PREDICTION OF GAUSSIAN VOLTERRA PROCESSES WITH COMPOUND POISSON JUMPS

. We consider a Gaussian Volterra process with compound Poisson jumps and derive its prediction law.


Introduction
We are interested in a mixed process X = G + J , where the continuous part G is a so-called Gaussian Volterra process.In older terminology these are processes that admit canonical representation of multiplicity one.A typical Gaussian Volterra process is the fractional Brownian motion as shown in Norros et al. [17].See Section 4 for the definition of fractional Brownian motion and for other examples.Intuitively, a Gaussian process G is a Gaussian Volterra process if one can construct a martingale M from by using a non-anticipative linear transformation and then represent the original process G in a non-anticipative way as a linear transformation of the martingale M .The motivation to use Gaussian Volterra processes is that for them one can calculate their prediction law in terms of the kernels that transfer the Gaussian Volterra process into its driving martingale M and vice versa.In the mixture X = G + J the jump part J will be an independent compound Poisson process with square-integrable jump distribution.
One motivation to study processes of the type X = G + J comes from mathematical finance.Indeed, it is well-known that the returns of financial assets do not follow Gaussian distribution [7,12,16] and the returns also exhibit jumps, or shocks [1,2,6,14,18].Also, there is evidence of long-range dependence in the returns also explain the presence of long-memory [5,9,13,15,19].Thus models where the returns are Gaussian with jumps seems more reasonable: The Gaussian part could take care of the long-range dependence with fractional Brownian motion (fBm) as the Gaussian Volterra process, and the shocks would come from the compound Poisson part.Then one can use the result of this paper to calculate imperfect hedges in the mixed model in the similar way as done in [20,22].Indeed, this is work in progress by the authors.
In this paper we derive the prediction law of the mixed process X = G + J .
The rest of the paper is organized as follows.In Section 2 we define Gaussian Volterra processes and derive their prediction laws.Section 3 is the main section of the paper where we introduce the Gaussian Volterra processes with compound Poisson jumps and derive their prediction laws.Finally, in Section 4 we provide examples of Gaussian Volterra processes.

Gaussian Volterra processes
A Gaussian Volterra process is a Gaussian process that has a canonical representation of multiplicity one with respect to a Gaussian martingale.The Gaussian Volterra process is defined in Definition 2.1 below in terms of covariance functions.The Gaussian Volterra representations follow from Definition 2.1 and are stated in Proposition 2.1.
For convenience we consider processes over the compact time-interval [0, T ] with an arbitrary but fixed time horizon T > 0.
A kernel K : [0, T ] 2 → R is a Volterra kernel if K(t, s) = 0 whenever t < s.For a Volterra kernel K we define its associated operator K as.
The adjoint associated operator K * of the Volterra kernel K is given by extending linearly the relation K It turns out that K * G for a Gaussian Volterra process with covariance extends to an isometry from Λ to L 2 ([0, T ], dv) where v is given in Definition 2.1(i) and Λ, the space of wiener integrands, is the closure of the indicator functions Remark 2.1.By Aòs et al. [3], if K is of bounded variation in its first argument, we can write for any elementary f Moreover, we have for elementary f and g that justifying the name "adjoint" associated operator.
For Gaussian Volterra representations we recall what is the co-called abstract Wiener integral (for more information on abstact Wiener integrals and their relation to conditioning we refer to [23])).
The linear space L is the closure of of the random variables G t , t ∈ [0, T ], in L 2 (Ω, F , P).The spaces Λ and L are isometric.Indeed, the mapping extends to an isometry.This isometry is called the abstract Wiener integral and we denote it Note that by Definition 2.1(ii) the operator K * is invertible and we have We note that due to Definition 2.1(i) for a Gaussian Volterra process the space Λ is isometric to L 2 ([0, T ], dv).Indeed, we have In particular, this means that the mapping K * in Definition 2.1(ii) is an isometry between Λ and L 2 ([0, t], dv).We also note that due to Definition 2.1(ii) the operator K * is invertible and we have, in particular, that The following representation proposition is a direct consequence of Definition 2.1.Indeed, Proposition 2.1 could have been taken as the definition of Gaussian Volterra process.Proposition 2.1 (Volterra representation).Let G be a Gaussian Volterra process.Let K −1 be the kernel in Definition 2.1(ii).Then the process where v and K are as in Definition 2.1(i).
Note that form Proposition 2.1 we immediately see that the filtrations F G and F M coincide.
The Volterra representations of Proposition 2.1 extend immediately to the following transfer principle for Wiener integrals.
In what follows we will use the following notation for the conditional mean, the conditional covariance and the conditional law of a stochastic process Y : We end this section by stating the prediction formula for Gaussian Volterra processes.The formula and its proof is similar to that given in [21].We give here the proof in detail for the convenience of the readers.

Proposition 2.3 (Volterra Prediction). Let G be a Gaussian Volterra process as in Def
Then Proof.It is well-known that conditional Gaussian processes are Gaussian.Therefore it is enough to identify the conditional mean and conditional covariance.
We consider first the conditional mean.Now Let us then consider the conditional covariance.Now

Gaussian Volterra process with jumps
In this section we prove our main result, the prediction formula for Gaussian Volterra processes with compound Poisson jumps.Indeed, we consider the process X = (X t ) t∈[0,T ] given by (3.1) where G is a continuous Gaussian Volterra process and J is an independent compound Poisson process with intensity λ and jump distribution F .In other words where N = (N t ) t∈[0,T ] is a Poisson process with intensity λ and the jumps ξ k , k ∈ N, are i.i.d. with common distribution F , and they are independent of the Poisson process N and the Gaussian Volterra process G.We denote We note that it is crucial that the Gaussian Volterra part is continuous since it implies that F X = F G,J , i.e. the signals G and J can be separated from the signal X .We refer to [4] and references therein on the continuity of Gaussian processes.
Our main theorem is the following.Theorem 3.1 (Mixed Prediction).Let X be given by (3.1).Then Here F * n is the n-fold convolution of the distribution F : Proof.Let us begin with the mean mX t (u).The conditional mean of G is already known.As for the conditional mean of J , we have, by independence, that The formula for the conditional mean follows from this.
Let us then consider the conditional variance RX (t, s|u).By independence we have RX (t, s|u) = RG (t, s|u) + RJ (t, s|u).
Now RG (t, s|u) is known and for RJ (t, s|u) we have Finally, let us consider the conditional law P X t (dx|u).By the law of total probability and independence we have The formula for the conditional law follows from this by plugging in P G t (dx|u) and the Poisson probabilities.Remark 3.1.It is interesting to note that just like in the Gaussian case, also in the mixed case, the conditional covariance is deterministic.

Examples
In this section we give examples for different Gaussian Volterra processes G for the prediction formula of Theorem 3.1.This means that we give the kernel K , the function v , and the kernel Ψ.
Example 4.1 (fBm).The fractional Brownian motion B H with Hurst index H ∈ (0, 1) is a centered Gaussian process with covariance By Norros et al. [17] (see also [21])) B H is a Gaussian Volterra process with v(t) = t and the Volterra kernel with a normalizing constant , and we have In other word, it has the following Volterra representation and so for all f ∈ Λ a,b,H , the space of all integrands from M , which is simply L 2 [0, T ] in this case.The (K * a,b,H ) −1 is the inverse operator of K * a,b,H , where from [11] ( for all f ∈ Λ, the space of all integrands from M , and ( K * H ) −1 is the inverse operator of K * H .