Introduction

VMSTA

Modern Stochastics: Theory and Applications

2351-6054 2351-6046

2351-6046

VTeX

Mokslininkų g. 2A, 08412 Vilnius, Lithuania

VMSTA149

10.15559/20-VMSTA149

Research Article

Pathwise asymptotics for Volterra processes conditioned to a noisy version of the Brownian motion

Pacchiarotti

Barbara

pacchiar@mat.uniroma2.it Dept. of Mathematics, University of Rome “Tor Vergata”, via della Ricerca Scientifica, 00133, Roma, Italy

2020

2722020

711741 3072019 30112019 1022020

2020

Open access article under the CC BY license.

In this paper we investigate a problem of large deviations for continuous Volterra processes under the influence of model disturbances. More precisely, we study the behavior, in the near future after T, of a Volterra process driven by a Brownian motion in a case where the Brownian motion is not directly observable, but only a noisy version is observed or some linear functionals of the noisy version are observed. Some examples are discussed in both cases.

Keywords Large deviations Volterra type Gaussian processes conditional processes

2010 MSC 60F10 60G15 60G22

MIUR

E83C18000100006

The author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

1 Introduction

In this paper we study the asymptotics of the regular conditional prediction law of a Gaussian Volterra process in a case where one does not observe the process directly, but instead observes a noisy version of it. More precisely we consider two different situations which generalize the results contained in [13] and [9], respectively. Let X=(Xt)t≥0 be a continuous real Volterra process.

Definition 1.

A centered Gaussian process X is a Volterra process if, for every T>0 0$]]>, it admits the representation (1) Xt=∫0TK(t,s)dBs, where B=(Bt)t≥0 is a Brownian motion and K is a square integrable function on [0,T]2 (the kernel) such that K(t,s)=0 for all s>t t$]]>.

For a Volterra process the covariance function is (2) k(t,s)=∫0s∧tK(t,u)K(s,u)dufort,s∈[0,T]. Let B˜=(B˜t)t≥0 be another Brownian motion independent of B and for α,α˜∈R define Wα,α˜=αB+α˜B˜ .

First case For fixed n∈N and T>0 0$]]>, we consider the conditioning of X on n linear functionals of the paths of Wα,α˜ , GT(Wα,α˜)=(GT1(Wα,α˜),…,GTn(Wα,α˜))⊺, more precisely, GT(Wα,α˜)=∫0Tg(t)dWtα,α˜=(∫0Tg1(t)dWtα,α˜,…,∫0Tgn(t)dWtα,α˜)⊺, where g=(g1,…,gn)⊺ is a suitable vectorial function defined on [0,T] . Informally the generalized conditioned process Xg;x , for x∈Rn , is the law of the Gaussian process X conditioned on the set {∫0Tg(t)dWtα,α˜=x}= ⋂i=1n{∫0Tgi(t)dWtα,α˜=xi}. We obtain a large deviation principle for the family of processes ((XT+εtg;x−XTg;x)t∈[0,1])ε>0 0}}$]]>.

Second case We are interested in the regular conditional law of the process X given the σ-algebra FTα,α˜ , where (Ftα,α˜)t≥0 is the filtration generated by the mixed Brownian motion Wα,α˜ , i.e. we want to condition the process to the past of the mixed Brownian motion up to a fixed time T>0 0$]]>. Informally the generalized conditioned process Xψ , for ψ being a continuous function, is the law of the Gaussian process X conditioned on the set {Wtα,α˜=ψt,t∈[0,T]}.

Here we obtain a large deviation principle for the family of processes ((XT+εtψ−XTψ)t∈[0,1])ε>0 0}}$]]>.

Since T,α and α˜ are fixed positive numbers the dependence (in the notations) from these quantities will be omitted.

The paper is organized as follows. In Section 2 we recall some basic facts on large deviation theory for continuous Gaussian processes and Volterra processes. Sections 3 and 4 are dedicated to the main results. Both are divided into three subsections. In the first one we give the conditional law, in the second one we prove the large deviation principle and in the third one we present some examples. Section 3 is dedicated to the conditioning on n functionals of the paths of the noisy process. Section 4 is dedicated to the conditioning on the past of the noisy process.

2 Large deviations for continuous Gaussian processes

We briefly recall some main facts on large deviations principles we are going to use. For a detailed development of this very wide theory we can refer, for example, to the following classical references: Chapter II in Azencott [1], Section 3.4 in Deuschel and Strook [6], Chapter 4 (in particular Sections 4.1, 4.2 and 4.5) in Dembo and Zeitouni [5].

Definition 2.

Let E be a topological space, B(E) be the Borel σ-algebra and (με)ε>0 0}}$]]> be a family of probability measures on B(E) . We say that the family of probability measures (με)ε>0 0}}$]]> satisfies a large deviation principle on E with the rate function I and the inverse speed ηε ( ηε>0 0$]]>, ηε→0 as ε→0 ) if, for any open set Θ, −infx∈ΘI(x)≤lim infε→0ηεlogμε(Θ) and for any closed set Γ, lim supε→0ηεlogμε(Γ)≤−infx∈ΓI(x).

A rate function is a lower semicontinuous mapping I:E→[0,+∞] . A rate function I is said to be good if the sets {I≤a} are compact for every a≥0 .

In this paper E will be the set of continuous functions on [0,1] and B(E) will be the Borel σ-algebra generated by the open sets induced by the uniform convergence. Therefore in this section we consider process in the interval [0,1] . Let U=(Ut)t∈[0,1] , be a continuous and centered Gaussian process on a probability space (Ω,F,P) . From now on, we will denote by C[0,1] the set of continuous functions on [0,1] , and by B(C[0,1]) the Borel σ-algebra generated by the open sets induced by the uniform convergence. Moreover, we will denote by M[0,1] its dual, that is, the set of signed Borel measures on [0,1] . The action of M[0,1] on C[0,1] is given by ⟨λ,h⟩=∫01h(t)dλ(t),λ∈M[0,1],h∈C[0,1].

Remark 1.

We say that a family of continuous processes ((Utε)t∈[0,1])ε>0 0}}$]]> satisfies a large deviation principle if the associated family of laws satisfy a large deviation principle on C[0,1] .

The following remarkable theorem (Proposition 1.5 in [1]) gives an explicit expression of the Cramér transform Λ∗ of a continuous centered Gaussian process (Ut)t∈[0,1] with covariance function k. Let us recall that for λ∈M[0,1] , Λ(λ)=logE[exp(⟨U,λ⟩)]=12∫01∫01k(t,s)dλ(t)dλ(s).

Theorem 1.

Let (Ut)t∈[0,1] be a continuous and centered Gaussian process with covariance function k. Let Λ∗ denote the Cramér transform of Λ, that is Λ∗(x)=supλ∈M[0,1](⟨λ,x⟩−Λ(λ))=supλ∈M[0,1](⟨λ,x⟩−12∫01∫01k(t,s)dλ(t)dλ(s)). Then, Λ∗(x)=12‖x‖H2,x∈H,+∞,otherwise, where H and ‖.‖H denote, respectively, the reproducing kernel Hilbert space and the related norm associated to the covariance function k.

Reproducing kernel Hilbert spaces are an important tool to handle Gaussian processes. For a detailed development of this wide theory we can refer, for example, to Chapter 4 in [10] (in particular Section 4.3) and Chapter 2 in [3] (in particular Sections 2.2 and 2.3). In order to state a large deviation principle for a family of Gaussian processes, we need the following definition.

Definition 3.

A family of continuous processes ((Utε)t∈[0,1])ε>0 0}}$]]> is exponentially tight at the inverse speed ηε , if for every R>0 0$]]> there exists a compact set KR such that lim supε→0ηεlogP(Uε∉KR)≤−R.

If the means and the covariance functions of an exponentially tight family of Gaussian processes have a good limit behavior, then the family satisfies a large deviation principle, as stated in the following theorem which is a consequence of the classic abstract Gärtner–Ellis Theorem (Baldi Theorem 4.5.20 and Corollary 4.6.14 in [5]) and Theorem 1.

Theorem 2.

Let ((Utε)t∈[0,1])ε>0 0}}$]]> be an exponentially tight family of continuous Gaussian processes at the inverse speed function ηε . Suppose that, for any λ∈M[0,1] , limε→0E[⟨λ,Uε⟩]=0 and the limit Λ(λ)=limε→01ηεVar(⟨λ,Uε⟩)=∫01∫01k(t,s)dλ(t)dλ(s) exists for some continuous, symmetric, positive definite function k, that is the covariance function of a continuous Gaussian process, then ((Utε)t∈[0,1])ε>0 0}}$]]> satisfies a large deviation principle on C[0,1] with the inverse speed ηε and the good rate function I(h)=12‖h‖H2,h∈H,+∞,otherwise, where H and ‖.‖H , respectively, denote the reproducing kernel Hilbert space and the related norm associated to the covariance function k.

In order to prove exponential tightness we shall use the following result (see Proposition 2.1 in [12]).

Proposition 1.

Let ((Utε)t∈[0,1])ε>0 0}}$]]> be a family of continuous Gaussian processes, where U0ε=0 for all ε>0 0$]]>. Suppose there exist constants β,M1,M2>0 0$]]> such that for ε>0 0$]]> sups,t∈[0,1],s≠t|E[Utε−Usε]||t−s|β≤M1 and (3) sups,t∈[0,1],s≠tVar(Utε−Usε)ηε|t−s|2β≤M2 then ((Utε)t∈[0,1])ε>0 0}}$]]> is exponentially tight at the inverse speed function ηε .

Remark 2.

Suppose ((Utε)t∈[0,1])ε>0 0}}$]]> is a family of centered Gaussian processes, defined on the probability space (Ω,F,P) , that satisfies a large deviation principle on C[0,1] with the inverse speed ηε and the good rate function I. Let (mε)ε>0⊂C[0,1] 0}}\subset C[0,1]$]]>, m∈C[0,1] be functions such that mε⟶C[0,1]m , as ε→0 . Then, the family of processes (mε+Uε)ε>0 0}}$]]> satisfies a large deviation principle on C[0,1] with the same inverse speed ηε and the good rate function Im(h)=I(h−m)=12‖h−m‖H2,h−m∈H,+∞,h−m∉H. In fact the two families (mε+Uε)ε>0 0}}$]]> and (m+Uε)ε>0 0}}$]]> are exponentially equivalent (at the inverse speed ηε ) and therefore as far as the large deviation principle is concerned, they are indistinguishable. See Theorem 4.2.13 in [5].

Our first aim is to study the behavior of the covariance function and of the mean function of the original process X in order to get a functional large deviation principle for the family ((XT+εt−XT)t∈[0,1])ε>0 0}}$]]>, as ε→0 .

Let (Xt)t≥0 be a continuous centered Gaussian processes and fix T>0 0$]]>. The next two assumptions guarantee that Theorem 2 is applicable to the family of processes ((XT+εt−XT)t∈[0,1])ε>0 0}}$]]>. Let γε>0 0$]]> be an infinitesimal function, i.e. γε→0 for ε→0 .

Assumption 1.

For any fixed T>0 0$]]> there exists an asymptotic covariance function k¯ defined as (4) k¯(t,s)=limε→0Cov(XT+εt−XT,XT+εs−XT)γε2=limε→0k(T+εt,T+εs)−k(T+εt,T)−k(T+εs,T)+k(T,T)γε2, uniformly in (t,s)∈[0,1]×[0,1] .

Remark 3.

Notice that k¯ is a continuous covariance function, being the (uniform) limit of continuous, symmetric and positive definite functions.

Remark 4.

Recall that the continuity of the covariance function is not a sufficient condition to identify a Gaussian process with continuous paths. We need some more regularity. Since we are investigating continuous Volterra processes, it would be useful to have a criterion to establish the regularity of the paths. A sufficient condition for the continuity of the trajectories of a centered Gaussian process can be given in terms of the metric entropy induced by the canonical metric associated to the process (for further details, see [7] and [8]). Such approach may be difficult to apply. However, in [2], a necessary and sufficient condition for the Hölder continuity of a centered Gaussian process is established in terms of the Hölder continuity of the covariance function. More precisely a Gaussian process (Xt)t∈[0,T] is Hölder continuous of exponent 0<a<A if and only if for every ε>0 0$]]>, s,t∈[0,T] , there exists a constant cε>0 0$]]> such that E[(Xt−Xs)2]≤cε|t−s|A−ε. Although, obviously, the Hölder continuity property of the process is stronger than continuity, in many cases of interest this is more easily established because the covariance function is not difficult to study. Recalling the form of the covariance of a Volterra process (2) we have the following sufficient condition for the Hölder continuity of a Volterra process: there exist constants c,A>0 0$]]> such that (5) M(δ)≤cδA for all δ∈[0,T] , where M(δ)=sup{t1,t2∈[0,T]:|t1−t2|≤δ}∫0T|K(t1,s)−K(t2,s)|2ds.

From now on with covariance regular enough we mean that the covariance function satisfies some sufficient condition to ensure that the associate process has continuous paths.

Assumption 2.

For any fixed T>0 0$]]> there exist constants M,τ>0 0$]]>, such that for ε>0 0$]]>, sups,t∈[0,1],s≠tVar(XT+εt−XT+εs)γε2|t−s|2τ=sups,t∈[0,1],s≠tk(T+εt,T+εt)−2k(T+εt,T+εs)+k(T+εs,T+εs)γε2|t−s|2τ≤M.

As an immediate application of Theorem 2 (take Utε=XT+εt−XT ), Assumptions 1 and 2 imply, if k¯ is regular enough, that the family ((XT+εt−XT)t∈[0,1])ε>0 0}}$]]> satisfies a large deviation principle on C[0,1] with the inverse speed γε2 and the good rate function given by JX(h)=12‖h‖H¯2,h∈H¯,+∞,otherwise, where H¯ is the reproducing kernel Hilbert space associated to the covariance function k¯ and the symbol ‖·‖H¯ denotes the usual norm defined on H¯ .

In fact Assumption 1 immediately implies that Λ(λ)=limε→0Var(⟨λ,XT+ε·−XT⟩)γε2=∫01∫01k¯(t,s)λ(dt)λ(ds). Furthermore, Assumption 2 implies that the family ((XT+εt−XT)t∈[0,1])ε>0 0}}$]]> is exponentially tight at the inverse speed function γε2 .

3 Conditioning to <italic>n</italic> functionals of the path 3.1 Conditional law

Let (Ω,F,(Ft)t≥0,P) be a filtered probability space. On this space we consider a Brownian motion B=(B)t≥0 , a continuous real Volterra process X=(Xt)t≥0 and another Brownian motion B˜=(B˜)t≥0 independent of B. For α,α˜∈R let us define the mixed Brownian motion Wα,α˜=αB+α˜B˜ .

For fixed n∈N and T>0 0$]]>, we consider the conditioning of X on n linear functionals of GT(Wα,α˜)=(GT1(Wα,α˜),…,GTn(Wα,α˜))⊺ of the paths of Wα,α˜ , GT(Wα,α˜)=∫0Tg(t)dWtα,α˜=(∫0Tg1(t)dWtα,α˜,…,∫0Tgn(t)dWtα,α˜)⊺, where g=(g1,…,gn)⊺ is a vectorial function and gk∈L2[0,T] , for k=1,…,n . We assume, without any loss of generality, that the functions gi , i=1,…,n , are linearly independent. The linearly dependent components of g can be simply removed from the conditioning. As we said in the Introduction, the generalized conditioned process Xg;x , for x∈Rn , is the law of the Gaussian process X conditioned on the set {∫0Tg(t)dWtα,α˜=x}= ⋂i=1n{∫0Tgi(t)dWtα,α˜=xi}. The law Pg;x of Xg;x is the regular conditional distribution on C[0,+∞) , endowed with the topology induced by the sup-norm on compact sets, Pg;x(X∈E)=P(Xg;x∈E)=P(X∈E|∫0Tg(t)dWtα,α˜=x). For more details about existence of such regular conditional distribution see, for example, [11].

Denote by Cg=(cijgigj)i,j=1,…,n the matrix defined by cijgigj=Cov(∫0Tgi(t)dWtα,α˜,∫0Tgj(t)dWtα,α˜)=(α2+α˜2)∫0Tgi(t)gj(t)dt.

The matrix Cg is invertible (since the functions gi , i=1,…,n , are linearly independent). Let us denote rigi(t)=Cov(Xt,∫0Tgi(u)dWuα,α˜)=α∫0t∧TK(t,u)gi(u)du, and rg(t)=(r1g1(t),…,rngn(t))⊺. The following theorem, similar to Theorem 3.1 in [17], gives mean and covariance function of the generalized conditioned process. Theorem 3.

The generalized conditioned process Xg;x can be represented as Xtg;x=Xt−rg(t)⊺(Cg)−1(∫0Tg(u)dWuα,α˜−x). Moreover, the conditioned process Xg;x is a Gaussian process with mean (6) mg;x(t)=E[Xtg;x]=rg(t)⊺(Cg)−1x, and covariance (7) kg(t,s)=Cov(Xtg;x,Xsg;x)=k(t,s)−κg(t,s), where (8) κg(t,s)=rg(t)⊺(Cg)−1rg(s).

Proof.

It is a classical result on conditioned Gaussian laws. See, e.g., Chapter II, §13, in [15]. □

Remark 5.

Let us note that the covariance function of the conditioned process depends on the conditioning functions g1,…,gn and on the time T, but not on the vector x .

Remark 6.

If the conditioning functions gi are the indicator functions of the interval [0,Ti) , for i=1,…,n , then the process is conditioned to the position of the noisy Brownian motion at the times T1,…,Tn , more precisely to the set ⋂i=1n{WTiα,α˜=xi} .

Remark 7.

If the conditioning functions are gi(s)=K(Ti,s)1[0,Ti)(s) , for i=1,…,n , and α=1 , α˜=0 , then the process is conditioned to its position at the times T1,…,Tn , more precisely to the set ⋂i=1n{XTi=xi} (this is a particular case of the conditioned process in [13]).

3.2 Large deviations

Let γε>0 0$]]> be an infinitesimal function, i.e. γε→0 for ε→0 . In this section (Xt)t≥0 is a continuous Volterra process as in (1). Now, in order to achieve a large deviation principle for the family of processes ((XT+εtg;x−XTg;x)t∈[0,1])ε>0 0}}$]]>, we have to investigate the behavior of the functions kg and mg;x (defined in (7) and (6), respectively) in a small time interval of length ε.

Now we give some conditions on the original process in order to guarantee that the hypotheses of Theorem 2 hold for the conditioned process. The next assumption (Assumption 3) implies the existence of a limit covariance.

Assumption 3.

For any T>0 0$]]> and for gi∈L2[0,T] , i=1,…,n , there exists a vectorial function r¯g=(r¯1g1,…,r¯ngn) , possibly r¯igi=0 for some i=1,…,n , such that (9) r¯igi(t)=limε→0Cov(XT+εt−XT,∫0Tgi(u)dWuα,α˜)γε=limε→0rigi(T+εt)−rigi(T)γε, uniformly in t∈[0,1] .

The next assumption (Assumption 4) implies the exponential tightness of the family of the centered processes.

Assumption 4.

For any fixed T>0 0$]]> there exist constants M,τˆ>0 0$]]>, such that for i=1,…,n and ε>0 0$]]>, sups,t∈[0,1],s≠t|Cov(XT+εt−XT+εs,∫0Tgi(u)dWuα,α˜)|γε|t−s|τˆ=sups,t∈[0,1],s≠t|rigi(T+εt)−rigi(T+εs)|γε|t−s|τˆ≤M.

Remark 8.

Let us observe that Assumption 3 implies that for any fixed T>0 0$]]> limε→0rigi(T+εt)−rigi(T)=0, uniformly in t∈[0,1] . Therefore, (10) limε→0mg;x(T+εt)=mg;x(T), uniformly in t∈[0,1] . In fact, one has mg;x(T+εt)−mg;x(T)=(rg(T+εt)−rg(T))⊺(Cg)−1x, and (10) immediately follows.

Remark 9.

Let us observe that Assumption 4 implies that there exists M>0 0$]]> such that the following estimate holds for the function κg defined in (8): (11) sups,t∈[0,1],s≠t|κg(T+εt,T+εt)−2κg(T+εt,T+εs)+κg(T+εs,T+εs)|γε2|t−s|2τˆ≤M.

In fact, straightforward computations show that κg(T+εt,T+εt)−2κg(T+εt,T+εs)+κg(T+εs,T+εs)=((rg(T+εt)−rg(T+εs)))⊺(Cg)−1((rg(T+εt)−rg(T+εs))).

Therefore (11) immediately follows from Assumption 4.

Proposition 2.

Under Assumptions 1 and 3, one has limε→0Cov(XT+εtg;x−XTg;x,XT+εsg;x−XTg;x)γε2=k¯g(t,s), uniformly in (t,s)∈[0,1]×[0,1] , with (12) k¯g(t,s)=k¯(t,s)−r¯g(t)⊺(Cg)−1r¯g(s), where r¯g(t)⊺=(r¯1g1(t),…,r¯ngn(t)) and r¯igi(t) is defined in (9) for i=1,…,n .

Proof.

Taking into account equation (7), simple computations show that for s,t∈[0,1] , (13) Cov(XT+εtg;x−XTg;x,XT+εsg;x−XTg;x)=(k(T+εt,T+εs)−k(T+εt,T)−k(T+εs,T)+k(T,T))+−((r¯g(T+εt)−r¯g(T)))⊺(Cg)−1((r¯g(T+εs)−r¯g(T))).

Therefore the claim easily follows from Assumptions 1 and 3. □

Remark 10.

Notice that k¯g is a continuous covariance function, being the (uniform) limit of continuous, symmetric and positive definite functions.

Proposition 3.

Under Assumptions 2 and 4 the family ((XT+εtg;x−XTg;x−E[XT+εtg;x−XTg;x])t∈[0,1])ε>0 0}}$]]> is exponentially tight at the inverse speed function γε2 .

Proof.

As ((XT+εtg;x−XTg;x−E[XT+εtg;x−XTg;x])t∈[0,1])ε>0 0}}$]]> is a family of centered processes, it is enough to prove that (3) is satisfied with an appropriate speed function. For ε>0 0$]]> the covariance of such process is given by (13). Therefore Var(XT+εtg;x−XT+εsg;x)=k(T+εt,T+εt)−2k(T+εt,T+εs)+k(T+εs,T+εs)+rg(T+εt)−rg(T)))⊺(Cg)−1((rg(T+εs)−rg(T)). From Assumption 2 we already know that sups,t∈[0,1],s≠tk(T+εt,T+εt)−2k(T+εt,T+εs)+k(T+εs,T+εs)γε2|t−s|2τ≤M. Furthermore, Assumption 4 implies that sups,t∈[0,1],s≠t|(rg(T+εt)−rg(T)))⊺(Cg)−1((rg(T+εs)−rg(T))|γε2|t−s|2τˆ≤M.

Therefore condition (3) holds with the inverse speed ηε=γε2 and β=τ∧τˆ . □

We are now ready to prove the main large deviation result of this section. Theorem 4.

Suppose (Xt)t≥0 satisfies Assumptions 1, 2, 3 and 4. Suppose, furthermore, that the (existing) covariance function k¯g defined in Proposition 2 is regular enough, then the family of processes ((XT+εtg;x−XTg;x)t∈[0,1])ε>0 0}}$]]> satisfies a large deviation principle on C[0,1] with the inverse speed γε2 and the good rate function (14) JXg(h)=12‖h‖H¯g2,h∈H¯g,+∞,otherwise, where H¯g is the reproducing kernel Hilbert space associated to the covariance function k¯g .

Proof.

Cosider the family of centered processes ((XT+εtg;x−XTg;x−E[XT+εtg;x−XTg;x])t∈[0,1])ε>0 0}}$]]>. Thanks to Proposition 3 this family of processes is exponentially tight at the inverse speed γε2 . Thanks to Proposition 2, for any λ∈M[0,1] , one has limε→0Var(⟨λ,XT+ε·g;x−XTg;x⟩)γε2=limε→0∫01dλ(v)∫01dλ(u)Cov(XT+εvg;x−XTg;x,XT+εug;x−XTg;x)γε2=∫01dλ(v)∫01dλ(u)k¯g(v,u), where k¯g is defined in (12). Since k¯g is the covariance function of a continuous Volterra process, a large deviation principle for ((XT+εtg;x−XTg;x−E[XT+εtg;x−XTg;x])t∈[0,1])ε>0 0}}$]]> actually holds from Theorem 2 with the inverse speed γε2 and the good rate function given by (14). From Equation (10) and Remark 2 the same large deviation principle holds for the noncentered family ((XT+εtg;x−XTg;x)t∈[0,1])ε>0 0}}$]]>. □

3.3 Examples

In this section we consider some examples to which Theorem 4 applies. Therefore we want to verify that Assumptions 1, 2, 3 and 4 are fulfilled. Let X be a continuous, centered Volterra process process with kernel K. Suppose g1(t)=1[0,T)(t) and g2(t)=T−tT1[0,T)(t) , that is, WTα,α˜=∫0Tg1(u)dWuα,α˜=x1 and by the integration by parts formula, 1T∫0TWuα,α˜du=∫0Tg2(u)dWuα,α˜=x2. Then the matrix (Cg)−1 is given by (Cg)−1=1det(Cg)c22g2g2−c12g1g2−c12g1g2c11g1g1, where c11g1g1=(α2+α˜2)T,c12g1g2=(α2+α˜2)T2,c22g2g2=(α2+α˜2)T3,det(Cg)=(α2+α˜2)2T212. Example 1 (Fractional Brownian Motion).

Let X be the fractional Brownian motion of the Hurst index H>1/2 1/2$]]>. The fractional Brownian motion with the Hurst parameter H∈(0,1) is the centered Gaussian process with covariance function k(t,s)=12(t2H+s2H−|t−s|2H). The fractional Brownian motion is a Volterra process with kernel, for s≤t , (15) K(t,s)=cH[(ts(t−s))H−1/2−(H−12)s1/2−H∫stuH−3/2(u−s)H−1/2du], where cH=(2HΓ(3/2−H)Γ(H+1/2)Γ(2−2H))1/2 . Notice that when H=1/2 we have K(t,s)=1[0,t](s) , and then the fractional Brownian motion reduces to the Wiener process.

First, let us prove that there exists a limit covariance and that it is regular enough. For s≤t , one has Cov(XT+εt−XT,XT+εs−XT)ε2H=Cov(Xt,Xs), because of the homogeneity and self-similarity properties holding for the fractional Brownian motion, so that the limit in (4) trivially exists and Assumption 1 holds with k¯(t,s)=k(t,s) and γε=εH . Now let us prove that Assumption 3 is fulfilled. Cov(XT+εt−XT,WTα,α˜)=∫0T(K(T+εt,u)−K(T,u))du=cH∫0T((T+εtu)H−1/2(T+εt−u)H−1/2−(Tu)H−1/2(T−u)H−1/2)du+−cH(H−1/2)∫0T1uH−1/2∫TT+εt(v−u)H−1/2vH−3/2dvdu.

Thanks to the Lagrange theorem, ((T+εtu)H−1/2(T+εt−u)H−1/2−(Tu)H−1/2(T−u)H−1/2)=H−1/2uH−1/2[(T+ξε)H−3/2(T+ξε−u)H−1/2+(T+ξε)H−1/2(T+ξε−u)H−3/2]εt, for ξε∈[0,εt] . Therefore from the Lebesgue theorem, limε→01ε∫0T((T+εtu)H−1/2(T+εt−u)H−1/2−(Tu)H−1/2(T−u)H−1/2)du=t(H−1/2)∫0T1uH−1/2(TH−3/2(T−u)H−1/2+TH−1/2(T−u)H−3/2)du, uniformly in t∈[0,1] . Furthermore, in a similar way, we have, limε→01ε∫0T1uH−1/2∫TT+εt(v−u)H−1/2vH−3/2dvdu=t∫0T1uH−1/2(T−u)H−1/2TH−3/2du. Therefore, r¯1g1(t)=limε→0Cov(XT+εt−XT,WTα,α˜)εH=0, uniformly in t∈[0,1] . Similar calculations show that r¯2g2(t)=limε→0Cov(XT+εt−XT,∫0TT−uTdWuα,α˜)εH=0. So, we have k¯g(t,s)=k(t,s) and therefore the limit covariance exists and is regular enough.

Now let us prove the exponential tightness of the family of processes. For s<t , Var(XT+εt−XT+εs)=ε2H(t−s)2H, then Assumption 2 holds with τ=H and γε=εH . For s<t , we have Cov(XT+εt−XT+εs,WTα,α˜)=∫0T(K(T+εt,u)−K(T+εs,u))du=cH∫0T((T+εtu)H−1/2(T+εt−u)H−1/2−(T+εtu)H−1/2(T+εs−u)H−1/2)du+−cH(H−1/2)∫0T1uH−1/2∫T+εsT+εt(v−u)H−1/2vH−3/2dvdu.

Thanks to the Lagrange theorem we can find M>0 0$]]> such that ∫0T((T+εtu)H−1/2(T+εt−u)H−1/2−(T+εsu)H−1/2(T+εs−u)H−1/2)du≤ε(t−s)(H−12)∫0T1uH−12[TH−32(T+1−u)H−12+(T+1)H−12(T−u)H−32]du≤εM(t−s) and ∫0T1uH−1/2∫T+εsT+εt(v−u)H−1/2vH−3/2dvdu=(t−s)∫0T1uH−1/2(T+ξε−u)H−1/2(T+ξε)H−3/2du≤εM(t−s). Therefore, a fortiori, sups,t∈[0,1],s≠t|Cov(XT+εt−XT+εs,WTα,α˜)|εH|t−s|≤M. Similar calculations show that sups,t∈[0,1],s≠t|Cov(XT+εt−XT+εs,∫0TT−uTdWuα,α˜)|εH|t−s|≤M. Thus, Assumption 4 is fulfilled with τˆ=1 . Therefore the family ((XT+εtg;x−XTg;x)t∈[0,1])ε>0 0}}$]]> satisfies a large deviation principle with the inverse speed function γε2=ε2H as the nonconditioned process. Note that the same result was obtained in [4] for the n-fold conditional fractional Brownian motion.

Example 2 (<italic>m</italic>-fold integrated Brownian motion).

For m≥1 , let X be the m-fold integrated Brownian motion, i.e. Xt=∫0tdu(∫0udum−1⋯∫0u2du1Bu1). It is a continuous Volterra process with kernel K(t,u)=1m!(t−u)m and covariance function k(t,s)=1(m!)2∫0s∧t(s−ξ)m(t−ξ)mdξ. First, let us prove that there exists a limit covariance and that it is regular enough. Assumption 1 is fulfilled. In fact, for s≤t , we have limε→0Cov(XT+εt−XT,XT+εs−XT)ε2=limε→01(m!)21ε2∫TT+εs(T+εt−u)m(T+εs−u)mdu+limε→01(m!)2ε2∫0T((T+εt−u)m−(T−u)m)((T+εs−u)m−(T−u)m)du. It is straightforward to show that k¯(t,s)=limε→0Cov(XT+εt−XT,XT+εs−XT)ε2=1(m!)2m22m−1T2m−1st, uniformly in (t,s)∈[0,1]×[0,1] . Furthermore, r¯1g1(t)=limε→0Cov(XT+εt−XT,WTα,α˜)ε=limε→0αm!1ε∫0T((T+εt−u)m−(T−u)m)du=αm!Tmt, and r¯2g2(t)=limε→0Cov(XT+εt−XT,∫0TT−uTdWuα,α˜)ε=limε→0αm!T1ε∫0T((T+εt−u)m−(T−u)m)(T−u)du=αm!mm+1Tmt, uniformly in t∈[0,1] . Therefore also Assumption 3 is fulfilled.

Thus, we have k¯g(t,s)=ast , where a=1(m!)2(m22m−1T2m−1−α2T2m(1,mm+1)(Cg)−1(1,mm+1)⊺). Note that k¯g is regular enough.

Now let us prove the exponential tightness of the family of processes. For s<t , there exists a constant M>0 0$]]>, such that Var(XT+εt−XT+εs)=1(m!)2(∫T+εsT+εt(T+εt−u)2mdu+∫0T+εs((T+εt−u)m−(T+εs−u)m)2du)≤Mε2(t−s)2. Then Assumption 2 holds with τ=1 and γε=ε . For s<t , |Cov(XT+εt−XT+εs,WTα,α˜)|=∫0T((T+εt−u)m−(T+εs−u)m)du= ∑k=0m−1mk1k+1Tk+1εm−k(t−s)m−k. Then we have sups,t∈[0,1],s≠t|Cov(XT+εt−XT+εs,WTα,α˜)|ε|t−s|≤M. Similar calculations show that sups,t∈[0,1],s≠t|Cov(XT+εt−XT+εs,∫0TT−uTdWuα,α˜)|ε|t−s|≤M. Thus, Assumption 4 is fulfilled with τˆ=1 . Therefore the family ((XT+εtg;x−XTg;x)t∈[0,1])ε>0 0}}$]]> satisfies a large deviation principle with the inverse speed γε2=ε2 .

Example 3 (Integrated Volterra Process).

Let Z be a Volterra process with kernel K satisfying condition (5) for some A>0 0$]]>. Let X be the integrated process, i.e. Xt=∫0tZudu. The process X is a continuous, Volterra process with kernel h(t,s)=∫stK(u,s)du,i.e.Xt=∫0th(t,s)dBs. First, let us prove that there exists a limit covariance and that it is regular enough. Assumption 1 is fulfilled, in fact, for s≤t , we have limε→0Cov(XT+εt−XT,XT+εs−XT)ε2=limε→0∫TT+εsh(T+εt,u)h(T+εs,u)duε2+limε→0∫0T(h(T+εt,u)−h(T))(h(T+εs,u)−h(T,u))duε2. Now, one has ∫TT+εsh(T+εt,u)h(T+εs,u)du=ε3∫0s(∫utK(T+εx,T+εu)dx∫usK(T+εx,T+εu)dx)du and ∫0T(h(T+εt,u)−h(T,u))(h(T+εs,u)−h(T,u))du=∫0T(∫TT+εsK(v,u)dv∫TT+εtK(v,u)dv)du. Therefore limε→0Cov(XT+εt−XT,XT+εs−XT)ε2=st∫0TK2(T,u)du, uniformly in (t,s)∈[0,1]×[0,1] . Furthermore, with similar calculations we have r¯1g1(t)=limε→0Cov(XT+εt−XT,WTα,α˜)ε=αt∫0TK(T,u)du, and r¯2g2(t)=limε→0Cov(XT+εt−XT,∫0TT−uTdWuα,α˜)ε=αtT∫0TK(T,u)(T−u)du, uniformly in t∈[0,1] . Therefore also Assumption 3 is fulfilled.

So, we have k¯g(t,s)=ast , where a=∫0TK2(T,u)du−α2A⊺(Cg)−1A, and A⊺=(∫0TK(T,u)du,1T∫0TK(T,u)(T−u)du) . Note that k¯g is regular enough. Let us now prove the exponential tightness. We have, for s<t , Var(XT+εt−XT+εs)=∫T+εsT+εth(T+εt,u)2du+∫0T+εs(h(T+εt,u)−h(T+εs,u))2. Now, recalling that K is a square integrable function, there exists a constant M>0 0$]]>, such that ∫T+εsT+εth(T+εt,u)2du=ε3∫st(∫utK(T+εx,T+εu)dx)2du≤ε3M∫st(t−u)du≤Mε3(t−s)2 and ∫0T+εs(h(T+εt,u)−h(T+εs,u)2du=ε2∫0T+εs(∫stK(T+εv,u)dv)2du≤Mε2(t−s)2, therefore Assumption 2 holds with τ=1 and γε=ε .

With similar computations we can prove that also Assumption 4 is fulfilled with τˆ=1 and γε=ε . For s<t , we have |Cov(XT+εt−XT+εs,WTα,α˜)|=|∫0T(h(T+εt,u)−h(T+εs))du|=|∫0T∫T+εsT+εtK(v,u)dvdu|≤Mε(t−s). Therefore sups,t∈[0,1],s≠t|Cov(XT+εt−XT+εs,WTα,α˜)|ε|t−s|≤M. Similar calculations show that sups,t∈[0,1],s≠t|Cov(XT+εt−XT+εs,∫0TT−uTdWuα,α˜)|ε|t−s|≤M. Therefore the family ((XT+εtg;x−XTg;x)t∈[0,1])ε>0 0}}$]]> satisfies a large deviation principle with the inverse speed function γε2=ε2 .

4 Conditioning to a path 4.1 Conditional law

We are interested in the regular conditional law of the process X given the σ-algebra FTα,α˜ , where (Ftα,α˜)t≥0 is the filtration generated by the mixed Brownian motion Wα,α˜ , i.e. we want to condition the process to the past of the mixed Brownian motion up to a fixed time T>0 0$]]>. To do this, consider the conditional law on C[0,+∞) endowed with the topology induced by the sup-norm on compact sets, P(X∈·∣FTα,α˜) . There exists a regular version of such conditional probability (see [11] and [14]), namely a version such that Γ↦P(X∈Γ∣FTa,b) is almost surely a Gaussian probability law.

The following theorem, Theorem 2.1 in [16], gives mean and covariance function of the Gaussian conditional law. Theorem 5.

For T>0 0$]]>, the regular conditional law of X∣FTa,b is a Gaussian measure with the random mean Ψt(Wα,α˜)=E[Xt|FTα,α˜]=α2α2+α˜2∫0TK(t,u)dWuα,α˜ and the deterministic covariance, (16) Υ(t,s)=∫0t∧s(1−α2α2+α˜21[0,T](v))2K(t,v)K(s,v)dv+α2α˜2(α2+α˜2)2∫0TK(t,v)K(s,v)dv.

Remark 11.

Observe that the mean process (α2α2+α˜2∫0TK(t,u)dWuα,α˜)t≥0 is a continuous process. Therefore for almost every continuous function ψ defined on [0,T] , (17) mtψ=Ψt(ψ) defines a continuous function mψ:[0,+∞)⟶R . Thus, we can consider the continuous Gaussian process (Xtψ)t≥0 with mean function mψ and covariance function Υ.

From continuity of mψ one has (18) limε→0mT+εtψ=mTψ uniformly for t∈[0,1] .

Remark 12.

Let us note that the covariance function of the conditioned process depends on the time T, but not on the function ψ as in the previous section.

Remark 13.

For s∧t≥T we have (19) Υ(t,s)=α2α2+α˜2∫Tt∧sK(t,v)K(s,v)dv+α˜2α2+α˜2k(t,s). For α˜=0 , i.e. Ftα,0=σ{Xu:u≤t} (for details about the filtrations generated by X and B, see, for example, [18]), we have the same conditioned variance as in [9].

4.2 Large deviations

Let γε>0 0$]]> be an infinitesimal function, i.e. γε→0 for ε→0 . In this section (Xt)t≥0 is a continuous Volterra process as in (1).

Now, in order to achieve a large deviation principle for the generalized conditioned process Xψ , we have to investigate the behavior of the functions Υ and mψ (defined in (16) and (17), respectively) in a small time interval of length ε. We want to investigate the behavior of the conditioned process (Xtψ)t≥0 in the near future after T.

For s≤t , taking into account equation (19), simple computations show that (20) Cov(XT+εtψ−XTψ,XT+εsψ−XTψ)=εα2α2+α˜2∫0t∧sK(T+εt,T+εu)K(T+εs,T+εu)du++α˜2α2+α˜2(k(T+εt,T+εs)−k(T+εt,T)−k(T,T+εs)+k(T,T)).

Now we give conditions on the kernel K of the original Volterra process in order to guarantee that the hypotheses of Theorem 2 hold for the conditioned process. The next assumption (Assumption 5) implies the existence of a limit covariance.

Assumption 5.

For any T>0 0$]]> there exists a square integrable function K¯ (possibly 0) such that limε→0εK(T+εt,T+εs)γε=K¯(t,s) uniformly in (t,s)∈[0,1]×[0,1] .

Remark 14.

Notice that we can choose γε so that limε→0γεε−12∈{0,1,+∞} .

The next assumption (Assumption 6) implies the exponential tightness of the family of processes.

Assumption 6.

For any T>0 0$]]> there exist constants c,τˆ>0 0$]]> such that sups,t∈[0,1],s≠t∫0t(K(T+εt,T+εu)−K(T+εs,T+εu))2duγε2|t−s|2τˆ≤c.

Remark 15.

Note that ∫0t(K(T+εt,T+εu)−K(T+εs,T+εu))2du=∫stK(T+εt,T+εu)2du+∫0s(K(T+εt,T+εu)−K(T+εs,T+εu))2du, therefore in order to prove that Assumption 6 is fulfilled we can prove that there exists c>0 0$]]> such that sups,t∈[0,1],s≠t∫stK(T+εt,T+εu)2duγε2|t−s|2τˆ≤c,sups,t∈[0,1],s≠t∫0s(K(T+εt,T+εu)−K(T+εs,T+εu))2duγε2|t−s|2τˆ≤c.

Proposition 4.

Under Assumptions 1 and 5 one has limε→0Cov(XT+εtψ−XTψ,XT+εsψ−XTψ)γε2=Υ¯(t,s) uniformly in (t,s)∈[0,1]×[0,1] , where (21) Υ¯(t,s)=α˜2α2+α˜2∫0t∧sK¯(t,u)K¯(s,u)du+α˜2α2+α˜2k¯(t,s).

Proof.

From (20), Assumption 1 and Assumption 5 the claim easily follows. □

Proposition 5.

Under Assumptions 2 and 6 the family ((XT+εtψ−XTψ−E[XT+εtψ−XTψ])t∈[0,1])ε>0 0}}$]]> is exponentially tight at the inverse speed γε2 .

Proof.

Since ((XT+εtψ−XTψ−E[XT+εtψ−XTψ])t∈[0,1])ε>0 0}}$]]> is a family of centered processes, it is enough to prove that (3) is satisfied with an appropriate speed function.

The covariance of such process is given by (20). Therefore Var(XT+εtψ−XT+εsψ)=εα2α2+α˜2∫0t(K(T+εt,T+εu)−K(T+εs,T+εu))2du++α˜2α2+α˜2k(T+εt,T+εt)−2k(T+εt,T+εs)+k(T+εs,T+εs). From Assumption 2 we already know that sups,t∈[0,1],s≠t|k(T+εt,T+εt)−2k(T+εt,T+εs)+k(T+εs,T+εs)|γε2|t−s|2τ≤M. Furthermore, Assumption 6 implies that sups,t∈[0,1],s≠t∫0t(K(T+εt,T+εu)−K(T+εs,T+εu))2duγε2|t−s|2τˆ≤M.

Therefore condition (3) holds with the inverse speed ηε=γε2 and β=τ∧τˆ . □

We are ready to state a large deviation principle for the conditioned Volterra process. Theorem 6.

Suppose Assumptions 1, 2, 5 and 6 are fulfilled. If the (existing) covariance function Υ¯ defined in Proposition 4 is regular enough, then the family of processes ((XT+εtψ−XTψ)t∈[0,1])ε>0 0}}$]]> satisfies a large deviation principle on C[0,1] with the inverse speed γε2 and the good rate function (22) I(h)=12‖h‖H¯2,h∈H¯,+∞,otherwise, where H¯ and ‖.‖H¯ , respectively, denote the reproducing kernel Hilbert space and the related norm associated to the covariance function Υ¯ given by (21).

Proof.

Cosider the family of centered processes ((XT+εtψ−XTψ−E[XT+εtψ−XTψ])t∈[0,1])ε . Thanks to Proposition 5 this family of processes is exponentially tight at the inverse speed γε2 . Thanks to Proposition 4, for any λ∈M[0,1] , one has limε→0Var(⟨λ,XT+εtψ−XTψ⟩)γε2=∫01∫01Υ¯(t,s)dλ(t)dλ(s) where Υ¯ is defined in (21) Since Υ¯ is the covariance function of a continuous Volterra process, a large deviation principle for ((XT+εtψ−XTψ−E[XT+εtψ−XTψ])t∈[0,1])t∈[0,1])ε>0 0}}$]]> actually holds from Theorem 2 with the inverse speed γε2 and the good rate function given by (22). From Equation (18) and Remark 2 the same large deviation principle holds for the noncentered family ((XT+εtψ−XTψ)t∈[0,1])ε>0 0}}$]]>. □

4.3 Examples

In this section we consider some examples to which Theorem 6 applies. Therefore we want to verify that Assumptions 1, 2, 5 and 6 are fulfilled. Let X be a continuous, centered Volterra process process with kernel K.

Example 4 (Fractional Brownian Motion).

Consider a fractional Brownian motion with H>1/2 1/2$]]> as in Example 1. We have already proved that Assumptions 1 and 2 are fulfilled with τ=H and γε=εH .

We want to show that Assumptions 5 and 6 are fulfilled with τˆ=1 , γε=εH . From Example 4.17 in [9], we have that limε→0εK(T+εt,T+εs)εH=cH(t−s)H−12 uniformly for t,s∈[0,1] . Therefore limε→01ε2H∫0t∧sK(T+εt,T+εu)K(T+εs,T+εu)du=cH2∫0t∧s(t−u)H−12(s−u)H−12du uniformly for (t,s)∈[0,1]×[0,1] . So, we have that Assumption 5 is fulfilled with Υ¯(t,s)=α˜2α2+α˜2k(t,s)+α2α2+α˜2cH2∫0t∧s(t−u)H−12(s−u)H−12du. Note that Υ¯ is regular enough. Let us now prove the exponential tightness.

Since (a+b)2≤2(a2+b2) for a,b∈R , from Equation (15), there exists a constant c>0 0$]]> such that, for s<t , K(T+εt,T+εu)2≤c(ε2H−1(t−u)2H−1+(∫T+εuT+εt(v−(T+εu))H−12dv)2)≤c(ε2H−1(t−u)2H−1+ε2H+1(t−u)2H+1). Thus, ε∫stK(T+εt,T+εu)2du≤c(ε2H(t−s)2H+ε2H+2(t−s)2H+2). Furthermore, (K(T+εt,T+εu)−K(T+εs,T+εu))2=(A(u)−B(u))2≤2(A2(u)+B2(u)), where A(u)=cH[(T+εtT+εu)H−12(t−u)H−12εH−12−(T+εsT+εu)H−12(s−u)H−12εH−12)]B(u)=cH(H−12)1(T+εu)H−12∫T+εsT+εtvH−32(v−(T+εu))H−12dv. Now, thanks to the Lagrange theorem, there exists x∈[s,t] such that A(u)≤c((T+εt)H−12(t−u)H−12εH−12−(T+εs)H−12(s−u)H−12εH−12)=c((T+εx)H−32(x−u)H−12+(T+εx)H−12(x−u)H−32)εH−12(t−s).

The estimation of B is easily done. There exists x∈[s,t] such that B(u)≤c∫T+εsT+εt(v−(T+εu))H−12dv=cεH+12∫st(v−u)H−12dv=cεH+12(x−u)H−32(t−s), and then ε∫0sA2(u)du≤cε2H(t−s)2,ε∫0sB2(u)du≤cε2H+2(t−s)2. From Remark 15 we have that ∫0t(K(T+εt,T+εu)−K(T+εs,T+εu))2du≤cε2H(t−s)2. Assumption 6 is then fulfilled with τˆ=1 and γε=εH .

A large deviation principle is then established for the family of processes ((XT+εtψ−XTψ)t∈[0,1])ε>0 0}}$]]>, with the inverse speed ε2H .

Example 5 (<italic>m</italic>-fold integrated Brownian motion).

Let X be the process defined in Example 2. We have already proved that Assumptions 1 and 2 are fulfilled with τ=1 and γε=ε . We want to show that Assumptions 5 and 6 are fulfilled with τˆ=1 , γε=ε . For s≤t , we have K(T+εt,T+εs)=(t−s)mεm, therefore limε→0εK(T+εt,T+εs)ε=0 uniformly in t,s∈[0,1] . Thus, Υ¯(t,s)=α˜2α2+α˜21(m!)2m22m−1T2m−1st, and Assumption 5 is verified. Note that Υ¯ is regular enough. Let us now prove the exponential tightness of the family of processes. For s<t , there exist positive constants c1,c2 such that ∫0t(K(T+εt,T+εu)−K(T+εs,T+εu))2du=1(m!)2∫0sε2m((t−u)m−(s−u)m)2du+1(m!)2∫stε2m((t−u)2mdu≤ε2mc1(t−s)2+ε2mc2(t−s)2m+1 and Assumption 6 is verified with τˆ=1 ad γε=ε . A large deviation principle is then established for the family of conditioned processes ((XT+εtψ−XTψ)t∈[0,1])ε>0 0}}$]]>, with the inverse speed ε2 .

Example 6 (Integrated Volterra Process).

Let X be the process defined in Example 3. We have already proved that Assumptions 1 and 2 are fulfilled with τ=1 , γε=ε . We want to show that Assumptions 5 and 6 are fulfilled with τˆ=1 and γε=ε . For s≤t , we have h(T+εt,T+εs)=∫T+εsT+εtK(v,T+εs)dv=ε∫stK(T+εv,T+εs)dv, therefore limε→0εh(T+εt,T+εs)ε=0 uniformly in t,s∈[0,1] . Thus, Υ¯(t,s)=α˜2α2+α˜2∫0TK2(T,u)dust, and Assumption 5 is verified. Note that Υ¯ is regular enough. Let us now prove the exponential tightness of the family of processes. For s<t , there exist a constant c>0 0$]]> such that ∫0t(h(T+εt,T+εu)−h(T+εs,T+εu))2du=∫0s(∫T+εsT+εtK(v,T+εu)dv)2du+∫st(∫T+εuT+εtK(v,T+εu)dv)2du≤ε2c(t−s)2 and Assumption 6 is verified with τˆ=1 and γε=ε . A large deviation principle is then established for the family of conditioned processes ((XT+εtψ−XTψ)t∈[0,1])ε>0 0}}$]]>, with the inverse speed ε2 .

Acknowledgments

The author wish to thank the Referees for their very useful comments which allowed me to greatly improve the presentation of the paper.

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