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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