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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 fYi gi2N separated by i.i.d. random waiting times (positive random variables) fJi gi2N. 4 of 27 Fractional Poisson Process Let fJi gi2N be a sequence of i.i.d. and positive β-stable rv's with the meaning of waiting times between jumps and β 2 (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) = maxfn : Tn 6 tg: which counts the number of jumps up to time t > 0. 5 of 27 Compound Fractional Poisson Process Let fYi gi2N 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 2 (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 fYi gi2N are rv's in the DOA of an α-stable process with index α 2 (0; 2]; • the waiting times fJi gi2N are rv's in the DOA of a β-stable process with index β 2 (0; 1); We want to study the limits of the following stochastic process 8 9 8 9 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 Under the following distributional assumptions, • the jumps fYi gi2N are rv's in the DOA of an α-stable process with index α 2 (0; 2]; • the waiting times fJi gi2N are rv's in the DOA of a β-stable process with index β 2 (0; 1); We want to study the limits of the following stochastic process 8 9 8 9 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 But limits in which sense? 7 of 27 α-stable L´evyprocesses A continuous-time process L = fLt gt>0 with values in R is called a L´evyprocess if its sample paths are c`adl`agat every time point t, and it has stationary, independent increments. An α-stable process is a real-valued L´evyprocess Lα = fLα(t)gt>0 with initial value Lα(0) that satisfies the self-similarity property 1 L (t) =L L (0); 8t > 0: t1/α α α If α = 2 then the α-stable L´evyprocess 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 2 [0; T ), x(t+) = lims#t x(s) exists and x(t+) = x(t). 2. For t 2 (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 fYi gi2N are rv's in the DOA of an α-stable process with index α 2 (0; 2]; • the waiting times fJi gi2N are rv's in the DOA of a β-stable process with index β 2 (0; 1); We want to study the Functional Central Limit Theorem of 8 9 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 fc Tct gt>0 ) fDβ(t)gt>0; as c ! +1: A β-stable subordinator fDβ(t)gt>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: fTn 6 tg = fNβ(t) 6 ng: Theorem (Meerschaert & Scheffler (2001)) −1/β J1−top −1 fc Nβ(ct)gt>0 ) fDβ (t)gt>0; as c ! +1: The functional inverse of fDβ(t)gt>0 can be defined as −1 Dβ (t) := inffx > 0 : Dβ(x) > tg: 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 2 (0; 2], then Theorem (Meerschaert & Scheffler (2004)) 8 [ct] 9 < −1/α X = J1−top c Yi ) fLα(t)gt>0; when c ! +1: : 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 8 9 Nβ (ct) < −β/α X = M1−top −1 c Yi ) fLα(Dβ (t))gt>0; when c ! +1; : i=1 ; t>0 (5) in the Skorokhod space D([0; +1); 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 (c) (c) ∆ = f(Yi ; Ji ): i > 1; c > 0g: It is assumed that ∆ is given so that 80 btc btc 19 < X (c) X (c) = J1−top Y ; J ) f(Lα(t); Dβ(t))g ; c ! 1: @ i i A 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 80 [nt] 19 < 1 X 2 1 = J1−top + @ Yi ; Tnt A ) f(L (t); Dβ(t))gt 0; n2/α n1/β n!+1 α/2 > : i=1 ; t>0 in D([0; +1); R+ × R+) . Moreover, we have also Nβ (nt) 1 X M1−top Y 2 ) L+ (D−1(t)); as n ! +1; n2β/α i α/2 β i=1 + α in D([0; +1); R+), where Lα/2(t) denotes an 2 -stable positive L´evy process.
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