Stability of hybrid Lévy systems 4 La´szlo´ Gerencsér and Máté Mánfay 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 returnprocessisaninfinitesimalLévyprocess. Lévyprocessesarealso usedtomodelthetrafficinatelecommunicationnetwork. Inthispaperwefocus ] on a particular technical problem: stability of time-varying stochastic systems R drivenormodulatedbyaLévyprocesswithdiscretetimeinterventions,suchas P parameter or state resetting. Such systems will be called hybrid Lévy 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 malLévyprocess. Forlongtermmodellingamoresuitablemodelisastochastic system with poles close to 1 driven by a Lévy process, see [4]. Lévy processes are also used to model the traffic in a telecommunication network. Other areas where Lévy 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 Lévy process will be called a Lévy system. Description of real data in terms of Lévy 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 Lévy systems, along the lines of [3]. The problem is the stability analysis of time-varying stochastic systems drivenormodulatedbyaLévyprocesswithdiscretetimeinterventions,suchas parameter or state resetting. Such systems will be called hybrid Lévy 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 Lévy 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. AfewbasicnotionsrelatedtoLévyprocesseswillbegiveninthe 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 isLévyprocessthathasfinitevariationandhasnocontinuouspartand 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 Lévy 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 Lévy 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 Lévy 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 Lévyterms 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 ε]ν(x) νε(x)= , ν(dx) |x|>ε R and let Lε be the Lévy process with Lévy 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 Lévy 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

Stability of hybrid Levy systems

Similar Stability of hybrid Levy systems