obtain
the compound fractional Poisson process, which is not Markov
Nβ (t)
XNβ (t) :=
X
Yi .
i=1
If β = 1, this is the Compound Poisson process.
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(4)
Our goal
Under the following distributional assumptions,
• the jumps {Yi }i∈N are rv’s in the DOA of an α-stable process with
index α ∈ (0, 2];
• the waiting times {Ji }i∈N are rv’s in the DOA of a β-stable process
with index β ∈ (0, 1);
We want to study the limits of the following stochastic process



β (nt)
 1 NX
Ti
Yi
G t−
 nβ/α

n
i=1
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t>0



β (nt)
 1 NX
Ti
and
Yi
Gv t −
 nβ/α

n
i=1
t>0
.
Our problem (decoupled case)
Under the following distributional assumptions,
• the jumps {Yi }i∈N are rv’s in the DOA of an α-stable process with
index α ∈ (0, 2];
• the waiting times {Ji }i∈N are rv’s in the DOA of a β-stable process
with index β ∈ (0, 1);
• Yi and Ji are independent for all i ∈ N.
We want to study the Functional Central Limit Theorem of



β (nt)
 1 NX
Ti
Yi
f
 nβ/α

n
i=1
for all f continuous and bounded function.
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,
t>0
The Skorokhod space
The Skorokhod space, denoted by D = D([0, T ], R) (with T > 0), is
the space of real functions x : [0, T ] → R that are right-continuous
with 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 càdlàg functions.
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Skorokhod topologies
The Skorokhod space provides a natural and convenient formalism
for describing the trajectories of stochastic processes with jumps:
• Poisson processes;
• Lévy processes;
• martingales and semimartingales;
• empirical distribution functions;
• discretizations of stochastic processes.
Topology: It can be assigned a topology that intuitively allows us to
wiggle space and time a bit (whereas the traditional topology of
uniform 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 approximating
process to accumulate into a large jump for the limit process
• J1 -topology requires a large jump for the limit process to be
approximated 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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Convergence to the β-stable subordinator
For t > 0, we define
Tt :=
btc
X
Ji .
i=1
We have
{c −1/β Tct }t>0
J1 −top
⇒ {Dβ (t)}t>0 ,
as
c → +∞.
A β-stable subordinator {Dβ (t)}t>0 is a real-valued β-stable Lévy
process 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/β Nβ (ct)}t>0
J1 −top
⇒ {Dβ−1 (t)}t>0 ,
as c → +∞.
The functional inverse of {Dβ (t)}t>0 can be defined as
Dβ−1 (t) := inf{x > 0 : Dβ (x) > t}.
It has a.s. continuous non-decreasing sample paths and without
stationary and independent increments.
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Convergence to the symmetric α-stable Lévy process
Assume the jumps Yi belong to the strict generalized DOA of some
stable law with α ∈ (0, 2], then
Theorem (Meerschaert & Scheffler (2004))


[ct]


X
c −1/α
Yi


i=1
J1 −top
⇒ {Lα (t)}t>0 ,
when
c → +∞.
t>0
M. Meerschaert, H. P. Scheffler.
Limit theorems for continuous-time random walks with infinite
mean 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 waiting
times Ji and the jumps Yi , we have


Nβ (ct)


X
M1 −top
c −β/α
Yi
⇒ {Lα (Dβ−1 (t))}t>0 , when c → +∞,


i=1
t>0
(5)
in the Skorokhod space D([0, +∞), R).
M. Meerschaert, H. P. Scheffler.
Limit theorems for continuous-time random walks with infinite
mean waiting times.
J. Appl. Probab., 41 (3), 623–638, 2004.
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Main ingredients of the proof
• Triangular array approach
(c)
∆ = {(Yi
(c)
, Ji ) : i > 1, c > 0}.
It is assumed that ∆ is given so that


btc
btc

 X
X
J1 −top
(c)
(c)

⇒ {(Lα (t), Dβ (t))}t>0 , c → ∞.
Yi ,
Ji 


i=1
i=1
t>0
• Continuous-Mapping Theorem (CMT)
W. Whitt,
Stochastic-Process Limits: An Introduction to
Stochastic-Process Limits and Their Application to Queues.
Springer, New York (2002).
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Example: Quadratic Variation
The quadratic variation of
Nβ (t)
X (t) =
X
Yi
i=1
is
Nβ (t)
[X ](t) = [X , X ](t) =
X
i=1
Nβ (t)
2
[X (Ti ) − X (Ti−1 )] =
X
Yi2 .
i=1
E. Scalas, N. Viles,
On the Convergence of Quadratic variation for Compound
Fractional 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


[nt]


X
1
1
J1 −top
2


Y
,
T
⇒ {(L+
i
α/2 (t), Dβ (t))}t>0 ,
1/β nt 
n→+∞
 n2/α
n
i=1
t>0
in D([0, +∞), R+ × R+ ) . Moreover, we have also
1
n2β/α
Nβ (nt)
X
Yi2
M1 −top
⇒
−1
L+
α/2 (Dβ (t)),
in D([0, +∞), R+ ), where L+
α/2 (t) denotes an
process.
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as
n → +∞,
i=1
α
2 -stable
positive Lévy
FCLT for certain stochastic integrals (decoupled case)
Distributional assumptions
• Jumps {Yi }i∈N : i.i.d. symmetric α-stable random variables such
that Y1 belongs to DOA of an α-stable random variable with
α ∈ (0, 2].
• Wating times {Ji }i∈N : i.i.d. random variables such that J1 belongs
to DOA of some β-stable random variables with β ∈ (0, 1).
• Yi and Ji are independent.
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FCLT for stochastic integrals
Theorem (Scalas & V.)
Let f ∈ Cb (R). Under the distributional assumptions and the scaling,



β (nt)
 1 NX
Ti
Yi
f

 nβ/α
n
i=1
M1 −top
⇒
n→+∞
t>0
in D([0, +∞), R) with M1 -topology.
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Z
t
f
0
(s)dLα (Dβ−1 (s))
,
t>0
Result 1
Proposition (Scalas & V.)
Let f ∈ Cb (R). Under the distributional assumptions and a suitable
scaling we have that


β
Z t
 1 bn
Xtc Ti 
J1 −top
f (Dβ (s))dLα (s)
,
f
Yi
⇒
 nβ/α

n
0
t>0
i=1
as n → +∞.
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t>0
Proof Result 1
•
X
(n)
(t) :=
1
nβ/α
bnβ tc
X
Yi
i=1
•
Ti
(n)
Yi , X
f
n
t>0
J1 −top
⇒ {(f (Dβ (s)), Lα (t))}t>0 .
n→∞
• Theorem by Kurtz and Protter


bnβ tc 

X
T
1
T
i
i
f
Yi , X (n) , β/α
f
Yi 


n
n
n
i=1
t>0
Z t
J1 −top
⇒
f (Dβ (s)), Lα (t),
f (Dβ (s))dLα (s)
n→∞
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0
t>0
Result 2
Proposition (Scalas & V.)
Let f ∈ Cb (R). Under the distributional assumptions and a suitable
scaling,


(Z −1
)

β (nt)
NX
Dβ (t)
Ti
M1 −top
Yi
f
⇒
f (Dβ (s))dLα (s)


n
0
i=1
t>0
t>0
as n → +∞, where
Z
Dβ−1 (t)
a.s.
Z
f (Dβ (s))dLα (s) =
0
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0
t
f (s)dLα (Dβ−1 (s)).
Proof Result 2
• Result 1
• Continuous Mapping Theorem using φ(x, y ) := (x, y −1 ):


bnβ tc 

X
Ti
 1
Yi , n−1/β Nβ (nt)
f
β/α
 n

n
i=1
t>0
J1 −top
t
Z
⇒
n→∞
0
f (Dβ (s))dLα (s), Dβ−1 (t)
t>0
• Continuous Mapping Theorem using φ(x, y ) := x ◦ y :



β (nt)
NX
Ti
f
Yi


n
i=1
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t>0
M1 −top
(Z
⇒
n→∞
Dβ−1 (t)
)
f (Dβ (s))dLα (s)
0
t>0
Application (Scalas & V.)
Let G (t) be the Green function of the equation (1) and Gv (t) its
derivative. Under the distributional assumptions and the suitable
scaling, it follows that



β (nt)
 1 NX
Ti
G t−
Yi

 nβ/α
n
i=1
M1 −top
t
Z
⇒
G (t −
0
s)dLα (Dβ−1 (s))
,
t>0
t>0
and

 1
 nβ/α
Nβ (nt)
X
i=1


Ti
Yi
Gv t −

n
M1 −top
⇒
n→+∞
t>0
in D([0, +∞), R) with M1 -topology.
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Z
t
Gv (t −
0
s)dLα (Dβ−1 (s))
,
t>0
Conclusions:
• We have studied the convergence of a class of stochastic integrals
with respect to the Compound Fractional Poisson Process.
• Under proper scaling hypotheses, these integrals converge to the
integrals w.r.t a symmetric α-stable process subordinated to the
inverse β-stable subordinator.
Extensions:
• It is possible to approximate some of the integrals discussed in
Kobayashi (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 in
Becker-Kern, Meerschaert and Schefller (2002) and Straka.
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