Fclts for the Quadratic Variation of a CTRW and for Certain Stochastic Integrals

FCLTs for the Quadratic Variation of a CTRW and for certain stochastic integrals

No`eliaViles Cuadros (joint work with Enrico Scalas)

Universitat de Barcelona

Sevilla, 17 de Septiembre 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 at time t, γ > 0 is the damping coeﬃcient, k > 0 is the spring constant and ξ(t) represents white L´evynoise.

I. M. Sokolov, Harmonic oscillator under L´evynoise: Unexpected properties in the phase space. Phys. Rev. E. Stat. Nonlin Soft Matter Phys 83, 041118 (2011).

2 of 27 The formal solution is Z t x(t) = F (t) + G(t − t0)ξ(t0)dt0, (2) −∞ where G(t) is the Green function for the homogeneous equation. The solution for the velocity component can be written as

Z t 0 0 0 v(t) = Fv (t) + Gv (t − t )ξ(t )dt , (3) −∞

d d where Fv (t) = dt F (t) and Gv (t) = dt G(t).

3 of 27 • Replace the white noise with a sequence of instantaneous shots of random amplitude at random times. • They can be expressed in terms of the formal derivative of compound renewal process, a random walk subordinated to a counting process called continuous-time random walk. A continuous time random walk (CTRW) is a pure jump process given 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.

4 of 27 Fractional Poisson Process

Let {Ji }i∈N be a sequence of i.i.d. and positive β-stable rv’s with the meaning of waiting times between jumps and β ∈ (0, 1). Let n X Tn = Ji i=1 represent the epoch of the n-th jump and the counting process associated is the fractional Poisson process

Nβ(t) = max{n : Tn 6 t}. which counts the number of jumps up to time t > 0.

5 of 27 Compound Fractional Poisson Process

Let {Yi }i∈N be a sequence of i.i.d. and α-stable rv’s (independent of Ji ) with the meaning of jumps taking place at every epoch and α ∈ (0, 2].

If we subordinate a CTRW to the fractional Poisson process, we obtain the compound fractional Poisson process, which is not Markov

Nβ (t) X XNβ (t) = Yi . (4) i=1

6 of 27 But limits in which sense?

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     Nβ (nt) Nβ (nt)  1 X Ti   1 X Ti  G t − Y and G t − Y . nβ/α n i nβ/α v n i  i=1   i=1  t>0 t>0

7 of 27 Our goal

But limits in which sense?

7 of 27 α-stable L´evyprocesses

A continuous-time process L = {Lt }t>0 with values in R is called a Lévyprocess if its sample paths are càdlàgat every time point t, and it has stationary, independent increments.

An α-stable process is a real-valued Lévyprocess Lα = {Lα(t)}t>0 with initial value Lα(0) that satisfies the self-similarity property 1 L (t) =L L (0), ∀t > 0. t1/α α α If α = 2 then the α-stable Lévyprocess is the Wiener process.

8 of 27 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 c`adl`ag functions.

9 of 27 Skorokhod topologies

The Skorokhod space provides a natural and convenient formalism for describing the trajectories of stochastic processes with jumps: • Poisson processes; • L´evyprocesses; • 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).

10 of 27 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.

11 of 27 Our problem

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 Functional Central Limit Theorem of   Nβ (nt)  1 X Ti  f Y , nβ/α n i  i=1  t>0 for all f continuous and bounded function.

12 of 27 Convergence to the β-stable subordinator

For t > 0, we deﬁne btc X Tt := Ji . i=1

We have

−1/β J1−top {c Tct }t>0 ⇒ {Dβ(t)}t>0, as c → +∞.

A β-stable subordinator {Dβ(t)}t>0 is a real-valued β-stable L´evy process with nondecreasing sample paths.

13 of 27 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))

−1/β J1−top −1 {c Nβ(ct)}t>0 ⇒ {Dβ (t)}t>0, as c → +∞.

The functional inverse of {Dβ(t)}t>0 can be deﬁned as

−1 Dβ (t) := inf{x > 0 : Dβ(x) > t}. It has a.s. continuous non-decreasing sample paths and without stationary and independent increments. 14 of 27 Convergence to the symmetric α-stable L´evyprocess

Assume the jumps Yi belong to the strict generalized DOA of some stable law with α ∈ (0, 2], then Theorem (Meerschaert & Scheﬄer (2004))

 [ct]   −1/α X  J1−top c Yi ⇒ {Lα(t)}t>0, when c → +∞.  i=1  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.

15 of 27 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)  −β/α X  M1−top −1 c Yi ⇒ {Lα(Dβ (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. 16 of 27 Main ingredients of the proof

• Triangular array approach

 btc btc   X (c) X (c)  J1−top Y , J ⇒ {(Lα(t), Dβ(t))} , c → ∞.  i i  t>0  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). 17 of 27 Example: Quadratic Variation

The quadratic variation of

Nβ (t) X X (t) = Yi i=1 is

Nβ (t) Nβ (t) X 2 X 2 [X ](t)=[ X , X ](t) = [X (Ti ) − X (Ti−1)] = Yi . i=1 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). 18 of 27 FCLT for the Quadratic Variation

Theorem (Scalas & V. (2012)) Under the distributional assumptions considered above, we have that

 [nt]   1 X 2 1  J1−top +  Yi , Tnt  ⇒ {(L (t), Dβ(t))}t 0, n2/α n1/β n→+∞ α/2 >  i=1  t>0 in D([0, +∞), R+ × R+) . Moreover, we have also

Nβ (nt) 1 X M1−top Y 2 ⇒ L+ (D−1(t)), as n → +∞, n2β/α i α/2 β i=1

+ α in D([0, +∞), R+), where Lα/2(t) denotes an 2 -stable positive L´evy process. 19 of 27 FCLT for certain stochastic integrals

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).

20 of 27 Our preliminary results

Proposition (Scalas & V.)

Let f ∈ Cb(R). Under the distributional assumptions and a suitable scaling we have that

 β  bn tc Z t  1 X Ti  J1−top f Y ⇒ f (D (s))dL (s) , nβ/α n i β α  i=1  0 t>0 t>0 as n → +∞.

21 of 27 Our preliminary results

Proposition (Scalas & V.)

Let f ∈ Cb(R). Under the distributional assumptions and a suitable scaling,   Nβ (nt) ( −1 ) Z Dβ (t)  X Ti  J1−top f Yi ⇒ f (Dβ(s))dLα(s) n 0  i=1  t 0 t>0 > as n → +∞, where

Z D−1(t) Z t β a.s. −1 f (Dβ(s))dLα(s) = f (s)dLα(Dβ (s)). 0 0

22 of 27 FCLT for stochastic integrals

Theorem (Scalas & V.) Let f ∈ Cb(R). Under the distributional assumptions and the scaling,   Nβ (nt) Z t  1 X Ti  M1−top −1 f Yi ⇒ f (s)dLα(D (s)) , nβ/α n n→+∞ β  i=1  0 t>0 t>0 in D([0, +∞), R) with M1-topology.

23 of 27 Theorem (Kurtz & Protter) for stochastic integrals

n n n For each n, let (H , X ) be an Ft -adapted process in n n DMkm×Rm ([0, ∞)) and let X be an Ft -semimartingale. Fix δ > 0 n,δ n n n,δ n,δ n,δ and deﬁne X = X − Kδ(X ). Let X = M + A be a n,δ n decomposition of X into an Ft -local martingale and a process with θ ﬁnite variation. Suppose for each θ > 0, there exist stopping times τn θ 1 such that P(τn 6 θ) 6 θ and furthermore h n,δ n,δ n,δ θ i sup E [M , M ]t∧τ θ + TV (A , t ∧ τn ) < +∞. n n

n n J1−top If (H , X ) ⇒ (H, X ) on DMkm×Rm ([0, ∞)), then X is a semimartingale w.r.t a ﬁltration to which H and X are adapted and

Z J −top Z Hn, X n, HndX n 1⇒ H, X , HdX , as n → +∞, on D km× m× k ([0, ∞)). 24 of 27M R R 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   Nβ (nt) Z t  1 X Ti  M1−top G t − Y ⇒ G(t − s)dL (D−1(s)) , nβ/α n i α β  i=1  0 t>0 t>0 and   Nβ (nt) Z t  1 X Ti  M1−top −1 Gv t − Yi ⇒ Gv (t − s)dLα(D (s)) , nβ/α n n→+∞ β  i=1  0 t>0 t>0 in D([0, +∞), R) with M1-topology.

25 of 27 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).

26 of 27 Thank you for your attention! Gracias por su atenci´on!

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