Pathwise asymptotics for Volterra processes conditioned to a noisy version of the Brownian motion
Barbara Pacchiarotti
math.PR



Research Article



Dept. of Mathematics, University of Rome “Tor Vergata”,
via della Ricerca Scientifica, 00133, Roma, Italy

B. PacchiarottiPathwise asymptotics for Volterra processes















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.



Large deviations
Volterra type Gaussian processes
conditional processes


[MSC2010]
60F10
60G15
60G22


30 7 2019
30 11 2019
10 2 2020
27 2 2020







§ INTRODUCTION


In this paper we study the asymptotics of the regular conditional prediction
law of a Gaussian Volterra processGaussian 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 <cit.> and <cit.>, respectively. Let X=(X_t)_t≥ 0 be a
continuous real Volterra process.continuous real Volterra process



A centered Gaussian processGaussian process X is a Volterra processVolterra process if, for every T>0, it
admits the representation



    X_t = ∫_0^T
    K(t,s)   dB_s,


where B=( B_t)_t≥ 0 is a Brownian motionBrownian motion and K is a square
integrable function on [0,T]^2 (the kernel) such that K(t,s)=0 for all
s > t.


For a Volterra processVolterra process the covariance function is



    k(t,s)= ∫_0^s∧ t
    K(t,u)K(s,u) du   t,s ∈ [0,T].


Let B̃=(B̃_t)_t≥ 0 be another Brownian motionBrownian motion
independent of B and for α , α̃∈ℝ define
W^α ,α̃= α B + α̃B̃.

First case For fixed n∈ℕ and T>0, we consider the
conditioning of X on n linear functionals of the paths of W^α
,α̃,


    G_T(W^α ,α̃)=
    (G_T^1(W^α
    ,α̃),…
    ,G_T^n(W^α ,α̃)
    )^⊺,


more precisely,


    G_T(W^α ,α̃)= ∫
    _0^T g(t) dW^α ,α̃_t=
    (∫_0^T g_1(t) 
    dW^α ,α̃_t,… ,∫_0^T
    g_n(t)   dW^α
    ,α̃_t)^⊺,


where g =(g_1,… ,g_n)^⊺ is a suitable
vectorial function defined on [0,T]. Informally the generalized conditioned
process X^ g;x, for x∈ℝ^n, is
the law of the Gaussian processGaussian process X conditioned on the set


    {∫_0^T
    g(t) dW^α ,α̃_t= x}=⋂
    _i=1^n{∫
    _0^T g_i(t) dW^α ,α̃_t=x_i}.


We obtain a large deviation principle for the family of processes
((X^ g;x_T+ε t- X^ g;x_T)_t ∈
[0,1])_ε >0.

Second case We are interested in the regular conditional law of the
process X given the σ-algebra ℱ^α ,α̃_T, where (ℱ^α ,α̃_t)_t≥ 0 is
the filtration generated by the mixed Brownian motionBrownian motion W^α
,α̃, i.e. we want to condition the process to the past of the
mixed Brownian motionBrownian motion up to a fixed time T>0. Informally the generalized
conditioned process X^ψ, for ψ being a continuous function, is the
law of the Gaussian processGaussian process X conditioned on the set


    { W^α ,α̃_t=ψ _t,   t∈ [0,T]
    }.


Here we obtain a large deviation principle for the family of processes
((X^ψ_T+ε t- X^ψ_T)_t∈ [0,1])_ε
>0.

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

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




§ LARGE DEVIATIONS FOR CONTINUOUS GAUSSIAN PROCESSESGAUSSIAN PROCESS


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
<cit.>, Section 3.4 in Deuschel and Strook <cit.>, Chapter 4 (in
particular Sections 4.1, 4.2 and 4.5) in Dembo and Zeitouni <cit.>.



Let E be a topological space, ℬ(E) be the Borel σ-algebra and (μ _ε)_ε >0 be a family of
probability measures on ℬ(E). We say that the family of
probability measures (μ _ε)_ε >0 satisfies a
large deviation principle on E with the rate function I and the inverse
speed η _ε (η _ε>0, η _ε→ 0 as ε→ 0) if, for any open set Θ,


    -inf_x ∈Θ I(x) ≤lim inf_ε→ 0η
    _εlogμ _ε(Θ )


and for any closed set Γ,


    lim sup_ε→ 0η _εlogμ
    _ε(Γ ) ≤ -inf_x ∈Γ I(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 = (U_t)_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 ⟩ =∫_0^1 h(t)
     dλ (t),   λ∈ℳ[0, 1],  h∈ C [0, 1].




We say that a family of continuous processes ((U_t^ε)_t ∈
[0,1])_ε >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 <cit.>) gives an
explicit expression of the Cramér transform Λ ^* of a
continuous centered Gaussian process (U_t)_t ∈ [0,1] with covariance
function k. Let us recall that for λ∈ℳ[0,1],


    Λ (λ )=log𝔼[exp(⟨ U, λ⟩)]= 1/2∫_0^1∫_0^1 k(t,s)   dλ (t)  d λ
    (s).




Let (U_t)_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)    =   sup_λ∈ℳ[0,1]( ⟨λ , x ⟩ - Λ (λ ) )
    
       =   sup_λ∈ℳ[0,1]( ⟨λ , x ⟩ -
    1/2∫_0^1∫
    _0^1 k(t,s)   dλ (t)  dλ (s)
    ).


Then,


    Λ ^*(x) = 
    1/2x _ℋ^2,     x ∈ℋ,
    
    
    +∞,    otherwise,


where ℋ and . _ℋ denote, respectively, the
reproducing kernelreproducing kernel Hilbert space Hilbert space and the related norm associated to the
covariance function k.


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



A family of continuous processes ((U^ε_t)_t ∈
[0,1])_ε >0 is exponentially tight at the inverse speed η
_ε, if for every R>0 there exists a compact set K_R such
that


    lim sup_ε→ 0η _εlogℙ(U^ε∉ K_R) ≤ -R.




If the means and the covariance functions of an exponentially tight family of
Gaussian processesGaussian process 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 <cit.>) and
Theorem <ref>.



Let ((U_t^ε)_t ∈ [0,1])_ε >0 be an
exponentially tight family of continuous Gaussian processesGaussian process at 
the inverse speed function η _ε. Suppose that, for any
λ∈ℳ[0,1],


    lim_ε→ 0𝔼[ ⟨λ ,
    U^ε⟩] = 0


and the limit


    Λ (λ ) = lim_ε→ 01/η _εVar( ⟨λ ,
    U^ε⟩) = ∫_0^1∫_0^1k(t,s)   dλ (t)  d
    λ (s)


exists for some continuous, symmetric, positive definite function k,
that is the covariance function of a continuous Gaussian process,Gaussian process then
((U_t^ε)_t ∈ [0,1])_ε >0 satisfies a large
deviation principle on C[0,1] with the inverse speed η _ε
and the good rate function


    I(h) = 
    1/2   h     ^2_ℋ,     h ∈ℋ,
    
    +∞,    otherwise,


where ℋ and .     _ℋ,
respectively, denote the reproducing kernelreproducing kernel Hilbert space Hilbert space and the related norm
associated to the covariance function k.


In order to prove exponential tightnessexponential tightness we shall use the following result
(see Proposition 2.1 in <cit.>).



Let ((U_t^ε)_t ∈ [0,1])_ε >0 be a family of
continuous Gaussian processes,Gaussian process where U_0^ε=0 for all
ε >0. Suppose there exist constants β ,M_1,M_2>0 such
that for ε >0


    sup_s,t∈ [0,1],s≠ t|𝔼[U^ε_t-U^ε_s]|/|t-s|^β≤ M_1


and



    sup_s,t∈ [0,1],s≠ tVar(U^ε_t-U^ε_s)/η _ε |t-s|^2β≤
    M_2


then ((U_t^ε)_t ∈ [0,1])_ε >0 is
exponentially tight at 
the inverse speed function η_ε.




Suppose ((U_t^ε)_t ∈ [0,1])_ε >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 η _ε and the
good rate function I. Let (m^ε)_ε >0⊂
C[0,1], m∈ C[0,1] be functions such that m^εC[0,1]⟶m, as ε→
0. Then, the family of processes (m^ε+U^ε)_ε >0 satisfies a large deviation principle on C[0,1] with
the same inverse speed η _ε and the good rate function


    I_m(h) =I(h-m)= 
    1/2   h-m     ^2_ℋ,     h- m ∈ℋ,
    
    +∞,     h- m ∉ℋ.


In fact the two families (m^ε+U^ε)_ε
>0 and (m +U^ε)_ε >0 are exponentially
equivalent (at the inverse speed η _ε) and therefore as far
as the large deviation principle is concerned, they are indistinguishable.
See Theorem 4.2.13 in <cit.>.


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 ((X_T+ε t-X_T)_t∈
[0,1])_ε >0, as ε→ 0.

Let (X_t)_t≥ 0 be a continuous centered Gaussian processes and fix
T>0. The next two assumptions guarantee that Theorem <ref>
is applicable to the family of processes ((X_T+ε t-X_T)_t∈
[0,1])_ε >0. Let γ _ε>0 be an infinitesimal
function, i.e. γ _ε→ 0 for ε→ 0.



For any fixed T>0 there exists an asymptotic covariance function k̅
defined as



    k̅(t,s)    =lim_ε→ 0Cov(X_T+ε t-X_T, X_T+ε
    s-X_T)/γ ^2_ε
       =lim_ε→ 0k(T+ε t,
    T+ε s) -k(T+ε t, T)-k( T+ε s,T) +k(T,
    T)/γ ^2_ε,


uniformly in (t,s)∈ [0,1]× [0,1].




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




Recall that the continuity of the covariance function is not a sufficient
condition to identify a Gaussian processGaussian process with continuous paths. We need some
more regularity. Since we are investigating continuous Volterra processes,Volterra process 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
<cit.> and <cit.>). Such approach may be difficult to apply.
However, in <cit.>, 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 processGaussian process (X_t)_t∈ [0,T] is Hölder continuous of
exponent 0<a<A if and only if for every ε >0, s,t∈ [0,T],
there exists a constant c_ε>0 such that


    𝔼[(X_t-X_s)^2] ≤ c_ε|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 processVolterra process
(<ref>) we have the following sufficient condition
for the Hölder continuity of a Volterra process:Volterra process there exist constants c,A
> 0 such that



    M(δ )≤ c δ ^A


for all δ∈ [0,T], where


    M(δ )= sup_{t_1,t_2∈ [0,T]: |t_1-t_2|≤δ}∫_0^T|K(t_1,s)-K(t_2,s)|^2 ds.


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.




For any fixed T>0 there exist constants M, τ >0, such that for
ε >0,


    sup_s,t∈ [0,1],s≠ tVar(X_T+ε
    t-X_T+ε s)/γ ^2_ε |t-s|^2τ
       =sup _s,t∈ [0,1],s≠ tk(T+ε t, T+ε t) -2
    k(T+ε t, T+ε s ) +k(T+ε s , T+ε
    s)/γ ^2_ε |t-s|^2τ≤ M.



As an immediate application of Theorem <ref> (take
U^ε_t=X_T+ε t-X_T),
Assumptions <ref> and <ref> imply, if k̅
is regular enough, that the family ((X_T+ε t-X_T)_t∈
[0,1])_ε >0 satisfies a large deviation principle on C[0,1]
with the inverse speed γ _ε^2 and the good rate
function given by


    J_X(h)=
    1/2h^2_ℋ̅,     h∈ℋ̅,
    
    
    +∞,


where ℋ̅ is the reproducing kernelreproducing kernel Hilbert space Hilbert space
associated to the covariance function k̅ and the symbol ·_ℋ̅ denotes the usual norm defined on
ℋ̅.

In fact Assumption <ref> immediately implies that


    Λ (λ )=lim_ε→ 0Var(⟨λ , X_T+ε·-X_T⟩ )/γ ^2_ε =∫_0^1∫ _0^1k̅(t,s)λ (dt)λ (ds).


Furthermore, Assumption <ref> implies that the family
((X_T+ε t-X_T)_t∈ [0,1])_ε >0 is
exponentially tight 
at the inverse speed function γ
^2_ε.




§ CONDITIONING TO N FUNCTIONALS OF THE PATH





 §.§ Conditional lawconditional law


Let (Ω , ℱ, (ℱ_t)_t≥ 0, ℙ)
be a filtered probability space. On this space we consider a Brownian motionBrownian motion
B=( B)_t≥ 0, a continuous real Volterra process X=(X_t)_t≥ 0continuous real Volterra process
and another Brownian motionBrownian motion B̃=(B̃)_t≥ 0 independent of
B. For α , α̃∈ℝ let us define the mixed
Brownian motionBrownian motion W^α ,α̃= α B + α̃B̃.

For fixed n∈ℕ and T>0, we consider the conditioning of X
on n linear functionals of G_T(W^α ,α̃)= (G_T^1(W^α ,α̃),… ,G_T^n(W^α
,α̃))^⊺ of the paths of W^α ,α̃,


    G_T(W^α ,α̃)= ∫ _0^T g(t) dW^α ,α̃_t= (∫_0^T
    g_1(t)  dW^α ,α̃_t,… ,∫_0^T g_n(t)  dW^α ,α̃_t)^⊺,


where g =(g_1,… ,g_n)^⊺ is a vectorial
function and g_k∈𝕃^2[0,T], for k=1,… ,n. We
assume, without any loss of generality, that the functions g_i,
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 X^
g;x, for x∈ℝ^n, is the law of the Gaussian
processGaussian process X conditioned on the set


    {∫_0^T g(t) dW^α ,α̃_t=
    x}=⋂ _i=1^n{∫ _0^T
    g_i(t) dW^α ,α̃_t=x_i}.


The law ℙ^ g;x of X^ g;x is the
regular conditional distribution on C[0,+∞ ), endowed with the
topology induced by the sup-norm on compact sets,


    ℙ^g;x(X∈ E)=ℙ(X^
    g;x∈ E )=ℙ(X∈ E |∫_0^T
    g(t) dW^α ,α̃_t= x).


For more details about existence of such regular conditional distribution
see, for example, <cit.>.

Denote by C^ g=(c^g_ig_j_ij)_i,j=1,… ,n the
matrix defined by


    c^g_ig_j_ij=Cov( ∫ _0^T g_i(t)  dW^α ,α̃_t,∫_0^T g_j(t)  dW^α ,
    α̃_t)=(α ^2+α̃^2)∫_0^T g_i(t)g_j(t) dt.


The matrix C^ g is invertible (since the functions g_i,
i=1,… ,n, are linearly independent). Let us denote


    r_i^ g_i(t)=Cov(X_t, ∫_0^T g_i(u)
     dW^α ,α̃_u)=α∫_0^t∧ T
    K(t,u) g_i(u) du,


and


    r^ g(t)=(r_1^g_1(t),…
    ,r_n^g_n(t))^⊺.


The following theorem, similar to Theorem 3.1 in <cit.>, gives mean
and covariance function of the generalized conditioned process.



The generalized conditioned process X^ g;x can be represented
as


    X^ g;x_t=X_t - r^ g(t)^⊺(C^ g)^-1( ∫_0^T
    g(u) dW^α ,α̃_u- x).


Moreover, the conditioned process X^g;x is a Gaussian
processGaussian process with mean



    m^ g;x(t)=𝔼[X^ g;x_t]
    =r^ g (t)^⊺(C^ g)^-1 x,


and covariance



    k^g(t,s)=Cov(X^ g;x_t,X^ g;x_s) =k(t,s)-κ ^g(t,s),


where



    κ ^g(t,s)=r^ g (t)^⊺(C^ g)^-1r^ g(s).





It is a classical result on conditioned Gaussian laws. See, e.g., Chapter II,
13, in <cit.>.




Let us note that the covariance function of the conditioned process depends
on the conditioning functions g_1,… ,g_n and on the time T, but
not on the vector x.




If the conditioning functions g_i are the indicator functions of the
interval [0,T_i), for i=1,… ,n, then the process is conditioned to
the position of the noisy Brownian motionBrownian motion at the times T_1,… ,
T_n, more precisely to the set ⋂_i=1^n{ W^α
,α̃_T_i=x_i}.




If the conditioning functions are g_i(s)=K(T_i,s) _[0,T_i)(s), for i=1,… ,n, and α =1, α̃=0, then the process is conditioned to its position at the times
T_1,… , T_n, more precisely to the set ⋂_i=1^n{
X_T_i=x_i} (this is a particular case of the conditioned process in
<cit.>).





 §.§ Large deviations


Let γ _ε>0 be an infinitesimal function, i.e. γ
_ε→ 0 for ε→ 0. In this section
(X_t)_t≥ 0 is a continuous Volterra processVolterra process as in
(<ref>). Now, in order to achieve a large
deviation principle for the family of processes ((X^
g;x_T+ε t- X^ g;x_T)_t ∈
[0,1])_ε >0, we have to investigate the behavior of the
functions k^g and m^ g;x (defined in
(<ref>) and (<ref>), 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 <ref> hold for the conditioned
process. The next assumption (Assumption <ref>) implies the
existence of a limit covariance.



For any T>0 and for g_i∈𝕃^2[0,T], i=1,… ,n,
there exists a vectorial function
r̅^g=(r̅_1^g_1,… ,r̅_n^g_n
), possibly r̅_i^g_i=0 for some i=1,… ,n, such that



    r̅_i^g_i(t)=lim _ε→ 0Cov
    (X_T+ε t-X_T, ∫_0^T g_i(u) dW^α ,α̃_u  )/γ _ε= lim_ε→ 0r_i^g_i(T+ε t)-r_i^g_i(T)/γ
    _ε,


uniformly in t∈ [0,1].


The next assumption (Assumption <ref>) implies the
exponential tightnessexponential tightness of the family of the centered processes.



For any fixed T>0 there exist constants M,τ̂>0, such that for
i=1,… ,n and ε >0,


    sup_s,t∈ [0,1],s≠ t  |Cov  (X_T+ε
    t-X_T+ε s, ∫_0^T g_i(u) dW^α ,α̃_u  )  |/γ _ε|t-s|^τ̂
         =sup_s,t∈ [0,1],s≠ t|r_i^g_i(T+ε
    t)-r_i^g_i(T+ε s)|/γ _ε|t-s|^τ̂≤ M.





Let us observe that Assumption <ref> implies that for any
fixed T>0


    lim_ε→ 0r_i^g_i(T+ε t)-r_i^g_i(T)=0,


uniformly in t∈ [0,1]. Therefore,



    lim_ε→ 0 m^ g;x(T+ ε t)=m^ g;x(T),


uniformly in t∈ [0,1]. In fact, one has


    m^ g;x(T+ε t)-m^ g;x(T)=(
    r^g(T+ε t)-r^g(T) )^⊺(C^ g)^-1 x,


and (<ref>) immediately follows.




Let us observe that Assumption <ref> implies that there
exists M>0 such that the following estimate holds for the function κ
^g defined in (<ref>):



    sup_s,t∈ [0,1],s≠ t|κ ^g(T+ε t,
    T+ε t)-2 κ ^g(T+ε t, T+ε s
    ) +κ ^g(T+ε s , T+ε s)|/γ
    _ε^2|t-s|^2τ̂
         ≤ M.


In fact, straightforward computations show that


    κ ^g(T+ε t, T+ε t) -2
    κ ^g(T+ε t, T+ε s ) +κ
    ^g(T+ε s , T+ε s)
         =(( r^g(T+ε
    t)-r^g(T+ ε s)))^⊺(C^ g)^-1((r^g(T+ε
    t)-r^g(T+ε s))).



Therefore (<ref>) immediately follows from
Assumption <ref>.



Under Assumptions <ref> and <ref>, one has


    lim_ε→ 0Cov(X^g;x_T+ε
    t-X^g;x_T, X^g;x_T+ε
    s-X^g;x_T)/γ ^2_ε=
    k̅^g(t,s),


uniformly in (t,s)∈ [0,1]× [0,1], with



    k̅^g(t,s)=k̅(t,s)-r̅^g(t)^⊺(C^ g)^-1r̅^ g(s),


where r̅^ g(t)^⊺=(r̅^g_1_1(t), … ,r̅^g_n_n(t)) and
r̅_i^g_i(t) is defined in (<ref>) for i=1,… ,n.



Taking into account equation (<ref>), simple computations show
that for s,t∈ [0,1],



    Cov( X^ g;x_T+ε t-
    X^ g;x_T, X^ g;x_T+ε s-
    X^ g;x_T)
    
         = (k(T+ε t, T+ε s) -k(T+ ε t, T)-k(T+
    ε s , T) +k(T, T))+
    
           -(( r̅^ g(T+ ε t)-
    r̅^ g(T)))^⊺(C^ g)^-1((r̅^ g(T+ε s)-r̅^ g(T)
    )).


Therefore the claim easily follows from Assumptions <ref> and
<ref>.




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




Under Assumptions <ref> and <ref> the
family ((X^g;x_T+ε t- X^g;x_T-
𝔼[X^g;x_T+ε t- X^g;x_T])_t∈ [0,1])_ε >0 is exponentially
tight at 
the inverse speed function γ _ε^2.



As ((X^g;x_T+ε t- X^g;x_T-
𝔼[X^g;x_T+ε t-
X^g;x_T])_t∈ [0,1])_ε >0 is a family of
centered processes, it is enough to prove that
(<ref>) is satisfied with 
an appropriate speed
function. For ε >0 the covariance of such process is given by
(<ref>). Therefore


    Var(X^g;x_T+ε t-
    X^g;x_T+ε s)
    
          = k(T+ε t, T+ε t) -2 k(T+ ε t, T+
    ε s ) +k(T+ε s , T+ε s)
    
           + r^g(T+ε t)-
    r^g(T)))^⊺(C^ g)^-1(( r^g(T+ε s)-
    r^g(T)).


From Assumption <ref> we already know that


    sup_s,t∈ [0,1],s≠ tk(T+ε t, T+ε t) -2
    k(T+ε t, T+ε s ) +k(T+ε s , T+ε
    s)/γ _ε^2 |t-s|^2τ≤ M.


Furthermore, Assumption <ref> implies that


    sup_s,t∈ [0,1],s≠ t|(r^g(T+ε
    t)-r^g(T)))^⊺(C^
    g)^-1((r^g(T+ε
    s)-r^g(T))|/γ _ε^2 |t-s|^2τ̂≤ M.


Therefore condition (<ref>) holds with the
inverse speed η _ε=γ _ε^2 and β
=τ∧τ̂.


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



Suppose (X_t)_t≥ 0 satisfies Assumptions <ref>,
<ref>, <ref> and <ref>.
Suppose, furthermore, that the (existing) covariance function
k̅^g defined in Proposition <ref> is
regular enough, then the family of processes
((X^g;x_T+ε t - X^g;x_T)_t ∈
[0,1])_ε >0 satisfies a large deviation principle on C[0,1]
with the inverse speed γ ^2_ε and the good rate
function



    J_X^g(h)=
    1/2 h^2_ℋ̅^g,      h∈ℋ̅^g,
    
    +∞,    otherwise,


where ℋ̅^g is the reproducing kernelreproducing kernel Hilbert space
Hilbert space associated to the covariance function
k̅^g.



Cosider the family of centered processes
((X^g;x_T+ε t- X^g;x_T-
𝔼[X^g;x_T+ε t- X^g;x_T])_t∈ [0,1])_ε >0. Thanks to
Proposition <ref> this family of processes is
exponentially tight at the inverse speed γ _ε^2. Thanks
to Proposition <ref>, for any λ∈ℳ[0,1], one has


    lim_ε→ 0Var(⟨λ , X^g;x_T+ε·-X^g;x_T⟩ )/γ ^2_ε
         =lim _ε→ 0∫_0^1 d λ
    (v)∫_0^1 dλ (u)
    Cov(X^g;x_T+ε
    v-X^g;x_T, X^g;x_T+ε
    u-X^g;x_T)/γ ^2_ε
         =∫_0^1 dλ (v)∫_0^1 dλ (u)
    k̅^g(v,u),


where k̅^g is defined in (<ref>). Since
k̅^g is the covariance function of a continuous
Volterra process,Volterra process a large deviation principle for
((X^g;x_T+ε t - X^g;x_T-
𝔼[X^g;x_T+ε t- X^g;x_T])_t∈ [0,1])_ε >0 actually holds from
Theorem <ref> with the inverse speed γ _ε^2 and the good rate function given by (<ref>). From
Equation (<ref>) and Remark <ref> the same
large deviation principle holds for the noncentered family
((X^g;x_T+ε t - X^g;x_T)_t ∈
[0,1])_ε >0.





 §.§ Examples


In this section we consider some examples to which Theorem <ref>
applies. Therefore we want to verify that Assumptions <ref>,
<ref>, <ref> and <ref>
are fulfilled. Let X be a continuous, centered Volterra process process
with kernel K. Suppose g_1(t)=_[0,T)(t) and
g_2(t)=T-t/T_[0,T)(t), that is,


    W^α ,α̃_T=∫_0^T g_1(u) dW^α ,
    α̃_u=x_1


and by the integration by parts formula,


    1/T∫_0^T W^α ,α̃_u du= ∫_0^T
    g_2(u)  dW^α ,α̃_u = x_2.


Then the matrix (C^ g)^-1 is given by


    (C^ g)^-1=1/(C^ g) (
    
    [  c^ g_2g_2_22 -c^ g_1g_2_12; -c^ g_1g_2_12   c^g_1g_1_11 ] ),


where


    c^g_1g_1_11   = (α ^2 + α̃^2) T,
         c^ g_1g_2_12= (α ^2 + α̃^2) T/2,
    
    
    c^ g_2g_2_22    =(α ^2 + α̃^2)T/3,     (C^ g)=(α ^2 + α̃^2)^2 T^2/12.



[Fractional Brownian Motion]
Let X be the fractional Brownian motionfractional Brownian motion of the Hurst index H>1/2. The
fractional Brownian motionfractional Brownian motion with the Hurst parameter H ∈ (0,1) is the centered
Gaussian process with covariance function


    k(t,s)=1/2(t^2H+s^2H-|t-s|^2H).


The fractional Brownian motionfractional Brownian motion is a Volterra processVolterra process with kernel, for s≤
t,



    K(t,s) = c_H[ (
    t/s (t-s) ) ^H-1/2 - ( H-
    1/2) s^1/2 - H∫_s^tu^H-3/2(u-s)^H-1/2 du ],


where c_H =   ( 2H  Γ (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 motionfractional 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(X_T+ε t-X_T, X_T+ε
    s-X_T)/ε ^2H =Cov(X_t, X_s),


because of the homogeneity and self-similarity properties holding for the
fractional Brownian motion,fractional Brownian motion so that the limit in (<ref>) trivially
exists and Assumption <ref> holds with k̅(t,s)=k(t,s)
and γ _ε=ε ^H. Now let us prove that
Assumption <ref> is fulfilled.


    Cov(X_T+ε t-X_T,
    W^α ,α̃_T)
    
       = ∫_0^T(K(T+ε t,
    u)- K(T,u) ) du
    
       = c_H∫_0^T(
    (T+ε t/u)^H-1/2(T+ ε t
    -u)^H-1/2- (T/u)^H-1/2(T-u)^H-1/2)  du+
    
         - c_H (H-1/2) ∫_0^T1/u^H-1/2∫_T^T+
    ε t
    (v-u)^H-1/2 v^H-3/2 dv   du.


Thanks to the Lagrange theorem,Lagrange theorem


    ( (T+ε t/u)^H-1/2(T+
    ε t -u)^H-1/2- (T/u)^H-1/2(T-u)^H-1/2)
    
       = H-1/2/u^H-1/2[ ( T+ ξ _ε)^H-3/2 (T+ξ _ε-u)^H-1/2+
    ( T+ ξ _ε)^H-1/2 (T+ξ _ε-u)^H-3/2]ε t,


for ξ _ε∈ [0,ε t]. Therefore from the Lebesgue
theorem,


    lim_ε→ 01/ε∫
    _0^T( (T+ε t/u)^H-1/2(T+ε t -u)^H-1/2- (
    T/u)^H-1/2(T-u)^H-1/2) du
    
       = t (H-1/2)∫_0^T1/u^H-1/2(T^H-3/2(T -u)^H-1/2+ T^H-1/2(T-u)^H-3/2) du ,


uniformly in t∈ [0,1]. Furthermore, in a similar way, we have,


    lim_ε→ 01/ε∫
    _0^T1/u^H-1/2∫ _T^T+ε t (v-u)^H-1/2
    v^H-3/2 dv   du
    
         =t ∫_0^T1/u^H-1/2 (T-u)^H-1/2 T^H-3/2 du.


Therefore,


    r̅_1^g_1(t)=lim_ε→ 0Cov
    (X_T+ε t-X_T, W^α ,α̃_T  )/ε
    ^H=0,


uniformly in t∈ [0,1]. Similar calculations show that


    r̅_2^g_2(t)=lim_ε→ 0Cov
    (X_T+ε t-X_T, ∫_0^TT-u/T dW^α
    ,α̃_u  )/ε ^H=0.,


So, we have k̅^g(t,s)= k(t,s) and therefore the limit
covariance exists and is regular enough.

Now let us prove the exponential tightnessexponential tightness of the family of processes. For
s< t,


    Var(X_T+ε t-X_T+ε s)=ε
    ^2H(t-s)^2H,


then Assumption <ref> holds with τ =H and
γ _ε=ε ^H. For s<t, we have


    Cov(X_T+ε t-X_T+ε s,
    W^α ,
    α̃_T)
       = ∫_0^T(K(T+ε t, u)- K(T+ ε s ,u) ) du
    
       = c_H∫_0^T( (T+ε t/u)^H-1/2  (T+
    ε t -u)^H-1/2 - ( T+ε t/u)^H-1/2 (T+ ε s -u)^H-1/2)  du+
    
         - c_H (H-1/2) ∫_0^T1/u^H-1/2∫_T+ε s^T+
    ε t
    (v-u)^H-1/2 v^H-3/2 dv   du.


Thanks to the Lagrange theoremLagrange theorem we can find M>0 such that


    ∫_0^T( (
    T+ε t/u)^H-1/2(T+ε t
    -u)^H-1/2- (T+ε s/u)^H-1/2(T+
    ε s -u)^H-1/2) du
    
       ≤ε (t-s)(H- 1/2)∫_0^T1/u^H-1/2[
    T^H-3/2 (T+1 -u)^H-1/2+ ( T+ 1)^H-1/2 (T
    -u)^H-3/2] du 
       ≤ε M (t-s)


and


    ∫_0^T1/u^H-1/2∫_T+ε s^T+ε t
    (v-u)^H-1/2 v^H-3/2 dv   du 
        =(t-s) ∫ _0^T1/u^H-1/2 (T+ξ _ε-u)^H-1/2
    (T+ξ _ε)^H-3/2 du≤ε M(t-s).


Therefore, a fortiori,


    sup_s,t∈ [0,1],s≠ t  |Cov  (X_T+ε
    t-X_T+ε s, W^α ,α̃_T  )  |/ε
    ^H|t-s|≤ M.


Similar calculations show that


    sup_s,t∈ [0,1],s≠ t  |Cov  (X_T+ε
    t-X_T+ε s, ∫_0^TT-u/T dW^α ,α̃_u  )  |/ε ^H|t-s|≤ M.


Thus, Assumption <ref> is fulfilled with τ̂=1.
Therefore the family ((X^g;x_T+ε t-
X^g;x_T)_t ∈ [0,1])_ε >0 satisfies a large
deviation principle with the inverse speed function
 γ _ε^2=ε ^2H as the
nonconditioned process. Note that the same
result was obtained in <cit.> for the n-fold conditional fractional
Brownian motion.fractional Brownian motion



[m-fold integrated Brownian motion]
For m≥ 1, let X be the m-fold integrated Brownian motion,Brownian motion i.e.


    X_t=∫_0^t du (∫ _0^u du_m-1⋯∫ _0^u_2 du_1 B_u_1).


It is a continuous Volterra processVolterra process with kernel K(t,u)=1/m!
(t-u)^m and covariance function


    k(t,s)=1/(m!)^2∫_0^s∧ t(s-ξ )^m(t-ξ )^m dξ.


First, let us prove that there exists a limit covariance and that it is
regular enough. Assumption <ref> is fulfilled. In fact, for
s≤ t, we have


    lim_ε→ 0Cov(X_T+ε t-X_T,
    X_T+ε s-X_T)/ε ^2
       = lim_ε→ 01/(m!)^21/ε
    ^2∫_T^T+ε s (T+
    ε t -u)^m (T+ε s -u)^m du
    
          +lim_ε→ 01/(m!)^2ε
    ^2∫
    _0^T((T + ε t - u)^m - (T - u)^m) ((T + ε s
     - u)^m -(T -u)^m)  du.


It is straightforward to show that


    k̅(t,s)=lim_ε→ 0Cov(X_T+ε t-X_T, X_T+ε
    s-X_T)/ε ^2= 1/(m!)^2m^2/2m-1T^2m-1
    st,


uniformly in (t,s)∈ [0,1]× [0,1]. Furthermore,


    r̅_1^g_1(t)   =   lim_ε→ 0Cov  (X_T+ε t-X_T, W^α
    ,α̃_T  )/ε
       =   lim_ε→ 0α/m!1/ε∫_0^T((T+ε t -u)^m -(T -u)^m)  du=
    α/m! T^m  t,


and


    r̅_2^g_2(t)   =   lim_ε→ 0Cov  (X_T+ε t-X_T, ∫_0^TT-u/T  dW^α ,α̃_u  )/ε
       =   lim_ε→ 0α/m!  T1/ε∫_0^T((T+ε t -u)^m -(T -u)^m) (T-u) du
    
       =   α/m!m/m+1 T^m  t,


uniformly in t∈ [0,1]. Therefore also Assumption
<ref> is fulfilled.

Thus, we have k̅^g(t,s)= a  st, where


    a=1/(m!)^2(m^2/2m-1 T^2m-1 - α ^2 T^2m(1,m/m+1) (C^ g)^-1(1,
    m/m+1)^⊺).


Note that k̅^g is regular enough.

Now let us prove the exponential tightnessexponential tightness of the family of processes. For
s< t, there exists a constant M>0, such that


    Var(X_T+ε t-X_T+ε s)
       =1/(m!)^2(∫_T+ε s ^T+ε t (T + ε t  - u)^2m du + ∫_0^T+ε s((T + ε t  - u)^m - (T + ε s
     - u)^m)^2 du )
       ≤ M ε ^2 (t-s)^2.


Then Assumption <ref> holds with τ =1 and γ
_ε=ε. For s<t,


    |Cov(X_T+ε t-X_T+ε s,
    W^α ,α̃_T) |   = ∫_0^T((T+ε t-u)^m-(T+ε s-u)^m) du
       = ∑_k=0^m-1mk1/k+1T^k+1ε
    ^m-k(t-s)^m-k.,


Then we have


    sup_s,t∈ [0,1],s≠ t  |Cov  (X_T+ε
    t-X_T+ε s, W^α ,α̃_T  )  |/ε
    |t-s|≤ M.


Similar calculations show that


    sup_s,t∈ [0,1],s≠ t  |Cov  (X_T+ε
    t-X_T+ε s, ∫_0^TT-u/T dW^α
    ,α̃_u  )  |/ε |t-s|≤ M.


Thus, Assumption <ref> is fulfilled with τ̂=1.
Therefore the family ((X^g;x_T+ε t-
X^g;x_T)_t ∈ [0,1])_ε >0 satisfies a large
deviation principle with the inverse speed γ _ε^2=ε ^2.



[Integrated Volterra Process]
Let Z be a Volterra processVolterra process with kernel K satisfying condition
(<ref>) for some A>0. Let X be the integrated process, i.e.


    X_t=∫_0^t Z_u du.


The process X is a continuous, Volterra processVolterra process with kernel


    h(t,s)=∫_s^t K(u,s)  du,   X_t=∫_0^t h(t,s)
     dB_s.


First, let us prove that there exists a limit covariance and that it is
regular enough. Assumption <ref> is fulfilled, in fact, for
s≤ t, we have


    lim_ε→ 0Cov(X_T+ε
    t-X_T, X_T+ε s-X_T)/ε ^2
            =lim_ε→ 0∫_T^T+ε s
    h(T+ε t, u) h(T+ε s, u)  du/ε ^2
              + lim_ε→ 0∫_0^T (
    h(T+ε t, u)- h(T) )(h(T+ε s, u)- h(T,
    u)) du/ε ^2.


Now, one has


    ∫_T^T+ε s h(T+ε t,
    u) h(T+ε s, u)  du
         = ε ^3∫ _0^s(∫_u^t K(T+ε x,
    T+ε u)  dx∫_u^s K( T+ε x, T+ε u) dx
    ) du


and


    ∫_0^T( h(T+
    ε t, u)- h(T,u) ) (h(T+ε s, u)- h(T, u)) du
    
         = ∫_0^T(∫_T^T+ε s  K(v,u) dv
    ∫_T^T+ε t  K(v,u) dv ) du.


Therefore


    lim_ε→ 0Cov(X_T+ε
    t-X_T, X_T+ε s-X_T)/ε ^2=st ∫ _0^T
    K^2(T,u)  du,


uniformly in (t,s)∈ [0,1]× [0,1]. Furthermore, with similar
calculations we have


    r̅_1^g_1(t)=lim_ε→ 0Cov
    (X_T+ε t-X_T, W^α ,α̃_T  )/ε= α  t ∫_0^T K(T,u)  du,


and


    r̅_2^g_2(t)=lim_ε→ 0Cov
    (X_T+ε t-X_T, ∫_0^TT-u/T  dW^α
    ,α̃_u  )/ε= α t/T∫_0^T
    K(T,u) (T-u)  du,


uniformly in t∈ [0,1]. Therefore also Assumption
<ref> is fulfilled.

So, we have k̅^g(t,s)= a  st, where


    a= ∫_0^T K^2(T,u)  du - α ^2 A^⊺(C^ g)^-1 A,


and  A^⊺=  (∫_0^T K(T,u)  du,1/T∫_0^T
K(T,u)(T-u)  du  ). Note that k̅^g is regular enough.
Let us now prove the exponential tightness.exponential tightness We have, for s< t,


    Var(X_T+ε t-X_T+ε s)
         = ∫_T+ε s^T+ε t h(T+ε t, u)^2 du+
    ∫_0^T+ε s( h(T+ε t, u)- h(T+ε s,u)
    )^2.


Now, recalling that K is a square integrable function, there exists a
constant M>0, such that


    ∫_T+ε s^T+ε t h(T+ε t, u)^2 du   =
    ε ^3∫_s^t(∫_u^t K(T+ε x, T+
    ε u)  dx )^2 du
       ≤ε ^3 M ∫_ s^ t  (t-u) du≤ Mε
    ^3(t-s)^2


and


    ∫_0^T+ε s ( h(T+ε
    t, u)-h(T+ε s, u)^2 du   = ε ^2∫
    _0^T+ε s(∫_ s^ t 
    K(T+ε v,u) dv )^2 du
       ≤ Mε
    ^2 (t-s)^2,


therefore Assumption <ref> holds with τ =1 and γ
_ε=ε.

With similar computations we can prove that also
Assumption <ref> is fulfilled with τ̂=1 and
γ _ε=ε. For s<t, we have


    |Cov(X_T+ε t-X_T+ε s,
    W^α ,α̃_T) |    = |∫_0^T( h(T+ε t,u)-h(T+ε s)) du |
       = |∫
    _0^T∫_T+ε s^T+ε t
    K(v,u)  dv  du | ≤ Mε (t-s).


Therefore


    sup_s,t∈ [0,1],s≠ t  |Cov  (X_T+ε
    t-X_T+ε s, W^α ,α̃_T  )  |/ε
    |t-s|≤ M.


Similar calculations show that


    sup_s,t∈ [0,1],s≠ t  |Cov  (X_T+ε
    t-X_T+ε s, ∫_0^TT-u/T dW^α
    ,α̃_u  )  |/ε |t-s|≤ M.


Therefore the family ((X^g;x_T+ε t-
X^g;x_T)_t ∈ [0,1])_ε >0 satisfies a large
deviation principle with the inverse speed function γ _ε^2=ε ^2.





§ CONDITIONING TO A PATH





 §.§ Conditional lawconditional law


Let (Ω , ℱ, (ℱ_t)_t≥ 0, ℙ)
be a filtered probability space. On this space we consider a Brownian motionBrownian motion
B=( B)_t≥ 0, a continuous real Volterra process X=(X_t)_t≥ 0continuous real Volterra process
and another Brownian motionBrownian motion B̃=(B̃)_t≥ 0 independent of
B. Fix α , α̃∈ℝ and define W^α
,α̃= α B + α̃B̃.

We are interested in the regular conditional law of the process X given the
σ-algebra ℱ^α ,α̃_T, where
(ℱ^α ,α̃_t)_t≥ 0 is the filtration
generated by the mixed Brownian motionBrownian motion W^α ,α̃, i.e. we
want to condition the process to the past of the mixed Brownian motionBrownian motion up to
a fixed time T>0. To do this, consider the conditional lawconditional law on C[0,+∞
) endowed with the topology induced by the sup-norm on compact sets,
ℙ(X ∈·|ℱ^α , α̃_T). There exists a regular version of such conditional probability (see
<cit.> and <cit.>), namely a version such that Γ↦ℙ(X ∈Γ|ℱ^a,b_T) is almost surely a
Gaussian probability law.

The following theorem, Theorem 2.1 in <cit.>, gives mean and covariance
function of the Gaussian conditional law.Gaussian conditional law



For T>0, the regular conditional law of X |ℱ^a,b_T is a
Gaussian measure with the random mean


    Ψ _t(W^α ,α̃)=𝔼[X_t|ℱ^α ,α̃_T]=
    α ^2/α ^2 + α̃^2∫_0^T K(t,u)
     dW^α ,α̃_u


and the deterministic covariance,



    Υ (t,s)    =∫_0^t∧ s( 1-
    α ^2/α ^2 +
    α̃^2_[0,T](v) )^2
    K(t,v)  K(s,v)  dv
    
         + α ^2 α̃^2/(α ^2 +
    α̃^2)^2∫_0^T
    K(t,v)  K(s,v)  dv.






Observe that the mean process (α ^2/α ^2 +
α̃^2∫_0^T K(t,u)  dW^α ,α̃_u)_t≥ 0 is a continuous process. Therefore for almost every
continuous function ψ defined on [0,T],



    m^ψ_t=Ψ _t(ψ )


defines a continuous function m^ψ: [0,+∞ ) ⟶ℝ. Thus, we can consider the continuous Gaussian process
(X_t^ψ)_t≥ 0continuous Gaussian process with mean function m^ψ and covariance
function Υ.

From continuity of m^ψ one has



    lim_ε→ 0m_T+ε t^ψ=m_T^ψ


uniformly for t ∈ [0,1].




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.




For s∧ t≥ T we have



    Υ (t,s) =α ^2/α ^2 + α̃^2∫_T^t∧ s K(t,v)  K(s,v)  dv + α̃^2/α
    ^2 + α̃^2  k(t,s).


For α̃=0, i.e. ℱ^α ,0_t=σ{X_u:
u≤ t} (for details about the filtrations generated by X and B, see,
for example, <cit.>), we have the same conditioned variance as in
<cit.>.





 §.§ Large deviations


Let γ _ε>0 be an infinitesimal function, i.e. γ
_ε→ 0 for ε→ 0. In this section
(X_t)_t≥ 0 is a continuous Volterra processVolterra process as in
(<ref>).

Now, in order to achieve a large deviation principle for the generalized
conditioned process X^ψ, we have to investigate the behavior of the
functions Υ and m^ψ (defined in (<ref>) and
(<ref>), respectively) in a small time interval of length
ε. We want to investigate the behavior of the conditioned
process (X_t^ψ)_t≥ 0 in the near future after T.

For s≤ t, taking into account equation (<ref>), simple
computations show that



    Cov( X^ψ_T+ε t-
    X^ψ_T, X^ψ_T+ε s-
    X^ψ_T)
    
          = εα ^2/α ^2 + α̃^2∫_0^t
     s
    K(T+ε t,T+ε u) K(T + ε s, T+ ε u)  du +
            + α̃^2/α ^2 + α̃^2( k(T+ ε t, T+ε s)
    -k(T+ε t, T)-k(T, T+ ε s) +k(T, T)).


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



For any T>0 there exists a square integrable function K̅
(possibly 0) such that


    lim_ε→ 0√(ε) K(T+ε t, T +
    ε s)/γ _ε  = K̅(t,s)


uniformly in (t,s)∈ [0,1]× [0,1].




Notice that we can choose γ _ε so that lim_ε→ 0γ _εε ^- 1/2∈{0,1,+∞}.


The next assumption (Assumption <ref>) implies the
exponential tightnessexponential tightness of the family of processes.



For any T>0 there exist constants c,τ̂> 0 such that


    sup_s,t ∈ [0,1],s≠ t ∫_0^ t (K(T+ε
    t,T+ε u)-K(T+ε s,T+ε u))^2  du/γ
    _ε^2|t-s|^2τ̂≤ c.






Note that


    ∫_0^ t(K(T+
    ε t,T+ε u)-K(T+ε s,T+ε u)
    )^2  du
    
       =∫_s^t
    K(T + ε t,T + ε u)^2  du + ∫
    _0^ s(K(T + ε t,T + ε
    u) - K(T + ε s,T + ε u))^2  du,


therefore in order to prove that Assumption <ref> is
fulfilled we can prove that there exists c>0 such that


    sup_s,t ∈ [0,1],s≠ t ∫_ s^t K(T+ε
    t,T+ε u)^2  du /γ _ε^2|t-s|^2τ̂≤ c,
    
          sup_s,t ∈ [0,1],s≠ t ∫_0^ s (K(T+ε
    t,T+ε u)-K(T+ε s,T+ε u))^2  du/γ
    _ε^2|t-s|^2τ̂≤ c.






Under Assumptions <ref> and <ref> one
has


    lim_ε→ 0Cov(X^ψ_T+ε
    t-X^ψ_T, X^ψ_T+ε s-X^ψ_T)/γ
    _ε^2= Υ̅(t,s)


uniformly in (t,s)∈ [0,1]× [0,1], where



    Υ̅(t,s)= α̃^2/α ^2 + α̃^2∫ _0^t  sK̅(t,u) K̅(s,u)   du+
    α̃^2/α ^2 + α̃^2 k̅(t,s).




From (<ref>), Assumption
<ref> and Assumption <ref> the claim
easily follows.




Under Assumptions <ref> and <ref> the
family ((X^ψ_T+ε t-X^ψ_T - 𝔼[X^ψ_T+ ε t-X^ψ_T])_t∈ [0,1])_ε >0 is
exponentially tight at 
the inverse speed γ _ε^2.



Since ((X^ψ_T+ε t-X^ψ_T - 𝔼[X^ψ_T+ ε t-X^ψ_T])_t∈ [0,1])_ε >0 is a
family of centered processes, it is enough to prove that
(<ref>) is satisfied with 
an appropriate
speed function.

The covariance of such process is given by
(<ref>). Therefore


    Var(X^ψ_T+ε t-X^ψ_T+
    ε s)
    
          = εα ^2/α ^2 + α̃^2∫_0^t(K(T+ ε t,T+ε u) - K(T+ε s,T+ε u)
    )^2 du +
    
            + α̃^2/α ^2 + α̃^2k(T+ ε t, T+ε t) -2 k(T+
    ε t, T+ε s ) +k(T+ε s , T+ε s).


From Assumption <ref> we already know that


    sup_s,t∈ [0,1],s≠ t|k(T+ε t, T+ε t) -2
    k(T+ε t, T+ε s ) +k(T+ε s , T+ε
    s)|/γ _ε^2 |t-s|^2τ≤ M.


Furthermore, Assumption <ref> implies that


    sup_s,t∈ [0,1],s≠ t∫_0^t (K(T+ε
    t,T+ε u) - K(T+ε s,T+ε u))^2 du/γ
    _ε^2 |t-s|^2τ̂≤ M.


Therefore condition (<ref>) holds with the
inverse speed η _ε=γ _ε^2 and β
=τ∧τ̂.


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



Suppose Assumptions <ref>, <ref>,
<ref> and <ref> are fulfilled. If
the (existing) covariance function Υ̅ defined in
Proposition <ref> is regular enough, then the family of
processes ((X^ψ_T+ε t-X^ψ_T)_t∈
[0,1])_ε >0 satisfies a large deviation principle on C[0,1]
with the inverse speed γ _ε^2 and the good rate
function



    I(h) = 
    1/2   h    _ℋ̅^2,     h ∈ℋ̅,
    
    
    +∞,    otherwise,


where ℋ̅ and .  
_ℋ̅, respectively, denote the reproducing kernelreproducing kernel Hilbert space Hilbert
space and the related norm associated to the covariance function
Υ̅ given by (<ref>).



Cosider the family of centered processes ((X^ψ_T+ε
t-X^ψ_T - 𝔼[X^ψ_T+ ε t- X^ψ_T])_t∈ [0,1])_ε. Thanks to
Proposition <ref> this family of processes is
exponentially tight at the inverse speed γ _ε^2. Thanks
to Proposition <ref>, for any λ∈ℳ[0,1], one has


    lim_ε→ 0Var( ⟨λ , X^ψ_T+ε t-X^ψ_T⟩ )/γ _ε^2 =
    ∫ _0^1∫_0^1Υ̅(t,s)   dλ (t)  d λ
    (s)


where Υ̅ is defined in (<ref>) Since
Υ̅ is the covariance function of a continuousVolterra
process,Volterra process a large deviation principle for ((X^ψ_T+ε
t-X^ψ_T- 𝔼[X^ψ_T+ ε t- X^ψ_T])_t∈ [0,1])_t∈ [0,1])_ε >0 actually holds from
Theorem <ref> with the inverse speed γ _ε^2 and the good rate function given by (<ref>). From
Equation (<ref>) and Remark <ref> the same
large deviation principle holds for the noncentered family ((X^ψ_T+ε t-X^ψ_T)_t∈ [0,1])_ε >0.





 §.§ Examples


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


[Fractional Brownian Motion]
Consider a fractional Brownian motionfractional Brownian motion with H>1/2 as in
Example <ref>. We have already proved that
Assumptions <ref> and <ref> are fulfilled with
τ =H and γ _ε=ε ^H.

We want to show that Assumptions <ref> and
<ref> are fulfilled with τ̂=1, γ
_ε=ε ^H. From Example 4.17 in <cit.>, we
have that


    lim_ε→ 0√(ε) K(T+ε
    t,T+ε s)/ε ^H = c_H (t-s)^H- 1/2


uniformly for t,s ∈ [0,1]. Therefore


    lim_ε→ 01/ε ^2H∫ _0^t  s
    K(T+ε t,T+ε u) K(T + ε s, T+ε u)  du
    
         = c_H^2∫_0^t  s (t-u)^H-1/2
    (s-u)^H-1/2  du


uniformly for (t,s)∈ [0,1]× [0,1]. So, we have that Assumption
<ref> is fulfilled with


    Υ̅(t,s)= α̃^2/α ^2 + α̃^2k(t,s) + α ^2/α ^2 + α̃^2
    c_H^2∫_0^t  s (t-u)^H-1/2 (s-u)^H-1/2 
    du.


Note that Υ̅ is regular enough. Let us now prove the
exponential tightness.exponential tightness

Since (a+b)^2≤ 2(a^2+b^2) for a,b ∈ℝ, from
Equation (<ref>), there exists a constant c>0 such that, for
s<t,


    K(T+ε t,T+ε u)^2
       ≤    c ( ε ^2H-1( t - u)^2H-1 +
    ( ∫_T+
    ε u^T+ε t(v-(T+ε u)
    )^H-1/2  dv )^2)
    
       ≤    c ( ε ^2H-1( t - u)^2H-1 +
    ε ^2H+1( t - u)^2H+1).


Thus,


    ε∫_s^t K(T+ε t,T+ ε u)^2  du ≤ c
    ( ε ^2H (t-s)^2H + ε ^2H+2 (t-s)^2H+2).


Furthermore,


    (K(T+ε t,T+ε u)-K(T+ε s,T+ε u)
    )^2= (A(u)- B(u))^2≤ 2( A^2(u)+ B^2(u)),


where


    A(u) = c_H[ ( T+ε t/T +ε u)^H-1/2 ( t - u)^H-1/2ε ^H-
    1/2
    - ( T+ε s/T +ε u)^H-
    1/2 (
    s-u)^H-1/2ε ^H-1/2 ) ]
    
       B(u) = c_H( H-1/2)
    1/(T+ε u)^H-1/2∫_T+ε s^T+
    ε t
    v^H-3/2(v-(T+ε u))^H-1/2  dv.


Now, thanks to the Lagrange theorem,Lagrange theorem there exists x∈ [s,t] such that


    A(u)    ≤ c ( (T+ε t)^H-1/2 ( t-u
    )^H-1/2ε ^H-1/2 - (T+ε
    s)^H-1/2 ( s-u)^H-1/2ε ^H-1/2)
    
       = c ((T+ε x)^H-3/2(x- u)^H-1/2 +(T+
    ε x)^H-1/2 (x- u)^H-3/2) ε
    ^H-1/2(t-s).


The estimation of B is easily done. There exists x∈ [s,t] such that


    B(u) ≤ c ∫_T+ε s^T+ε t(v-(T+ε u))^H-1/2  dv   =c ε
    ^H+1/2∫_ s^ t (v-
    u)^H-1/2  dv
    
       = c ε ^H+1/2(x-
    u)^H-3/2(t-s),


and then


    ε∫_0^ s A^2(u)   du ≤ c ε ^2H (t-s)^2,
        ε∫_0^ s B^2(u)   du≤ c ε ^2H+2
    (t-s)^2.


From Remark <ref> we have that


    ∫_0^ t(K(T+ ε t,T+ε u)-K(T+ε
    s,T+ε u) )^2  du≤ c ε ^2H (t-s)^2.


Assumption <ref> is then fulfilled with τ̂=1
and γ _ε=ε ^H.

A large deviation principle is then established for the family of processes((X^ψ_T+ε t-X^ψ_T)_t∈ [0,1])_ε
>0, with the inverse speed ε ^2H.



[m-fold integrated Brownian motion]
Let X be the process defined in Example <ref>. We have
already proved that Assumptions <ref> and <ref>
are fulfilled with τ =1 and γ _ε=ε. We
want to show that Assumptions <ref>
and <ref> are fulfilled with τ̂=1,
γ_ε=ε. For s≤ t, wee have


    K(T+ε t, T + ε s) =(t-s)^mε ^m,


therefore


    lim_ε→ 0√(ε) K(T+ε
    t,T+ε s)/ε = 0


uniformly in t,s ∈ [0,1]. Thus,


    Υ̅(t,s)= α̃^2/α ^2 + α̃^21/(m!)^2m^2/2m-1T^2m-1 st,


and Assumption <ref> is verified. Note that
Υ̅ is regular enough. Let us now prove the exponential
tightnessexponential tightness of the family of processes. For s< t, there exist positive
constants c_1,c_2 such that


    ∫_0^ t(K(T+
    ε t,T+ε u)-K(T+ε s,T+ε u)
    )^2  du
         = 1/(m!)^2∫_0^sε ^2m((t-u)^m-
    (s-u)^m)^2 du 
            + 1/(m!)^2∫_s^tε
    ^2m ((t-u)^2m du ≤ε ^2m c_1 (t-s)^2 + ε
    ^2mc_2 (t-s)^2m+1


and Assumption <ref> is verified with τ̂=1 ad
γ _ε=ε. A large deviation principle is then
established for the family of conditioned processes ((X^ψ_T+ε t-X^ψ_T)_t∈ [0,1])_ε >0, with the
inverse speed ε ^2.



[Integrated Volterra Process]
Let X be the process defined in Example <ref>. We have already
proved that Assumptions <ref> and <ref> are
fulfilled with τ =1, γ _ε=ε. We want to
show that Assumptions <ref> and
<ref> are fulfilled with τ̂=1 and γ
_ε=ε. For s≤ t, wee have


    h(T+ε t, T + ε s) =∫_T+ε s^T+
    ε t K(v, T+ε s)  dv=ε∫_s^ t K(T+
    ε v, T+ε s)  dv,


therefore


    lim_ε→ 0√(ε) h(T+ε t, T +
    ε s)/ε = 0


uniformly in t,s ∈ [0,1]. Thus,


    Υ̅(t,s)= α̃^2/α ^2 + α̃^2∫_0^T K^2(T,u)  du  st,


and Assumption <ref> is verified. Note that
Υ̅ is regular enough. Let us now prove the exponential
tightnessexponential tightness of the family of processes. For s< t, there exist a constant
c>0 such that


    ∫_0^ t(h(T+
    ε t,T+ε u)-h(T+ε s,T+ε u)
    )^2  du
    
         =∫_0^ s(∫_T+ε s^T+ε t K(v, T+ε u)  dv
    )^2 du+∫_s^t(
    ∫_T+ε u^T+ε t K(v, T+ε u)  dv
    )^2 du
    
         ≤ε ^2c
    (t-s)^2


and Assumption <ref> is verified with τ̂=1 and
γ _ε=ε. A large deviation principle is then
established for the family of conditioned processes ((X^ψ_T+ε t-X^ψ_T)_t∈ [0,1])_ε >0, with the
inverse speed ε ^2.







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



The author acknowledges the [id=GS1,sponsor-id=501100003407]MIUR Excellence Department Project awarded to the
Department of Mathematics, University of Rome Tor Vergata, CUP
[refid=GS1]E83C18000100006.





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