Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico Scalas BCAM Seminar - 26 July 2012 Noèlia Viles (BCAM) BCAM seminar 26 July 2012 1 / 43
Outline 1 Introduction 2 FCLT for the quadratic variation of Compound Renewal Processes 3 FCLT for the stochastic integrals driven by a time-changed symmetric α-stable Lévy process Noèlia Viles (BCAM) BCAM seminar 26 July 2012 2 / 43
Scaling Limits Consider a sequence of i.i.d. centered random variables ξ i { 1, 1}: 1, 1, 1, 1, 1, 1, 1... Define the centered random walk: S n := n ξ i. (a) How does S n behave when n is large? (b) What is the limit after rescaling? Central Limit Theorem (CLT) If E[ξ 1 ] = µ and Var(ξ 1 ) := σ 2 (0, ), then S n nµ n L N(0, σ 2 ). Noèlia Viles (BCAM) BCAM seminar 26 July 2012 3 / 43
Donsker s Theorem The classical CLT was generalized to a FCLT by Donsker (1951). Donsker s Theorem (1951) Suppose (ξ n ) n N is an i.i.d. sequence with E[ξ 1 ] = 0 and Var(ξ 1 ) = σ 2 defined by X n (t) = 1 σ n S nt + (nt nt ). Then, (X n (t), t [0, T ]) L (B(t), t [0, T ]) where B is a standard Brownian motion. This means that for all bounded, continuous functions f : C([0, T ]) R, lim E[f (S nt ) = E[f (B)]. n Noèlia Viles (BCAM) BCAM seminar 26 July 2012 4 / 43
Idea of the proof The proof of the Donsker s Theorem follows the next steps: Step 1: First we show the existence of a continuous version of S n and the tightness of the measures associated to S n. Step 2: Secondly, we prove the convergence of the finite-dimensional distributions. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 5 / 43
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+) = lim s t x(s) exists and x(t+) = x(t). 2 For t (0, T ], x(t ) = lim s t x(s) exists. Functions satisfying these properties are called càdlàg functions. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 6 / 43
Skorokhod topology The Skorokhod space provides a natural and convenient formalism for describing the trajectories of stochastic processes with jumps: Poisson process, Lévy processes, martingales and semimartingales, empirical distribution functions, discretizations of stochastic processes, etc. 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). Skorokhod (1965) proposed four metric separable topologies on D, denoted by J 1, J 2, M 1 and M 2. A. Skorokhod. Limit Theorems for Stochastic Processes. Theor. Probability Appl. 1, 261 290, 1956. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 7 / 43
The Skorokhod J 1 -topology For T > 0, let Λ := {λ : [0, T ] [0, T ], strictly increasing and continuous}. If λ Λ, then λ(0) = 0 and λ(t ) = T. For x, y D, the Skorokhod J 1 -metric is d J1 (x, y) := inf { sup λ(t) t, sup x(t) y(λ(t)) } (1) λ Λ t [0,T ] t [0,T ] Definition (J 1 -topology) The sequence x n (t) D converges to x 0 (t) D in the J 1 topology if there exists a sequence of increasing homeomorphisms λ n : [0, T ] [0, T ] such that as n. sup λ n (t) t 0, sup x n (λ n (t)) x 0 (t) 0, (2) t [0,T ] t [0,T ] Noèlia Viles (BCAM) BCAM seminar 26 July 2012 8 / 43
The Skorokhod M 1 -topology For x D, the completed graph of x is Γ (a) x = {(t, z) [0, T ] R : z = ax(t ) + (1 a)x(t) for some a [0, 1]}, where x(t ) is the left limit of x at t and x(0 ) := x(0). A parametric representation of Γ (a) x is a continuous nondecreasing function (r, u) : [0, 1] Γ (a) x, with r being the time component and u being the spatial component. Denote Π(x) the set of parametric representations of Γ (a) x. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 9 / 43
The Skorokhod M 1 -topology For x 1, x 2 D, the Skorokhod M 1 -metric on D is d M1 (x 1, x 2 ) := inf (r i,u i ) Π(x i ),2 { r 1 r 2 [0,T ] u 1 u 2 [0,T ] }. (3) Definition (M 1 -topology) The sequence x n (t) D converges to x 0 (t) D in the M 1 -topology if lim d M n + 1 (x n (t), x 0 (t)) = 0. (4) In other words, we have the convergence in M 1 -topology if there exist parametric representations (y(s), t(s)) of the graph Γ x0 (t) and (y n (s), t n (s)) of the graph Γ xn(t) such that lim (y n, t n ) (y, t) [0,T ] = 0. (5) n Noèlia Viles (BCAM) BCAM seminar 26 July 2012 10 / 43
Characterization for the M 1 -convergence Theorem (Silvestrov(2004)) {X n (t)} t 0 M 1 top {X (t)} t 0, when n if : (i) {X n (t)} t A L {X (t)}t A as n +, where A [0, + ) dense in this interval and contains 0, (ii) the condition on M 1 -compactness is satisfied lim δ 0 lim sup w(x n, δ) = 0, (6) n + where the modulus of M 1 -compactness is defined as w(x n, t, δ) := w(x n, δ) := sup w(x n, t, δ), (7) t A sup { X n (t 2 ) [X n (t 1 ), X n (t 3 )] }. 0 (t δ) t 1 <t 2 <t 3 (t+δ) T Noèlia Viles (BCAM) BCAM seminar 26 July 2012 11 / 43
Some remarks For x, y R denote the standard segment as [x, y] := {ax + (1 a)y, a [0, 1]}. The modulus of M 1 -compactness plays the same role for càdlàg functions as the modulus of continuity for continuous functions. D.S Silvestrov. Limit Theorems for Randomly Stopped Stochastic Processes. Probability and its Applications. Springer, New York, 2004. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 12 / 43
Continuous-Time Random Walks (CTRW) A continuous time random walk (CTRW) is a pure jump process given by a sum of i.i.d. random jumps (Y i ) i N separated by i.i.d. random waiting times (positive random variables) (J i ) i N. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 13 / 43
Compound Poisson Process Let X n = n Y i denote the position of a diffusing particle after n jumps and T n = n J i be the epoch of the n-th jump. The corresponding counting process N(t) is defined by N(t) def = max{n : T n t}. (8) Then the position of a particle at time t > 0 can be expressed as the sum of the jumps up to time t N(t) def X (t) = X N(t) = Y i. (9) It is called compound Poisson process. It is a Markov and Lévy process. The functional limit is an α-stable Lévy process. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 14 / 43
α-stable Lévy processes A continuous-time process L = {L t } t 0 with values in R is called a Lévy process if its sample paths are càdlàg at every time point t, and it has stationary, independent increments, that is: (a) For all 0 = t 0 < t 1 < < t k, the increments L ti L ti 1 are independent. (b) For all 0 s t the random variables L t L s and L t s L 0 have the same distribution. An α-stable process is a real-valued Lévy process L α = {L α (t)} t 0 with initial value L α (0) that satisfies the self-similarity property 1 t 1/α L α(t) L = L α (1), t > 0. If α = 2 then the α-stable Lévy process is the Wiener process. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 15 / 43
Compound Fractional Poisson Process Consider a CTRW whose i.i.d. jumps (Y i ) i N have symmetric α-stable distribution with α (1, 2], and whose i.i.d waiting times (J i ) i N satisfy for β (0, 1], where P(J i > t) = E β ( t β ), (10) E β (z) = + j=0 denotes the Mittag-Leffler function. z j Γ(1 + βj), If β = 1, the waiting times are exponentially distributed with parameter λ = 1 and the counting process is the Poisson process. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 16 / 43
Compound Fractional Poisson Process The semi-markov extension of the compound Poisson process is the compound fractional Poisson process. The counting process associated is called the fractional Poisson process N β (t) = max{n : T n t}. For β (0, 1), the FPP is semi-markov and it is not Lévy. If we subordinate a CTRW to the fractional Poisson process, we obtain the compound fractional Poisson process, which is not Markov X Nβ (t) = N β (t) Y i. (11) The functional limit of the compound fractional Poisson process is an α-stable Lévy process subordinated to the fractional Poisson process. These processes are possible models for tick-by-tick financial data. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 17 / 43
β-stable subordinator and its functional inverse A β-stable subordinator {D β } t 0 nondecreasing sample paths. The functional inverse of {D β } t 0 can be defined as is a real-valued β-stable Lévy process with D 1 β (t) := inf{x 0 : D β(x) > t}. It has almost surely continuous non-decreasing sample paths and without stationary and independent increments. Magdziard & Weron, 2006 Noèlia Viles (BCAM) BCAM seminar 26 July 2012 18 / 43
Convergence to the inverse β-stable subordinator For t 0, we define We have t T t := J i. {c 1/β T ct } t 0 L {Dβ (t)} t 0, as c +. For any integer n 0 and any t 0: {T n t} = {N β (t) n}. Theorem (Meerschaert & Scheffler (2001)) {c 1/β N β (ct)} t 0 L {D 1 β (t)} t 0, as c +. Theorem (Meerschaert & Scheffler (2001)) {c 1/β J 1 top N β (ct)} t 0 {D 1 β (t)} t 0, as c +. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 19 / 43
References M. Meerschaert, H. P. Scheffler. Limit Theorems for continuous time random walks. Available at http://www.mathematik.uni-dortmund.de/lsiv/scheffler/ctrw1.pdf., 2001. M. Meerschaert, H. P. Scheffler. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley Series in Probability and Statistics., 2001. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 20 / 43
Convergence to the symmetric α-stable Lévy process Assume the jumps Y i belong to the strict generalized domain of attraction of some stable law with α (0, 2), then a n > 0 such that a n n L Y i Lα, as c +. Theorem (Meerschaert & Scheffler (2001)) [ct] c 1/α Y i t 0 Corollary (Meerschaert & Scheffler (2004)) [ct] c 1/α Y i t 0 L {L α (t)} t 0, when c +. J 1 top {L α (t)} t 0, when c +. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 21 / 43
Functional Central Limit Theorem Theorem (Meerschaert & Scheffler (2004)) Under the distributional assumptions considered above for the waiting times J i and the jumps Y i, we have c β/α N β (t) Y i t 0 M 1 top {L α (D 1 β (t))} t 0, when c +, (12) in the Skorokhod space D([0, + ), R) endowed with the M 1 -topology. 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. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 22 / 43
Idea of the proof Apply {c 1/β J 1 top N β (ct)} t 0 {D 1 β (t)} t 0, as c +. and [ct] c 1/α Y i t 0 J 1 top {L α (t)} t 0, when c +. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 23 / 43
Idea of the proof Continuous Mapping Theorem (Whitt 2002) Suppose that (x n, y n ) (x, y) in D([0, a], R k ) D 1 (where D1 is the subset of functions nondecreasing and with x i (0) 0). If y is continuous and strictly increasing at t whenever y(t) Disc(x) and x is monotone on [y(t ), y(t)] and y(t ), y(t) / Disc(x) whenever t Disc(y), then x n y n x y in D([0, a], R k ), where the topology throughout is M 1 or M 2. In that case we can only ensure the convergence in the weaker topology, M 1 instead of J 1 because {D 1 β (t)} t 0 is not strictly increasing. W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York (2002). Noèlia Viles (BCAM) BCAM seminar 26 July 2012 24 / 43
FCLT for the quadratic variation of Compound Renewal Processes Noèlia Viles (BCAM) BCAM seminar 26 July 2012 25 / 43
Quadratic Variation Let {Y i } be a sequence of i.i.d. random variables (also independent of the J i s) then the compound process X (t) defined by The quadratic variation of X is [X ](t) = [X, X ](t) = X (t) = N β (t) N β (t) [X (T i ) X (T i 1 )] 2 = Y i (13) N β (t) Y 2 i. (14) Noèlia Viles (BCAM) BCAM seminar 26 July 2012 26 / 43
FCLT for the Quadratic Variation Theorem (Scalas & V. (2012)) Under the distributional assumptions considered above for the waiting times J i and the jumps Y i, we have 1 [nt] n 2/α Yi 2 1, n 1/β T J nt 1 top c + {(L+ α/2 (t), D β(t))} t 0, (15) in the Skorokhod space D([0, + ), R + R + ) endowed with the J 1 -topology. Moreover, we have also N β (nt) Yi 2 M 1 top n 2β/α t 0 L + α/2 (D 1 β (t)), as c +, in the Skorokhod space D([0, + ), R + ) with the M 1 -topology, where L + α/2 (t) denotes an α/2-stable positive Lévy process. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 27 / 43
E. Scalas, N. Viles, On the Convergence of Quadratic variation for Compound Fractional Poisson Processes. Fractional Calculus and Applied Analysis, 15, 314 331 (2012). Noèlia Viles (BCAM) BCAM seminar 26 July 2012 28 / 43
FCLT for the stochastic integrals driven by a time-changed symmetric α-stable Lévy process Noèlia Viles (BCAM) BCAM seminar 26 July 2012 29 / 43
Damped harmonic oscillator subject to a random force The equation of motion is informally given by ẍ(t) + γẋ(t) + kx(t) = ξ(t), (16) 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 (formal derivative symmetric L α (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). Noèlia Viles (BCAM) BCAM seminar 26 July 2012 30 / 43
The formal solution is x(t) = F (t) + t G(t t )ξ(t )dt, (17) where G(t) is the Green function for the homogeneous equation. The solution for the velocity component can be written as t v(t) = F v (t) + G v (t t )ξ(t )dt, (18) where F v (t) = d dt F (t) and G v (t) = d dt G(t). Noèlia Viles (BCAM) BCAM seminar 26 July 2012 31 / 43
Motivation Replace the white noise with a sequence of instantaneous shots of random amplitude at random times and then with an appropriate functional limit of this process. The sequence of instantaneous shots of random amplitude at random times can be expressed in terms of the formal derivative of a compound renewal process given by X (t) = and the corresponding noise Ξ(t) = dx (t)/dt = N β (t) N β (t) Y i = i 1 Y i 1 {Ti t} Y i δ(t T i ) = i 1 Y i δ(t T i )1 {Ti t}. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 32 / 43
Our goal To study the convergence of the integral of a deterministic continuous and bounded function with respect to a properly rescaled CTRW. We aim to prove that under a proper scaling and distributional assumptions: N β (nt) ( G t T ) i Yi { t M 1 top n and N β (nt) n β/α ( G v t T ) i Yi n n β/α t 0 t 0 M 1 top 0 { t 0 G(t s)dl α (D 1 β (s)) } G v (t s)dl α (D 1 β (s)) } when n +, in the Skorokhod space D([0, + ), R) endowed with the M 1 -topology. t 0, t 0, Noèlia Viles (BCAM) BCAM seminar 26 July 2012 33 / 43
Rescaled CTRW Let h and r be two positive scaling factors such that lim h,r 0 h α = 1, (19) r β with α (1, 2] and β (0, 1]. We rescale the duration J and the jump by positive scaling factors r and h: The rescaled CTRW denoted: J r := rj, Y h := hy. X r,h (t) = N β (t/r) hy i, where N β = {N β (t)} t 0 is the fractional Poisson process. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 34 / 43
FCLT for stochastic integrals driven by a time-changed symmetric α-stable Lévy process Theorem (Scalas & V.) Let f C b (R). Under the distributional assumptions and the scaling, N β (nt) ( ) Ti Yi { t } M 1 top f f (s)dl n n β/α α (D 1 n + β (s)), 0 t 0 t 0 in D([0, + ), R) with M 1 -topology. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 35 / 43
Sketch of the proof Check M 1 -compactness condition for the integral process N β (nt) I ( ) Tk Yk n(t) := f. n k=1 n β/α t 0 Prove the convergence in law of the family of processes {I n (t)} t 0 when n +. {X (n) (t)} t 0 is uniformly tight or a good sequence. Apply the Continuous Mapping Theorem (CMT) taking as a continuous mapping the composition function. Apply Characterization of the M 1 -convergence. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 36 / 43
M 1 -compactness condition Lemma (Scalas & V.) Let f C b (R). Let {Y i } i N be i.i.d. symmetric α-stable random variables. Assume that Y 1 belongs DOA of S α, with α (1, 2]. Let {J i } i N be i.i.d. such that J 1 belongs to the strict DOA of S β withβ (0, 1). Consider I n (t) := N β (nt) k=1 f ( ) Tk Yk. (20) n nβ/α If where X n (t) := N β (nt) k=1 lim δ 0 Y k n β/α lim δ 0 lim sup w s (X n, δ) = 0, n + = k 1 Y k n β/α 1 {τk t}. Then, lim sup w s (I n, δ) = 0. (21) n + Noèlia Viles (BCAM) BCAM seminar 26 July 2012 37 / 43
Good sequence Let (X n ) n N be an R k -valued process defined on (Ω n, F n, P n ) s.t. it is Ft n -semimartingale. Assume that (X n L ) n N X in the Skorokhod topology. The sequence (X n ) n N is said to be good if for any sequence (H n ) n N of M km -valued, càdlàg processes, H n F n t -adapted, such that (H n, X n ) L (H, X ) in the Skorokhod topology on D M km R m([0, )), a filtration F t such that H is F t -adapted, X is an F t -semimartingale, and when n. t 0 H n (s )dx n s t L H(s )dx s, 0 Noèlia Viles (BCAM) BCAM seminar 26 July 2012 38 / 43
(X (n) ) n N uniformly tight Lemma (Scalas & V.) Assume that (Y i ) i N be i.i.d. symmetric α-stable random variables, with α (1, 2]. Let X (n) (t) := n β t Y i n β/α (22) be defined on the probability space (Ω n, F n, P n ). Then X n (t) is a Ft n -martingale (with respect the natural filtration of X (n) ) and ] sup E [sup n X (n) (s) < +, n s t for each t < +, where denotes the increment of X (n). X (n) (s) := X (n) (s) X (n) (s ) (23) Noèlia Viles (BCAM) BCAM seminar 26 July 2012 39 / 43
Convergence in law Proposition (Scalas & V.) Let f C b (R). Under the distributional assumptions and the scaling considered above we have that n β t f ( ) Ti Yi n n β/α t L 0 f (D β (s))dl α (s), n +. Proposition (Scalas & V.) Let f C b (R). Under the distributional assumptions and scaling, N β (nt) ( ) { Ti } D 1 L β (t) f Y i f (D β (s))dl α (s) n t 0 as n +, where D 1 β (t) 0 f (D β (s))dl α (s) a.s. = t 0 f (s)dl α(d 1 β (s)). 0 Noèlia Viles (BCAM) BCAM seminar 26 July 2012 40 / 43 t 0
Applications Corollary (Scalas & V.) N β (nt) G and N β (nt) ( t T ) i Yi n n β/α ( G v t T ) i Yi n n β/α t 0 t 0 M 1 top in D([0, + ), R) with M 1 -topology. { t 0 G(t s)dl α (D 1 β (s)) } t 0 { t } M 1 top G v (t s)dl α (D 1 n + β (s)), 0 t 0, Noèlia Viles (BCAM) BCAM seminar 26 July 2012 41 / 43
Summary: 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. Noèlia Viles (BCAM) BCAM seminar 26 July 2012 42 / 43
Thank you for your attention Noèlia Viles (BCAM) BCAM seminar 26 July 2012 43 / 43