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Stability of hybrid L´evy systems 4 La´szlo´ Gerencs´er and M´at´e M´anfay 1 0 2 Continuous-timestochasticsystemshaveattractedalotofattentionrecently, n a due to their wide-spread use in finance for modelling price-dynamics. More J recently models taking into accounts shocks have been developed by assuming 6 thatthe returnprocessisaninfinitesimalL´evyprocess. L´evyprocessesarealso usedtomodelthetrafficinatelecommunicationnetwork. Inthispaperwefocus ] on a particular technical problem: stability of time-varying stochastic systems R drivenormodulatedbyaL´evyprocesswithdiscretetimeinterventions,suchas P parameter or state resetting. Such systems will be called hybrid L´evy systems. . h They are hybrid in the sense that jumps both in the dynamics may occur. The t a peculiarity of our systems is that the jump-times are defined by a more or less m arbitrarypointprocess,butthere exists anasymmetry inthe systemdynamics. [ The novelty of our model relative to the theory of switching stochastic systems istwo-fold. First,weallowslowtimevariationoftheparameters,inastochastic 1 sense,withoutanystatisticalpattern,inthespiritoftheclassicalstabilityresult v 9 of Desoer, see [2]. Secondly, we allow certain jumps (resetting) in the system 6 parameters almost without any a priori condition. 0 1 . 1 Introduction 1 0 4 Continuous-time stochastic systems have attracted a lot of attention recently, 1 due to their wide-spread use in finance for modelling price-dynamics. A widely : v used model for continuous-time returns has been, since the works of L. Bache- i lier,Gaussianwhitenoisewithdrift. Morerecentlymodelstakingintoaccounts X shockshavebeendevelopedbyassumingthatthereturnprocessisaninfinitesi- r a malL´evyprocess. Forlongtermmodellingamoresuitablemodelisastochastic system with poles close to 1 driven by a L´evy process, see [4]. L´evy processes are also used to model the traffic in a telecommunication network. Other areas where L´evy processes are used in modeling: robotics, mechanicalsystems,biology[5],[11]. Afurtherpotentialapplicationismodelling the shocks received by the wheel of a car due to the irregularity of the road surface. Stochastic processes driven or modulated by a L´evy process will be called a L´evy system. Description of real data in terms of L´evy systems is far from being settled. In this paper we focus on a particular technical problem that provedtobefundamentalinthestatisticalanalysisofcontinuous-timestochastic systems driven by Gaussian white noise, see [1]. Ultimately it is hoped that 1 this technical result may contribute to the development of a continuous-time recursive maximum likelihood method for finite dimensional linear stochastic L´evy systems, along the lines of [3]. The problem is the stability analysis of time-varying stochastic systems drivenormodulatedbyaL´evyprocesswithdiscretetimeinterventions,suchas parameter or state resetting. Such systems will be called hybrid L´evy systems. They are hybrid in the sense that jumps both in the dynamics and the state may occur. The peculiarity of our systems is that the jump-times are defined by a more or less arbitrary point process, but there exists an asymmetry in the system dynamics, inasmuch jumps can occur only one-way, after a period of slow variation, namely, back to a fixed point. We note that the well-developed theory of switching stochastic systems, see [5], does not cover the problem that we consider. The novelty of our model relative to the theory of switching stochastic systems is two-fold. First, we allow slow time variation of the parameters, in a stochastic sense, without any statisticalpattern,inthe spiritofthe classicalstabilityresultofDesoer,see[2]. Secondly, we allow certain jumps (resetting) in the system parameters almost without any a priori condition. The structure of the paper is as follows: in Section II we develop the ba- sic technical tools, such as the geometric drift condition and the associated Lyapunov-function method, for the analysis of time-invariant L´evy systems, and provide estimates for higher order moments of the Lyapunov-function. In SectionIII we presentthe simplestversionofanextensionofDesoer’s theorem. In Sections IV we prove a stability result under parameter resetting. AfewbasicnotionsrelatedtoL´evyprocesseswillbegiveninthe Appendix. Throughout the paper we use the following notations: |·| stands for the Eu- clidean norm, k·k stands for the induced matrix norm. [X,Y] denotes the t quadratic variation of semi-martingales X and Y . t t 2 Preliminaries Consider the time-invariant linear stochastic system dX =AX dt+BdL +CdW (1) t t− t t whereL isL´evyprocessthathasfinitevariationandhasnocontinuouspartand t W is a standard Wiener process. Assume that A and B are time-independent t constant matrices. X ∈ Rn,L ∈ Rl,W ∈ RkA ∈ Rn×n,B ∈ Rn×l,C ∈ Rn×k. t t t We will denote the ith component of a vector V with V(i). We will use the fact thatifL isoffinitevariationthenitcanbewrittenasL =L +bt+ ∆L , t t 0 s≤t s and the quadratic covariance of two coordinates of such vector proPcesses is of the form ∆L(i)∆L(j). s s Xs≤t 2 Definition 2.1 We say that a vector L = (L(1),...,L(l)) with independent t t t components that are L´evy processes satisfies the moment condition of order Q if |x|qν(i)(dx) < ∞ holds for all 1 ≤ i ≤ l, and for 1 ≤ q ≤ Q, where the L´evy R Rmeasure of L(i) is denoted by ν(i)(x). t Thenextdefinitionismotivatedbythegeometricdriftconditionintroduced in[1].ToestimatethemomentsofX wewilluseaquadraticLyapunovfunction t V . t Definition 2.2 Let L(i),1 ≤ i ≤ l be independent L´evy processes with finite t variation. Let f be a polynomial with coefficients bounded uniformly in t and degf ≤Q. Given a process V satisfying with some ε>0 t dV =V u dt+dM + t t− t t (cid:16) (cid:17) (2) V1−εf(∆L(1),...,∆L(l)), t t t where ∆L(i) denotes the size of the jump of L(i) at t. We say that V satisfies t t the modified geometric drift condition of order Q if there exist α,γ > 0, such that u ≤ −α t d[M] t ≤ γ. (3) dt Without loss generality may always assume that decomposition of L con- t tains no drift term. Any possible drift term can be incorporatedinto u dt. The t following lemma will be used several times in the paper. Lemma 2.1 Assume Condition 1, let D ⊂ D compact and θ ∈ D . Then, 0 0 0 there exists a smooth function P(θ),θ ∈D and α>0 such that P(θ)≥P(θ )≥ 0 I for all θ ∈D and P(θ)A(θ)+AT(θ)P(θ)≤−αP(θ), for all θ ∈D . 0 0 For its proof see [1]. ThenexttwolemmasshowthatourLyapunovfunctionV anditsqth power t satisfy the modified geometric drift condition. Lemma 2.2 Let X be defined via (1), and P given by Lemma 2.1. Define t V = 1 +XTPX , then V satisfies the modified geometric drift condition of t t t t order two. Lemma 2.3 Let X be defined via (1). Define V = 1+XTPX , then Vq sat- t t t t t isfies the modified geometric drift condition of order 2q. The result of the next Lemma will be used in the proof of Theorem 2.1. Lemma 2.4 Let us suppose that V satisfies the modified geometric drift con- t dition of order Q, and suppose that L satisfies the moment condition of order t Q . Then E[V ]<∞ t holds. 3 Proof: Take a general process of the form (2). Then V satisfies t ∆V =V Z , (4) t t− t with Z =u dt+f(∆L(1),...,∆L(l)). t t t t Using the Doleans-Dade exponential formula, see [12], for processes with finite variation yields the solution for V : t Vt =eZt(c)−Z0(c) (1+∆Zs), sY≤t where.(c) denotesthe continuouspartofa process. This V is alsocalledas the t stochasticexponentialofU . Let c andM be uniformbounds for u andfor the t t coefficientsoff.Increasingbothu andthecoefficientsoff andtakingabsolute t value of the jumps we obtain a bound on the solution V : t l Vt ≤ect 1+M ∆Ls(i) ji≤ sY≤t 0≤j1+X...+jl≤QiY=1(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)  (5) l Q j ect 1+M ∆L(i)  s Yi=1sY≤t Xj=1(cid:12)(cid:12) (cid:12)(cid:12)  (cid:12) (cid:12)  Since L(i)-s are independent processes it is sufficient show that t Q j E 1+M ∆L(i) <∞. s sY≤t Xj=1(cid:12)(cid:12) (cid:12)(cid:12)   (cid:12) (cid:12)  Since L satisfies the moment condition of order Q t Q j E1+M ∆L(i) <∞. s Xj=1(cid:12)(cid:12) (cid:12)(cid:12)  (cid:12) (cid:12)  Hence applying Lemma 6.1, see Appendix, concludes the proof. The next theorem implies the stability of X defined in (1). t Theorem 2.1 Let us suppose that V satisfies the modified geometric drift con- t dition of order Q, and suppose that L satisfies the moment condition of order t Q . Then supE[V ]<∞ t t≥0 holds for 1≤q ≤Q. The proof will be given in the Appendix. Corollary: Since V = 1+ XTPX , satisfies the modified geometric drift t t t conditionso does Vq. Hence, by Theorem2.1sup E[Vq]<∞, which implies t t≥0 t that sup E[|X|q]<∞ holds under conditions seen in the theorem above. t≥0 t 4 3 A stochastic Desoer’s Theorem Consider a parametric family of linear stochastic state-space systems given by the state space equations: dX =A(θ )X dt+B(θ )dW +C(θ )dL (6) t t t t t t t Condition 1: A(θ) is stable for each θ ∈ D, where D ⊂ Rp is an open set, and A(θ),B(θ) and C(θ) are smooth in D. Definition 3.1 We say that θ is slowly varying in a stochastic sense if t dθ =β dt+σ dW +ρ dL , (7) t t t t t t with |β|2+||σ||2+||ρ||2 <δ, for some δ >0 and all t. t t t Theorem 3.1 Assume that L satisfies the moment condition of order Q, θ is t t slowly varying in the stochastic sense above, furthermore assume that θ ∈ Rp t is an adapted process taking its values in a compact set D ⊂ D, and that 0 Condition 1 holds. Then for a sufficiently small δ. supE[Vq]<∞, (8) t t≥0 for 1≤q ≤Q. Proof: The casewhenno L´evyterms arepresentinthe dynamics ofx and t θ hasbeensettledinTheorem1of[1].WemaythereforeassumethatB(θ)=0 t and σ =0. t For a given θ, let P(θ) ∈ C2 be a symmetric, positive definite matrix that solves P(θ)A(θ)+A(θ)TP(θ)≤−αP(θ), (9) with some α > 0, and P(θ) ≥ I. Let P = P(θ ), and consider V = (1 + t t t XTP X )q/2. ItisenoughtoprovethatV satisfiesthemodifiedgeometricdrift t t t t condition. By Lemma 2.4 we only need to check that 1+XTP X satisfies the t t t modifiedgeometricdriftcondition. Wecanwritethedynamicsof1+XTP X = t t t 1+Tr(P Z ), with Z =X XT as t t t t t dTr(P Z )=Tr(P dZ )+Tr(dP Z )+ t t t t t t dP d[X(i),X(j)] (10) i,j t Xi,j The first term can be handled using Lemma 2.2. The dynamics of P is given t by dP =u dt+Σ dL , (11) t t t t with ||u||2+||Σ||2 <cδ, with some c. Thus, the second and the third term give t t drifttermsthatdonotspoilthemodifiedgeometricdriftcondition. Thetypical 5 formofthecontributionofthesecondtermuptoaboundedconstantmultiplier is X(i)X(j)dL(k), (12) and that of the third term is dL(k)d[L(i),L(j)] . (13) t t Hence,1+XTP X indeedsatisfiesthemodifiedgeometricdriftcondition. Thus, t t t applying Theorem 2.1 concludes the proof. This result implies the stability of the parameter varying system defined by (6). Corollary: Under conditions seen in the previous theorem sup E[|X|q]< t≥0 t ∞ holds. 4 Jumps in the dynamics of the parameter We now assume that the slowly parameter varying process θ resets at random t times defined by a point process with counting process N . t dθ =β dt+σ dW +dL +(θ −θ )dN , (14) t t t t t 0 t− t where |β |2+||σ ||2 <δ. t t Theorem 4.1 AssumethatCondition1holds, andthatL satisfiesthemoment t condition of order Q, and let X be defined via (6). Then t supE[|X|q]<∞ t t≥0 holds for 1≤q ≤Q. Proof: We may assume that there is no diffusion part in the dynamics of X and θ . Let P(θ) be defined by Lemma 2.1 so that it attains its minimum t t on D in θ . Define V =(1+XTP(θ )X )q/2. Let ξ be the size of the jump at 0 t t t t t t induced by the jump of θ, ie. ξ =(1+XTP(θ )X )q/2−(1+XTP(θ )X )q/2, (15) t t 0 t t t t using this notation the dynamics of V can be written as t dV =V U +ξ dN , (16) t t− t t t with l (i) ji U =u dt+ c ∆L . t t j1,...,jl t 0≤j1+X...+jl≤Q iY=1(cid:16) (cid:17) By the minimality of P(θ ), the jump term in (16) causes a non-positive jump 0 in V . Let ψ be the stochastic exponential of U , then t t t t V =ψ V + ψ ψ−1ξ dN ≤ψ V . (17) t t 0 Z t s s s t 0 0 Since E[ψ V ]<∞ is implied by Theorem 2.1, we conclude the proof. t 0 6 5 Discussion: State resetting for jump processes Consider the hybrid linear system with jumps dX =AX +BdW +CdL +(X −X )dN , (18) t t t t 0 t− t where W is a Wiener process, and N is a counting process. t t Conjecture 5.1 Suppose that L satisfies the moment condition of order Q, t then for X defined by (18) t supE[|X|q]<∞, t t≥0 holds for 1≤q ≤Q. Our future work will focus on proving this conjecture. Although the main ideas of the proof are established some technical issues are still to be proven. 6 Appendix L´evy processes with finite variation are formally obtained via t Z = xN(ds,dx), (19) t Z0 ZR1 whereN(dt,dx)isatime-homogeneous,space-timePoissonpointprocess,count- ing the number of jumps of size x at time t. L´evy processes are characterized by their L´evy measures that can be defined via the intensity of N(dt,dx): E[N(dt,dx)]=dt·ν(dx), where ν(dx) is the L´evy-measure. The quadratic variation of semimartingales X and Y is defined by the following process: d[X,Y] =d(XY) −X dY −Y dX . t t t− t t− t If X =Y we get the quadratic variation of X. Proof of Lemma 2.2: Proof: This lemma is an extension of Lemma 8 in [1], where no L´evy processes are present in defining the dynamics of V . Thus, for the sake of t simplicity, we may omit the martingale M from (2). t Write V =1+XTPX =1+Tr(PZ ), where Z =X XT. The dynamic of t t t t t t t Z can be written as t dZ =X dXT +dX XT +Bd[L,L] BT, (20) t t− t t t− t whered[L,L] isanl×l matrixwithentriesrepresentingquadraticcovariances, t that is d[L,L](i,j) =d[L(i),L(j)]. Equation (20) reads as t X XT ATdt+dLTBT +(AX dt+BdL )XT t− t− t t− t t (21) (cid:0) (cid:1) +Bd[L,L] BT t 7 Thus the dynamics of V =1+Tr(PZ ) can be written as t t PX XT AT +PAX XT dt+ t− t− t− t− (22) +PX dL B(cid:0)T +PBdL XT +PBd[L,L](cid:1)BT. t− t t t− t So the dt terms in the dynamics of V =1+Tr(PZ ) we can write t t Tr XT ATPX +XT PAX t− t− t− t− (23) ≤(cid:0) −αXT PX =−αV +α(cid:1), t− t− t− for the terms having dL t Tr P(X dL BT +BdL XT ) = t− t t t− Tr(cid:0)(P +PT)BdL XT = (cid:1) (24) t t− 2X(cid:0)T (P +PT)BdL =(cid:1)ψTdL , t− t t t with |ψ |2 = 4XT (P +PT)BBT(P +PT)X ≤ 4KV , with some fixed K. t t− t− t− Finally for the term with d[L,L] t l Tr PBd[L,L] BT = c d[L(i),L(j)] = t i,j t (cid:0) (cid:1) iX,j=1 (25) l c ∆L(i)∆L(j), i,j t t iX,j=1 with some c ,1 ≤ i,j ≤ l constants. It follows that the dynamic of V can be i,j t written as dV =V u dt+ t t− t l ψ(i) l c (26) V1/2 t ∆L(i)+ i,j ∆L(i)∆L(j), t− Xi=1 Vt1−/2 t iX,j=1Vt1−/2 t t   with uniformly bounded u , ψt(i) , ci,j for any 1≤i,j ≤l. t Vt1−/2 Vt1−/2 Proof of Lemma 2.3: Proof: The dynamics of Vq can be written as t dVq =qVq−1dV +Vq−Vq = t t− t,(c) t t− qVq−1u dt+(V +∆V )q−Vq = t− t t− t t− (27) q q qVq u /V dt+ (∆V )kVq−k, t− t t−1 (cid:18)k(cid:19) t t− Xk=1 with u /V <α. Using that t t−1 ∆V =V1−εf(∆L(1),...,∆L(l)), t t− t t 8 we obtain that a typical jump term in (27) reads as up to constant multiplier Vq−kεf(∆L(1),...,∆L(l))k (28) t− t t ThisimpliesthatVq satisfiesthemodifiedgeometricdriftconditionoforder2q. t The next two technical Lemmas will be used in the proof of Theorem 2.1. Lemma 6.1 Let L be a L´evy process with L´evy measure ν. Suppose that a t function f satisfies f(x)ν(dx)<∞, ZR then E (1+f(∆Ls))=etRRf(x)ν(dx) sY≤t   holds for any t. Proof: First suppose that L is a compound Poisson process with rate λ, t then the expected value of ψ = f(∆L ) t s sY≤t can be estimated by conditioning on the number of jumps of L . Let N ,J t t t denote the number of jumps of L on [0,t], and the set of time indices when L t jumps on [0,t]. Define Dn ={(t ,...,t ):0≤t ≤t, for all 1≤i≤n}. t 1 n i ∞ E[Ψ ]= E[Ψ |N =n]P(N =n)= t t t t nX=0 E[Ψ |N =n,J ={t ,...,t }] t t t 1 n ZDn t P(N =N)dt ...dt = t 1 n ∞ (λt)n (m+1)ne−λt =eλmt, n! nX=0 where m = E[f(∆L )|L jumps at t], and P is the joint probability density of t the jump times. For the general case define the truncated L´evy measure 1[|x|>ε]ν(x) νε(x)= , ν(dx) |x|>ε R and let Lε be the L´evy process with L´evy measure νε. Then L is the weak t t limit of Lε as ε tends to zero. t mε = f(x)νε(dx) (29) f Z mε =E[f(∆Lε)|Lε jumps at t] (30) t λε = ν(dx) (31) Z |x|>ε 9 writing (6) for Lε yields E (1+f(∆Lε))=eλε(mε−1)t s sY≤t   Notethatλεmε = f(x)ν(dx), itfollowsthateλεmεt hasfinitelimitasε→ k |x|>ε 0+ provided Rf(xR)ν(dx) <∞ which is the case. Hence, E[ψt]=etRRf(x)ν(dx) follows. R Lemma 6.2 Let the one dimensional process L with L´evy measure ν satisfy t the moment condition of order Q. Let f be a polynomial with degf ≤ Q, and f(0)=0. Then t 1−e−αt E e−α(t−s)f(∆L ) = f(x)ν(dx) s (cid:20)Z0 (cid:21) α ZR Proof: FirstconsiderthecasewhenL is acompoundPoissonprocesswith t intensity λ. Let N ,J denote the number of jumps of L on [0,t], and the set t t t of time indices when L jumps on [0,t]. Define Dn = {(t ,...,t ) : 0 ≤ t ≤ t 1 n i t, for all 1≤i≤n}. t E e−α(t−s)f(∆L ) = s (cid:20)Z (cid:21) 0 ∞ t E e−α(t−s)f(∆L )|N =n P(N =n). (32) s t t (cid:20)Z (cid:21) nX=0 0 Calculating one term in the sum above t E e−α(t−s)f(∆L )|N =n,J ={t ,...,t } s t t 1 n Z (cid:20)Z (cid:21) Dn 0 t dt ...dt 1 n P(N =n) = t tn n e−α(t−ti)E[f(∆L )|N =n,J ={t ,...,t }] s t t 1 n Z DtnXi=1 e−λt(λt)ndt1...dtn = (33) n! tn t (λt)n n e−α(t−tn)E[f(∆L )|t ∈J ]e−λt s 1 t ZDn−1Z0 n! t dt dt ...dt n 1 n−1 = t tn−1 (λt)n 1−e−αt ne−λt E[f(∆L )|t ∈J ] . s 1 t n! αt 10

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