Recall that a family {Xε} is called exponentially tight with the speed function r(ε) if for each Q>0 0$]]>, there exists a compact set K⊂X such that lim supε→0r(ε)logP(Xε∉K)≤−Q. For an exponentially tight family, the LDP is equivalent to the weak LDP; the latter by definition means that the upper bound (2) holds for all compact sets F, whereas the lower bound (3) still holds for all open sets G. An equivalent formulation of the weak LDP is the following: for each x∈X , (5) limδ→0lim supε→0r(ε)logP(Xε∈Bδ(x))=limδ→0lim infε→0r(ε)logP(Xε∈Bδ(x))=−I(x), where Bδ(x) denotes the open ball with center x and radius δ.

To prove (5), we will use a certain extension of the contraction principle, which in its classical form (e.g., [8], Section 3.1, and [7], Section 4.2.1) states the LDP for a family Xε=F(Yε) , where Yε is a family of random elements in a Polish space Y that satisfies an LDP with a good rate function J, and F:Y→X is a continuous mapping. The rate function for Xε in this case has the form I(x)=infy:F(y)=xJ(y). In the sequel, we use two different representations of our particular family Xε as an image of certain family whose LDP is well understood; however, the functions F in these representations fail to be continuous. Within such a framework, the following general lemma appears quite useful. We denote by ρX,ρY the metrics in X,Y and by ΛF the set of continuity points of a mapping F:Y→X . Note that ΛF is Borel measurable; see Appendix II in [3].

Lemma 1 is a simplified and more precise version of Lemma 4 in [12]. The functions Iupper , Ilower are lower semicontinuous: we can show this easily using that, for any sequence xn→x , the sets Θδ/2(xn) are embedded into Θδ(x) for n large enough (see, e.g., Proposition 3 in [12]). In fact, Lemma 1 says that for an arbitrary image of a family {Yε} , one part of an LDP (the upper bound) holds with one rate function, whereas the other part (the lower bound) holds with another rate function. This is our reason to call (6) and (7) the upper and the lower semicontraction principles. To prove (5), it suffices to verify the inequalities (8) Ilower(x)≤I(x),I(x)≤Iupper(x),x∈X. We refer to [12] for a more discussion and an example where the pair of semicontraction principles do not provide an LDP.

In this section, we prove directly that the functional I specified in Theorem 1 is lower semicontinuous, that is, it is indeed a rate functional. This will explain the particular choice of the modified functions a¯,σ¯ . In addition, this will simplify the proofs, where we will use the representation for I(x) presented further.

Define S(x)=∫0xa¯(z)σ¯2(z)dz,x∈R. Then I(f) , if it is finite, can be represented as (9) I(f)=12∫0T(f˙t)2σ¯2(ft)dt+[S(fT)−S(f0)]+12∫0Ta¯2(ft)σ¯2(ft)dt=:I1(f)+I2(f)+I3(f). The function S is continuous; hence, the functional I2 is just continuous. The function a¯2/σ¯2 is lower semicontinuous by the choice of a¯,σ¯ ; thus, the functional I3 is lower semicontinuous. Finally, we can represent I1 in the form I1(f)=I0(Σ(f·)) , where the function Σ(x)=∫0x1σ¯(z)dz is continuous, and the functional I0(f)=12∫0T(f˙t)2dt is known to be lower semicontinuous (this is just the rate functional for the family {εW} ). Hence, I1 is lower semicontinuous, which completes the proof of the statement.

In this section, we prove that the family {Xε} is exponentially tight with the speed function r(ε)=ε2 . Note that Mtε:=∫0tσ2(Xsε)dWs is a continuous martingale with the quadratic characteristics ⟨Mε⟩t=∫0tσ2(Xsε)ds≤Ct; see (4). Recall that Mε can be represented as a Wiener process with the time change t↦⟨Mε⟩t ; see, for example, [10], II. §7. Then, for each R, lim supε→0ε2logP(supt∈[0,T]ε|Mtε|>R)≤lim supε→0ε2logP(supt∈[0,CT]ε|Wt|>R)=−R22CT. R\Big)\\{} & \displaystyle \hspace{1em}\le \underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\log \mathbf{P}\Big(\underset{t\in [0,CT]}{\sup }\varepsilon |W_{t}|>R\Big)=-\frac{{R}^{2}}{2CT}.\end{array}\]]]> On the other hand, for each ω∈Ω such that ε|Mtε(ω)|>R R$]]>, the corresponding trajectory of Xε satisfies |Xtε|≤|x0|+C∫0t(1+|Xsε|)ds+R, and therefore, by the Gronwall inequality, supt∈[0,T]|Xtε|≤(|x0|+CT+R)eCT. Therefore, for any Q>0 0$]]>, there exists R such that lim supε→0ε2logP(supt∈[0,T]|Xtε|>R)≤−Q. R\Big)\le -Q.\]]]>

Next, recall the Arzelà–Ascoli theorem: for a closed set K⊂C(0,T) to be compact, it is necessary and sufficient that it is bounded and equicontinuous. The family εMε is represented as a time changed family εWε , where each Wε is a Wiener process, and the derivative of ⟨Mε⟩t is bounded by C. Using these observations, it is easy to deduce the exponential tightness for {εMε} using the well-known fact that the family {εW} is exponentially tight. On the other hand, for any ω such that the trajectory of Xtε is bounded by R, the corresponding trajectory of the process Xtε−εMtε satisfies the Lipschitz condition w.r.t. t with the constant C(1+R) . Combined with the previous calculation, this easily yields the required exponential tightness.

In what follows, we proceed with the proof of (5). Since now the state space X=C(0,T) is specified, we change the notation and denotes the points in this space by f,g,… . Since the set B1(f) is bounded, the law of Xε restricted to any Bδ(f) does not change if we change the coefficients a,σ on the intervals (−∞,−R],[R,∞) with R>0 0$]]> large enough. Hence, we furthermore assume the coefficients a,σ to be constant on such intervals for some R.

We proceed with the further proof in a step-by-step way, increasing gradually the classes of the coefficients a,σ for which the corresponding LDP is proved. First, let a,σ be constant on the intervals (−∞,z) and (z,∞) with some z∈R . Without loss of generality, we can assume that z=0 . Then we can use Theorem 2.2 [11], where the LDP with the speed function r(ε)=ε2 is established for the pair (Xε,Zε) with Ztε=∫0t1(0,∞)(Xsε)ds,t∈[0,T]. The corresponding rate function in [11] is given in the following form. Denote a±=a(0±),σ±=σ(0±) and define the class H(f) of functions ψ∈AC(0,T) such that ψ˙t=0,ft<0;=1,ft>0;∈[0,1],ft=0. 0;\\{} \in [0,1],& f_{t}=0.\end{array}\right.\]]]> Then the rate functional for (Xε,Zε) equals I(f,ψ)=12∫0TL(ft,f˙t,ϕt)dt with L(x,y,z)=(y−a(x))2σ2(x),x≠0;(a+z+a−(1−z))2σ+2z+σ−2(1−z),x=0,a−σ−2>a+σ+2;a+2σ+2z+a−2σ−2(1−z),x=0,a−σ−2≤a+σ+2 \frac{a_{+}}{{\sigma _{+}^{2}}};\\{} \frac{{a_{+}^{2}}}{{\sigma _{+}^{2}}}z+\frac{{a_{-}^{2}}}{{\sigma _{-}^{2}}}(1-z),& x=0,\hspace{0.1667em}\frac{a_{-}}{{\sigma _{-}^{2}}}\le \frac{a_{+}}{{\sigma _{+}^{2}}}\end{array}\right.\]]]> for all pairs (f,ψ) such that f∈AC(0,T),f0=x0,ψ∈H(f) and, for all other pairs, I(f,ψ)=∞ .

From this result, using the contraction principle (see Section 2), we easily derive the LDP for Xε with the rate function I(f)=infψ∈H(f)I(f,ψ)=∫0TL(ft,f˙t)dt,L(x,y)=infz∈[0,1]L(x,y,z) for f∈AC(0,T) , f0=x0 and I(f)=∞ otherwise. Now only a minor analysis is required to show that this rate function actually coincides with that specified in Theorem 1. First, we observe that L(0,y)=a¯2(0)σ¯2(0). This is obvious if either a−/σ−2≤a+/σ+2 or a−>0 0$]]>, a+<0 . In the case where a−/σ−2>a+/σ+2 a_{+}/{\sigma _{+}^{2}}$]]> and a− , a+ have the same sign, we can verify directly that Lz′(0,y,z) have the same sign for z∈[0,1] , which completes the proof of the required identity.

We will use repeatedly the following fact, which follows easily from the change-of-variables formula: for any f∈AC(0,T) and any set A⊂R with zero Lebesgue measure, the Lebesgue measure of the set (10) ∫0T1ft∈A,f˙t≠0dt=0; see, for example, Lemma 1 in [13]. Applying (10) with A={0} , we conclude that in the above expression for I(f) , the function L can be changed to L(x,y)=(y−a¯(x))2σ¯2(x), which completes the proof of Theorem 1 in this case.

Let, for some z1<⋯<zm , the functions a,σ be constant on the intervals (−∞,z1) , (z1,z2), …,(zm,∞) . Assume that x0∉{zk,k=1,…,m} , which does not restrict the generality of the construction given further, and define the functions ak,σk,k=0,…,m by a0(x)=a(x0),σ0(x)=σ(x0),x∈R, ak(x)=a(zk−),x<zk,a(zk+),x≥zk,σk(x)=σ(zk−),x<zk,σ(zk+),x≥zk,k=1,…,m. Consider a family of independent processes Y0,ε,Yn,k,ε , k=1,…,m , n≥1 , such that Y0,ε solves SDE (1) with the coefficients a0,σ0 and each Yn,k,ε solves a similar SDE with the coefficients ak,σk and the initial value zk . Define iteratively the process X˜ε in the following way: put X˜ε equal Y0,ε until the time moment τ1:=inf{t:Yt0,ε∈{zk,k=1,…,m}}. Define the random index κ1∈{1,…,m} such that Yτ10,ε=zκ1 . Then put X˜tε=Yt−τ11,κ1,ε until the first time moment τ2 when this process hits {zk,k=1,…,m}∖{zκ1} . Iterating this procedure, we get a process X˜tε with Xtε=Yt0,ε,t≤τ1,Xtε=Yt−τnn,κk,ε,t∈[τn,τn+1],n≥1. It follows from the strong Markov property of Xε that X˜ε has the same law with Xε . Hence, the given construction in fact represents the law of Xε as the image of the joint law family of independent processes Y0,ε,Yn,k,ε , k=1,…,m , n≥1 . Each of these processes is a solution to (1) with corresponding coefficients having at most one discontinuity point; hence, the LDP for them is provided in the previous section. Our idea is to deduce the LDP Xε via a version of the contraction principle. With this idea in mind, we first perform a simplification of the above representation. For some N (the choice of N will be discussed below), we consider the space Y=C(0,T)1+mN and construct a function F:Y→X=C(0,T) in the following way. For y=(y0,yn,k,k=1,…,m,n=1,…,N) , we first define τ1(y)=inf{t:yt0∈{zk,k=1,…,m}} with the usual convention that inf∅=T. The function [F(y)]t , t∈[0,T] , is defined to be equal to yt0 for t≤τ1(y) . If τ1(y)<T , then the construction is iterated: we define κ1(y) by yτ1(y)0=xκ1(y) and put, for t≥τ1(y) , [F(y)]t equal to yt−τ1(y)1,κ1(y) up to the first moment when this function hits {zk,k=1,…,m}∖{xκ1(y)} . We iterate this procedure at most N times; that is, if τN(y)<T , then we put [F(y)]t=yt−τN(y)N,κN(y),t∈[τN(y),T]. Denote Δ:=mink=2,…,m(zk−zk−1). For any fixed f∈C(0,T) , we can choose δf>0 0$]]> small enough and Nf large enough so that each g∈Bδ(f) has less than N Δ-oscillations on [0,T] . Hence, if in the above construction, N is taken equal to Nf , then the restriction of the law of Xε to any ball Bδ(f),δ≤δf , equals to the same restriction of the image of the joint law of the finite family Y0,ε , Yn,k,ε , k=1,…,m , n=1,…,N , under the mapping F specified before.

We aim to verify (5), and we argue in the following way. We fix f and choose N=Nf as before, so that the laws of Xε , restricted to Bδ(f) for δ small, can be obtained as the image under F specified before. Then we prove (8) at this particular point x=f , with Ilower , Iupper being constructed by this particular F. This yields the required weak LDP (5).

Within such an argument, we have to treat for any N the image under the corresponding F of the family of laws in Y=C(0,T)1+mN , which, according to the result proved in the previous section, satisfies the LDP with the rate function J(y)=12∫0T(y˙t0−a0(yt0))2σ02(yt0)dt+12 ∑k=1m ∑n=1N∫0T(y˙tk,n−a¯k(ytk,n))2σ¯k2(ytk,n)dt for y=(y0,yn,k)k≤m,n≤N such that y0,yn,k∈AC(0,T) , y00=x0 , y0n,k=zk and J(y)=∞ otherwise. To apply Lemma 1 in this setting, we first analyze the structure and the properties of the corresponding F.

Each trajectory f=F(y)∈C(0,T) is actually a patchwork, which consists of pieces of trajectories y0,yn,k , k=1,…,m , n=1,…,N : the pasting points are τ1(y),…,τr(y), r=r(y)≤N , and after τn(y) , the (part of the) new trajectory is used with the number κn(y) . For a yl→y in Y , the corresponding sequence of trajectories fl=F(yl) may fail to converge to f because the functionals τn(·),κn(·) are not continuous. However, the above “patchwork representation” easily yields the following two facts:

Using the first fact, now it is easy to prove the second inequality in (8). If it fails, then for a given f, there exists a sequence {yl} such that F(yl)→f and J(yl)≤c<I(f) . Since the level set {y:J(y)≤c} is compact, we can assume without loss of generality that yl converge to some y; recall that J is lower semicontinuous and thus J(y)≤c . The function f possesses the above patchwork representation with the trajectories taken from y, some pasting points τ1∗,…,τr∗ , and some numbers κ1∗,…,κr∗ . From this representation it is clear that f∈AC(0,T) and f0=x0 : if this fails, then the same properties fail at least for one trajectory from the family y and thus J(y)=∞ , which contradicts to J(y)≤c . Hence, we have I(f)=12∫0T(f˙t−a¯(ft))2σ¯2(ft)dt=12 ∑n=1r+1∫τn−1∗τn∗(f˙t−a¯(ft))2σ¯2(ft)dt, where we put τ0∗=0,τr+1∗=T . Let x0 be located on some interval (zk−1,zk) , k=2,…,m , say, x0∈(z1,z2) . Then, on the interval (0,τ1∗) , the trajectory f is contained in the segment [z1,z2] . The functions a0,σ0 are constant and coincide with a¯,σ¯ on (z1,z2) . In addition, a0=a(z1+)=a(z2−) , σ0=σ(z1+)=σ(z2−) ; hence, by the choice of a¯,σ¯ we have a¯2(z)σ¯2(z)≤a02(z)σ02(z),z=z1,z2. Then by (10) with A={z1,z2} we have ∫0τ1∗(f˙t−a¯(ft))2σ¯2(ft)dt=∫0τ1∗((f˙t−a0(ft))2σ02(ft)1ft∉A+a¯2(ft)σ¯2(ft)1ft∈A)dt≤∫0τ1∗((f˙t−a0(ft))2σ02(ft)1ft∉A+a02(ft)σ02(ft)1ft∈A)dt=∫0τ1∗(f˙t−a0(ft))2σ02(ft)dt=∫0τ1∗(y˙t0−a0(yt0))2σ02(yt0). Analogous inequalities hold on each of the time intervals (τn∗,τn+1∗), n=1,…,r , with a0,σ0 changed to a¯κn∗,σ¯κn∗ (the proof is similar and omitted). Thus, (11) I(f)≤12∫0τ1∗(y˙t0−a0(yt0))2σ02(yt0)dt+12 ∑n=1r∫τn∗τn+1∗(y˙tκn∗,n−a¯κn∗(ytκn∗,n))2σ¯κn∗2(ytκn∗,n)dt≤12∫0T(y˙t0−a0(yt0))2σ02(yt0)dt+12 ∑k=1m ∑n=1N∫0T(y˙tk,n−a¯k(ytk,n))2σ¯k2(ytk,n)dt=J(y). This gives a contradiction with inequalities J(y)≤c and I(f)>c c$]]>, which completes the proof of the second inequality in (8).

The first inequality in (8) holds immediately for f such that I(f)=∞ . We fix f with I(f)<∞ and γ>0 0$]]> and construct yγ such that F(yγ)=f , the functions τ1(·),τ2(·),… are continuous at yγ , and J(yγ)≤I(f)+γ . This completes the proof of (8).