Asymptotic Analysis and Extinction In

7/30/2019 Asymptotic Analysis and Extinction In

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The Annals of Applied Probability2001, Vol. 11, No. 4, 12631291

ASYMPTOTIC ANALYSIS AND EXTINCTION INA STOCHASTIC LOTKAVOLTERRA MODEL

By F. C. Klebaner1 and R. Liptser1

University of Melbourne and Tel Aviv University

A stochastic LotkaVolterra model is formulated by using the semi-

martingale approach. The large deviation principle is established, and is

used to obtain a bound for the asymptotics of the time to extinction of prey

population. The bound is given in terms of past-dependent ODEs closely

related to the dynamics of the deterministic LotkaVolterra model.

1. Introduction. Main result.

1.1. Deterministic LotkaVolterra system. The LotkaVolterra system of

ordinary differential equations [Lotka (1925) and Volterra (1926)],

xt = xt xtytyt = xtyt yt

(1.1)

with positive x0 y0 and positive parameters describes a behavior ofa predatorprey system in terms of the prey and predator intensities xt andyt. Here, is the rate of increase of prey in the absence of predator, is a rateof decrease of predator in the absence of prey while the rate of decrease in

prey is proportional to the number of predators yt, and similarly the rate ofincrease in predator is proportional to the number of prey xt [see, e.g., May(1976)]. The system (1.1) is one of the simplest nonlinear systems.

Since the population numbers are discrete, a description of the predator

prey model in terms of continuous intensities xt yt is based implicitly ona natural assumption that the numbers of both populations are large, andthe intensities are obtained by a normalization of population numbers by a

large parameter K. Thus (1.1) is an approximation, an asymptotic descriptionof the interaction between the predator and prey. Although this model may

capture some essential elements in that interaction, it is not suitable to answer

questions of extinction of populations, as the extinction never occurs in the

deterministic model; see Figure 1 for the pair xt yt in the phase plane.We introduce here a probabilistic model which has as its limit the determin-

istic LotkaVolterra model, evolves in continuous time according to the same

local interactions and allows evaluating asymptotically the time for extinction

of prey species.

There is a vast amount of literature on the LotkaVolterra model, and a his-

tory of research on stochastic perturbations of this system both exact, approx-

Received June 2000; revised January 2001.1Supported in part by Australian Research Council.

AMS 2000 subject classifications. Primary 60I27, 60F10.

Key words and phrases. Predatorprey models, large deviations, semimartingales, extinction

1263


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1264 F. C. KLEBANER AND R. LIPTSER

Fig. 1. rx y when = 5, = 1, = 5, = 1.

imate and numerical; see, for example, Goel, Maitra and Montroll (1971),

Turelli (1977), Kesten and Ogura (1981), Hitchcock (1986), Watson (1987),

Roozen (1989) and references therein. We approach the problem of extinc-

tion via the theory of large deviations, thus obtaining new theoretical results,

which previously were studied numerically.

The system (1.1) possesses the first integral which is a closed orbit in the

first quadrant of phase plane x y. It is given by

rx y = cx logx + y logy + r0(1.2)where r0 is an arbitrary constant. It depends only on the initial points x0 y0(see Figure 1).

1.2. Stochastic LotkaVolterra system. In this paper, we introduce and

analyze a probabilistic model of preypredator population related to the clas-

sical LotkaVolterra equations.

Let Xt, and Yt be numbers of prey and predators at time t. We start withsimple balance equations for preypredator populations

Xt = X0 + t t Yt = Y0 + t t

(1.3)

where:t is a number of prey born up to time t.t is a number of prey killed up to time t.t is a number of predators born up to time t.t is a number of predators died up to time t.


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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1265

We assume that t t

t

t , are double Poisson processes with the following

random rates: Xt,

KXtYt,

KXtYt, Yt, respectively and disjoint jumps [the

latter assumption reflects the fact that in a short time interval t t + tonly one prey might be born and only one might be killed, only one predator

might be born and only one might die, with the above-mentioned intensities;

moreover all these events are disjoint in time].

The existence of such a model is not obvious; therefore in Section 2 we give

its detailed probabilistic derivation.

Assume X0 = Kx0 and Y0 = Kx0 for some fixed positive x0 y0 and alarge integer parameter K. Introduce the normalized by K prey and predatorpopulations

xKt =XtK

and yKt =YtK

In terms ofxKt and yKt the introduced intensities for double Poisson processes

can be written as

KxKt x

Kt y

Kt x

Kt y

Kt y

Kt

We justify the choice of the probabilistic model given in (1.3) by Theorem 2

which states that the solution of the LotkaVolterra equations is the limit (in

probability) for xKt yKt ,xKt y

Kt

xt yt K Such an approximation is known as the fluid approximation. Results on the

fluid approximation for Markov discontinuous processes can be found in Kurtz

(1981) and are adapted to the case considered here, despite that in our case

the two intensities do not satisfy the linear growth condition in xKt yKt .

1.3. Fomulation of the problem. Here, we are interested in evaluation of

the prey extinction, namely, the asymptotics in K of

P

TKext T inftT

yKt > 0

= 0

where TKext = inft > 0 xKt = 0 is the prey extinction time. Unfortunately,the fluid approximation does not provide much information on the extinction

time TKext. Since for x0 > 0, y0 > 0 the fluid limit xt yt remains positivefor any t > 0 (see Figure l), we have

limK

P

TKext T inftT

yKt > 0

= 0(1.4)

A historical comment onT

K

extof evaluation is due. There is a large amount of

literature on the subject of extinction mostly using some simplification of the

original problem, such as linearization, and numerical studies. For a somewhat

different model with a state-dependent noise, Hitchcock (1986) showed that

ultimate extinction is certain, and derived exact probabilities for the predators

to become eventually extinct when the prey birth rate is zero. A power series


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approximation for the extinction probabilities as well as the number of steps

to extinction in special cases were given. Numerical studies of probabilities of

extinction were also done in Smith and Mead (1979, 1980), and Watson (1987)where a rough approximation (based on the normal approximation) was also

given.

Due to (1.4), the rate of convergence in (1.4) is of interest, and is the subject

of this paper. Neither fluid nor even diffusion approximations for the stretched

differences

K

xKt xt

K

yKt yt

are effective for such analysis.

Freidlin (1998), Freidlin and Weber (1998) carried out an effective asymp-

totic analysis for randomly perturbed oscillators and other Hamiltonian

systems. Their approach is based on the approximation of the first integral

process [in our case rxKt yKt ] by a scalar diffusion. However, in our case thisapproach does not seem to be of use, since at the time of extinction the first

integral process rxKt yKt explodes; see (1.2).A large deviation (LD) type evaluation yields results in our case. The ran-

dom process xKt yKt is a vector semimartingale, so that it appears that onecan have the large deviation principle (LDP) by using a general results from

Pukhaskii (1999). However, the method from Pukhaskii does not serve the

model studied here, since the intensities of two double Poisson processes are

of quadratic form in xKt yKt . For the same reason we could not find adequate

methods for proving the (LDP) in the literature, for example, Wentzell (1989),

Dupuis and Ellis (1997), Freidlin and Wentzel (1984), Dembo and Zeitouni

(1993).

It is well known that the main helpful tool in verification of the LDP with

unbounded intensities, satisfying a linear growth condition, is the exponen-

tial negligibility for sets suptT xKt C, suptT yKt C, for large K, Cand every T > 0. In our case two from four intensities do not satisfy the lin-ear growth condition. Nevertheless, specifics of the model and the fact that

xKt yKt are nonnegative processes with bounded jumps allow establishing the

above-mentioned exponential negligibility (Lemma 2) and deriving the LDP

similarly to Liptser and Pukhalskii (1992).

Although in principle the LDP allows finding the logarithmic rate with

norming 1K

for

P

TKext T inftT

yKt > 0

a realistic procedure for determining the required rate depends heavily on the

structure of the LDP rate function. In our case we deal with purely discontin-

uous process and the rate function is extremely inconvenient for this purpose;

therefore, we restrict ourselves to finding only a lower bound for the required

rate.


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Fig. 2. Trajectory that attains the lower bound. Parameters are the same as for Figure 1.

1.4. Main result. To formulate the main result, we need to introduce a

system of past-dependent differential equations parameterized by q > 0,

qt = qt

qt

qe t0 qt dsqt qt

qt = qt

qt

(1.5)

subject to the initial condition q0 = y0, q0 = x0, and denote

Tq = inft > 0 qt = 0(1.6)

Theorem 1. For every T > 0,

liminfK

1

Klog P

TKext T inf

tTyKt > 0

x

20

2Tq

0 e2t0 qt dsqt qt dt

where Tq is associated with the smallest q = q for which Tq T.1.5. Example. We give here an example with = 5, = 1, = 5, = 1 and

T = 15. The trajectories for qt qt , on which the lower bound is attainable, aregiven in the phase plane (see Figure 2). Parameter q 00023 and Tq 15,while the value of the rate function defined later in (4.2) is Jq q 00018.

1.6. Remark. We can show for a stochastic LotkaVolterra model with a

different noise structure (much simpler and somewhat artificial) the exact

asymptotic relation in Theorem 1 (rather than a lower bound), as well as that

the minimum of the rate function in the LDP occurs on the solution similar


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to (1.5). Therefore, loosely speaking, (1.5) (see Figure 2) gives a likely path to

extinction in the stochastic LotkaVolterra model.

The article is organized as follows. In Section 2 we give description of thestochastic model, in Section 3 we show the fluid approximation, in Section 4

we formulate the LDP and prove the main result. The verification of the LDP,

which is quite technical, is done in the Appendix.

2. The model: description of stochastic dynamics.

2.1. Existence. In this section, we show that the random process Xt Ytis well defined by (1.3). To this end, let us introduce four independent

sequences of processes,

t =

t 1 t 2

/K

t = /Kt 1 /Kt 2 /Kt =

/Kt 1 /Kt 2

t =

t 1 t 2

Each of them is a sequence of i.i.d. Poisson processes characterized by rates

, K

, K

, , respectively. Define the processes Xt Yt by the system of Itoequations

Xt = X0 +t

0

n1

IXs n dsn t

0

n1

IXsYs n d/Ks n

Yt = Y0 + t

0 n1IXsYs n d/Ks n

t

0 n1IYs n dsn

(2.1)

governed by these Poisson processes, which obviously has a unique solution

on the time interval 0 T, whereT = inft > 0 Xt Yt =

The double Poisson processes involved in (1.3) are obtained then in the follow-

ing way:

t =t

0

n1

IXs n dsn

t =

t

0 n1IXsYs n d/Ks n

t =t

0

n1

IXsYs n d/Ks n

t =t

0

n1

IYs n dsn

(2.2)


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Let us show that t, t ,

t,

t defined in 22 satisfy the required prop-

erties. Since all Poisson processes are independent, their jumps are disjoint,

so that the jumps of t, t , t, t are disjoint as well. To describe structureof intensities for t,

t ,

t,

t , let us introduce a stochastic basis F =

tt0 P supplied by the filtration F generated by all Poisson processes andsatisfying the general conditions. Then, obviously, the random process

At =t

0

n1

IXs nds

is the compensator of t. On the other hand, since Xs is an integer-valuedrandom variable, we have

n1 IXs n = Xs; that is, At =

t0 Xs ds and

the intensity of t is Xt. Analogously, others compensators are seen to be

At =t

0

KXsYs ds

At =

t0

KXsYs ds

At =

t0Ys ds

and thus all other intensities have the required form.

We now show that the process Xt Ytt0 does not explode.Lemma 1.

PT = = 1Proof. Set TXn = inft > 0 Xt n, n 1 and denote by TX =

limn TXn . Due to (2.1) it holds that

EXtTXn X0 +t

0EXsTXn ds

and so, by the GronwallBellman inequality EXTXn T X0eT for every T > 0.Hence by the Fatou lemma EXTXT X0eT. Consequently,

P

TX

T =

0

T > 0

Set TY = inft Yt , 1 and denote by TY = lim TY . Due to (2.1)it holds that

EYtTXn TYt Y0 +t

0

KEXsTXn TY YsTXn TY ds

Y0 +t

0

KnEYsTXn TY ds

Hence, by the GronwallBellman inequality for every T > 0 we haveEYTTxnTy Y0enT and by the Fatou lemma,

EYTTXn TY Y0e/KnT n 1Consequently, P

TY

TXn

T

=0

T > 0, n

1 and, since TXn

,

n , we obtainPTY T = 0 T > 0

Since T = TX TY, PT T = PTX T + PTY T, and wehave PT T = 0 for any T > 0.


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Corollary 1. For Tn = inft Xt Yt n,

limn PTn T = 0 T > 0

The above description of the model allows claiming that Xt Yt is acontinuous-time pure jump Markov process with jumps of two possible sizes

in both coordinates: 1 and 1 and infinitesimal transition probabilities (ast 0,

PXt+t = Xt + 1Xt Yt = Xtt + ot

PXt+t = Xt 1Xt Yt =

KXtYtt + ot

PYt+t = Yt + 1Xt Yt =

KXtYtt + ot

PYt+t = Yt 1Xt Yt = Ytt + ot

2.2. Semimartingale description for xKt yKt . Let At, At , At, At be thecompensators of t,

t ,

t,

t defined above. Introduce martingales

Mt = t At Mt = t At Mt = t At Mt = t At and also normalized martingales

mKt =Mt Mt

Kand mKt =

Mt MtK

(2.3)

Then, from (1.3) it follows that the process xKt yKt admits the semimartin-gale decomposition

xKt = x0 +t

0

xKs Ks xKs yKs

ds + mKt (2.4)

yKt = y0 +t

0

xKs y

Ks yKs

ds + mKt (2.5)

which is a stochastic analogue (in integral form) of (1.1)

In the sequel we need quadratic variations of the martingales in (2.4) and

(2.5). It is well known [see, e.g., Liptser and Shiryaev (1998), Chapter 18 or

Klebaner (1998), Theorem 9.3] that all martingales are locally square inte-

grable and possess the predictable quadratic variations

Mt = At Mt = At and Mt = At Mt = At (2.6)and zero mutual predictable quadratic variations M Mt 0 M Mt 0, implied by the disjointness of jumps for t, t , t, t .


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Hence

mK

m

K

t 0mKt =

1

K

t0

xKs + xKs yKs

ds

mKt =1

K

t0

xKs y

Ks + yKs

ds

(2.7)

2.3. Stochastic exponential and cumulant function. In the large deviation

theory we use stochastic exponential, and to this end the following represen-

tation is more convenient:

xKt = x0 =tK

t

K

yKt = y0 = tK tK (2.8)

tK

,tK

,tK

,tK

are counting process with jumps of the unit size K1 and com-

pensatorsAtK

,AtK

,AtK

,AtK

, respectively. With every pair tK

,AtK

, , tK

,AtK

and a predictable process t such that for any T > 0 and K large enough,T

0et/K

xKt + yKt + xKt yKt

dt < P-a.s.

we associate nonnegative processes

zt

K = expt

0

sK

ds

es/K

1

dAs

(2.9)

zt

K

= exp

t0

sK

ds es/K 1 d AsApplying the Ito formula to zt K, we find

dzt

K

= zt

K

et/K 1 dt Atthat is, zt K is a local martingale [analogously zt K, zt K, zt K are localmartingales as well]. All these martingales are nonnegatives, so that they

are supermartingales [see Problem 1.4.4 in Liptser and Shiryayev (1989) or

Theorem 7.20 in Klebaner (1998)]. Hence for any Markov time we haveEz K 1 Ezt K 1.

Introduce also (for predictable processes and )

zKt = zt

K

zt

K

zt

K

zt

K

(2.10)


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Applying the Ito formula to zKt and taking into account that jumps oft,

t ,

t,

t are disjoint we find

dZKt = zKt et/K 1 dt At + et/K 1 dt At + et/K 1 d t At + et/K 1 d t At

so that ZKt is a positive local martingale as well as a supermartingalewith

EZK 1for any Markov time .

Set

G u v= vu + e 1 u + e 1 + uv

G u v= u v + e

1 uv + e

1 + v(2.11)

and define the so-called cumulant function,

G u v = G u v + G u v(2.12)Using the cumulant function, (2.10) can be written as

ZKt = expt

0

s dxKs +

t0

s dyKs

expt

0KG

s

K

sK

xKs yKs

ds

(2.13)

Hence an equivalent multiplicative decomposition holds for the exponential

semimartingale

expt

0

s dxKs + s dyKs = ZKt VKt

with the positive local martingale ZKt and a positive predictable process

VKt = expt

0KG

s

K

sK

xKs yKs

ds

3. Fluid approximation. In this section we justify the choice of thestochastic dynamics by showing that the LotkaVolterra equations describe

a limit (fluid approximation) for the family

xKt y

Kt

K

In what follows xt yt is a solution of the LotkaVolterra equation (1.1).Theorem 2. For any T > 0 and > 0,

limK

P

suptT

xKt xt + yKt yt

>

= 0


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Proof. Set

TKn =

inft xKt yKt n(3.1)Since TKn = TnK (see Corollary 1 to Lemma 1),

limn

limsupK

P

TKn T = 0(3.2)

Hence, it suffices to show that for every n 1,

limK

P

sup

tTKn T

xKt xt + yKt yt

>

= 0(3.3)

Since suptTKn TxKt yKt n+ 1, there is a constant Ln, depending on n andT, such that for t TKn T,

xKt xKt yKt xt xtyt LnxKt xt+ yKt ytxKt yKt yKt xtyt yt LnxKt xt + yKt ytThese inequalities and (1.1), (2.4) implyxKTKn T xTKn T+ yKTKn T yTKn T

2Lnt

0

xKsTKn xsTKn + yKsTKn ysTKn ds+ sup

tTKn T

mKt + suptTKn T

mKt Now, by the GronwallBellman inequality we find

suptTKn T

xKt xt + yKt yt e2LnT suptTKn T

mKt + suptTKn T

mKt

Therefore (3.3) holds, if both suptTKn T mKt and suptTKn T mKt convergein probability to zero as K . By the Doob inequality for martingales (see,e.g., Theorem 1.9.1(3) and Problem 1.9.2 in Liptser and Shiryayev (1989) the

required convergence takes place provided that both mKTKn T and mKTKn Tconverge to zero in probability as K . But by (2.7) mKTKn T constK , mKTKn T constK , and the proof is complete.

4. Formulation of the LDP and the proof of Theorem 1.

4.1. LDP. The random process xKt yKt t0 has nonnegative paths fromthe Skorokhod space 2 = 20 . Since we are going to apply the LDPfor asymptotic analysis of the extinction time on a finite interval 0 T, weexamine the LDP in 20 T. Because of the fluid approximation, the limit forxKt yKt tT is xt yttT with continuous differentiable functions xt and yt.


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This fact allows examining the LDP in the metric space 20 T suppliedby the uniform metric

x x

2

0 T

and

x x =2

j=1suptT

xtj xt j(4.1)

More exactly, since xKt 0, yKt 0 andKx = inft > 0 xKt = 0 Ky = inft > 0 yKt = 0

are also absorption points for processes xKt and yKt , respectively, let us intro-

duce a subspace 2 +0 T of nonnegative function from

20 T. It is obvious that

2 +0 T is a closed subset of

20 T in the metric .

Therefore we formulate the LDP in metric space 2 +0 T .

Theorem 3. For every T > 0, the family xK

t yK

t tT, K obeys theLDP in the metric space 2 +0 T with the rate of speed 1K and the (good)rate function

JT=

T

0sup

t +t Gtt

dt dtdtdtdt

0=x00=y0 ,

otherwise.(4.2)

The proof of the LDP is given in the Appendix.

4.2. Proof of Theorem 1. Denote by o the interior of the set

2 +0 T inf

t

T

t = 0 inft

T

t > 0Due to the LDP,

lim infK

1

KlogP

inftT

xKt = 0 inftT

yKt > 0

inf

JT (4.3)

It is clear that

inf o

JT inf o JT

where is a subset of o of absolutely continuous functions t, t with0 = x0, 0 = y0 and for any = ttT,

t = tt (4.4)To emphasize the fact that t is a solution of (4.4) we use the notation

t .

The function t remains positive for any finite time interval and moreover,

sup

t + t G t t

= sup

t G t t

= sup

t t t e 1 t e 1 + tt


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Taking into account that functions ts from have to be absorbed on 0 T,we choose a subset


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that is,

t

x0 pt. Now, for any T > 0, the choice p =

x0T

guarantees T T.If some ut is chosen and associated with T < T, it is obvious that ut 0

for t T T. Then, particularly, in (4.6) the integral T0

u2t dt is replacedby

T0

u2t dt. On the other hand, due to T = 0 we have

0 = eT

0 s ds

x0

T0

et

0 s dsut

t

t dt

that is, x0 =T

0e

t0

s dsut

t

t dt. Further, the CauchySchwarz

inequality implies

T

0 u

2

t dt x20T

0e2 t0 s dstt dt (4.7)

Take any positive q and ut = qet

0 s ds

t

t then the inequality given

in (4.7) becomes an equality, and only for

q = x0T0

e2t

0 s dst

t dt

(4.8)

we have T = 0.Let q Tq be a pair such that (1.6) holds with q and T replaced by q and

Tq respectively and Tq

T. Denote by

qt and

qt the solution of (1.5) on

0 Tq corresponding to the pair q Tq. Then we get the lower bound

lim infK

1

Klog P

TKext T inf

tTyKt > 0

inf x

20

2Tq

0 e2 t0 s dsutqt qt dt

where inf is taken over all Tq T .The final step of the proof uses the property

q < q x2

0Tq0 e

2 t0 qs dsqt q dt C

= (A.1)

lim0

lim supK

1

Klog sup P

supt

xKt+ xK + supt

yKt+ yK > =

(A.2)

for any > 0, where sup is taken over all stopping times T. C local LDPis valid, if for any 20 T

lim sup0 limsupK

1

K logPxK yK JT (A.3)

lim inf0

lim infK

1

KlogPxK yK JT (A.4)

Since the linear growth condition for intensities of counting processes is

lost, a verification of (A.1) requires a different technique from the one used in

Liptser and Pukhalskii (1992). We verify (A.1) in the next subsection.

A.2. Exponential negligibility of suptT xKt , suptT y

Kt .

Lemma 2. For every T > 0,

limL

lim supK

1

KlogP

suptT

xKt L =

limC

limsupK

1

KlogP

suptT

yKt C

=


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Proof. Recall that xKt = x0 + ttK

and Mt = t At; that is, dxKt dtK

=xKt dt

+dMt

Kand so xKt

et

x0

+1

K t

0es dMs

. Consequently, the

first statement of the lemma is valid provided that

limL

lim supK

1

KlogP

suptT

t0

es dMs KL

=

It is clear that with KL = inftt

0es dMs KL it suffices to establish

limL

lim supK

1

KlogP

KL T

= (A.5)

To this end, with r > 0 and s = res we define zt by the first formula in(2.9) with

sK

replaced by s and use an obvious modification of this formula,

zt = expt0 s dMs t0 es 1 s dAs(A.6)Recall that zt is a supermartingale with EzKL T 1. We use thisinequality for the next one,

1 EzKL T

IKL T(A.7)We sharpen that inequality, by evaluating below z

KL T on the set KL T,

logzKL T

= rKL T

0es dMs

KL T0

ere

s 1 resKxKs ds

rKL

KL T

0 er

1

r

KxKs ds

rKL T

0er 1 rKxK

sKLds

To continue this evaluation we find an upper bound for xKsKL

. Since

KL T0

esdMs KL + 1

(recall that jumps oft

0esdMs is bounded by 1), we claim

xKKL t

etx0 + 1 + L

Hence, with p =eT

x0

+1

+L

L , we arrive at the lower boundlogz

KL T KLr er 1 rTp

It is clear that r > 0 can be chosen so that r er 1 rTp = > 0 andtherefore logz

KL T KL > 0.


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Thus, (A.7) with s = res replaced by sres, implies1

K logPKL T L L

that is, (A.5) holds.

The proof of the second statement of the lemma uses (A.5) heavily. SincesuptT

yKt C

suptKL T

yKt C

KL Tand thereby

P

suptT

yKt C

2

P

sup

tKL TyKt C

P

KL T

by virtue of (A.5) it suffices to check that for every fixed L,

limC

lim supK

1K

logP suptKL T

yKt + C = (A.8)The verification of (A.8) is similar to the proof of the first statement of the

lemma. It is clear that there is a positive constant RL so that IKL txKt RL, t 0. Further, due to dyKt xKt yKt dt + 1K dMt we have

dyKtKL

IKL txKt yKt dt +1

KdM

tKL

RLyKt dt +1

KdM

tKL

and hence yKt

KL

eRLt

y0

+ tKL

0 eRLd

Ms

Now, introduce

KL C = inf

ttKL

0eRL dMs C

and note that (A.8) holds provided that

limC

lim supK

1

KlogP

KL C T

= L > 0(A.9)The proof of (A.9) is similar to the proof of (A.5), so here we give only a sketch

of it. Set = RL and for positive r take s = res. Introduce a super-martingale

zt = expt

0s d

Ms

t

0 es 1 s

d

As

Due to EzKL TKL CT

1 write 1 EzKL TKL CT

IKL C T and evaluatefrom below z

KL TKL CT on the set KL C T as

log zKL TKL CT

KCr er 1 rTp


18/29


With r such that r er 1rTp = > 0 similarly to the proof of thefirst statement of the lemma we obtain 1

KlogPKL C T C and (A.9)

holds.

A.3. C-Exponential tightness. (A.1) is implied by Lemma 2.To verify (A.2), let us note that

xKt+ xK =1

K

t+

+ 1K

t+

yKt+ yK =1

K

t+

+ 1K

t+

Evidently, it suffices to establish the validity of (A.2) for every

1

Kt+

1

Kt+

separately.Recall that the Markov time TKn is defined in (3.1). By virtue of Lemma 2

we claim that limn lim supK1

KlogPTKn T = . Hence, with any

n 1 we have to verify (A.2) only for

limK

1

Klog sup

TP

+TKn TKn > K

= 0

limK

1

Klog sup

TP

+TKn TKn > K

= 0

The proofs for the above relations are similar. So, we give below only one of

them for t . With r > 0 set

s

=rI

TKn

+

TKn

s

. The random process

s is bounded and left continuous (and thereby predictable). Consequently[compare (2.9)], the random process

zt = expt

0s ds

t0

es 1dAs

is a supermartingale with EzT+ 1. Therefore,

1 EzT+I

+TKn TKn > K

(A.10)

Since A+TKn ATKn

Kn + 12, on the set +TKn

TKn > K,

logzT+ = r

+TKn

TKn

er 1

A+TKn A

TKn

Kr er 1n + 12Now, with small enough and r = log

n+12 ,

zT+ exp

K

log

n + 12

+ n + 12


19/29


Hence

1

K log supT P+TKn TKn > K log

n + 12

+ n + 12 0

A.4. C-Local-LDP. Upper bound. Obviously, for 0 = x0 or 0 = y0 theupper bound is equal to . The proof that for 0 = x0 and 0 = y0 and 20T\20T the upper bound is equal to under the C-exponentialtightness can be found in Theorem 1.3 in Liptser and Pukhalskii (1992).

Thus, the proof is concentrated on the case 20 T with 0 = x0,0 = y0. Let ZKt K K be defined in (2.13) with piecewise constant (deter-ministic) functions

t

and

t

. Since EZK

T K K

1, write

1 EZKT

K

K

I

xK yK

(A.11)

On the set xK yK , we evaluate ZKt K K from below. Since are piecewise constant functions notations,

T0

t dtT

0t dt will be

used for tjT

tj1tj tj1 and

tiTti1ti ti1

respectively. Obviously, a positive constant c can be chosen such that

logZKT K K K

c +

T0

t dt + t dt Gt t t t dt

Therefore, this lower bound jointly with (A.11) imply

lim sup0

lim supK

1

KlogPxK yK

T

0t dt + t dt Gt t t t dt

Since the left side of this inequality is independent oft t

the required

upper bound is obtained by minimization of the right side in t t. As inLiptser and Pukhalskii [(1992), Theorem 6.1], we find that for not absolutely

continuous t or t the minimal value of the right-hand side is equal to ,while if both t t are absolutely continuous functions the minimal valuecoincides with JT .


20/29


A.5. C-Local-LDP. Lower bound. It suffices to verify (A.4) only for JT, < . That supposes the verification of (A.4) for absolutely continuousfunctions t ttT from

2+0 T with 0 = x0 0 = y0. Moreover, to satisfy

JT < functions t t have to be absorbed on axis x = 0; y = 0(recall that xKt y

Kt are absorbed on these axes). For definiteness we denote

the class of the above-mentioned functions t t by .Set T = inft > 0 t = 0 and T = inft > 0 t = 0 inf = and

introduce

JT =

TT0

sup

t G t t dt(A.12)

JT =

TT0

sup

t G t t dt(A.13)

Then, obviously, we have JT = JT + JT . Below we givedetailed description of t = argmaxt G t t and t =argmaxt G t t for T T = ,

t= log

t2t

+

2t4t2

+ t

t= log

t2tt

+

2t4tt2

+ t

(A.14)

A.5.1. Main Lemma.

Lemma 3. Assume

(i) .(ii) T T = .

(iii) t const t const

Then there is a positive constant L , depending on L , such thatfor > 0 small enough,

lim infK

1

K

logP

xK yK

JT

L

Proof. Set K = inft xKt t and K = inft yKt t .Since jumps of xKt y

Kt are bounded by

1K

, by the triangular inequality we

have xKt t xKt t + 1K and yKt t yKt t + 1K . Consequently


21/29


with K 4

the inclusion holds:

suptT xKt t + suptT yKt t

suptT

xKt t + suptT

yKt t

2

=

suptK T

xKt t + suptK T

yKt t

2

Hence, the first lower bound we use is the following:

PxK yK

P

sup

tK

T

xKt t + supt

K

T

yKt t

2

(A.15)

For K large enough set

Kt= log

t

2xKt+

2t

xKt2+ x

Kty

Kt

xKt

t K T

Kt= log

t

2xKtyKt

+

2t

2xKtyKt2yKt

xKtyKt

t K T

(A.16)

[Kt Kt are defined similarly with an obvious change]. It is clear thatfor K large enough Kt Kt are bounded and strictly positive.

We define now the positive supermartingale [compare (2.13)]

ZKt K K= exp

KtK T

0

Kt dxKt GKt xKt yKt dt

exp

K

tK T0

Kt dyKt GKt xKt yKt dt

(A.17)

We show that ZKt K KtT is the square integrable martingale withEZKT K K = 1

To prove this property we have to check that EZKT K K2 < . (A.17)implies

ZKT

K K

2

=ZKT

2K 2K

exp

K

K T0

G

2Kt xKt yKt

2GKt xKt ykt dt exp

K

K T0

G

2Kt xKt yKt

2GKt xKt ykt dt


22/29


The process ZKT 2K 2KtT is a supermartingale as well; that is

EZ

K

T 2K

2

K

1and at the same time under assumptions of the lemma G2Kt xKt yKt ,GKt xKt yKt and G2Kt xKt yKt GKt xKt yKt are boundedfunctions on the time intervals 0 K T and 0 K T, respectively. Con-sequently, ZKT K K2 has a finite expectation.

Now, F = tt0 QK with dQK = ZKT K K dP is the stochasticbasis. Due to the positiveness of ZKT K K not only QK P but also P QK with dP = ZKT K K1 dQK. We apply this formula for the right sideof (A.15). With the notation

K =

sup

tK TxKt t + sup

tK TyKt t

2

we have

PK =

K

ZKT K K1 dQK(A.18)

The random process ZKtTK K is the martingale with respect to P andsince P QK it is a semimartingale with respect to QK [see, e.g., Liptser andShiryayev (1989), Chapter 4, Section 5]. For further analysis QK semimartin-gale description of ZKtTK K is required. To this end,we use the fact thatP-semimartingales xKt y

KT are Q

K semimartingales as well [see Liptser and

Shiryayev (1989), Chapter 4, Section 5] and find their semimartingale decom-

positions. Let us note that t t

t

t are counting processes with respect to

both measures P and QK. Recall that At

At

At At are their compensatorswith respect to P and denote by AQt AQt AQt AQt their compensators withrespect to QK. Then the P-martingale mKt = 1Kt At t At obeysthe semimartingale decomposition (P-and QK-a.s.)

mKt =1

K

t AQt

t AQt 1

K

At AQt

At AQt = mKQt

1

K

At AQt

At AQt (A.19)

with QK- square integrable martingales mK Qt with the predicatable quadratic

variation mK Qt =1

KAQ

t + AQ

t . Analogously, the QK-semimartingaledecomposition mKt = mK Qt 1KAt AQt At AQt holds with thesquare integrable martingale m

K Qt with mK Qt = 1KAQt + AQt and mK Q,

mK Qt 0 (recall that disjointness for jumps of counting processes remainsvalid with respect to QK).


23/29


We give now formulas for AQt A

Qt

A

Qt

A

Qt . With

t = t t1 ,

t = t t we have

ZKt K K= XKtK K expKtt T+Ktt t

(A.20)

AQt : It is well known that AQt is defined by integral equation: for anybounded and predictable function ut ,

T0

ut dt dQK =

T0

ut dAQt dQK

Taking into account that dQK = ZKT K KP and ZKtTK K is the mar-tingale with respect to P we get

T0

ut dt dQK =

T0

ut ZKT K K dt dP

=

T0

ut ZKt K K dt dP

Since jumps of counting processes are disjoint the right side of the above equal-

ity is equal to [see (A.20)]

T0

ut ZKtK KeKtt dtdP or, what is

the same,

T0

ut ZKtK KeKt dt dP. Now, since Z

KtK K is

the predictable process, the latter integral coincides with

T0 ut ZKtK KeKt dAt dPwhich, due to the martingale property of ZKt K K, is equal to

T0

ut ZKT K KeKt dAt dP =

T0

ut eKt dAt dQK

Consequently,

dAQt = e

Kt dAt

and dAQt = eKt dAt d

AQt = eKt d

At d A

Qt = eKtd

At are

obtained in the same way.Hence

dmKt = dmKQt +

eKt 1xKt eKt 1xKt yKt dt

dmKt = dmKQt +

eKt 1xKt yKt eKt 1yKt dt


24/29


and thereby we find the semimartingale decomposition with respect to QK:

dxKt= xKt yKt +eKt 1xKt eKt 1xKt yKt dt + dmKQt=

G

Kt xKt yKt

dt + dmKQt = t dt + dmKQt

dyKt =

yKtxKt

+eKt 1xKt yKt eKt 1yKt dt + dmKQt

=

G

Kt xKt yKt

dt + dmKQt = t dt + dmKQt

(A.21)

The use of this decomposition allows to obtain a description for 1K

log ZKT K,K with respect to QK:

1K

logZKT K K

=K T

0Kt dmKQt +

K T0

Ktt GKt xKt yKt

dt

+K T

0Kt dmKQt +

K T0

Ktt GKt xKt yKt

dt

Let us estimate above the right side of this equality on the set K . There are

positive constants c and c so that

suptK T

Kt t + suptK T

GKt xKt yKt Gt t t c

suptK T

K

t

t +

suptK T

G

K

t

xK

t yK

t G

t

t

t c

Further, since

tt Gt t t = sup

t G t t

tt Gt t t = sup

t G t t

and

JT =K T

0sup

t G t t dt

+

K T

0sup

t G t t dt

we find

1

KlogZKT K K

K T0

Kt dmKQt+ K T

0Kt dmKQt

+c + cT + JT


25/29


This upper bound implies the lower one,

ZK

T KK

exp

K

J

T

+ c

+c

T

exp

K

K T0

Kt dmKQtKK T

0Kt dmKQt

which, being applied in (A.18), gives

PK expKJ + c + cT

K

exp

K K

0Kt dmKQt

K

K T0

Kt mKQt

dQK

(A.22)

(A.22) can be sharpened as

PK eKJT+c+cT+2QK

K K K

where

K =

K T0

Kt dmKQt < and K = K T

0Kt dmKQt

< Hence

liminfK

1

Klog PK JT c + cT 2

+ lim infK

1

Klog QK

K K K

The latter inequality jointly with (A.15) imply the statement of the lemma

provided that for fixed and it holds lim infK QKK K K = 1.We derive that property from

limK

QK

\K = 0 lim

KQK

\K

= 0 limK

QK

\K = 0(A.23)

The first part of (A.23) follows from the Chebyshev inequality (EK is theexpectation with respect to QK),

QK

\K = QK K T

0Kt dmKQt

1

2EK

K T

0 K

t

2 d

mKQ

t

1K22

EKK T

0Kt2eKt dAt + e

Kt dAt

constK2

0 K


26/29


The validity of the second part of (A.23) is verified similarly. The third part

in (A.23) is verified similarly to the first and second ones since by virtue of

(A.21),

\K =

suptK T

mKQt + suptK T

mKQt 2

A.5.2. The lower bound under T T = .

Lemma 4. Assume

(i) .(ii) T T = .Then

lim inf0 lim infK1

K logPxK yK JT Proof. Set nt = x0 +

r0

sIs n ds. SinceT

0t dt < , it holds

limn

suptT

t nt T

0It > nt dt = 0

Similarly, for nt = y0+r

0 sIs n ds we have limn suptT tnt = 0. Letus choose n0 so that for n n0 it holds suptT t nt + suptT t nt 2 .Since by the triangular inequality,

supt

T

xKt t+ sup

t

T

yKt t+ sup

t

T

xKt nt+ sup

t

T

yKt nt+

2

for any n n0 we get

PxK yK P

xK yK n n 2

Therefore, by Lemma 3,

lim infK

1

KlogP

xK yK n n

2

JTn n Ln n

Since inftT t > 0 and inftT t > 0 and t t and approximated by nt

nt

uniformly in t T, for n large enough a majorant L for Ln n can bechosen, so that

lim infK

1K

logPxK yK 2 JTn n L

We show that

lim supn

JTn n JT (A.24)


27/29


Since for n large enough inftT nt > 0, inftT

nt > 0, introduce

nt= log nt2nt + n2t4nt 2 + nt t= log

nt

2nt nt

+

n2t4nt nt 2

+ nt

(A.25)

and note that

JTn n =T

0It n

ntt G

nt nt nt

dt

+T

0It > n

G

nt nt nt

dt

+ T0

It nntt Gnt nt nt dt+

T0

It > nGnt n ndt

JT +T

0

Gt t t Gnt nt nt dt+

T0

Gt t t Gnt nt nt dtThe second and third terms in the right side of the above inequality converge

to zero, as n , by virtue of the uniform (in t T) convergence n n

and T0 nt nt tt dt 0 and T0 nt nt tt dt 0;that is, (A.24) holds.

Thus

lim infK

1

KlogP

xK yK

2

JT L (A.26)

and the statement of the lemma holds.

A.5.3. The lower bound under T T < . For and h smallenough we have

lim inf

0

lim infK

1

KlogPxK yK

= lim inf0

liminfK

P

suptTT

xKt t+ suptTT

yKt t lim inf

0liminf

KP

sup

tThT

xKt t+ suptThT

yKt t


28/29


On the other side, for ht t, t T h, ht t, t T h, and fort > T h, t > T h,

ht = ht ht ht = ht ht respectively, we have

liminf0

lim infK

P

sup

tThT

xKt t + suptThT

yKt t = lim inf

0liminf

KPxK yK h h JTh h

where the latter inequality follows from Lemma 4.

Finally,

JYh h =

ThT

0sup

t G t t dt

+ ThT0

sup

t G t t dt

TT

0sup

t G t t dt

+TT

0sup

t G t t dt

= JT Now combining the obtained results above we arrive at the required lower

bound,

lim inf0 lim infK 1K logPxK yK JT

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Department of Mathematics

and Statistics

University of Melbourne

Parkville, Victoria 3052

AustraliaE-mail: [email protected]

Department of Electrical

Engineering Systems

Tel Aviv Univeristy

69978 Tel Aviv

IsraelE-mail: [email protected]

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