Stability of hybrid Levy systems

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by  L. Gerencser

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