aa r X i v : . [ m a t h . P R ] M a y Moderate Deviations and Extinction of anEpidemic ´E. PardouxMay 23, 2019
Abstract
Consider an epidemic model with a constant flux of susceptibles, in a sit-uation where the corresponding deterministic epidemic model has a uniquestable endemic equilibrium. For the associated stochastic model, whose lawof large numbers limit is the deterministic model, the disease free equilibriumis an absorbing state, which is reached soon or later by the process. However,for a large population size, i.e. when the stochastic model is close to its de-terministic limit, the time needed for the stochastic perturbations to stop theepidemic may be enormous. In this paper, we discuss how the Central LimitTheorem, Moderate and Large Deviations allow us to give estimates of theextinction time of the epidemic, depending upon the size of the population.
We consider epidemic models where there is a constant flux of susceptibleindividuals, either because the infected individuals become susceptible im-mediately after healing, or after some time during which the individual isimmune to the illness, or because there is a constant flux of newborn orimmigrant susceptibles.In the above three cases, for certain values of the parameters, there isan endemic equilibrium, which is a stable equilibrium of the associated de-terministic epidemic model. The deterministic model can be considered asthe Law of Large Numbers limit (as the size of the population tends to ∞ )1f a stochastic model, where infections, healings, births and deaths happenaccording to Poisson processes whose rates depend upon the numbers of in-dividuals in each compartment.Since the disease free states are absorbing, it follows from an irreducibil-ity property which is clearly valid in our models, that the epidemic will stopsoon or later in the more realistic stochastic model. However, the time whichthe stochastic perturbances will need to stop the epidemic may be enormouswhen the size N of the population is large. The aim of this paper is todescribe, based upon the Central Limit Theorem, Large and Moderate De-viations, the time it takes for the epidemic to stop in the stochastic model.The law of large numbers and central limit theorems are rather old. Theycan be found e.g. in chapter 11 of Ethier and Kurtz [3]. There are alsopresented, in the framework of epidemic models, in Britton and Pardoux[1]. The Large Deviations results are close to those presented in Shwartzand Weiss [9], [10], although their assumptions are not quite satisfied in ourmodels. Derivations adapted to our setup can be found in Kratz and Pardoux[5], Pardoux and Samegni–Kepgnou [6], and Britton and Pardoux [1]. Theresults concerning moderate deviations are new and constitute the core of thispaper. Our derivation is essentially based upon an infinite generalizationof the G¨artner–Ellis Theorem, Corollary 4.6.14 from Dembo and Zeitouni[2]. Our main results are Theorem 4.10 and Theorem 4.13. We also giveexpressions for the rate function in our three models of interest, and in caseof the simplest model we give an explicit formula for the quasi–potential. Wealso compare in that case the upper bound of fluctuations given respectivelyby the central limit theorem, moderate deviations, and large deviations.The paper is organized as follows. In section 2, we describe the threedeterministic and stochastic models which we have in mind, namely the SIS,SIRS and SIR model with demography. In section 3, we give the generalformulation of the stochastic models, and recall the Law of Large Numbers,the Central Limit Theorem and the Large Deviations, and their application tothe time of extinction of an epidemic. In section 4, we establish the moderatedeviations result and explain how it can be used to predict the time takenfor an epidemic to cease, depending upon the size of the population. Finallyan Appendix establishes an estimate of exponential moments of the integralwith respect to a compensated Poisson random measure. This estimate isused several times in our proofs.In this paper, the same letter C denotes an arbitrary constant, whosevalue may change from line to line. 2 The three models
The deterministic SIS model is the following. Let s ( t ) (resp. i ( t )) denotethe proportion of susceptible (resp. infectious) individuals in the population.Given an infection parameter λ , and a recovery parameter γ , the determin-istic SIS model reads ( s ′ ( t ) = − λs ( t ) i ( t ) + γi ( t ) ,i ′ ( t ) = λs ( t ) i ( t ) − γi ( t ) . Since clearly s ( t ) + i ( t ) ≡
1, the system can be reduced to a one dimensionalODE. If we let z ( t ) = i ( t ), we have s ( t ) = 1 − z ( t ),and we obtain the ODE z ′ ( t ) = λz ( t )(1 − z ( t )) − γz ( t ) . It is easy to verify that this ODE has a so–called “disease free equilibrium”,which is z = 0. If λ > γ , this equilibrium is unstable, and there is an endemicstable equilibrium z ∗ = 1 − γ/λ .The corresponding stochastic model is as follows. Let S Nt (resp. I Nt )denote the proportion of susceptible (resp of infectious) individuals in a pop-ulation of total size N . S Nt = S N − N P inf (cid:18) λN Z t S Nr I Nr dr (cid:19) + 1 N P rec (cid:18) γN Z t I Nr dr (cid:19) ,I Nt = I N + 1 N P inf (cid:18) λN Z t S Nr I Nr dr (cid:19) − N P rec (cid:18) γN Z t I Nr dr (cid:19) . Here P inf ( t ) and P rec ( t ) are two mutually independent standard (i.e. rate1) Poisson processes. Let us give some explanations, first concerning themodeling, then concerning the mathematical formulation.Let S Nt (resp. I Nt ) denote the number of susceptible (resp. infectious)individuals in the population. The equations for those quantities are theabove equations, multiplied by N . The argument of P inf ( t ) reads λ Z t S Nr N I Nr dr . The formulation of such a rate of infections can be explained as follows. Eachinfectious individual meets other individuals in the population at some rate3 . The encounter results in a new infection with probability p if the partnerof the encounter is susceptible, which happens with probability S Nt /N , sincewe assume that each individual in the population has the same probabilityof being that partner, and with probability 0 if the partner is an infectiousindividual. Letting λ = βp and summing over the infectious individuals attime t gives the above rate. Concerning recovery, it is assumed that eachinfectious individual recovers at rate γ , independently of the others. In the SIRS model, contrary to the SIS model, an infectious who heals is firstimmune to the illness, he is “recovered”, and only after some time does heloose his immunity and turn to susceptible. The deterministic SIRS modelreads s ′ ( t ) = − λs ( t ) i ( t ) + ρr ( t ) ,i ′ ( t ) = λs ( t ) i ( t ) − γi ( t ) ,r ′ ( t ) = γi ( t ) − ρr ( t ) , while the stochastic SIRS model reads S Nt = S N − N P inf (cid:18) λN Z t S Nr I Nr dr (cid:19) + 1 N P loim (cid:18) ρN Z t R Nr dr (cid:19) ,I Nt = I N + 1 N P inf (cid:18) λN Z t S Nr I Nr dr (cid:19) − N P rec (cid:18) γN Z t I Nr dr (cid:19) R Nt = R N + 1 N P rec (cid:18) γN Z t I Nr dr (cid:19) − N P loim (cid:18) ρN Z t R Nr dr (cid:19) . These two models could be reduced to two–dimensional models for z ( t ) =( i ( t ) , s ( t )) (resp. for Z Nt = ( I Nt , S Nt )). In this model, recovered individuals remain immune for ever, but there is aflux of susceptibles by births at a given rate multiplied by N , while individualsfrom each of the three compartments die at rate µ . Thus the deterministicmodel s ′ ( t ) = µ − λs ( t ) i ( t ) − µs ( t ) i ′ ( t ) = λs ( t ) i ( t ) − γi ( t ) − µi ( t ) r ′ ( t ) = γi ( t ) − µr ( t ) , S Nt = S N − N P inf (cid:18) λN Z t S Nr I Nr dr (cid:19) + 1 N P birth ( µN t ) − N P ds (cid:18) µN Z t S Nr dr (cid:19) ,I Nt = I N + 1 N P inf (cid:18) λN Z t S Nr I Nr dr (cid:19) − N P rec (cid:18) γN Z t I Nr dr (cid:19) − N P di (cid:18) µN Z t I Nr dr (cid:19) ,R Nt = R N + 1 N P rec (cid:18) γN Z t I Nr dr (cid:19) − N P dr (cid:18) µN Z t R Nr dr (cid:19) . Remark 2.1.
One may think that it would be more natural to decide thatbirths happen at rate µ times the total population. The total population pro-cess would be a critical branching process, which would go extinct in finitetime a.s., which we do not want. Next it might seem more natural to replacein the infection rate the ratio S Nt /N by S Nt / ( S Nt + I Nt + R Nt ) , which is theactual ratio of susceptibles in the population at time t . It is easy to show that S Nt + I Nt + R Nt is close to N , so we choose the simplest formulation. Again, we can reduce these models to two–dimensional models for z ( t ) =( i ( t ) , s ( t )) (resp. for Z Nt = ( I Nt , S Nt )), by deleting the r (resp. R N ) compo-nent. The three above stochastic models are of the following form. Z Nt = z N + 1 N k X j =1 h j P j (cid:18) N Z t β j ( Z Ns ) ds (cid:19) = z N + Z t b ( Z Ns ) ds + 1 N k X j =1 h j M j (cid:18) N Z t β j ( Z Ns ) ds (cid:19) , (3.1)where { P j ( t ) , t ≥ } ≤ j ≤ k are mutually independent standard Poisson pro-cesses, M j ( t ) = P j ( t ) − t , and b ( z ) = P kj =1 β j ( z ) h j . Z Nt takes its values inIR d . 5n the case of the SIS model, d = 1, k = 2, h = 1, β ( z ) = λz (1 − z ), h = − β ( z ) = γz .In the case of the SIRS model, d = 2, k = 3, h = (cid:18) − (cid:19) , β ( z ) = λz z , h = (cid:18) − (cid:19) , β ( z ) = γz and h = (cid:18) (cid:19) , β ( z ) = ρ (1 − z − z ).In the case of the SIR model with demography, we can restrict ourselvesto d = 2, while k = 4, h = (cid:18) − (cid:19) , β ( z ) = λz z , h = (cid:18) − (cid:19) , β ( z ) =( γ + µ ) z , h = (cid:18) (cid:19) , β ( z ) = µ , h = (cid:18) − (cid:19) , β ( z ) = µz .While the above expressions has the advantage of being concise, we shallrather use the following equivalent formulation of (3.1). Let {M j , ≤ j ≤ k } be mutually independent Poisson random measures on IR with meanmeasure the Lebesgue measure, and let M j ( ds, du ) = M j ( ds, du ) − ds du ,1 ≤ j ≤ k . We can rewrite (3.1) in the form Z Nt = z N + 1 N k X j =1 h j Z t Z Nβ j ( Z Ns )0 M j ( ds, du )= z N + Z t b ( Z Ns ) ds + 1 N k X j =1 h j Z t Z Nβ j ( Z Ns )0 M j ( ds, du ) , (3.2)The joint law of { Z N , N ≥ } is the same law of a sequence of randomelements of the Skorohod space D ([0 , T ]; IR d ), whether we use (3.1) or (3.2)for its definition.Let us state the assumptions which we will need in section 4 below. Thoseare more than necessary for the results of the present section to hold, see [1]for the proofs.( H. β j is bounded , ≤ j ≤ k ;( H. b ∈ C (IR d ; IR d ) , and ∇ b : IR d IR d × d is bounded and Lipshitz . Remark 3.1.
In practice, in our models, either the process Z Nt takes itsvalues in a compact subset of IR d (this is the case for all models with aconstant population size), or else we restrict ourselves to such a situation,by stopping the process when the total population exceeds a given large value,see section 4.2.7 in [1]. z ∈ [0 , d , z N = [ N z ] /N , where [ N z ] ∈ Z d + is the vector whose i –th component is theinteger part of the real number N z i . We have a Law of Large Numbers
Theorem 3.2.
Let Z Nt denote the solution of the SDE (3.1) . Then Z Nt → z t a.s. locally uniformly in t , where { z t , t ≥ } is the unique solution of theODE dz t dt = b ( t, z t ) , z = x. The main argument in the proof of the above theorem is the fact that,locally uniformly in t , P ( N t ) N → t a.s. as N → ∞ . We also have a Central Limit Theorem. Let U Nt := √ N ( Z Nt − z ( t )). Theorem 3.3. As N → ∞ , { U Nt , t ≥ } ⇒ { U t , t ≥ } for the topology oflocally uniform convergence, where { U t , t ≥ } is a Gaussian process of theform U t = Z t ∇ x b ( s, z s ) U s ds + k X j =1 h j Z t q β j ( s, z s ) dB j ( s ) , t ≥ , where { ( B ( t ) , B ( t ) , . . . , B k ( t )) , t ≥ } are mutually independent standardBrownian motions. We denote by AC T,d the set of absolutely continuous functions from [0 , T ]into IR d . For any φ ∈ AC T,d , let A k ( φ ) denote the (possibly empty) set of7unctions c ∈ L (0 , T ; IR k + ) such that c j ( t ) = 0 a.e. on the set { t, β j ( φ t ) = 0 } and dφ t dt = k X j =1 c j ( t ) h j , t a.e . We define the rate function I T ( φ ) := ( inf c ∈A k ( φ ) I T ( φ | c ) , if φ ∈ AC T,A ; ∞ , otherwise.where as usual the infimum over an empty set is + ∞ , and I T ( φ | c ) = Z T k X j =1 g ( c j ( t ) , β j ( φ t )) dt with g ( ν, ω ) = ν log( ν/ω ) − ν + ω . We assume in the definition of g ( ν, ω ) thatfor all ν >
0, log( ν/
0) = ∞ and 0 log(0 /
0) = 0 log(0) = 0. The collection Z N obeys a Large Deviations Principle, in the sense that Theorem 3.4.
For any open subset O ⊂ D ([0 , T ]; IR d ) , lim inf N →∞ N log IP (cid:0) Z N,z N ∈ O (cid:1) ≥ − I T,z ( O ) . For any closed subset F ⊂ D ([0 , T ]; IR d ) , lim sup N →∞ N log IP( Z N,z N ∈ F ) ≤ − I T,z ( F ) . A slight reinforcement of this theorem allows us to conclude a Wentzell–Freidlin type of result. In what follows, we assume that the first componentof Z Nt (resp. of z ( t )) is I Nt (resp. i ( t )). Assume that the deterministic ODEwhich appears in Theorem 3.2 has a unique stable equilibrium z ∗ whose firstcomponent satisfies z ∗ >
0. We define V := inf T > inf φ ∈AC T,d ,φ (0)= z ∗ ,φ ( T )=0 I T ( φ ) . Let now T N,z
Ext = inf { t > , Z N ( t ) = 0 , if Z N (0) = z N } . We have the 8 heorem 3.5.
Given any η > , for any z with z > , lim N →∞ P (cid:0) exp { N ( V − η ) } < T N,z
Ext < exp { N ( V + η ) } (cid:1) = 1 . Moreover, for all η > and N large enough, exp { N ( V − η ) } ≤ E ( T N,z
Ext ) ≤ exp { N ( V + η ) } . We refer for the proof of this Theorem to [5] and [1].It is important to evaluate the quantity V . Note that it is the valuefunction of an optimal control problem. In case of the SIS model, which isone dimensional, one can solve this control problem explicitly with the helpof Pontryagin’s maximum principle, see [8], and deduce in that case that V = log λγ − γλ . For other models, one can compute numerically a goodapproximation of the value of V for each given value of the parameters. The discussion of this subsection, which motivates the moderate deviationsapproach of this paper, is taken from section 4.1 in [1]. Consider the SIRwith demography. i ′ ( t ) = λi ( t ) s ( t ) − γi ( t ) − µi ( t ) ,s ′ ( t ) = − λi ( t ) s ( t ) + µ − µs ( t ) . We assume that λ > γ + µ , in which case there is a unique stable endemicequilibrium, namely z ∗ = ( i ∗ , s ∗ ) = ( µγ + µ − µλ , γ + µλ ). We can study the ex-tinction of an epidemic in the above model using the CLT. We note thatthe basic reproduction number R and the expected relative time of a life anindividual is infected, ε , are given by R = λγ + µ ε = 1 / ( γ + µ )1 /µ = µγ + µ . The rate of recovery γ is much larger than the death rate µ (52 compared to1/75 for a one week infectious period and 75 year life length) so we use theapproximations R ≈ λ/γ and ε ≈ µ/γ . Denote again by I Nt the fractionof the population which is infectious in a population of size N . The lawof large numbers tells us that for N and t large, I Nt is close to i ∗ . The9entral limit theorem tell us that √ N ( I Nt − i ∗ ) converges to a Gaussianprocess, whose asymptotic variance can be shown to well approximated by R − . This suggests that for large t , the number of infectious individualsin the population is approximately Gaussian, with mean N i ∗ and standarddeviation p N/R . If N i ∗ and p N/R are of the same order, i.e. N is ofthe same order as i ∗ ) R , it is likely that the fluctuations described by thecentral limit theorem explain that the epidemic might cease in time of orderone. This gives a critical population size roughly of the order of N c ∼ i ∗ ) R = 1 ǫ (1 − R − ) R , in fact probably a bit larger than that.Consider measles prior to vaccination. In that case it is known that R ≈
15, and ε ≈ / / (1 / / ≈ / N c ∼ (3750) / . So, if the population is at most a million (or perhaps acouple of millions), we expect that the disease will go extinct quickly, whereasthe disease will become endemic (for a rather long time) in a significantlylarger population. This confirms the empirical observation that measles wascontinuously endemic in UK whereas it died out quickly in Iceland (and waslater reintroduced by infectious people visiting the country). If the CLT allows to predict extinction of an endemic disease for populationsizes under a given threshold N c , and Large Deviations gives predictions forarbitrarily large population sizes, it is fair to look at Moderate Deviations,which describes ranges of fluctuations between those of the CLT and thoseof the LD.The assumptions ( H.
1) and ( H.
2) are assumed to hold throughout thissection.
We shall use the general model written in the form (3.2). We assume thatthe limiting law of large numbers ODE z ( t ) = z + Z t b ( z ( s )) ds z ∗ such that z ∗ >
0, called the endemicequilibrium, which is such that, provided z (0) > z ( t ) → z ∗ as t → ∞ .For the sake of simplifying many formulas below, we chance our coordi-nates, and let z ∗ = 0. The reader should be aware of the fact that there isa price to pay for that translation of the origin. Indeed, since in the originalcoordinate system, the process Z Nt was living on the set of vectors whosecoordinates are integer multiples of N − (this is essential for the process toremain in the set where it makes sense, i.e. for proportions to remain between0 and 1), the new origin generically does not belong to the set of point in IR d which our process Z Nt may visit. The grid on which Z Nt lives is translated bythe vector z ∗ − { z ∗ } N , where here and below { z } N := [ N z ] /N , [ N z ] denotingthe vector whose i –th component is the integer part of the i –th componentof N z . However, this minor complexity will appear only in the formula forthe initial condition of the SDE. Once the SDE starts on the correct grid,the solution remains there.From now on 0 will be the endemic equilibrium (of course in the translatedcoordinate system), while z ∗ = 0 will denote that endemic equilibrium in theoriginal coordinates (we shall need it for the formula of the initial conditionof the SDE).We want to study the moderate deviations at scale α of Z Nt , where 0 <α < /
2. Note that α = 0 would correspond to the large deviations, and α = 1 / z ∗ = 0, namely we shall consider thefunction { z N ( t ) , ≤ t ≤ T } , solution of the ODE z N ( t ) = N − α z + Z t b ( z N ( s )) ds, where z ∈ IR d is arbitrary. In fact, we shall be more interested in z N ( t ) := N α z N ( t ), which solves (below we exploit the fact that b (0) = 0) z N ( t ) = z + N α Z t b ( z N ( s )) ds = z + Z t Z ∇ b (0) z N ( s ) ds + Z t Z [ ∇ b ( θz N ( s )) − ∇ b (0)] dθ z N ( s ) ds. It is not hard to prove that, under our standing assumption ( H.
2) that b is of class C and ∇ b is bounded, as N → ∞ , z N ( t ) → z ( t ) uniformly for11 ≤ t ≤ T , where z ( t ) solves the linearized ODE near the endemic equilibrium0 : z ( t ) = z + Z t ∇ b (0) z ( s ) ds . We want to study the moderate deviations of the process Z Nt solution ofthe SDE (3.1) with the initial condition z N := { z ∗ + N − α z } N − z ∗ . Thisamounts to study the large deviations of Z N,αt := N α Z Nt at speed a N = N α − . We define Y Nt = 1 N k X j =1 h j Z t Z Nβ j ( Z Ns )0 M j ( ds, du ) , and Y N,αt = N α Y Nt . With these notations, the SDE for Z N,αt reads Z N,αt = N α (cid:0) { z ∗ + N − α z } N − z ∗ (cid:1) + Z t N α b (cid:0) N − α Z N,αs (cid:1) ds + Y N,αt = N α (cid:0) { z ∗ + N − α z } N − z ∗ (cid:1) + Z t ∇ b (0) Z N,αs ds + Z t V N,αs ds + Y N,αt , where V N,αs = N α b (cid:0) N − α Z N,αs (cid:1) − ∇ b (0) Z N,αs = (cid:20)Z (cid:0) ∇ b (cid:0) θN − α Z N,αs (cid:1) − ∇ b (0) (cid:1) dθ (cid:21) Z N,αs . If we let K := sup z k∇ b ( z ) k , we have k Z N,αt k ≤ k z k + √ dN α − + K Z t k Z N,αs k ds + k Y N,αt k . This combined with Gronwall’s Lemma yields k Z N,αt k ≤ e Kt (cid:18) k z k + √ dN α − + sup ≤ s ≤ t k Y N,αs k (cid:19) . From the boundedness and Lipschitz property of ∇ b , and the formula for V N,α , we deduce that k V N,αt k . k Z N,αt k , and k V N,αt k . N − α k Z N,αt k . We deduce from the last three inequalities(4.1) (cid:13)(cid:13)(cid:13) V N,αt (cid:13)(cid:13)(cid:13) . N α − + (cid:18) k z k + sup ≤ s ≤ t k Y N,αs k (cid:19) ∧ N − α (cid:18) k z k + sup ≤ s ≤ t k Y N,αs k (cid:19) .
12e now define e Y N,αt = Z t V N,αs ds + Y N,αt , t ≥ , so that(4.2) Z N,αt = N α (cid:0) { z ∗ + N − α z } N − z ∗ (cid:1) + Z t ∇ b (0) Z N,αs ds + e Y N,αt . We will see below that the large deviations of Z N,α will follow from thoseof e Y N,α by a variant of the contraction principle. We first consider the simplerprocesses Y Nt := 1 N k X j =1 h j Z t Z Nβ j (0)0 M j ( ds, du ) , and Y N,αt = N α Y Nt which are similar to Y N and Y N,α , but with Z Ns replaced by 0. Y N,α
We note that writing the integral over [0 , N β j (0)] as the sum from ℓ = 1 to ℓ = N of integrals over (( ℓ − β j (0) , ℓβ j (0)], we can rewrite Y N,α as follows. Y N,α = 1 N − α N X ℓ =1 Q ℓ ( t ) , where Q ℓ ( t ) = k X j =1 h j Z t Z ℓβ j (0)( ℓ − β j (0) M j ( ds, du ) . The processes Q , Q , . . . , Q N are i.i.d., and their law is that of(4.3) Q ( t ) = Q ( t ) = k X j =1 h j Z t Z β j (0)0 M j ( ds, du ) . Now let ν = ( ν , . . . , ν d ) be a vector of signed measures on [0 , T ]. Lemma 4.1. As N → ∞ , (recall that a N = N α − ) a N log IE exp n a − N ν ( Y N,α ) o →
12 IE (cid:0) ν ( Q ) (cid:1) . roof We use in an essential way the above decomposition of Y N,α . a N Λ N ( a − N ν ) = N α − log IE exp { N − α ν ( Y N ) } = N α log IE exp { N − α ν ( Q ) } = N α log IE { N − α ν ( Q ) + N − α ν ( Q ) + N − α R N } = N α log (cid:26) N − α ν ( Q ) ] + N − α IE[ R N ] (cid:27) →
12 IE[ ν ( Q ) ] , provided(4.4) sup N ≥ | IE[ R N ] | < ∞ , which we will check below. From this it follows that the argument of thelogarithm on the before last line is greater than or equal to 1, at least for N large enough, and the final conclusion follows easily from the fact that forany x ≥ x − x / ≤ log(1 + x ) ≤ x . Let us now check (4.4). It followsfrom an exact Taylor formula that | R N | ≤ | ν ( Q ) | N − α | ν ( Q | ) . But ν ( Q ) is an affine combination of mutually independent Poisson randomvariables, so that (4.4) follows easily by an explicit computation. (cid:3) e Y N,α
We want to study the large deviations of e Y N,α . The main step will be toprove that Lemma 4.1 remains valid if we replace Y N,α by e Y N,α , which willfollow from the next Proposition.
Proposition 4.2.
For any
C > , ν = ( ν , . . . , ν d ) a vector of signed mea-sures, as N → ∞ , a N log IE exp h Ca − N ν ( e Y N,α − Y N,α ) i → . Proposition 4.3.
Given Lemma 4.1, if Proposition 4.2 holds true, then forany signed measure ν on [0 , T ] , as N → ∞ , a N log IE exp n a − N ν ( e Y N,α ) o →
12 IE[ ν ( Q ) ] . Proof
For any δ >
0, we deduce from H¨older’s inequality a N log IE exp { a − N ν ( e Y N,α ) } = a N log IE h exp { a − N ν ( Y N,α ) } exp { a − N ν ( e Y N,α − Y N,α ) } i ≤ a N δ log IE exp { (1 + δ ) a − N ν ( Y N,α ) } + a N δ δ log IE exp (cid:26) δδa N ν ( e Y N,α − Y N,α ) (cid:27) , so that, if we combine Lemma 4.1 and Proposition 4.2, we deduce thatlim sup N a N log IE exp { ν ( a − N e Y N,α ) } ≤ (1 + δ )2 IE[ ν ( Q ) ] , and letting δ →
0, we conclude thatlim sup N a N log IE exp { ν ( a − N e Y N,α ) } ≤
12 IE[ ν ( Q ) ] . For the inequality in the other direction, we note that, by similar arguments, a N log IE exp (cid:26) a − N δ ν ( Y N,α ) (cid:27) ≤ a N δ log IE exp { a − N ν ( e Y N,α ) } + a N δ δ log IE exp { ( δa N ) − µ ( e Y N,α − Y N,α ) } , with µ = − ν , which implies thatlim inf N a N log IE exp { a − N ν ( e Y N,α ) } ≥ δ ) IE[ ν ( Q ) ] , hence, letting δ → N a N log IE exp { a − N ν ( Y N,α ) } ≥
12 IE[ ν ( Q ) ] . The remaining of this subsection will be devoted to the proof of Proposi-tion 4.2.We note that Proposition 4.2 is a consequence of the following two Propo-sitions.
Proposition 4.4.
For any
C > , as N → ∞ , (4.5) a N log IE exp h Ca − N ν ( Y N,α − Y N,α ) i → . Proposition 4.5.
For any
C > , as N → ∞ , a N log IE exp (cid:20) Ca − N sup ≤ t ≤ T (cid:13)(cid:13)(cid:13)(cid:13)Z t V N,αs ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:21) → . We start with the
Proof of Proposition 4.4
The exponents in the expressions entering(4.5) are sums over the indices 1 ≤ i ≤ d and 1 ≤ j ≤ k . Using repeat-edly Schwartz’s inequality, it is sufficient to prove the results with the sumreplaced by each of the summands. Therefore in this proof we do as if d = 1,we fix 1 ≤ j ≤ k and for the sake of simplifying the notations, we drop theindex j . We note that a − N ( Y N,α − Y N,α ) = N − α Z t Z N [ β ( Z Ns ) ∨ β (0)] Nβ (0) M ( ds, du ) − N − α Z t Z N [ β ( Z Ns ) ∨ β (0)] Nβ ( Z Ns ) M ( ds, du )It is not hard to see that one can treat each of the two terms on the rightseparately, and we treat only the first term, the treatment of the secondone being quite similar. We note that there exists a compensated standardPoisson process M ( t ) on IR + such that the factor of N − α in this first termcan be rewritten as W Nt := M (cid:18) N Z t ( β ( Z Ns ) − β (0)) + ds (cid:19) . We need to estimate IE exp[ CN − α ν ( W N )]. If we decompose the signed mea-sure ν as the difference of two measures as follows ν = ν + − ν − , we again have16wo terms, and it suffices to treat one of them, say ν + . Of course it sufficesto treat the case where ν + = 0. Since the positive constant C is arbitrary,we can w.l.o.g. assume that ν + is a probability measure on [0 , T ]. It is thenclear that exp (cid:20) CN − α Z T W Nt ν + ( dt ) (cid:21) ≤ exp (cid:20) CN − α sup ≤ t ≤ T W Nt (cid:21) . We choose a new parameter 0 < γ < α , and we write the expressionwhose expectation needs to be estimated as a sum of two terms as follows.exp (cid:26) CN − α sup ≤ t ≤ T W Nt (cid:27) = exp (cid:26) CN − α sup ≤ t ≤ T W Nt (cid:27) sup ≤ t ≤ T k Z Nt k≤ N − γ + exp (cid:26) CN − α sup ≤ t ≤ T W Nt (cid:27) sup ≤ t ≤ T k Z Nt k >N − γ . (4.6)We now estimate the first term on the right hand side of (4.6). For that sake,we define the stopping time σ N = inf { ≤ t ≤ T ; k Z Nt k > N − γ } and note thatexp (cid:26) CN − α sup ≤ t ≤ T M (cid:18) N Z t ( β ( Z Ns ) − β (0)) + ds (cid:19)(cid:27) sup ≤ t ≤ T k Z Nt k≤ N − γ ≤ exp (cid:26) CN − α sup ≤ t ≤ T M (cid:18) N Z t ∧ σ N ( β ( Z Ns ) − β (0)) + ds (cid:19)(cid:27) Consequently the expectation of the first term on the right of (4.6) is boundedfrom above byIE exp (cid:26) CN − α sup ≤ t ≤ T M (cid:18) N Z t ∧ σ N ( β ( Z Ns ) − β (0)) + ds (cid:19)(cid:27) ≤ IE exp (cid:26) ( e CN − α − − CN − α ) N Z T ∧ σ N ( β ( Z Nt ) − β (0)) + dt (cid:27) ≤ exp (cid:8) CN − α − γ (cid:9) , β . Consider now thesecond term on the right hand side of (4.6).IE (cid:18) exp (cid:26) N − α sup ≤ t ≤ T M (cid:18) N Z t ( β ( Z Ns ) − β (0)) + ds (cid:19)(cid:27) sup ≤ t ≤ T k Z Nt k >N − γ (cid:19) ≤ (cid:18) IE exp (cid:26) N − α sup ≤ t ≤ T M (cid:18) N Z t ( β ( Z Ns ) − β (0)) + ds (cid:19)(cid:27)(cid:19) / × IP (cid:18) sup ≤ t ≤ T k Z Nt k > N − γ (cid:19) / ≤ exp (cid:8) CN − α (cid:9) IP (cid:18) sup ≤ t ≤ T (cid:13)(cid:13) Y Nt (cid:13)(cid:13) > cN − γ (cid:19) / , for some c, C >
0, where the second inequality follows from Proposition 5.1and the boundedness of β . Estimating the second factor in the last expressionamounts to estimating the two probabilities (with another c > (cid:18) sup ≤ t ≤ T M (cid:18) N Z t β ( Z Ns ) ds (cid:19) > cN − γ (cid:19) andIP (cid:18) sup ≤ t ≤ T (cid:18) − M (cid:18) N Z t β ( Z Ns ) ds (cid:19)(cid:19) > cN − γ (cid:19) . (4.7)We estimate the first probability. For any a > (cid:18) sup ≤ s ≤ t M (cid:18) N Z s β ( Z Nr ) dr (cid:19) > cN − γ (cid:19) = IP (cid:18) sup ≤ s ≤ t exp (cid:26) aM (cid:18) N Z s β ( Z Nr ) dr (cid:19)(cid:27) > exp { acN − γ } (cid:19) ≤ e − acN − γ IE (cid:18) sup ≤ s ≤ t exp (cid:26) aM (cid:18) N Z s β ( Z Nr ) dr (cid:19)(cid:27)(cid:19) . e − acN − γ (cid:18) IE exp (cid:26) ( e a − − a ) N Z t β ( Z Ns ) ds (cid:27)(cid:19) / ≤ exp {− acN − γ + ( e a − − a ) N Ct }≤ exp (cid:26) − c Ct N − γ (cid:27) , (4.8) 18here the second inequality follows from Proposition 5.1 and the last in-equality by optimizing over a >
0. One can easily convince oneself that asimilar result holds for the second line of (4.7), making use of Proposition 5.1with a negative a . Note also for further use that the same result also holdsin case γ = 0. In that case, the probability on the second line of (4.7) is zerofor large enough c , in which case the anounced estimate is of course true.The expectation of the second term of the right hand side of (4.6) is thusdominated by (with c and c two positive constants)exp { c N − α − c N − γ } → , as N → ∞ . Finally IE exp (cid:26) N − α sup ≤ s ≤ t M (cid:18) N Z t ( β ( Z Ns ) − β ( z s )) + ds (cid:19)(cid:27) ≤ exp (cid:8) CN − α − γ (cid:9) + exp { c N − α − c N − γ } It follows readily from the inequality log( a + b ) ≤ log(2) + log( a ∨ b ) that for N large enough a N log IE exp (cid:26) N − α sup ≤ s ≤ t M (cid:18) N Z t ( β ( Z Ns ) − β (0)) + ds (cid:19)(cid:27) ≤ a N log(2) + CN − γ , which establishes (4.5). (cid:3) We now turn to the second proof.
Proof of Proposition 4.5
Recalling assumption ( H. β j := sup z ∈ IR d β j ( z ), ξ N,jt := 1 N Z t Z Nβ j M j ( ds, du ) 1 ≤ j ≤ k, the event A Nb := (cid:26) sup ≤ t ≤ T (cid:13)(cid:13)(cid:13) Y Nt (cid:13)(cid:13)(cid:13) ≤ b (cid:27) \ k \ j =1 (cid:26) sup ≤ t ≤ T ξ N,jt ≤ (1 + b ′ ) β j T (cid:27) , and the stopping time¯ τ b := inf n t > , (cid:13)(cid:13)(cid:13) Y Nt (cid:13)(cid:13)(cid:13) > b o ^ k ^ j =1 inf n t > , ξ N,jt > (1 + b ′ ) β j T o , b > b ′ > a N log IE (cid:20) exp (cid:26) a − N C sup ≤ t ≤ T (cid:13)(cid:13)(cid:13)(cid:13)Z t V N,αs ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:27)(cid:21) . N α − + a N log IE (cid:20) exp (cid:26) a − N C (cid:18) k z k + N α sup ≤ t ≤ T k Y Nt k (cid:19) ( A Nb ) c (cid:27)(cid:21) (4.9) + a N log IE (cid:20) exp (cid:26) a − N C (cid:18) N − α k z k + N α sup ≤ t ≤ T k Y Ns k (cid:19) A Nb (cid:27)(cid:21) , (4.10)We take the limit successively in the two terms of the above right hand side. Step 1 : Estimate of (4.9) We haveIE (cid:20) exp (cid:26) a − N C (cid:18) k z k + N α sup ≤ t ≤ T k Y Nt k (cid:19) ( A Nb ) c (cid:27)(cid:21) . exp (cid:8) CN − α k z k (cid:9) IP (cid:0) ( A Nb ) c (cid:1) + IE (cid:20) exp (cid:26) CN − α sup ≤ t ≤ T k N Y Ns k (cid:27) ( A Nb ) c (cid:21) + 1 , We first note that the arguments used in the proof of (4.8), in the particularcase γ = 0, yieldIP( (cid:0) A Nb ) c (cid:1) ≤ IP (cid:18) sup ≤ t ≤ T (cid:13)(cid:13)(cid:13) Y Nt (cid:13)(cid:13)(cid:13) > b (cid:19) + k X j =1 IP (cid:18) sup ≤ t ≤ T ( ξ N,jt − ¯ βt ) > b ′ ¯ βT (cid:19) . e − CN , (4.11)for some constant C >
0. We next estimate the productIE (cid:20) exp (cid:26) CN − α sup ≤ t ≤ T k N Y Ns k (cid:27)(cid:21) IP (cid:0) ( A Nb ) c (cid:1) . For the same reason as in the previous proof, we need only consider the case d = k = 1. It follows from Proposition 5.1 that the first factor satisfiesIE " exp ( CN − α sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z Nβ ( Z Ns )0 M ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) . e CN − α . Finally there exist two positive constants C and C such thatIE (cid:20) exp (cid:26) a − N C (cid:18) k z k + N α sup ≤ t ≤ T k Y Nt k (cid:19) ( A Nb ) c (cid:27)(cid:21) . { C N − α − C N }≤ , N large enough. So a N log of the above tends to 0, as N → ∞ . Step 2 : Estimate of (4.10) We first note that a N log IE (cid:20) exp (cid:26) a − N C (cid:18) N − α k z k + N α sup ≤ t ≤ T k Y Ns k (cid:19) A Nb (cid:27)(cid:21) ≤ CN − α k z k + a N log IE (cid:20) exp (cid:26) a − N CN α sup ≤ t ≤ T k Y Ns k A Nb (cid:27)(cid:21) . The first term on the right tends to 0 as N → ∞ . It remains to take care ofthe second term. Since Y Nt is a martingale, it is clear that the process (cid:26) exp (cid:18) a − N C N α k Y Nt k (cid:19) , t ≥ (cid:27) is a submartingale. Consequently, from Doob’s L submartingale inequality,IE (cid:20) sup ≤ t ≤ T exp n a − N CN α k Y Nt k A Nb o(cid:21) ≤ IE (cid:20) sup ≤ t ≤ T ∧ ¯ τ b exp (cid:8) a − N CN α k Y Nt k (cid:9)(cid:21) ≤ (cid:2) exp (cid:8) a − N CN α k Y NT ∧ ¯ τ b k (cid:9)(cid:3) NextIE (cid:2) exp (cid:8) CN − α k Y NT ∧ ¯ τ b k (cid:9)(cid:3) ≤ r IE h exp n CN − α (cid:16) k Y NT ∧ ¯ τ b k − k Y NT ∧ ¯ τ b k (cid:17)oi × r IE h exp n CN − α k Y NT ∧ ¯ τ b k oi (4.12)Consider first the first factor on the right hand side of (4.12). We deducefrom the definition of ¯ τ b that k Y NT ∧ ¯ τ b k − k Y NT ∧ ¯ τ b k = (cid:16) Y NT ∧ ¯ τ b + Y NT ∧ ¯ τ b , Y NT ∧ ¯ τ b − Y NT ∧ ¯ τ b (cid:17) ≤ ( c T + b + 2 N − sup j k h j k ) (cid:13)(cid:13)(cid:13) Y NT ∧ ¯ τ b − Y NT ∧ ¯ τ b (cid:13)(cid:13)(cid:13) , with c T = P kj =1 k h j k (1 + b ′ ) β j T . Consequently the square of the first factoron the right of (4.12) is bounded from above byIE h exp n CN − α (cid:13)(cid:13)(cid:13) Y NT ∧ ¯ τ b − Y NT ∧ ¯ τ b (cid:13)(cid:13)(cid:13)oi ≤ IE h exp n CN − α (cid:13)(cid:13)(cid:13) Y NT − Y NT (cid:13)(cid:13)(cid:13)oi , d = 1, note thatexp n CN − α (cid:12)(cid:12)(cid:12) Y NT − Y NT (cid:12)(cid:12)(cid:12)o ≤ exp n CN − α (cid:16) Y NT − Y NT (cid:17)o + exp n CN − α (cid:16) Y NT − Y NT (cid:17)o and exploit Proposition 4.4 in order to conclude concerning a N log of the firstfactor on the right of (4.12).We next note that (cid:13)(cid:13)(cid:13) Y NT ∧ ¯ τ b (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) Y NT (cid:13)(cid:13)(cid:13) {k Y NT k≤ b } + ( b + N − sup j k h j k ) { ¯ τ b 3. We have 22E exp n CN − α | ξ N | {| ξ N |≤√ aN/ } o = ⌊ aN/ ⌋ X k = ⌈ aN/ ⌉ exp (cid:26) CN − α ( k − aN ) aN (cid:27) e − aN ( aN ) k k ! . Z √ aN/ −√ aN/ exp (cid:8) CN − α x (cid:9) e − aN ( aN ) aN + x √ aN ( aN + x √ aN )! √ aN dx . √ π Z √ aN/ −√ aN/ exp (cid:8) CN − α x (cid:9) e x √ aN (cid:18) x √ aN (cid:19) − ( aN + x √ aN ) dx ≤ √ π Z √ aN/ −√ aN/ exp (cid:8) CN − α x (cid:9) × exp (cid:26) x √ aN − ( aN + x √ aN ) (cid:20) x √ aN − x aN + x aN ) / x< (cid:21)(cid:27) dx ≤ √ π Z √ aN/ −√ aN/ exp (cid:26) CN − α x − x x √ aN x> (cid:27) dx ≤ √ π Z √ aN/ −√ aN/ exp (cid:26) CN − α x − x (cid:27) dx We have proved that the second factor on the right of (4.13) remainsbounded, as N → ∞ . We next consider the first factor on the right of(4.13). We first note thatexp (cid:8) C ′ N − α { τ b We note that the full strength of (4.1) is necessary for theproof of Proposition 4.5. Indeed, while a N log IE exp { CN − α sup ≤ t ≤ T k Y Nt k} certainly does not converge to as N → ∞ , clearly with high probability k Y Nt k is smaller than k Y Nt k , but IE exp { CN − α k Y Nt k } = ∞ . .4 Large deviations of e Y N,α We first define the Fenchel–Legendre transform ofΛ( ν ) = 12 IE[ ν ( Q ) ]= k X j =1 β j (0)2 Z [0 ,T ] s ∧ t < h j , ν > ( ds ) < h j , ν > ( dt ) , where Q has been defined by (4.3), ν = ( ν , . . . , ν d ) is a vector of signedmeasures and < h j , ν > ( dt ) = P di =1 h ij ν i ( dt ), h ij being the i –th coordinate ofthe vector h j . We have exploited the fact that ν ( Q ) is the sum over j of zeromean mutually independent random variables. For each φ ∈ D ([0 , T ]; IR d ),we define Λ ∗ ( φ ) = sup ν ∈ ( D ([0 ,T ];IR d )) ∗ { ν ( φ ) − Λ( ν ) } . The next step will consist in proving that the sequence of processes { e Y N,α } N ≥ satisfies a Large Deviation Principle. Theorem 4.7. The sequence { e Y N,α , N ≥ } satisfies the Large DeviationPrinciple in D ([0 , T ]; IR d ) equipped with the supnorm topology, with the con-vex, good rate function Λ ∗ and with speed a N , in the sense that for any Borelsubset Γ ⊂ D ([0 , T ]; IR d ) , − inf φ ∈ ˚Γ Λ ∗ ( φ ) ≤ lim inf N a N log IP( e Y N,α ∈ Γ) ≤ lim sup N a N log IP( e Y N,α ∈ Γ) ≤ − inf φ ∈ Γ Λ ∗ ( φ ) . Since there is a difficulty with having a topology on D ([0 , T ]; IR d ) whichmakes it a topological vector space, and allows for a simple characterizationof the class of compact sets, we shall use a small detour for the proof of theabove Theorem. Recall that e Y N,αt = Y N,αt + Z t V N,αs ds, where Y N,αt is piecewise constant, with jumps of size h j N α − . Let Y N,α,ct de-note the continuous piecewise linear approximation of Y N,αt , which is defined24s follows. Let 0 = τ N < τ N < τ N < · · · denote the successive jump timesof the process Y N,αt . For i ≥ 0, on the interval [ τ Ni , τ Ni +1 ], Y N,α,ct = τ Ni +1 − tτ Ni +1 − τ Ni Y N,ατ Ni + t − τ Ni τ Ni +1 − τ Ni Y N,ατ Ni +1 . Next we define ee Y N,αt = Y N,α,ct + Z t V N,αs ds . We note that(4.14) sup ≤ t ≤ T (cid:13)(cid:13)(cid:13)(cid:13) e Y N,αt − ee Y N,αt (cid:13)(cid:13)(cid:13)(cid:13) ≤ sup j k h j k N α − , hence for any δ > 0, for N large enough,IP (cid:18) sup ≤ t ≤ T (cid:13)(cid:13)(cid:13)(cid:13) e Y N,αt − ee Y N,αt (cid:13)(cid:13)(cid:13)(cid:13) ≥ δ (cid:19) = 0 . This implies clearly Lemma 4.8. The two sequences { e Y N,α } N ≥ and n ee Y N,α o N ≥ are exponen-tially equivalent in D ([0 , T ]; IR d ) , equipped with the supnorm topology, in thesense that for each δ > , lim sup N →∞ a N log IP (cid:18) sup ≤ t ≤ T (cid:13)(cid:13)(cid:13)(cid:13) e Y N,αt − ee Y N,αt (cid:13)(cid:13)(cid:13)(cid:13) ≥ δ (cid:19) = −∞ . We shall prove below the following. Proposition 4.9. The sequence n ee Y N,α o N ≥ is exponentially tight in C ([0 , T ]; IR d ) , the space of continuous functions from [0 , T ] into IR d , whichstart from at t = 0 , in the sense that for any R > , there exists a compactsubset K R ⊂⊂ C ([0 , T ] : IR d ) such that lim sup N →∞ a N log IP (cid:18) ee Y N,α K R (cid:19) ≤ − R. Proof of Theorem 4.7 From (4.14), we deduce that a N log IE exp (cid:26) Ca − N ν (cid:18) e Y N,αt − ee Y N,αt (cid:19)(cid:27) . N α − → , as N → ∞ . Consequently, again by the argument of Proposition 4.3, wededuce from that same Proposition that for any signed measure ν on [0 , T ],as N → ∞ , a N log IE exp (cid:26) a − N ν (cid:18) ee Y N,α (cid:19)(cid:27) → 12 IE[ ν ( Q ) ] . This, together with Proposition 4.9, allows us to apply Corollary 4.6.14from [2], to conclude that the sequence n ee Y N,α o N ≥ satisfies a LDP in C ([0 , T ]; IR d ) with the good rate function Λ ∗ , and speed a N . Since C ([0 , T ]; IR d ) is closed in D ([0 , T ]; IR d ) equipped with the supnorm topol-ogy, it follows from Lemma 4.1.5 in [2] that the same LDP holds in thelatter space, with the same rate function Λ ∗ , extended to that space byΛ ∗ ( φ ) = + ∞ for φ ∈ D ([0 , T ]; IR d ) \ C ([0 , T ]; IR d ). The result now followsfrom Lemma 4.8, in view of Theorem 4.2.13 from [2]. (cid:3) We now turn to the Proof of Proposition 4.9 Clearly it suffices to prove both that(4.15) lim R →∞ lim sup N →∞ a N log IP (cid:18) sup ≤ t ≤ T k V N,αt k ≥ R (cid:19) = −∞ , and that the sequence { Y N,α,c } N ≥ is exponentially tight in C ([0 , T ]; IR d ).Let us first establish (4.15). It follows from (4.1) that k V N,αt k ≤ C (cid:18) k z k + 1 + sup ≤ t ≤ T k Y N,αt k (cid:19) . Consequently, if R > C ( k z k + 1), with R ′ = (2 C ) − R , a N log IP( sup ≤ t ≤ T k V N,αt k > R ) ≤ a N log IP( sup ≤ t ≤ T k Y N,αt k > R ′ ) ≤ a N log (cid:18) e − a − N R ′ IE sup ≤ t ≤ T exp { a − N k Y N,αt k} (cid:19) ≤ − R ′ + a N log IE (cid:18) sup ≤ t ≤ T exp { a − N k Y N,αt k} (cid:19) . 26t follows from Doob’s submartingale inequality and a combination of Lemma4.1 and Proposition 4.4 that the lim sup as N → ∞ of the second term ofthe last right hand side is finite. (4.15) clearly follows.It remains to consider Y N,α,c . Define the modulus of continuity of anelement x ∈ C ([0 , T ]; IR d ) as w x ( δ ) = sup ≤ s,t ≤ T, | s − t |≤ δ k x ( t ) − x ( s ) k . Itfollows from Ascoli’s theorem that for any sequence { δ ℓ , ℓ ≥ } of positivenumbers, the following is a compact subset of C ([0 , T ]; IR d ): \ ℓ ≥ { x : w x ( δ ℓ ) ≤ ℓ − } . Suppose that for each ℓ ≥ R > 0, we can find δ R,ℓ > N ≥ w Y N,α,c ( δ R,ℓ ) ≥ ℓ − ) ≤ exp {− a − N ( R + ℓ ) } . From this we deduce thatIP( ∪ ℓ ≥ { w Y N,α,c ( δ R,ℓ ) ≥ ℓ − } ) ≤ X ℓ ≥ e − a − N ( R + ℓ ) ≤ e − a − N R , so that lim sup N →∞ a N log IP( ∪ ℓ ≥ { w Y N,α,c ( δ R,ℓ ) ≥ ℓ − } ) ≤ − R, from which the result follows. A sufficient condition for (4.16) to be true isthat for any b > δ → lim sup N →∞ a N log IP ( w Y N,α,c ( δ ) > b ) = −∞ . In turn a sufficient condition for this is that(4.17) lim δ → lim sup N →∞ a N log IP ( w Y N,α ( δ ) > b ) = −∞ , which we now prove. It is not hard to see thatIP ( w Y N,α ( δ ) > b ) ≤ (cid:18) Tδ + 1 (cid:19) sup ≤ t ≤ T IP (cid:18) sup t ≤ s ≤ t +2 δ k Y N,αs − Y N,αt k ≥ b/ (cid:19) ≤ (cid:18) Tδ + 1 (cid:19) sup ≤ t ≤ T IP (cid:18) sup t ≤ s ≤ t +2 δ exp { a − N δ − / k Y N,αs − Y N,αt k} ≥ exp { ba − N / √ δ } (cid:19) ≤ (cid:18) Tδ + 1 (cid:19) exp {− ba − N / √ δ } sup ≤ t ≤ T IE exp { a − N δ − / k Y N,αt +2 δ − Y N,αt k} , { a − N δ − / k Y N,αt +2 δ − Y N,αt k} ≤ k Y j =1 exp ( N − α δ − / k h j k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t +2 δt Z Nβ j ( Z Ns )0 M j ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) Using repeatedly Cauchy–Schwartz’s inequality, we see that it suffices toestimate for each j IE exp ( CN − α δ − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t +2 δt Z Nβ j ( Z Ns )0 M j ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) ≤ IE exp ( CN − α δ − / Z t +2 δt Z Nβ j ( Z Ns )0 M j ( ds, du ) ) + IE exp ( − CN − α δ − / Z t +2 δt Z Nβ j ( Z Ns )0 M j ( ds, du ) ) ≤ { C N − α ¯ β j } , where ¯ β j = sup z β j ( z ), we have used Proposition 5.1 and the inequality e x − − x ≤ x , valid for x ≤ log(2), which we have applied with x =2 CN − α δ − / and x = − CN − α δ − / (recall that we will first let N → ∞ ).Putting together the last estimates yieldslim sup N →∞ a N IP ( w Y N,α ( δ ) > b ) ≤ − b √ δ + C. (4.17) follows, and the Proposition is proved. (cid:3) Λ ∗ Let us compute Λ ∗ in the three examples which we discussed above in section2. Here we do not translate z ∗ to the origin. Λ ∗ for the SIS model Recall that in this case d = 1, k = 2, h = 1, β ( z ) = λz (1 − z ), h = − β ( z ) = γz . If λ > γ , there is a unique stable endemic equilibrium z ∗ =1 − γ/λ . We first computeΛ( ν ) = 12 IE Z [0 ,T ] × [0 ,T ] Q ( s ) Q ( t ) ν ( ds ) ν ( dt ) , Q ( t ) = Z t Z β ( z ∗ )0 M ( ds, du ) − Z t Z β ( z ∗ )0 M ( ds, du ) . It is easy to check that IE[ Q ( t ) Q ( s )] = σ ( z ∗ ) s ∧ t , where σ ( z ∗ ) = β ( z ∗ ) + β ( z ∗ ) = 2 γλ ( λ − γ ) . Consequently Λ( ν ) = σ ( z ∗ )2 Z [0 ,T ] × [0 ,T ] s ∧ t ν ( ds ) ν ( dt ) . We now need to compute Λ ∗ ( φ ) in case φ ∈ C ([0 , T ]). We should takethe supremum over the signed measures ν on [0 , T ] of the quantity Z [0 ,T ] φ ( t ) ν ( dt ) − σ ( z ∗ )2 Z [0 ,T ] × [0 ,T ] s ∧ t ν ( ds ) ν ( dt ) . The supremum is achieved at the signed measure ν which makes the gradientwith respect to ν of the above zero, if any. We first note that for such a ν toexist, we need that φ (0) = 0, unless Λ ∗ ( φ ) = + ∞ . Now the optimal ν mustsatisfy φ ( t ) = σ ( z ∗ ) Z [0 ,T ] s ∧ t ν ( ds )= σ ( z ∗ ) Z [0 ,t ] s ν ( ds ) + σ ( z ∗ ) t Z ( t,T ] ν ( dt ) . So necessarily ν ( dt ) = − φ ′′ ( t ) σ ( z ∗ ) dt + φ ′ ( T ) σ ( z ∗ ) δ T ( dt ) . Substituting this signed measure ν in the above formula, we obtain that Z [0 ,T ] φ ( t ) ν ( dt ) = φ ′ ( T ) σ ( z ∗ ) φ ( T ) − Z T φ ′′ σ ( z ∗ ) ( t ) φ ( t ) dt = 1 σ ( z ∗ ) Z T | φ ′ ( t ) | dt . ConsequentlyΛ( φ ) = ( σ ( z ∗ ) R T | φ ′ ( t ) | dt, if φ (0) = 0 and φ is absolutely continuous;+ ∞ , otherwise . .5.2 Computation of Λ ∗ for the SIRS model In this model, d = 2 and k = 3. We have h = (cid:0) − (cid:1) , β ( z ) = λz z , h = (cid:0) − (cid:1) , β ( z ) = γz , and h = (cid:0) (cid:1) , β ( z ) = ρ (1 − z − z ). In the case λ > γ , thereis a unique stable endemic equilibrium, namely z ∗ = (cid:0) ργ + ρ ( − γλ ) γλ (cid:1) . In order tosimplify the notations, we shall write a = β ( z ∗ ), b = β ( z ∗ ), c = β ( z ∗ ) and A = ab + ac + bc . We haveΛ( ν ) = a Z [0 ,T ] × [0 ,T ] s ∧ t ( ν − ν )( ds )( ν − ν )( dt )+ b Z [0 ,T ] × [0 ,T ] s ∧ tν ( ds ) ν ( dt ) + c Z [0 ,T ] × [0 ,T ] s ∧ tν ( ds ) ν ( dt )The functional to be maximized with respect to ν if < ν , φ > + < ν , φ > − a Z [0 ,T ] × [0 ,T ] s ∧ t ( ν − ν )( ds )( ν − ν )( dt ) − b Z [0 ,T ] × [0 ,T ] s ∧ tν ( ds ) ν ( dt ) − c Z [0 ,T ] × [0 ,T ] s ∧ tν ( ds ) ν ( dt )Writing that the gradient w.r.t. ν and ν of this functional is zero leads tothe identities φ ( t ) = a Z [0 ,T ] s ∧ t ( ν − ν )( ds ) + b Z [0 ,T ] s ∧ tν ( ds ) ,φ ( t ) = a Z [0 ,T ] s ∧ t ( ν − ν )( ds ) + c Z [0 ,T ] s ∧ tν ( ds ) . This implies the identities ν ( dt ) = − (cid:18) a + cA φ ′′ ( t ) + aA φ ′′ ( t ) (cid:19) dt + (cid:18) a + cA φ ′ ( T ) + aA φ ′ ( T ) (cid:19) δ T ,ν ( dt ) = − (cid:18) aA φ ′′ ( t ) + a + bA φ ′′ ( t ) (cid:19) dt + (cid:18) aA φ ′ ( T ) + a + bA φ ′ ( T ) (cid:19) δ T . Finally we deduce that Λ ∗ ( φ ) is + ∞ unless φ is absolutely continuous and φ (0) = 0, in which caseΛ ∗ ( φ ) = a A Z T | φ ′ ( t ) + φ ′ ( t ) | dt + c A Z T | φ ′ ( t ) | dt + b A Z T | φ ′ ( t ) | dt . .5.3 Computation of Λ ∗ for the SIR model with demography In this case, d = 2, k = 4, h = (cid:0) − (cid:1) , β ( z ) = λz z , h = (cid:0) − (cid:1) , β ( z ) = ( γ + µ ) z , h = (cid:0) (cid:1) , β ( z ) = µ and h = (cid:0) − (cid:1) , β ( z ) = µz . In the case λ > γ + µ ,there is a unique stable endemic equilibrium, namely z ∗ = (cid:0) µ ( γ + µ − λ ) γ + µλ (cid:1) . Weshall use the notations a = β ( z ∗ ), b = β ( z ∗ ), c = β ( z ∗ ) + β ( z ∗ ) and A = ab + ac + bc We haveΛ( ν ) = a Z [0 ,T ] × [0 ,T ] s ∧ t ( ν − ν )( ds )( ν − ν )( dt )+ b Z [0 ,T ] × [0 ,T ] s ∧ tν ( ds ) ν ( dt ) + c Z [0 ,T ] × [0 ,T ] s ∧ tν ( ds ) ν ( dt )Formally the functional Λ( ν ) has exactly the same form as in the case ofthe SIRS model, only the constants have different values. The same com-putations as in the previous subsection lead to the same result, namely thatΛ ∗ ( φ ) is + ∞ unless φ is absolutely continuous and φ (0) = 0, in which caseΛ ∗ ( φ ) = a A Z T | φ ′ ( t ) + φ ′ ( t ) | dt + c A Z T | φ ′ ( t ) | dt + b A Z T | φ ′ ( t ) | dt . Z N We again equip D ([0 , T ]; IR d ) with the supnorm topology. Let for z ∈ IR d F z : D ([0 , T ]; IR d ) D ([0 , T ]; IR d ) be the continuous map which to x associates y solution of the ODE y ( t ) = z + Z t ∇ b (0) y ( s ) ds + x ( t ) , and for each N ≥ F z,N : D ([0 , T ]; IR d ) D ([0 , T ]; IR d ) be the continuousmap which to x associates y N solution of the ODE y N ( t ) = N α ( { z ∗ + N − α z } N − z ∗ ) + Z t ∇ b (0) y N ( s ) ds + x ( t ) . We have(4.18) F z,N ( x )( t ) − F z ( x )( t ) = exp[ ∇ b (0) t ] (cid:0) N α ( { z ∗ + N − α z } N − z ∗ ) − z (cid:1) , N → ∞ , uniformly in t ∈ [0 , T ] and x ∈ D ([0 , T ]; IR d ). We want to study the moderate deviations of Z N , or in otherwords the large deviations of Z N,α = N α Z N . In what follows, we shall denoteby Z N,αz the process Z N,α starting from Z N,α (0) = N α ( { z ∗ + N − α z } N − z ∗ ).From (4.2), Z N,αz = F z,N ( e Y N,α ), hence the following statement is a conse-quence of Theorem 4.7, (4.18) and Corollary 4.2.21 from [2]. Theorem 4.10. Assume that ( H. and ( H. hold. The collection of pro-cesses { Z N,αz ( t ) , ≤ t ≤ T } N ≥ satisfies a large deviations principle withspeed a N = N α − and the good rate function I z,T ( φ ) = Λ ∗ ( F − z ( φ ))= ( Λ ∗ (cid:0) φ ( · ) − z − ∇ b (0) R · φ ( s ) ds (cid:1) if φ (0) = z ;+ ∞ , otherwise.More precisely, for any Borel subset Γ ⊂ D ([0 , T ]; IR d ) , − inf φ ∈ ˚Γ I z,T ( φ ) ≤ lim inf N a N log IP( Z N,αz ∈ Γ) ≤ lim sup N a N log IP( Z N,αz ∈ Γ) ≤ − inf φ ∈ Γ I z,T ( φ ) . Since the mapping F z has the nice property that F z ( x )( t ) − F z ′ ( x )( t ) =exp[ ∇ b (0) t ]( x − x ′ ), it follows readily again from Corollary 4.2.21 in [2] thatthe above result can be extended to the following statement. Theorem 4.11. Assume that ( H. and ( H. hold. For any closed set F ⊂ D ([0 , T ]; IR d ) , for any sequence z N → z , lim sup N →∞ a N log IP( Z N,αz N ∈ F ) ≤ − inf φ ∈ F I z,T ( φ ) . For any open set G ⊂ D ([0 , T ]; IR d ) , for any sequence z N → z , lim inf N →∞ a N log IP( Z N,αz N ∈ G ) ≥ − inf φ ∈ G I z,T ( φ ) . From this last Theorem, we can deduce, with the same proof as that ofCorollary 5.6.15 in [2], the following Corollary.32 orollary 4.12. Assume that ( H. and ( H. hold. Let K denote an arbi-trary compact subset of IR d .For any closed set F ⊂ D ([0 , T ]; IR d ) , lim sup N →∞ a N log sup z ∈ K IP( Z N,αz ∈ F ) ≤ − inf φ ∈ F,z ∈ K I z,T ( φ ) . For any open set G ⊂ D ([0 , T ]; IR d ) , lim inf N →∞ a N log inf z ∈ K IP( Z N,αz ∈ G ) ≥ − sup z ∈ K inf φ ∈ G I z,T ( φ ) . We now define V ( z, z ′ , t ) = inf φ, φ (0)= z,φ ( t )= z ′ I z,t ( φ ) ,V ( z, z ′ ) = inf t> V ( z, z ′ , t ) ,V a = inf z, z = − a V (0 , z ) , where a > 0, and we recall that we have translated the endemic equilibrium z ∗ at the origin.We can now state our main result. Theorem 4.13. Assume that ( H. and ( H. hold. For some a > , let T Nz,a := inf { t > , Z N,αz, ( t ) ≤ − a } , where Z N,αz, ( t ) denotes the first coordinateof the process Z N,αz ( t ) . The following hold.For any z ∈ IR d such that z > − a , and any η > , lim N →∞ IP (cid:16) e a − N ( V a − η ) < T Nz,a < e a − N ( V a + η ) (cid:17) = 1 , and lim N →∞ a N log IE( T Nz,a ) = V a . Given Corollary 4.12, the proof of the above result follows the exact samesteps as that of Theorem 5.7.11 in [2], with some minor modifications, toadapt to the fact that our processes have discontinuous trajectories, see theproof of Theorem 7.14 in [5], or of Theorem 4.2.17 in [1].Recall that a − N = N − α . In the CLT regime, α = 1 / a − N = 1, while inthe LD regime, α = 0, a − N = N . 33 .7.1 Interpretation. The critical population size Going back to the original coordinates, i.e. z ∗ = 0, we should interpret Z N,α ( t ) as Z N,α ( t ) = N α ( Z N ( t ) − z ∗ ). So (dropping the index for the startingpoint in order to simplify our notations), T Na is the first time when Z N ( t ) ≤ z ∗ − aN − α . For T Na to be finite, we need to have z ∗ − aN − α ≥ 0, since Z N ( t )cannot become negative. This is of course no problem for the limit theorem,since aN − α → N → ∞ , while z ∗ is fixed. However, a deviation of theorder of − aN − α is enough for Z N ( t ) to hit zero, if z ∗ is of the order of N − α ,which means that N is of the order of ( z ∗ ) − /α . e N − α V a is the order ofmagnitude of the time needed for Z Nt − z t to make a deviation of size aN − α .This is sufficient to extinguish an epidemic, provided z ∗ is of the same order,so that the corresponding critical size is N c,α ∼ (1 /z ∗ ) /α , which is roughlythe CLT critical population size raised to the power 1 / α . V a in the SIS model In the particular case of the SIS model, we can compute explicitly the valueof the quasi–potential V a . In this case, d = 1, the linearized ODE aroundthe endemic equilibrium translated at 0 reads˙ x = − ( λ − γ ) x + u, and the cost functional to minimize is I T ( u ) = λ γ ( λ − γ ) Z T u ( t ) dt . We are looking for the minimal cost for driving x from 0 to − a . We nowexploit the Pontryagin maximum principle, see [8]. The Hamiltonian reads H ( x, p, u ) = − ( λ − γ ) px + pu − λ γ ( λ − γ ) u . The optimal control ˆ u must maximize the Hamiltonian, so it satisfies ˆ u = γ ( λ − γ ) λ p . Since the final time is free and the system is autonomous, theHamiltonian vanishes along the optimal trajectory, so that along such a tra-jectory, either p = 0, in which case ˆ u = 0, or else x = γλ p , hence ˆ u = 2( λ − γ ) x .Finally the pieces of optimal trajectory which move towards the origin cor-respond to u ≡ 0, those which move away from the origin (this is the case we34re interested in) satisfy the time reversed ODE ˙ x = ( λ − γ ) x . There is nooptimal trajectory from x = 0 to x = − a . However, if we start from x = − ε ,the optimal trajectory is x ( t ) = − e ( λ − γ ) t ε , so ˆ u ( t ) = − λ − γ ) e ( λ − γ ) t ε , thefinal state − a is reached at time ( λ − γ ) − log( a/ε ), and the optimal costis λ γ ( a − ε ). A possible sub–optimal control starting from 0 is as follows.Choose u = − ε , until x ( t ) reaches − ε , whose cost is ofthe order of ε , and then choose the optimal feedback, until − a is reached.Letting ε → 0, the total cost converges to V a = λγ a . We do that comparison in case of the SIS model, for which we have explicitexpressions for the rate functions and the quasi–potentials. We still translate z ∗ at the origin, and start our process at the origin : Z N = 0. To make achange with the above, we fix a > t large) theupper bounds for IP( N α Z Nt ≥ a ) in the three cases α = 1 / < α < / α = 0 (large deviations).We start with the central limit theorem . It is easy to see that U t =lim N →∞ √ N Z Nt solves the SDE U t = − ( λ − γ ) Z t U s ds + p γ ( λ − γ ) /λB t , so that the asymptotic variance of U t is γ/λ . Consequently for a > η > 0, there exist t and N large enough such that we have thefollowing upper bound for the probability of a positive deviation of √ N Z Nt IP( √ N Z Nt ≥ a ) ≤ exp (cid:26) − λa γ + η (cid:27) . This bound follows from the following estimate, valid for a N (0 , σ ) randomvariable ξ : IP( ξ > a ) ≤ e − λa e λ σ after optimizing over λ > moderate deviations . Theorem 4.10 combined withthe computation from the last subsection indicates that for 0 < α < / 2, any η > 0, there exists t and N large enough such thatIP( N α Z Nt ≥ a ) ≤ exp (cid:26) − N − α (cid:18) λa γ − η (cid:19)(cid:27) . 35e finally consider the large deviations . Here we need to assume that a < γ/λ . We exploit the computations from sections 4.2.6 and A.6 in [1].The optimal trajectory to go from ε to a is the original ODE, but timereversed, i.e. it follows the ODE ˙ x t = β ( x t ) − β ( x t ). The running cost is( β ( x t ) − β ( x t )) log (cid:16) β ( x t ) β ( x t ) (cid:17) , so the total cost is Z T a T ε log (cid:18) β ( x t ) β ( x t ) (cid:19) ( β ( x t ) − β ( x t )) dt = Z T a T . eps log (cid:18) β ( x t ) β ( x t ) (cid:19) ˙ x t dt = Z aε log (cid:18) β ( x ) β ( x ) (cid:19) dx → a + (cid:16) γλ − a (cid:17) log (cid:18) − a λγ (cid:19) , as ε → . Consequently, from Theorem 3.4, for any η > 0, there exists t and N largeenough such thatIP( Z Nt ≥ a ) ≤ exp (cid:26) − N (cid:20) a + (cid:16) γλ − a (cid:17) log (cid:18) − a λγ (cid:19) − η (cid:21)(cid:27) . We note that Moderate Deviations resembles much more the Central LimitTheorem than Large Deviations. The fact that the discontinuity in the formof the rate function is exactly at α = 0 is typical of random variables withlight tails. The situation would be quite different with heavy tails, see e. g.section VIII.4 in Petrov [7].Note however that for small a , a + (cid:16) γλ − a (cid:17) log (cid:18) − a λγ (cid:19) ∼ λa γ , which is not too surprising, and in a sense reconciles Large Deviations andModerate Deviations. Were our driving noises Brownian, then the LD ratefunction would be quadratic as that of MD, but the LD quasi–potential is theminimal cost when controlling the LLN ODE, while the MD quasi–potentialis the minimal cost when controlling the linearized ODE around the endemicequilibrium. In this Appendix, we establish the following technical result.36 roposition 5.1. Let M be a standard Poisson random mesure on IR , and M ( dt, du ) = M ( dt, du ) − dt du the associated compensated measure. If ϕ isan IR + –valued predictable process such that R T ϕ t dt has exponential momentsof any order, and a ∈ IR , then for any ≤ t ≤ T , IE (cid:20) sup ≤ s ≤ t exp (cid:26) a Z s Z ϕ r M ( dr, du ) (cid:27)(cid:21) . (cid:18) IE exp (cid:26) ( e a − − a ) Z t ϕ s ds (cid:27)(cid:19) / . Proof Consider with b ≥ X t = a Z t Z ϕ s M ( ds, du ) − b Z t ϕ s ds . It follows from Itˆo’s formula that e X t = 1 − b Z t e X s ϕ s ds + a Z t Z ϕ s e X s − M ( ds, du )+ ( e a − − a ) Z t Z ϕ s e X s − M ( ds, du ) . From Lemma 5.2 below, M t = R t R ϕ s e X s − M ( ds, du ) is a martingale. Hence e X is a martingale if b = ( e a − − a ), a submartingale if we replace = by < , anda supermartingale if we replace = by > . Consequently if b ≥ ( e a − − a ),IE e X t ≤ 1. Now, using first Doob’s L inequality for submartingales, andlater Schwartz’s inequality, we haveIE (cid:20) sup ≤ s ≤ t exp (cid:26) a Z s Z ϕ r M ( dr, du ) (cid:27)(cid:21) . IE exp (cid:26) a Z t Z ϕ s M ( ds, du ) (cid:27) = IE (cid:18) exp (cid:26) a Z t Z ϕ s M ( ds, du ) − b Z t ϕ s ds (cid:27) exp (cid:26) b Z t ϕ s ds (cid:27)(cid:19) ≤ (cid:18) IE exp (cid:26) a Z t Z ϕ s M ( ds, du ) − b Z t ϕ s ds (cid:27)(cid:19) / × (cid:18) IE exp (cid:26) b Z t ϕ s ds (cid:27)(cid:19) / If 2 b = e a − − a , it follows from the previous argument that the firstfactor on the second right hand side is less than or equal to 1, hence theresult follows. (cid:3) In order to complete the proof of Proposition 5.1, we still need to establish37 emma 5.2. The process ϕ satisfying the same assumptions as in Propo-sition 5.1, and X t being given by (5.1) , M t = R t R ϕ s e X s − M ( ds, du ) is amartingale. Proof It is plain that M t is a local martingale, whose predictable quadraticvariation is given as < M > t = Z t e X s ϕ s ds ≤ exp n a R t R ϕ s M ( ds, du ) o R t ϕ s ds, if a > ≤ exp n − a + b ) R t ϕ s ds o R t ϕ s ds, if a ≤ . All we need to show is that the above quantity is integrable. It is clearlya consequence of the assumption in case a < 0. In case a > 0, the secondfactor of the right hand side has finite exponential moments, so is squareintegrable, and all we need to show is thatIE exp (cid:26) a Z t Z ϕ s M ( ds, du ) (cid:27) < ∞ . Using Itˆo’s formula we have Y t = exp (cid:26) a Z t Z ϕ s M ( ds, du ) − ( e a − Z t ϕ s ds (cid:27) = 1 + ( e a − Z t Z ϕ s Y s − M ( ds, du ) . The same computation with ϕ s replaced by ϕ ns = ϕ s ∧ n , and then Y s replacedby Y ns would show that Y nt is a martingale satisfying IE Y nt = 1. But 0 ≤ Y nt → Y t a.s., hence Fatou’s Lemma implies that IE Y t ≤ 1. Since4 a Z t Z ϕ s M ( ds, du ) = 4 a Z t Z ϕ s M ( ds, du ) − e a − Z t ϕ s ds + e a − Z t ϕ s ds, it follows from Schwartz’s inequality thatIE exp (cid:26) a Z t Z ϕ s M ( ds, du ) (cid:27) ≤ p IE Y t s IE exp (cid:26) ( e a − Z t ϕ s ds (cid:27) , and the result follows from our assumption on ϕ . 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