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

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.


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 = (X t ) t≥0 be a continuous real Volterra process. Definition 1. A centered Gaussian process X is a Volterra process if, for every T > 0, it admits the representation where B = (B t ) 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.
For a Volterra process the covariance function is k(t, s) = s∧t 0 K(t, u)K(s, u) du for t, s ∈ [0, T ].
(2) LetB = (B t ) t≥0 be another Brownian motion independent of B and for α,α ∈ R define W α,α = αB +αB. 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,α , G T W α,α = G 1 T W α,α , . . . , G n T W α,α ⊺ , more precisely, 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 process X conditioned on the set We obtain a large deviation principle for the family of processes ((X g;x T +εt − 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,α T , where (F α,α t ) t≥0 is the filtration generated by the mixed Brownian motion W α,α , i.e. we want to condition the process to the past of the mixed Brownian 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 process X conditioned on the set Here we obtain a large deviation principle for the family of processes ((X ψ T +εt − X ψ T ) t∈[0,1] ) ε>0 . Since T, α andα 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.

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 (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 and for any closed set Γ, 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 Remark 1. We say that a family of continuous processes ((U ε t ) 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 [1]) 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], Theorem 1. 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 Then, 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.
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. exists for some continuous, symmetric, positive definite function k, that is the covariance function of a continuous Gaussian process, then ((U ε t ) 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 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] then ((U ε t ) t∈[0,1] ) ε>0 is exponentially tight at the inverse speed function η ε .
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 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 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 [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 ((X T +εt − 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 2 is applicable to the family of processes ((X T +εt − X T ) t∈[0,1] ) ε>0 . Let γ ε > 0 be an infinitesimal function, i.e. γ ε → 0 for ε → 0. Assumption 1. For any fixed T > 0 there exists an asymptotic covariance functionk defined as Remark 3. Notice thatk 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 (X t ) t∈[0,T ] is Hölder continuous of exponent 0 < a < A if and only if for every ε > 0, 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 such that 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.
As an immediate application of Theorem 2 (take U ε t = X T +εt − X T ), Assumptions 1 and 2 imply, ifk is regular enough, that the family ((X T +εt − 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 given by whereH is the reproducing kernel Hilbert space associated to the covariance functionk and the symbol · H denotes the usual norm defined onH . In fact Assumption 1 immediately implies that Furthermore, Assumption 2 implies that the family ((X T +εt − X T ) t∈[0,1] ) ε>0 is exponentially tight at the inverse speed function γ 2 ε .
3 Conditioning to n functionals of the path

Conditional law
Let (Ω, F , (F t ) 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 = (X t ) t≥0 and another Brownian motionB = (B) t≥0 independent of B. For α,α ∈ R let us define the mixed Brownian motion W α,α = αB +αB. For fixed n ∈ N and T > 0, we consider the conditioning of X on n linear 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 process X conditioned on the set 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, For more details about existence of such regular conditional distribution see, for example, [11]. Denote by C g = (c gigj ij ) i,j=1,...,n the matrix defined by The matrix C g is invertible (since the functions g i , i = 1, . . . , n, are linearly independent). Let us denote and r g (t) = r g1 1 (t), . . . , r gn n (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 X g;x can be represented as Moreover, the conditioned process X g;x is a Gaussian process with mean and covariance where 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 g 1 , . . . , g n and on the time T , but not on the vector x.
Remark 6. 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 motion at the times T 1 , . . . , T n , more precisely to the set Remark 7. If the conditioning functions are g i (s) = K(T i , 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 T 1 , . . . , T n , more precisely to the set n i=1 {X Ti = x i } (this is a particular case of the conditioned process in [13]).

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 process as in (1). Now, in order to achieve a large deviation principle for the family of processes ((X g;x T +εt − 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 (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. Cov(X T +εt − X T , The next assumption (Assumption 4) implies the exponential tightness of the family of the centered processes.
Proof. Taking into account equation (7), simple computations show that for s, t ∈ [0, 1], Therefore the claim easily follows from Assumptions 1 and 3.

Remark 10.
Notice thatk g is a continuous covariance function, being the (uniform) limit of continuous, symmetric and positive definite functions.
is exponentially tight at the inverse speed function γ 2 ε .
is a family of centered processes, it is enough to prove that (3) is satisfied with an appropriate speed function. For ε > 0 the covariance of such process is given by (13). Therefore .
Therefore condition (3) holds with the inverse speed η ε = γ 2 ε and β = τ ∧τ . We are now ready to prove the main large deviation result of this section. Theorem 4. Suppose (X t ) t≥0 satisfies Assumptions 1, 2, 3 and 4. Suppose, furthermore, that the (existing) covariance functionk g defined in Proposition 2 is regular enough, then the family of processes ((X g;x T +εt − 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 otherwise, whereH g is the reproducing kernel Hilbert space associated to the covariance functionk g .
Proof. Cosider the family of centered processes ((X g;x T +εt − X g;x T − E[X g;x T +εt − X g;x T ]) t∈[0,1] ) ε>0 . Thanks to Proposition 3 this family of processes is exponentially tight at the inverse speed γ 2 ε . Thanks to Proposition 2, for any λ ∈ M [0, 1], one has wherek g is defined in (12). Sincek g is the covariance function of a continuous Volterra process, a large deviation principle for ((X g;x actually holds from Theorem 2 with the inverse speed γ 2 ε and the good rate function given by (14). From Equation (10) and Remark 2 the same large deviation principle holds for the noncentered family ((X g;x T +εt −X g;x T ) t∈[0,1] ) ε>0 .

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 g 1 (t) = 1 [0,T ) (t) and and by the integration by parts formula, Then the matrix (C g ) −1 is given by Example 1 (Fractional Brownian Motion). Let X be the fractional Brownian motion of the Hurst index H > 1/2. The fractional Brownian motion with the Hurst parameter H ∈ (0, 1) is the centered Gaussian process with covariance function The fractional Brownian motion is a Volterra process with kernel, for s ≤ t, 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 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 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 γ ε = ε H . Now let us prove that Assumption 3 is fulfilled.
Thanks to the Lagrange theorem, for ξ ε ∈ [0, εt]. Therefore from the Lebesgue theorem, uniformly in t ∈ [0, 1]. Furthermore, in a similar way, we have, So, we havek 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, then Assumption 2 holds with τ = H and γ ε = ε H . For s < t, we have Thanks to the Lagrange theorem we can find M > 0 such that Therefore, a fortiori,

Similar calculations show that
Thus, Assumption 4 is fulfilled withτ = 1. Therefore the family ((X g;x T +εt − X g;x T ) t∈[0,1] ) ε>0 satisfies a large deviation principle with the inverse speed function γ 2 ε = ε 2H as the nonconditioned process. Note that the same result was obtained in [4] for the n-fold conditional fractional Brownian motion.
Example 2 (m-fold integrated Brownian motion). For m ≥ 1, let X be the m-fold integrated Brownian motion, i.e.
It is a continuous Volterra process with kernel K(t, u) = 1 m! (t − u) m and covariance function 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 It is straightforward to show that uniformly in t ∈ [0, 1]. Therefore also Assumption 3 is fulfilled. Thus, we havek g (t, s) = a st, where Note thatk 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, such that Then Assumption 2 holds with τ = 1 and γ ε = ε. For s < t, Then we have

Similar calculations show that
Thus, Assumption 4 is fulfilled withτ = 1. Therefore the family ((X g;x T +εt − X g;x T ) t∈[0,1] ) ε>0 satisfies a large deviation principle with the inverse speed γ 2 ε = ε 2 . Example 3 (Integrated Volterra Process). Let Z be a Volterra process with kernel K satisfying condition (5) for some A > 0. Let X be the integrated process, i.e.
The process X is a continuous, Volterra process with kernel 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  1]. Furthermore, with similar calculations we havē r g1 Cov(X T +εt − X T , uniformly in t ∈ [0, 1]. Therefore also Assumption 3 is fulfilled. So, we havek g (t, s) = a st, where . Note thatk g is regular enough. Let us now prove the exponential tightness. We have, for s < t, Now, recalling that K is a square integrable function, there exists a constant M > 0, such that therefore Assumption 2 holds with τ = 1 and γ ε = ε.
4 Conditioning to a path
We are interested in the regular conditional law of the process X given the σalgebra F α,α T , where (F α,α t ) t≥0 is the filtration generated by the mixed Brownian motion W α,α , i.e. we want to condition the process to the past of the mixed Brownian motion up to a fixed time T > 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 ∈ · | F α,α T ). There exists a regular version of such conditional probability (see [11] and [14]), 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 [16], gives mean and covariance function of the Gaussian conditional law.
Theorem 5. For T > 0, the regular conditional law of X | F a,b T is a Gaussian measure with the random mean T 0 K(t, u) dW α,α u and the deterministic covariance, Remark 11. Observe that the mean process ( α 2 α 2 +α 2 T 0 K(t, u) dW α,α u ) t≥0 is a continuous process. Therefore for almost every continuous function ψ defined on [0, T ], defines a continuous function m ψ : [0, +∞) −→ R. Thus, we can consider the continuous Gaussian process (X ψ t ) t≥0 with mean function m ψ and covariance function Υ.
From continuity of m ψ one has 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 Forα = 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, [18]), we have the same conditioned variance as in [9].

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 process as in (1). 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 (16) and (17), respectively) in a small time interval of length ε. We want to investigate the behavior of the conditioned process (X ψ t ) t≥0 in the near future after T .
Proposition 5. Under Assumptions 2 and 6 the family is exponentially tight at the inverse speed γ 2 ε .
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 functionῩ defined in Proposition 4 is regular enough, then the family of processes ((X ψ T +εt − X ψ 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 whereH and . H , respectively, denote the reproducing kernel Hilbert space and the related norm associated to the covariance functionῩ given by (21).