Filtering of Gaussian processes in Hilbert spaces
Vit Kubelka Bohdan Maslowski





Filtering of Gaussian processes in Hilbert spaces
    V. KubelkaFaculty of Mathematics and Physics, Charles University,
Sokolovská 83, 186 75 Prague 8, Czech Republic 
 
 B. Maslowski[1] 
 
 
    
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Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain.



Keywords: Kalman - Bucy filter, stochastic evolution equations, Gaussian processes 

 



§ INTRODUCTION

The aim of this paper is to study the linear filtering problem for infinite-dimensional Gaussian processes with finite-dimensional observation. Typically, the signal process may be governed by a linear SPDE driven by noise that is not white in time, like a Gaussian Volterra noise or, in particular, fractional Brownian motion (FBM).  



An analogous problem for finite-dimensional (or scalar) processes have been studied by Kleptsyna and Le Breton <cit.> for the case of general Gaussian process observed through a linear channel driven by standard Brownian motion; this problem has been revisited in <cit.>. A rather general approach to filtering with fractional Brownian motion is presented in <cit.> and specified in more concrete situations, for example, in <cit.>.

 

In infinite dimensions, a pioneering result belongs to Falb <cit.>, where Kalman-Bucy  (KB) type theorem has been established. In this case, both observed process and observation live in a Hilbert space and are governed by linear evolution equation with a Q-Wiener process. We are not aware of any analogous result in infinite dimensions for general Gaussian processes that would cover, for example, linear SPDEs driven by fractional noise (however, we would like to point out that a "dual" LQ control problem has been treated, for instance, in <cit.> and <cit.>, while related statistical inference problems were addressed in numerous papers, like  <cit.>, <cit.> or <cit.>).




In the present paper, integral equations for the filter and for covariance of observation error on a rigged Hilbert space are derived and the general results are applied to stochastic parabolic equation perturbed by Gauss-Volterra noise, observed at finitely many points of the domain. In this case, comparing to the classical KB theorem, there are two major obstacles: the fact that the noise does not have independent increments and the need to apply the results to linear SPDEs with space-dependent noise and observation at specific points in the domain of the equation. While the first problem is treated similarly as in the finite-dimensional papers quoted above, the second one is overcome by posing the equation on a rigged Hilbert space. The larger space H (which is usually a Lebesgue space on the domain) is suitable for the definition of the noise term and the stochastic integral, while the smaller space V is contained in the space of continuous functions (for which values at given points are well defined).


 

The paper is divided into four Sections. In section 1 the problem is posed and the main result (Theorem <ref>) is stated and proved. It is shown that the filter and observation error satisfy certain integral equations. In Section 2, uniqueness of solutions to the (nonlinear) integral equation for the error covariance (Theorem <ref>) is shown. Section 3 is devoted to Examples. At first it is demonstrated that if the signal process is governed by linear evolution equation driven by  a standard cylindrical Wiener process, our result stated in Theorem <ref> reduces to an infinite-dimensional analogue of the classical Kalman-Bucy Theorem (Theorem <ref>). Then, following <cit.>, <cit.> and <cit.>, some basic concepts concerning the infinite-dimensional Gauss-Volterra processes and SPDEs driven by them are recalled. Finally, in Example <ref> the general results are applied to 2m-th order stochastic parabolic equation on a bounded domain (and further specified in the case of stochastic heat equation). In Corollary <ref> the main results is specified to the case of pointwise observation of solution to such equation.  




Bounded linear operator mapping a Banach space X to a a Banach space Y is denoted as ℒ(X,Y),  ℒ(X): = ℒ(X,X). The space of Hilbert-Schmidt operators on a Hilbert space H is denoted by ℒ_2 (H).




§ SOLUTION OF THE FILTERING PROBLEM

Let (H,V) be a rigged separable Hilbert space, where H = (H, ⟨·,·⟩_H , ·_H ) and V = (V, ⟨·,·⟩_V , ·_V ) are separable Hilbert spaces such that V ⊂ H, V is dense in H and identifying H with the dual H^* the embeddings

    V ↪ H = H^* ↪ V^*

are continuous and dense. The duality pairing between V and V^* is defined by the inner product on H, that is ⟨ u,v ⟩_ = ⟨ u,v ⟩_H for u ∈ V ⊂ H and v ∈ H ⊂ V^*.

For arbitrary x, y ∈ V we define tensor product x ∘ y ∈ℒ(V^*,V), (x ∘ y)v = x ⟨ y , v ⟩_, v ∈ V^*.


Let us consider stochastic basis (Ω , F,P,(F_t)) and the signal θ = {θ_t, t ∈ [0,T] } that is a centered Gaussian mean - square  continuous measurable process in V. Let ξ = {ξ_t, t ∈ [0,T] } denote an ℝ^n - valued observation process given as 

    ξ_t = ∫^t_0  A(s) θ_s  s + W_t,
 
where ( A(s) )_s ∈ [0,T] is a family of linear operators V →ℝ^n such that mapping t → A(t) is strongly measurable and A(t) _ℒ(V,ℝ^n)≤ K, t ∈ [0,T] for some K < ∞. Here W = { W_t, t ∈ [0,T]  } is a standard ℝ^n - valued Wiener process independent of the signal θ.

Further, assume that for each t ∈ [0,T] operator A(t) can be decomposed into functionals A_1(t),…,A_n(t) ∈ V^* such that A(t)b = (⟨ b , A_1(t) ⟩_ , …, ⟨ b , A_n(t) ⟩_)^T for all b ∈ V. Note that the dual operator A^*(t): ^n → V^* then satisfies A^*(t)z = ∑_i=1^n z_i A_i (t) for all z ∈^n.  
   
We are dealing with the optimal filter θ_t, which is defined as

    θ_t = 𝔼[θ_t| F^ξ_t],

where ( F^ξ_t)_t ∈ [0,T] is the filtration generated by the observation process ξ.

Set K^θ(t,s) = 𝔼[ θ_t ∘θ_s ], t,s ∈ [0,T]. Notice that the mean - square  continuity of the process {θ_t, t ∈ [0,T] } implies that the mapping K^θ: [0,T]^2 →ℒ(V^*,V) is strongly continuous and bounded.


Let Δ = { (t,s) ∈ [0,T]^2 ; 0 ≤ s ≤ t ≤ T }.
The filter θ satisfies the stochastic integral equation
  
    θ_t = ∫^t_0 Φ(t,s)A^*(s) ξ_s -  ∫^t_0 Φ(t,s)A^*(s)A(s) θ_s  s,   t ∈ [0,T],

where operator Φ: Δ→ℒ(V^*,V) defined as Φ(t,s) = 𝔼[θ_t ∘ (θ_s -  θ_s)] for all (s,t) ∈Δ is strongly continuous and satisfies the integral equation
    
    Φ(t,s) = K^θ(t,s) - ∑_j=1^n ∫^s_0 ( Φ(t,r)A_j(r) ) ∘( Φ(s,r) A_j(r) )  r.

Moreover, for all t ∈ [0,T], Φ(t,t) is the covariance of the estimation error at time  t ∈ [0,T], that is,
  
    Φ(t,t) = 𝔼[(θ_t -  θ_t) ∘ (θ_t -  θ_t) ]

 holds.


In the proof of theorem (<ref>)the following lemma will be useful.
 
Let us consider an ℝ^n - valued process of diffusion type  ξ = {ξ_t, t ≥ 0 } on probability space ( Ω , F, P, (F_t) ) with the differential

    ξ_t = a_t  t +  W_t,   ξ_0 = 0,
 
where  a = { a_t, t ≥ 0 } is an (F^ξ_t) - progresively measurable ℝ^n - valued process and 
 W = { W_t, t ≥ 0 } is a standard (F_t) - Wiener process in  ℝ^n.
If  

    ∫^T_0 𝔼| a_s |^2  s < ∞
 
 for every T < ∞, then
any one - dimensional (F_t^ξ) - martingale X, forming together with ( ξ, W ) a Gaussian system can be represented in the form

    X_t = 𝔼 X_0 + ∑_j = 1^n ∫^t_0 f_j (s)  W^j_s,

where f_1, … , f_n are deterministic square integrable measurable functions.



Note that by (<ref>) W is also (F^ξ_t) - Wiener process.

According to Theorem 3.1 in <cit.> there is a representation

    X_t = 𝔼 X_0 + ∑_j = 1^n ∫^t_0 f_j (s)  W^j_s,

where f = (f_1, … , f_n) is (F^ξ_t) - progresively measurable ^n - valued process and

    ∫^T_0 𝔼| f (s) |^2  s  < ∞

 for every T < ∞.
 
Using Gaussianity of the process (X, ξ, W) and Theorem on Normal Correlation (cf. Theorem 13.1 in <cit.>) it can be shown, analogously to the proof of Theorem 5.21 in <cit.>, that for all j = 1, … , n and for all 0 ≤ s ≤ t < ∞ 

    𝔼[ ( X_t - X_s ) ( W^j_t - W^j_s ) | F^ξ_s] 
    = 𝔼[ ( X_t - X_s ) ( W^j_t - W^j_s ) ]

holds.

Furter, using Itô formula and the fact that mixed variation ⟨  W^j , W^k ⟩_t = 0 for j ≠ k and ⟨  W^j , W^k ⟩_t = t for j = k we have

    ( X_t - X_s ) ( W^j_t - W^j_s )
    =  ∫^t_s ( X_r - X_s )   W^j_r +  ∑_k = 1^n ∫^t_s ( W^j_r - W^j_s ) f_k (r)  W^k_r + ∫^t_s f_j (r)  r.

Now, using (<ref>), (<ref>) and martingale property of Itô integral for all j = 1, … , n we get 

    𝔼[ ∫^t_s f_j (r)  r ]
    = 𝔼[ ∫^t_s f_j (r)  r  | F^ξ_s].

Taking into account (<ref>) we can use Fubini's theorem to obtain

    ∫^t_s 𝔼[f_j (r) ]  r
    = ∫^t_s 𝔼[ f_j (r) | F^ξ_s]  r

for all j = 1, … , n.

We can use (<ref>) piecewise on a sequence of decompositions
 { 0 = t_0^(n) < ⋯ < t_n^(n) = t, n ∈} of interval [0,t] such that 
max_i= 1, … n| t_i^(n) - t_i-1^(n)| 0
to obtain

    ∫_0^t 𝔼[f_j (r) ]  r = ∫_0^t f_j,n (r)  r,    j = 1, … , n,

where f_j,n (r) = 𝔼[ f_j (r) | F^ξ_t_i^(n)], t_i^(n)≤ r < t_i+1^(n).

Analogously to the proof of Theorem 5.21 in <cit.>, using continuity of the filtration (F^ξ_s, 0 < s < t) which follows from Theorem 5.19 in <cit.> and the uniform integrability of { f_j,n, n ∈}  we get

    |∫^t_s {[f_j (r) ] - f_j (r) } r | 0,    j = 1, … , n.

From this, for all j = 1, … , n and each t, 0 ≤ t ≤ T, we have

    ∫^t_s 𝔼[f_j (r) ]  r
    = ∫^t_s  f_j (r)  r

and, therefore, for all j = 1, … , n and for almost all  t, 0 ≤ t ≤ T

    f_j (t) = 𝔼[f_j (t) ].

Equality (<ref>) together with (<ref>) proves the representation (<ref>).




Now, let us prove of theorem (<ref>).


According to Lemma 2.2 in <cit.> process {W̃_t, t ∈ [0,T] } defined as 

    W̃_t = ξ_t -  ∫^t_0 𝔼 [ A(r) θ_r | F_t^ξ ]  r
    = ξ_t -  ∫^t_0 A(r) θ_r  r.

is ℝ^n - valued ( F^ξ_t ) - standard Wiener process called innovation process. The formula (<ref>) reads

    ξ_t = A(t) θ_t  t + W̃_t,   ξ_0 = 0

and the process { A(t) θ_t, t ≥ 0 } satisfies the condition (<ref>) of Lemma <ref> hence the observation process takes the form (<ref>).

Further, define the square integrable V - valued process M^s =  { M^s_t, t ∈ [0,T] } as 

    M^s_t = 𝔼[θ_s| F^ξ_t]

for all s ∈ [0,T].
Note that θ_t = M^t_t and the proces M^s is (F^ξ_t) - martingale.

Let { e_i, i ∈ I } be an orthonormal basis on V. Then we have

    M^s_t = ∑_i ∈ I e_i M^s_i, t

where M^s_i, t = 𝔼[⟨θ_s , e_i ⟩_H | F^ξ_t]. Process M^s_i is one - dimensional ( F^ξ_t ) - martingale for all s ∈ [0,T] and all i ∈ I. The triple (M^s_i, ξ, W̃) forms a Gaussian system, therefore, in virtue of Lemma <ref>

    M^s_i, t = ∑_j = 1^n ∫^t_0 F^s_i,j (r) W̃^j_r,

where F^s_i,1, … , F^s_i,n are deterministic square integrable measurable functions, holds for every s ∈ [0,T] and every i ∈ I.
From equations (<ref>) and (<ref>) we obtain

    M^s_t = ∑_i ∈ I e_i ∑_j = 1^n ∫^t_0 F^s_i,j (r) W̃^j_r.

Using Itô isometry

    ∞   > 𝔼 M^s_t ^2_V  = ∑_i ∈ I𝔼 ( M^s_i,t )^2 = ∑_i ∈ I𝔼[    
    ∑_j = 1^n ∫^t_0 F^s_i,j (r) W̃^j_r ]^2
     = ∑_i ∈ I∑_j = 1^n ∫^t_0  ( F^s_i,j (r) )^2  r 
       = ∑_j = 1^n ∫^t_0 ∑_i ∈ I( F^s_i,j (r) )^2  r
     =  ∑_j = 1^n  F^s_j _L^2(V),

where F^s(r) = (F^s_1(r), … , F^s_n(r)) is square integrable V^n - valued deterministic function such that F^s_j (r) = ∑_i ∈ I e_i  F^s_i,j (r) for all j = 1, … , n.

Therefore, swaping the sums and the integral in (<ref>) we finally obtain

    M^s_t = ∫^t_0 F^s (r) W̃_r = ∑_j = 1^n ∫^t_0 F^s_j (r) W̃^j_r.



Now, for every s ∈ [0,T], we can consider an arbitrary square integrable measurable V^n - valued deterministic function  f^s(r) = (f^s_1 (r), … , f^s_n (r)) to show

    𝔼[ (θ_s - M^s_t ) ∘(∫^t_0 f^s (r) W̃_r )] = 0.


Indeed, using (<ref>) for arbitrary v ∈ V^* we have

    𝔼[ (θ_s - M^s_t ) ∘(∫^t_0 f^s (r) W̃_r )] v 
    = 𝔼[ (θ_s - M^s_t ) ⟨∫^t_0 f^s (r) W̃_r , v ⟩_] 
       =𝔼[ 𝔼[ (θ_s - M^s_t )  ⟨∫^t_0 f^s (r) W̃_r , v ⟩_|  F^ξ_t] ] 
       = 𝔼[ ( 𝔼[ θ_s |  F^ξ_t] - M^s_t )  ⟨∫^t_0 f^s (r) W̃_r , v ⟩_] = 0.


Further, we show that

    𝔼[ M^s_t ∘(∫^t_0 f^s (r) W̃_r )] =
    ∑_j = 1^n ∫^t_0 F^s_j (r) ∘f^s_j (r) r.

Note that to prove equality of two arbitrary operators P,Q ∈ℒ(V^*,V) it is sufficient to show ⟨ P v , b ⟩_V = ⟨ Q v , b ⟩_V for all elements v ∈ V^*, b ∈ V. We have that

    ⟨𝔼[ M^s_t ∘(∫^t_0 f^s (r) W̃_r )] v , b ⟩_V
    = ⟨𝔼[ M^s_t  ⟨∫^t_0 f^s (r) W̃_r , v ⟩_] , b ⟩_V 
       = 𝔼[ ⟨  M^s_t  ⟨∫^t_0 f^s (r) W̃_r , v ⟩_, b ⟩_V ]
    = 𝔼[  ⟨∫^t_0 f^s (r) W̃_r , v ⟩_⟨  M^s_t  , b ⟩_V ] 
       = 𝔼[  ⟨∫^t_0 f^s (r) W̃_r , v ⟩_⟨∫^t_0 F^s (r) W̃_r ,  b ⟩_V ] 
       = 𝔼[  ⟨∑_j = 1^n ∫^t_0 f^s_j (r) W̃^j_r , v ⟩_⟨∑_j = 1^n ∫^t_0 F^s_j (r) W̃^j_r  , b ⟩_V ] 
       = 𝔼[  ∑_j = 1^n ∫^t_0 ⟨  f^s_j (r) , v  ⟩_W̃^j_r ∑_j = 1^n ∫^t_0 ⟨ F^s_j (r) , b ⟩_V W̃^j_r ].

Using  Itô isometry and the fact that mixed variation ⟨W̃^i ,W̃^j ⟩ = 0, i ≠ j we obtain

    𝔼[  ∑_j = 1^n ∫^t_0 ⟨ f^s_j (r) , v ⟩_W̃^j_r ∑_j = 1^n ∫^t_0 ⟨  F^s_j (r) , b ⟩_V W̃^j_r ] 
       = ∑_j = 1^n ∫^t_0 ⟨  f^s_j (r) , v ⟩_⟨  F^s_j (r) , b ⟩_V  r 
    = ∑_j = 1^n ⟨( ∫^t_0 F^s_j (r) ∘f^s_j (r) r ) v ,  b ⟩_V 
       = ⟨( ∑_j = 1^n ∫^t_0 F^s_j (r) ∘f^s_j (r) r ) v ,  b ⟩_V

which concludes the proof of equality (<ref>).

Next, we show equality

    𝔼[ θ_s ∘(∫^t_0 f^s (r) W̃_r )] 
    =  ∑_j = 1^n ∫^t_0 ( 𝔼[ θ_s ∘( θ_r -  θ_r )] A_j (r) ) ∘f^s_j (r) r.

Using (<ref>), (<ref>) and independence of θ and W we have 

    𝔼[ θ_s ∘(∫^t_0 f^s (r) W̃_r )] 
    = 𝔼[ θ_s ∘(∫^t_0 f^s (r) ξ_r - ∫^t_0 f^s (r) A(r) θ_r  r )] 
       = 𝔼[ θ_s ∘(∫^t_0 ∑_j=1^n f^s_j (r) ⟨θ_r ,  A_j(r) ⟩_ r + ∫^t_0 f^s (r)  W_r - ∫^t_0 ∑_j=1^n f^s_j (r) ⟨θ_r ,  A_j(r) ⟩_ r )] 
       = 𝔼[ θ_s ∘(∫^t_0 ∑_j = 1^n f^s_j (r) ⟨θ_r - θ_r , A_j (r) ⟩_ r )]
       = ∑_j = 1^n ∫^t_0 𝔼[ θ_s ∘(  f^s_j (r) ⟨θ_r - θ_r , A_j (r) ⟩_)] r.

To complete the proof of equality (<ref>) it is sufficient to show

    θ_s ∘(  f^s_j ⟨(r) θ_r - θ_r ,  A_j (r) ⟩_)
    = [ ( θ_s ∘ ( θ_r - θ_r ) ) A_j (r) ] ∘f^s_j (r) ,   j = 1, … , n.

By the definition of tensor product we obtain

    ⟨θ_s ∘(  f^s_j (r) ⟨θ_r - θ_r , A_j(r) ⟩_) v , b ⟩_V
    = ⟨θ_s ⟨  f^s_j (r) ⟨θ_r - θ_r , A_j (r) ⟩_, v ⟩_, b ⟩_V 
       = ⟨θ_r - θ_r , A_j(r) ⟩_⟨  f^s_j (r), v ⟩_⟨θ_s   , b ⟩_V

for all v ∈ V^*, b ∈ V.
Similarly we get

    ⟨( [ ( θ_s ∘ ( θ_r - θ_r ) ) A_j (r) ] ∘f^s_j (r) ) v , b ⟩_V
    = ⟨( θ_s ∘ ( θ_r - θ_r ) ) A_j (r) ⟨ f^s_j (r) ,  v ⟩_, b ⟩_V 
       = ⟨ f^s_j (r) , v ⟩_⟨θ_s ⟨θ_r - θ_r , A_j (r) ⟩_, b ⟩_V  
    = ⟨θ_r - θ_r , A_j (r) ⟩_⟨  f^s_j (r), v ⟩_⟨θ_s   , b ⟩_V.

Which proves equality (<ref>) and, therefore, completes the proof of (<ref>).

Combining equalities (<ref>), (<ref>) and (<ref>) we obtain

    ∑_j = 1^n ∫^t_0 F^s_j (r) ∘ f^s_j (r)  r
    = ∑_j = 1^n ∫^t_0 ( 𝔼[ θ_s ∘ ( θ_r -  θ_r ) ] A_j (r) ) ∘f^s_j (r) r.

The formula (<ref>) holds for any arbitrary square integrable V^n - valued deterministic function  f^s(r) = (f^s_1 (r), … , f^s_n (r)) hence

    F^s_j (r) =  𝔼[ θ_s ∘( θ_r -  θ_r )] A_j (r) = Φ(s,r) A_j (r)

for all s ∈ [0,T] and for almost all r ∈ [0,T].

From (<ref>) and (<ref>), by the choice s=t and taking into account (<ref>) we have 

    θ_t = M^t_t =  ∑_j = 1^n ∫^t_0 F^t_j (r) W̃^j_r
    = ∑_j = 1^n ∫^t_0 Φ(t,r) A_j (r) W̃^j_r 
       = ∑_j = 1^n ∫^t_0 Φ(t,r)A_j(r) ξ^j_r -  ∑_j = 1^n ∫^t_0 Φ(t,r)A_j(r) ⟨θ_r , A_j(r) ⟩_ r 
       = ∫^t_0 Φ(t,s)A^*(s) ξ_s -  ∫^t_0 Φ(t,s)A^*(s)A(s) θ_s  s

which concludes the proof of stochastic integral equation (<ref>).

Let us verify the formula (<ref>). Notice that

    Φ(t,s) = 𝔼[θ_t ∘ (θ_s -  θ_s) ] 
    = 𝔼[ θ_t ∘θ_s ] -  𝔼[ θ_t  ∘θ_s ].

Using the above proved representation of θ_s, (<ref>), (<ref>) and independence of θ and W, we have

    𝔼[ θ_t ∘θ_s ]
    = 𝔼[ θ_t ∘( ∫^s_0 Φ(s,r)A^*(r) ξ_r -  ∫^s_0 Φ(s,r)A^*(r)A(r) θ_r  r ) ]  
       = 𝔼[ θ_t ∘( ∫^s_0 Φ(s,r) A^* (r) A(r) θ_r  r + ∫^s_0 Φ(s,r) A^* (r)  W_r - ∫^s_0 Φ(s,r) A^* (r) A(r) θ_r  r ) ]  
       = 𝔼[ θ_t ∘( ∫^s_0 Φ(s,r) A^* (r) A(r) ( θ_r - θ_r )  r ) ] 
    = ∫^s_0  𝔼[ θ_t ∘( Φ(s,r) A^* (r) A(r) ( θ_r - θ_r ) ) ]  r 
       = ∫^s_0  𝔼[ ∑_j = 1^n  θ_t  ∘[ Φ(s,r) A_j(r) ⟨θ_r - θ_r , A_j(r) ⟩_] ]  r.

For arbitrary v ∈ V^*, b ∈ V we have that

    ⟨𝔼[ ∑_j = 1^n  θ_t  ∘[ Φ(s,r) A_j(r) ⟨θ_r - θ_r , A_j(r) ⟩_] ]   v , b ⟩_V 
       =  𝔼[ ∑_j = 1^n ⟨θ_r - θ_r , A_j(r) ⟩_⟨Φ(s,r) A_j(r) , v ⟩_⟨θ_t , b ⟩_V ] 
       =   𝔼[ ∑_j = 1^n  ⟨θ_t ⟨θ_r - θ_r , A_j(r) ⟩_  , b ⟩_V ⟨Φ(s,r) A_j(r) , v ⟩_] 
       = ⟨∑_j = 1^n   𝔼[ θ_t ∘θ_r - θ_r ] A_j(r)  ⟨Φ(s,r) A_j(r) , v ⟩_  , b ⟩_V 
       = ⟨∑_j = 1^n  [ Φ(t,r)  A_j(r) ∘Φ(s,r) A_j(r) ] v  , b ⟩_V.

By (<ref>) it follows that

    𝔼[ θ_t ∘θ_s ]
    = ∑_j = 1^n ∫^s_0 Φ(t,r)  A_j(r) ∘Φ(s,r) A_j(r)  r.

This, together with (<ref>), proves the formula (<ref>).

By (<ref>) the mapping t ↦Φ(t,s) is strongly continuous on [0,T] uniformly with respect to s ∈ [0,t] hence the representation (<ref>) implies the mean - square continuity of the proces θ̂ in V. By (<ref>) and (<ref>) the mapping Φ: Δ→ℒ(V^*,V) is strongly continuous.

It remains to prove equality (<ref>). For every v ∈ V^* it holds

    𝔼[ θ_t ∘ (θ_t -  θ_t) ]v
    = 𝔼[ 𝔼[ θ_t ⟨θ_t -  θ_t , v ⟩_| F^ξ_t] ] 
       = 𝔼[ θ_t ( ⟨𝔼[ θ_t | F^ξ_t] , v  ⟩_  - ⟨θ_t , v ⟩_) ] = 0,

 therefore, we obtain

    𝔼[(θ_t -  θ_t) ∘ (θ_t -  θ_t) ] 
    = Φ(t,t) + 𝔼[ θ_t ∘ (θ_t -  θ_t) ] = Φ(t,t)

which completes the proof of Theorem <ref>.






§ UNIQUENESS OF THE COVARIANCE EQUATION

In the present section we prove uniqueness of solutions to the integral equation (<ref>) in the class of operators 
    𝕄 = {Ψ: Δ→ℒ(V^*,V); Ψ; Ψ (t,t) ≥ 0 ∀ t ∈ [0,T] }.


More generally, we prove the following.


Let K: Δ→ℒ(V^*,V) be a strongly continuous mapping such that K(t,t) ≥ 0 for each t ∈ [0,T].

Then the integral equation

    Ψ(t,s) = K(t,s) - ∑_j=1^n ∫^s_0 ( Ψ(t,r)A_j(r) ) ∘( Ψ(s,r) A_j(r) )  r,  (t,s) ∈Δ

has at most one solution in the class 𝕄.



First, we show that every Ψ∈𝕄 satisfying the equation (<ref>) is bounded on Δ.

Using Cauchy - Schwarz inequality we obtain for arbitrary x,y ∈ V^*

    | ⟨(∑_j=1^n ∫^s_0 ( Ψ(t,r)A_j(r) ) ∘( Ψ(s,r) A_j(r) )  r ) x , y ⟩_| 
       ≤∑_j=1^n ∫^s_0 | ⟨[ ( Ψ(t,r)A_j(r) ) ∘( Ψ(s,r) A_j(r) ) ] x , y ⟩_|  r 
         = ∑_j=1^n ∫^s_0 | ⟨Ψ(t,r)A_j(r) ⟨Ψ(s,r) A_j(r) , x ⟩_, y ⟩_|  r  
         = ∑_j=1^n ∫^s_0 | ⟨Ψ(t,r)A_j(r) , y ⟩_|  | ⟨Ψ(s,r) A_j(r) , x ⟩_ |  r 
       ≤∑_j=1^n  ( ∫^s_0 ( ⟨Ψ(t,r)A_j(r) , y ⟩_)^2   r )^1/2( ∫_0^s ( ⟨Ψ(s,r) A_j(r) , x ⟩_)^2 )^1/2
        = ∑_j=1^n  ( ∫^s_0 ⟨[ ( Ψ(t,r)A_j(r) ) ∘( Ψ(t,r) A_j(r) ) ] y ,y ⟩_ r  )^1/2
                          ( ∫^s_0 ⟨[ ( Ψ(s,r)A_j(r) ) ∘( Ψ(s,r) A_j(r) ) ] x , x ⟩_ r )^1/2.

In the last inequality we can increase the upper bound of the first integral from s to t because the integrand is nonnegative. Thus all we need is to estimate the term
 
    ∫^t_0 ⟨[ ( Ψ(t,r)A_j(r) ) ∘( Ψ(t,r) A_j(r) ) ] x , x ⟩_ r

 for all j = 1, … , n and x ∈ V^*. Using (<ref>) and the boundedness of the family ( K(t,s), t,s ∈ [0,T] ) in ℒ(V^*,V) (which follows by the Resonance Theorem) we obtain

    0 ≤∫^t_0 ⟨[ ( Ψ(t,r)A_j(r) ) ∘( Ψ(t,r) A_j(r) ) ] x , x ⟩_ r 
       ≤∑_j=1^n ∫^t_0 ⟨[ ( Ψ(t,r)A_j(r) ) ∘( Ψ(t,r) A_j(r) ) ] x , x ⟩_ r
    = ⟨ K(t,t) x , x ⟩_ - ⟨Ψ(t,t) x , x ⟩_
       ≤⟨ K(t,t) x , x ⟩_≤ C_1(T)  x ^2_V^*,   t ∈ [0,T].

Now, by (<ref>), (<ref>), (<ref>) and again by the boundedness of the family ( K(t,s), t,s ∈ [0,T] ) we have that

    |⟨Ψ(t,s) x , y ⟩_|  
       ≤ K(t,s)_ℒ(V^*,V)x_V^*y_V^*  + | ⟨(∑_j=1^n ∫^s_0 ( Ψ(t,r)A_j(r) ) ∘( Ψ(s,r) A_j(r) )  r ) x , y ⟩_| 
       ≤ C_2 (T)  x _V^* y _V^*

for all x,y ∈ V^* and (t,s) ∈Δ, which proves that the family of operators ( Ψ (t,s), (t,s) ∈Δ) is bounded in ℒ(V^*,V).

Assume that Ψ_1, Ψ_2 ∈𝕄 are solutions to the equation (<ref>) and using essential supremum set φ:  [0,T] → [0,∞), φ (s) = sup_t ∈ [s,T]Ψ_1(t,s) - Ψ_2(t,s) _ℒ(V^*,V). In virtue of (<ref>) we have


    φ (s) = sup_t ∈ [s,T]Ψ_1(t,s) - Ψ_2(t,s) _ℒ(V^*,V)
       = sup_t ∈ [s,T]∑_j=1^n ∫^s_0 ( Ψ_1(t,r)A_j(r) ) ∘( Ψ_1(s,r) A_j(r) )  - ( Ψ_2(t,r)A_j(r) ) ∘( Ψ_2(s,r) A_j(r) )  r  _ℒ(V^*,V)
       ≤∑_j=1^n ∫^s_0  sup_t ∈ [s,T]( Ψ_1(t,r)A_j(r) ) ∘( Ψ_1(s,r) A_j(r) )  - ( Ψ_2(t,r)A_j(r) ) ∘( Ψ_2(s,r) A_j(r) )  _ℒ(V^*,V) r 
       = ∑_j=1^n ∫^s_0  sup_t ∈ [s,T]( Ψ_1(t,r)A_j(r) ) ∘( Ψ_1(s,r) A_j(r) )  - ( Ψ_2(t,r)A_j(r) ) ∘( Ψ_2(s,r) A_j(r) )
       =                     ±( Ψ_1(t,r)A_j(r) ) ∘( Ψ_2(s,r) A_j(r) )_ℒ(V^*,V) r 
       = ∑_j=1^n ∫^s_0  sup_t ∈ [s,T]( Ψ_1(t,r)A_j(r) ) ∘[ (  Ψ_1(s,r) - Ψ_2(s,r) ) A_j(r) ] _ℒ(V^*,V)
                 + sup_t ∈ [s,T][ (  Ψ_1(t,r) - Ψ_2(t,r) ) A_j(r) ]  ∘( Ψ_2(s,r)A_j(r) ) _ℒ(V^*,V) r 
       ≤∑_j=1^n ∫^s_0  sup_t ∈ [s,T]Ψ_1(t,r) _ℒ(V^*,V) A_j(r) _V^*Ψ_1(s,r) - Ψ_2(s,r) _ℒ(V^*,V) A_j(r) _V^*
                 + sup_t ∈ [s,T]Ψ_1(t,r) - Ψ_2(t,r)
    _ℒ(V^*,V) A_j(r) _ℒ(V^*,V)Ψ_2(s,r) _ℒ(V^*,V) A_j(r) _V^* r 
       ≤ C(T) ∫^s_0 sup_t ∈ [s,T](Ψ_1(s,r) -  Ψ_2(s,r) ) _ℒ(V^*,V) + sup_t ∈ [s,T]( Ψ_1(t,r) - Ψ_2(t,r) ) _ℒ(V^*,V) r
       ≤ 2C(T) ∫^s_0 sup_t ∈ [s,T](Ψ_1(t,r) -  Ψ_2(t,r) ) _ℒ(V^*,V) r 
       ≤ 2C(T) ∫^s_0 sup_t ∈ [r,T](Ψ_1(t,r) -  Ψ_2(t,r) ) _ℒ(V^*,V) r = 2C(T) ∫^s_0 φ (r)  r.

Now, by the Gronwall lemma we obtain
 sup_t ∈ [s,T]Ψ_1(t,s) - Ψ_2(t,s) _ℒ(V^*,V) = 0 for all 0 ≤ s ≤ T and the proof is complete.




Note that if the signal θ = {θ_t, t ∈ [0,T] } takes its values in the Hilbert space H, the family  ( A(s) )_s ∈ [0,T] of observation operators H →ℝ^n can be characterised by n functionals from the dual space of H and no embedding is needed. In this case the equation (<ref>) can be expressed using adjoint operators A^* and Φ^* as

    Φ(t,s) = K^θ(t,s) - ∫^s_0 Φ(t,r)A^*(r) A(r)Φ^*(s,r)  r.

Indeed we have

    ⟨( ∑_j=1^n ∫^s_0 ( Φ(t,r)A_j(r) ) ∘( Φ(s,r) A_j(r) )  r ) h , k ⟩_H
       = ∑_j=1^n ∫^s_0 ⟨Φ(t,r)A_j(r) ⟨Φ(s,r) A_j(r) , h ⟩_H , k ⟩_H  r
    
       = ∑_j=1^n ∫^s_0 ⟨Φ(t,r)A_j(r) ⟨Φ^*(s,r) h , A_j(r) ⟩_H , k ⟩_H  r
       = ⟨( ∫^s_0 Φ(t,r)A^*(r) A(r)Φ^*(s,r)  r ) h , k ⟩_H

for all h,k ∈ H and all (t,s) ∈Δ.





§ EXAMPLES



Linear Stochastic Evolution Equation driven by Wiener Process

Let the signal θ = {θ_t, t ∈ [0,T] } be an H - valued random process defined by stochastic evolution equation

    θ_t = 𝒜θ_s  s + G 𝒲_t,    θ_0 = 0,

where 𝒜 is the infinitesimal generator of a strongly continuous semigroup (S(t)) _t ∈_+ in H, G ∈ℒ(H) and {𝒲_t, t ∈ [0,T] } is an H - valued standard cylindrical Wiener process defined on a stochastic basis ( Ω , F, P, (F_t) ).

Assume ∀ t > 0, S(t)G ∈ℒ_2 (H) and ∫_0^T_0| S(r)G |^2_ℒ_2 (H) r < ∞ for some T_0 > 0. Then the equation (<ref>) has a unique mean - square continuous solution {θ_t, t ∈ [0,T] } (cf. <cit.>).
The observation ξ = {ξ_t, t ∈ [0,T] } is given by the equation (<ref>) where V=H, { W_t, t ∈ [0,T] } is a ^n-valued Wiener process on  ( Ω , F, P, (F_t) ) independent of {𝒲_t, t ∈ [0,T] } and A: [0,T] →ℒ(H,^n) is strongly measurable and bounded.

For all 0 ≤ r ≤ t ≤ T the process θ_t satisfies

    θ_t = S(t-r) θ_r + γ_t,r,

where γ_t,r = ∫_r^t S(t-s) G 𝒲_s is a well defined centered Gaussian variable in H and is independent of θ_r (see <cit.>).

In this case the equations (<ref>) and (<ref>) simplify to the infinte - dimensional analogue of standard Kalman - Bucy filter as shown in the following theorem.


The filter θ satisfies the stochastic differential equation
  
    θ_t = 𝒜θ_t  t + Φ(t) A^*(t) ( ξ_t - A(t) θ_t  t ),   t ∈ [0,T],

where the solution to (<ref>) is understood in the mild sense, i.e. θ_t solves the equation

    θ_t = ∫^t_0 S(t-s) Φ(s) A^*(s) ξ_s -   ∫^t_0 S(t-s) Φ(s) A^*(s)A(s) θ_s  s.

The family of operators ( Φ(t) )_t ∈ [0,T]: H → H defined as Φ(t) =𝔼[(θ_t -  θ_t) ∘ (θ_t -  θ_t) ] for all t ∈ [0,T] satisfies the differential equation
    
    Φ̇(t) = 𝒜Φ(t) +  Φ(t) 𝒜^* -  Φ(t)A^*(t) A(t)Φ(t) + Q,    Q = G G^*

in the weak sense, that is

    t⟨Φ(t) x , y ⟩
     = ⟨Φ(t) x , 𝒜^* y ⟩ +   ⟨𝒜^* x , Φ(t) y ⟩ -  ⟨ A(t)Φ(t) x , A(t)Φ(t) y ⟩ + ⟨ Qx , y ⟩

for all x,y ∈ Dom(𝒜^*).


Note that the operator Φ(t) is in fact the operator Φ(t,s) defined in theorem (<ref>) when s = t and is selfadjoint. Set Φ(t) = Φ(t,t) and K(t) = K^θ(t,t) on t ∈ [0,T].


Using (<ref>), (<ref>), the independence of γ_t,s and θ_s for all 0 ≤ s ≤ t and the definition of operator Φ(t,s) in theorem (<ref>) we obtain 

    θ_t 
    = ∫^t_0 𝔼[ ( S(t-s) θ_s + γ_t,s) ∘ (θ_s - θ_s) ] A^*(s) ξ_s 
                    
     -  ∫^t_0 𝔼[ ( S(t-s) θ_s + γ_t,s) ∘ (θ_s - θ_s) ] A^*(s)A(s) θ_s  s 
       = ∫^t_0 S(t-s) 𝔼[ θ_s ∘ (θ_s - θ_s) ] A^*(s) ξ_s -  ∫^t_0 S(t-s) 𝔼[ θ_s ∘ (θ_s - θ_s) ] A^*(s)A(s) θ_s  s 
        =  ∫^t_0 S(t-s) Φ(s) A^*(s) ξ_s -   ∫^t_0 S(t-s) Φ(s) A^*(s)A(s) θ_s  s

for all t ∈ [0,T] which is exactly (<ref>).

Using the same arguments as above, equation (<ref>) and self - adjointness of the operator Φ(t), t ∈ [0,T] we have

    Φ(t) = 
        K(t) - ∫^t_0 𝔼[ ( S(t-r) θ_r + γ_t,r) ∘ (θ_r - θ_r) ]  A^*(r) A(r) ( 𝔼[ ( S(t-r) θ_r + γ_t,r) ∘ (θ_r - θ_r) ] )^*  r 
       = K(t) - ∫^t_0  S(t-r) 𝔼[ θ_r ∘ (θ_r - θ_r) ]  A^*(r) A(r) ( S(t-r) 𝔼[  θ_r ∘ (θ_r - θ_r) ] )^*  r
       = K(t) - ∫^t_0 S(t-r) Φ(r) A^*(r) A(r)Φ(r) S^*(t-r) r .

Since the covariance operator K(t),  t ∈ [0,T]  is a mild solution to the equation

    K̇(t) = 𝒜 K(t) + K(t) 𝒜^* + Q,     K(0) = 0,

which takes the form

    K(t) = ∫_0^t S(t-r) Q S^*(t-r)  r

we obtain

    Φ(t) =∫^t_0 S(t-r) ( Q - Φ(r) A^*(r) A(r)Φ(r) ) S^*(t-r) r

which is known to be equivalent to the weak form of the equation (<ref>) (cf. <cit.>).




Now, let us consider a signal defined by linear stochastic evolution equation driven by a regular Gauss - Volterra noise. First, we summarise a few basic facts from the infinte - dimensional theory of such processes (see <cit.>, <cit.>, <cit.> for more details, see also <cit.> for a brief survey).

By scalar Gauss - Volterra process (called also α - regular Volterra process) we understand a centered Gaussian process { b_t , t ∈ [0,T] }, b_0 = 0, the covariance of which takes the form

    [b_t b_s] = R(s,t) = ∫_0^s ∧ t K(s,r)K(t,r)  r,    s,t ≥ 0,

where the kernel  ( K(t,r) , (t,r) ∈Δ) satisfies

    (i)   K(0,0) = 0,
       (ii)    K( · , r) ∈ C^1 ([r,T]),     0 ≤ r ≤ T,
       (iii)    ∃ α∈( 0,12);   | K u(u,r) | < C(u-r)^α-1( ur)^α.

Note that in virtue of the Kolmogorov continuity criterion the process ( b_t , t ∈ [0,T] ) has an

ϵ - Hölder version for ϵ < α +12   (cf. <cit.>, Remark 2.1).

An important example of α - regular Gauss - Volterra process is the fractional Brownian motion with the Hurst parameter h > 12, in which case α = h - 12 and

    K(t,s) = K^h(t,s) = C_h ∫_s^t (us)^h - 1/2 (u-s)^h-3/2 u,

where c_h is a suitable constant (see <cit.>).

Another example is the multifractional Brownian motion where the Hurst parameter h = h(t) is a function of time. For the definition and conditions on h which ensure that the process satisfies the above assumptions see <cit.>, Example 2.14.

Now, given a separable Hilbert space H and a stochastic basis (Ω , F,P,(F_t) ), the cylindrical Gauss - Volterra process  { B_t , t ∈ [0,T] } on H is defined by the formal series

    B_t = ∑_n = 1^∞β_n(t) e_n

where { e_n, n ∈} is an orthonormal basis in H and {β_n (t) , t ∈ [0,T] }_n ∈ is a sequence of independent scalar α - regular Gauss - Volterra processes with the same kernel K. The series (<ref>) does not converge in the space H but defines the system of scalar processes { B(h) , h ∈ H }, 

    B_t (h) = ∑_n=1^∞⟨ e_n, h ⟩β_n (t)
 (for definitions and basic properties of cylindrical Volterra processes and stochastic integrals driven by them see <cit.>, <cit.>).

Suppose that the signal satisfies the equation

    θ_t = 𝒜θ_s  s + G  B_t,    t ∈ [0,T],    θ_0 = 0,

where 𝒜 is infinitesimal generator of a strongly continuous semigroup (S(t)) _t ∈+ in H and G ∈ℒ(H).

By the analyticity of semigroup S there exists λ∈ such that the operator (λ I -  𝒜) is strictly positive. Therefore, for  δ≥ 0, we can define the Hilbert space

    V_δ = Dom((λ I -  𝒜)^δ)

equipped with the graph norm topology.

The solution {θ_t, t ∈ [0,T] } to the equation (<ref>) is understood in the mild sense, that is,

    θ_t = ∫_0^t S(t-r) G  B_r,    t ∈ [0,T].


As a particular case of Corollary 4.1. in <cit.> we obtain the following statement.

Assume S(u)G ∈ℒ_2(H), u ∈ [0,T], and let there exist a γ∈ [0, α + 1/2) such that 

    | S(u)G |_ℒ_2(H)≤ cu^-γ,    u∈ [0,T]

for a constant c>0. Then {θ_t, t ∈ [0,T] } has a continuou version in the space V_δ for

    0 ≤δ < α + 1/2 - γ.

If G ∈ℒ_2(H) (which corresponds to the case when the driving noise in (<ref>) may be represented by a genuine H-valued Gauss-Volterra process), the condition (<ref>) is satisfied with γ = 0 and (<ref>) reads 0 ≤δ < α + 1/2 (for the fractional Brownian motion 0 ≤δ < h).


Note that for fractional Brownian motion the proposition holds true even if h < 1/2 (cf. <cit.>).


Consider the signal given by the following parabolic equation 

    ∂_t u = L_2m u + η

on [0,T]  × 𝒟 with initial condition u(0, ·) = 0 and with the Dirichlet boundary condition

    ∂^k u∂𝐱^k|_[0,T]  × ∂𝒟 = 0,
       k ∈{0, ⋯ , m-1 },

where ∂∂𝐱^k denotes the conormal derivative. The domain 𝒟⊂^d is open and bounded with smooth boundary and L_2m is a differential operator uniformly elliptic of order 2m,

    L_2m = ∑_|k| ≤ 2m a_k(·) ∂^k

with a_k ∈ C^∞_b(𝒟). The noise η is Gauss - Volterra in time and may be white (if G is identity operator) or correlated in space. This system can be reformulated as the stochastic evolution equation (<ref>) in H = L^2(𝒟). Indeed, the noise η is formally given as

    η(t,·) = G  t B_t,

where { B_t , t ∈ [0,T] } is a Gauss - Volterra process on H and 𝒜 = L_2m|_Dom(𝒜) where

    Dom(𝒜) = { f ∈ W^2m,2(𝒟) : ∂^k f∂𝐱^k = 0 on ∂𝒟 for k ∈{0, ⋯ , m-1 }}.

The operator 𝒜 generates an analytic semigroup (S(t), t∈ [0,T]) on H.

If the observation is given by the equation (<ref>) where V=H=L^2(𝒟), { W_t, t ∈ [0,T] } is independent of { B_t, t ∈ [0,T] } and A: [0,T] →ℒ(L^2(𝒟),^n) is strongly measurable and bounded, Theorems <ref>, <ref> and Remark <ref> may be applied.

However, it may be interesting to consider the case when the only accessible information comes from observation of the signal at given points z_1, …, z_n ∈𝒟. Then the signal process has to be more regular.

For simplicity, assume that m = 1, i.e. (<ref>) is a stochastic heat equation. If the condition (<ref>) is satisfied and, moreover 

    δ > d4

then the signal θ has a continuous version in V_δ (cf. <cit.>, <cit.>) that is continuously embedded into C(𝒟) by the Sobolev theorem. This follows from the fact that V_δ⊂ W^2δ,2(𝒟) (cf.  <cit.>). Such a choice of δ is possible if

    α + 1/2 - γ > d/4

(note that if G ∈ℒ_2(H) we may put γ = 0 and, on the other hand, for γ = d/4 we may consider arbitrary G ∈ℒ(H)).

Then in Theorems <ref> and <ref> we may put V = V_δ↪ C(𝒟) and as an example of observation operator we may take A(t) = A defined as

    A θ_t = (θ_t(z_1), … , θ_t(z_n)),

where z_1, … ,z_n ∈𝒟, which corresponds to pointwise observation of the signal process θ.

In this case the equations (<ref>) and (<ref>) can be rewritten according to the following theorem.


Let the signal θ = {θ_t, t ∈ [0,T] } satisfy the stochastic evolution equation

    θ_t = 𝒜θ_s  s + G  B_t,    θ_0 = 0,

where { B_t , t ∈ [0,T] } is a Gauss - Volterra process on L^2(𝒟), G ∈ℒ(L^2(𝒟)) and 𝒜 = L_2|_Dom(𝒜) is given by (<ref>) and (<ref>). Further assume that condition (<ref>) holds. Consider the observation process ξ = {ξ_t, t ∈ [0,T] } given by (<ref>) with operator A(t) = A defined by (<ref>).
Then the filter θ satisfies stochastic integral equation

    θ_t = ∑_j=1^n ∫^t_0 Φ_z_j(t,s) ξ^j_s -  ∑_j=1^n ∫^t_0 Φ_z_j(t,s) θ_s (z_j)  s,   t ∈ [0,T],

where Φ_z_i: Δ→ C(𝒟) is defined as Φ_z_i(t,s) = 𝔼[(θ_s -  θ_s)(z_i) θ_t] for all (t,s) ∈Δ, i = 1, … , n and integral equation

    Φ_z_i(t,s) =  [θ_s(z_i) θ_t] - ∑_j=1^n ∫^s_0 Φ_z_j(s,r)(z_i) Φ_z_j(t,r)  r,    i = 1, … , n

is satisfied.



From (<ref>), the definition of operator Φ, the continuous embedding V_δ↪ C(𝒟) and (<ref>) we have

    θ_t = ∫^t_0 Φ(t,s)A^*ξ_s -  ∫^t_0 Φ(t,s)A^*A θ_s  s
       = ∑_j=1^n  ∫^t_0 [θ_t ⟨θ_s -  θ_s , A_j ⟩_] ξ_s - ∑_j=1^n ∫^t_0 [θ_t ⟨θ_s -  θ_s, A_j ⟩_] ⟨θ_s , A_j ⟩_ s
       =  ∑_j=1^n  ∫^t_0 [(θ_s -  θ_s) (z_j) θ_t ] ξ_s - ∑_j=1^n ∫^t_0 [(θ_s -  θ_s) (z_j) θ_t ] θ_s (z_j)  s

which concludes the proof of (<ref>).

Analogously, using (<ref>) we obtain

    Φ_z_i(t,s) = Φ(t,s) A_i
       =  [ θ_t ⟨θ_s , A_i ⟩_] - ∑_j=1^n ∫^s_0 [θ_t ⟨θ_r -  θ_r , A_j ⟩_] ⟨[θ_s ⟨θ_r -  θ_r , A_j ⟩_] , A_i ⟩_ r
       =  [θ_s(z_i) θ_t] - ∑_j=1^n ∫^s_0 [ (θ_r -  θ_r ) (z_j) θ_t ] [ ( [(θ_r -  θ_r ) (z_j) θ_s ] )(z_i) ]  r
       =   [θ_s(z_i) θ_t]- ∑_j=1^n ∫^s_0 Φ_z_j(t,r) [ Φ_z_j(s,r) (z_i ) ]  r

which concludes the proof of (<ref>).


Note that if the driving process B=B^h is a fractional Brownian motion the condition (<ref>) reads 

    h > d/4 + γ,    h ∈ (0,1),

(cf. <cit.>).

It may be interesting to specify the covariances [θ_s(z_i) θ_t] that appear in the equation (<ref>). Suppose, for simplicity, that the driving process B_t = B^h_t is a fractional Brownian motion with the Hurst parameter h > 1/2, n=1 (i.e. the process θ = {θ_t, t ∈ [0,T] } is observed at a single point z_1 ∈𝒟) and the noise term G is Hilbert-Schmidt, i.e. it may be expressed as

    [G(f)](ξ) = ∫_𝒟 k(ξ,η)f(η) η,    f ∈ H = L^2(𝒟),

where k ∈ L^2 (𝒟×𝒟). It is also well known that the semigroup (S(t), t∈) may be represented by a Green function g: [0,T] ×𝒟×𝒟→, that is,

    [S(t)(f)](ξ) = ∫_𝒟 g(t,ξ,η)f(η) η,    f ∈ H,  t>0.

Therefore, the composition S(t)G may be written as

    [S(t)G(f)](ξ) = ∫_𝒟g(t,ξ,η)f(η) η,    f ∈ H,  t>0,

where the composition kernel g is given by

    g(t,ξ,η) = ∫_𝒟 g(t,ξ,λ)k(λ,η) λ.

Now it is standard to compute the covariance 

    [ θ_s(z_1) θ_t ] (η) = ∫_0^s ∫_0^t ϕ_h(λ,r) ∫_𝒟g(s-r,z_1,ξ) g(t-λ,η,ξ) ξλ r,

for (s,t) ∈Δ, η∈𝒟, where ϕ_h(λ,r) = h(2h-1) |λ - r |^2h-2 and (η,λ) ∈Δ.






Acknowledgements: This research was partially suported by GAUK Grant no. 980218, Czech Science Foundation (GAČR) Grant no. 19-07140S  and by the SVV Grant No. 260454.

mybst


