From persistent random walk to the telegraph noise P . Vallois - PowerPoint PPT Presentation

From persistent random walk to the telegraph noise P . Vallois Institut Élie Cartan Nancy, Nancy-Université joint work with C. Tapiero (NYU) and S. Herrmann (Institut Élie Cartan Nancy) Roscoff, March 22, 2010 P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 1 / 24

Outline Introduction Results at a fixed time and applications From discrete to continuous time Few extensions P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 2 / 24

1. Introduction Let ( Y n ) n ≥ 0 be a Markov chain taking its values in {− 1 , 1 } with transition matrix : � 1 − α � α 0 < α < 1 , 0 < β < 1 . π = 1 − β β Associated with ( Y n ) consider the process n ≥ 0 . X n := Y 0 + Y 1 + · · · + Y n , ( X n ) is said to be a persistent random walk . Two particular cases are interesting : β = 1 − α : ( X n ) is a classical random walk whose increment is distributed as ( 1 − α ) δ − 1 + αδ 1 . β = α , ( X n ) is a Kac random walk : Y n + 1 = Y n with probability 1 − α and − Y n otherwise. P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 3 / 24

2. Study at a fixed time and applications Proposition 1 Let ρ := 1 − α − β the asymmetry factor. Then : 2 α E [ X t | Y 0 = − 1 ] = α − β ( 1 − ρ ) 2 ( 1 − ρ t + 1 ) . 1 − ρ ( t + 1 ) − 2 β E [ X t | Y 0 = + 1 ] = α − β ( 1 − ρ ) 2 ( 1 − ρ t + 1 ) . 1 − ρ ( t + 1 ) − Remark 1) In the classical random walk case, we have : ρ = 0 . 2) it is actually possible to compute explicitly the second moment of X t , see C. Tapiero and P .V. : Memory-based persistence in a counting random walk process, Physica A, 2007 P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 4 / 24

Let us introduce : Φ( λ, t ) = E [ λ X t ] , ( λ > 0 ) . Proposition 2 The generating function of X t equals: Φ( λ, t ) = a + θ t + + a − θ t − with a + = 1 − α + λ ( λα − θ − ) a − = 1 when X 0 = Y 0 = − 1 λ 2 √ and λ − a + D � 1 − α � θ ± := 1 √ + ( 1 − β ) λ ± D 2 λ � 1 − α � 2 + ( 1 − β ) λ − 4 ( 1 − α − β ) . D = λ P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 5 / 24

Sketch of the proof of Proposition 2 We decompose Φ( λ, t ) as follows : Φ( λ, t ) = Φ − ( λ, t ) + Φ + ( λ, t ) , with Φ − ( λ, t ) = E [ λ X t 1 { Y t = − 1 } ] , Φ + ( λ, t ) = E [ λ X t 1 { Y t = 1 } ] . The, we obtain the recursive relations : 1 − α Φ − ( λ, t ) + β Φ − ( λ, t + 1 ) = λ Φ + ( λ, t ) λ Φ + ( λ, t + 1 ) αλ Φ − ( λ, t ) + ( 1 − β ) λ Φ + ( λ, t ) = � P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 6 / 24

An application to insurance - A "normal claim" is labelled 0 and its values at time i is Z 0 i ; - an "unusual claim" (for instance "large") is labelled 1 and equals Z 1 i at time i . � � � � Z 0 Z 1 - The claims i , i ≥ 1 are i.i.d, i , i ≥ 1 are i.i.d and the two families of r.v.’s are independent. - The process which attributes labels is ( Y ′ i ) . So, Y ′ i = 0 if at time i a normal claim occurs. Note that Y ′ i ∈ { 0 , 1 } . Set Y i = 2 Y ′ i − 1. Then Y i ∈ {− 1 , 1 } and Y ′ i = 1 ⇔ Y i = 1 . - We suppose that : * ( Y ′ i ) is a Markov chain (then ( Y i ) is a Markov chain as above); � � Z j * all the claims i , j = 0 , 1 , i ≥ 1 and ( Y ′ i ) are independent. P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 7 / 24

The sum of claims at time t is : t t � � Z 1 Z 0 t 1 { Y ′ t 1 { Y ′ ξ t = i = 1 } + i = 0 } . i = 0 i = 0 Proposition 3 1) The first moment of ξ t is : � � E ( ξ t ) = ( t + 1 ) E ( Z 0 E ( Z 1 1 ) − E ( Z 0 E ( X ′ 1 ) + 1 ) t ) t � where X ′ Y ′ t = i . i = 0 2) The Laplace transform of ξ t equals : � 1 �� t + 1 � � e − λξ t � � e − λ Z 0 λ > 0 E = E Φ( z , t ) where � 1 � e − λ Z 1 � t � z := E � z X ′ 1 � , Φ( z , t ) := E . � e − λ Z 0 E P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 8 / 24

Remark Reference : C. Tapiero and P .V. A claims persistence process and Insurance, Insurance : Mathematics and Economics (2009). 2) Recall that : X t = 2 X ′ t − ( t + 1 ) . Then, t ) = 1 � � � t � � z X t / 2 � t + 1 E ( X ′ E ( X t ) + t + 1 z X ′ 2 E , E = z 2 P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 9 / 24

3. From discrete to continuous time S. Herrmann and P .V. : From persistent random walks to the Telegraph noise. Accepted in Stochastics and Dynamics (2009). 3.1 Notations a) Denote α 0 , β 0 two real numbers : 0 < α 0 ≤ 1 , 0 < β 0 ≤ 1. b) ∆ x is a "small" parameter such that : α := α 0 + c 0 ∆ x ∈ [ 0 , 1 ] , β := β 0 + c 1 ∆ x ∈ [ 0 , 1 ] . c) ( Y t , t ∈ N ) is a Markov chain which takes its values in {− 1 , 1 } with transition matrix : � 1 − α 0 − c 0 ∆ x � α 0 + c 0 ∆ x π ∆ = 1 − β 0 − c 1 ∆ x β 0 + c 1 ∆ x P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 10 / 24

d) The re-normalized random walk associated with ( Y t ) is defined as : Z ∆ (∆ t > 0 ) . s = ∆ x X s / ∆ t , s ∈ ∆ t N e) ( � Z ∆ s , s ≥ 0 ) is the continuous time process which is obtained by linear interpolation from ( Z ∆ s ) . f) Set ρ 0 = 1 − α 0 − β 0 ρ takes into account the "distance" of the persistent random walk to the classical r. w. Remark Note that : ρ 0 = 1 ⇔ 1 − α 0 − β 0 = 0 ⇔ α 0 + β 0 = 0 ⇔ α 0 = β 0 = 0 . P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 11 / 24

3.2 Convergence to the Brownian motion with drift, ρ 0 � = 1 Theorem 4 We assume that α 0 , β 0 > 0 (i.e. ρ 0 � = 0 ) and r ∆ t = ∆ 2 ( r > 0 ) . x Then the processes √ r η 0 t ξ ∆ = � Z ∆ √ ∆ t + t t 1 − ρ 0 converge in distribution to the process ( ξ 0 t , t ≥ 0 ) , as ∆ x → 0 , with : � � � � � η 2 r ( 1 + ρ 0 ) − c η 0 c ξ 0 0 1 − t = r + t + W t , 1 − ρ 0 ( 1 − ρ 0 ) 2 1 − ρ 0 ( 1 − ρ 0 ) 2 where (W t , t ≥ 0 ) stands for a standard Brownian motion and : η 0 = β 0 − α 0 , c = c 0 + c 1 , c = c 1 − c 0 . Remark η 0 = 0 corresponds to the Kac random walk. P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 12 / 24

Theorem 4 is a consequence of the central limit theorem : � Y 1 + · · · + Y n � √ (∆ t = 1 1 Z ∆ = n n , ∆ x = √ n ) 1 n �� � � � � = √ n Y 1 − E ( Y 1 ) + ··· + Y n − E ( Y n ) + R n , n � E ( Y 1 ) + · · · + E ( Y n ) � √ with R n := n . n β α Since ν := α + β δ 1 is the invariant probability measure α + β δ − 1 + associated with the Markov chain ( Y n ) we have � x ν 0 ( dx ) = α 0 − β 0 η 0 lim n →∞ E ( Y n ) = = . 1 − ρ 0 α 0 + β 0 � P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 13 / 24

3.3 Convergence when ρ 0 = 1 In this case, the transition matrix of ( Y t ) equals � 1 − c 0 ∆ x � c 0 ∆ x π ∆ = ( c 0 , c 1 > 0 ) . 1 − c 1 ∆ x c 1 ∆ x Consider a sequence ( e n , n ≥ 1 ) of independent r.r.v.’s such that : ( e 2 n , n ≥ 1 ) (resp. ( e 2 n − 1 ; n ≥ 1 ) ) are iid with common exponential distribution with parameter 1 1 c 1 (resp. c 0 ) i.e. E [ e 2 n ] = c 1 (resp. E [ e 2 n − 1 ] = c 0 ). Let � N c 0 , c 1 1 { e 1 + ... + e k ≤ t } , t ≥ 0 . = t k ≥ 1 be the counting process associated with ( e n ; n ≥ 1 ) . P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 14 / 24

Theorem 5 We suppose : α 0 = β 0 = 0 , Y 0 = − 1 , ∆ x = ∆ t . Then, the interpolated persistent random walk ( � Z ∆ s , s ≥ 0 ) converges � � − Z c 0 , c 1 in distribution, as ∆ x → 0 ,to the process where : s � s c 0 , c 1 Z c 0 , c 1 ( − 1 ) N s ≥ 0 . = du u s 0 � � N c 0 , c 1 In the case where c 0 = c 1 , then is the Poisson process with u parameter c 0 . P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 15 / 24

Remarks In the symmetric case, Theorem 5 is a stochastic version of analytical approaches developed by Kac (1974). See for instance the G. Weiss book (1994). � � Z c , c The process has been already introduced by D. Stroock s (1982). The convergence in terms of continuous processes allows to obtain for instance the convergence in distribution of max Z ∆ � s to 0 ≤ s ≤ 1 � � − Z c 0 , c 1 the r.v. max , as ∆ x → 0 . s 0 ≤ s ≤ 1 P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 16 / 24

Sketch of the proof of Theorem 5 We only consider Y 0 = − 1. Let : � � T 1 = inf n ≥ 1 ; Y n = 1 . � � Then T 1 ∼ G . c 0 ∆ x Consequently : � � inf s ; � s > � T ′ Z ∆ Z ∆ = 1 s − � � inf n ∆ t ; Y n = 1 = = T 1 ∆ t . it is easy to deduce the convergence in distribution of T ′ 1 to e 1 , as ∆ x = ∆ t → 0. Recall that e 1 is exponentially distributed with parameter 1 / c 0 . P . Vallois (IECN) persistent random walk Roscoff, March 22, 2010 17 / 24