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Affine Volterra processes with jumps

The theory of affine processes has been recently extended to the framework of stochastic Volterra equations with continuous trajectories. These so-called affine Volterra processes overcome modeling shortcomings of affine processes because they can have trajectories whose regularity is different from the regularity of the paths of Brownian motion. More specifically, singular kernels yield rough affine processes. This paper extends the theory by considering affine stochastic Volterra equations with jumps. This extension is not straightforward because the jump structure together with possible singularities of the kernel may induce explosions of the trajectories.


Introduction
Affine stochastic processes constitute unquestionably the most popular multi-factor framework to model rich and flexible stochastic dependence structures. Semi-explicit formulas for the Fourier-Laplace transform of affine processes make them numerically tractable since Fourier transform-based methods can be used to perform fast calculations.
We recall that a square-integrable conservative regular affine process X with state space E ⊂ R m is a special semimartingale whose semimartingale characteristics (B, C, ν), with respect to the "truncation function" h(ξ) = ξ, are of the form a(X s ) ds, ν(dt, dξ) = η(X t , dξ) dt (1) where In (2) we take A k ∈ R m×m , b k ∈ R m , and ν k (dξ) signed measures on R m such that ν k ({0}) = 0 and to guarantee existence and invariance results depending on the state space E. See for instance [11] for E = R k + × R l and [8] for E equal to the space of positive semidefinite matrices. The Fourier-Laplace transform of such an affine process X is given by with ψ a C m −valued function that solves the Riccati equation where and φ the C−valued function given by The identity (3) is only valid under additional hypotheses on the C m −valued function f and t, T ≥ 0 that imply appropriate conditions on the functions φ and ψ. 1 The theory of affine processes was recently extended in [5,14] to the framework of stochastic Volterra equations with continuous trajectories, where in general the semimartingale and Markov properties do not hold. These so-called affine Volterra processes overcome modeling shortcomings of affine processes because they can have trajectories whose regularity is different from the regularity of the paths of Brownian motion. More specifically, singular kernels yield rough processes in the spirit of [6,13,12]. The goal of this paper is to extend the results in [5] by considering affine stochastic Volterra equations with jumps. This extension is not straightforward because the jump structure together with possible singularities of the kernel may induce explosions of the trajectories.
Our study can be motivated by financial models for stock volatility. In particular the observation in [24] that a complete description of volatility should take into account both path roughness and jumps; see also [25] for an interesting discussion on the topic. In this paper, however, we concentrate on the mathematical properties of this family of processes and we address their possible applications in a separate article [7].
We summarize in this introduction the framework and the main results of our study. Suppose that X is a predictable solution to a stochastic Volterra equation of the form X t = g 0 (t) + t 0 K(t − s) dZ s , P ⊗ dt-a.e.
defined on a filtered probability space (Ω, F , (F t ) t≥0 , P) satisfying the usual conditions and with trajectories in L 1 loc (R + ; E) for some state space E ⊂ R m . In (7) we take g 0 ∈ L 1 loc (R + ; R m ), K ∈ L 2 loc (R + ; R m×d ) a matrix-valued kernel, and Z a d−dimensional semimartingale whose characteristics depend on X. In order to have an affine structure we suppose that Z has characteristics of the form (1)- (2), with A k ∈ R d×d , b k ∈ R d , and ν k signed measures on R d such that ν k ({0}) = 0 and R d |ξ| 2 |ν k |(dξ) < ∞. In this case we call X an affine Volterra process. When E = R m , existence of weak solutions to (7) with trajectories in L 2 loc (R + ; R m ) has been established in [2,Theorem 1.2]. For E = R + and for a Volterra CIR-type of process with positive jumps results in this direction can be found in [1,Theorem 2.13].
Section 2 contains more details about this setting as well as important results regarding stochastic convolutions with respect to processes with jumps. These results, which play a crucial role in our arguments, extend and are inspired by those in [5] where the authors study stochastic convolutions with respect to processes with continuous trajectories.
Fix T > 0, f ∈ C (R + ; C m ). By analogy with (4) assume that ψ ∈ C(R + ; C d ) solves the Riccati-Volterra equation with F as in (5), and let φ be given by (6). In Section 3 we show our first main result, namely Theorem 5, which is a generalization of [5,Theorem 4.3] and provides a semi-explicit formula for the Fourier-Laplace transform of X. This theorem shows that under the above-mentioned framework if we define where (g t (·)) t≥0 denotes the adjusted forward process 2 then M is a local martingale, and if M is a martingale then one has the exponential-affine formula As a consequence, under these conditions, uniqueness in law holds for the stochastic Volterra equation (7). Section 4 contains our second main result which is Theorem 7. This theorem establishes, under the assumption m = d and additional conditions on the kernel K, an alternative formula for the local martingale M t in (9) in terms of (X s ) s≤t and Z t only, namely with φ as in (6) and π h ∈ L 1 loc (R + ; C d ), h > 0, a deterministic function that depends on K and ψ. This expression is a corollary of a similar expression for the adjusted forward process (10), shown in Lemma 6. The identity (12) can be used to show that (3) is a particular instance of (11) when g 0 is constant, and K is constant and equal to the identity matrix.
In Section 5, using our first two main results, we give a complete proof in Theorems 10 and 11 of the exponential-affine formula (11) in the particular case m = d = 1, E = R + and for a Volterra CIRtype processes with positive jumps. The argument hinges on a novel comparison result between solutions of Riccati-Volterra equations, namely between a solution of (8) and a solution of an analogous equation in which the functions ψ and F are substituted with the corresponding real parts. This comparison result, together with the affine with respect to the past formula (12) of Theorem 7, yields the desired conclusion because we can bound the complex-valued local martingale M (9) of Theorem 5 with a real-valued martingale. 3 Appendix A contains some basic results regarding the classical forward process and Appendix B results regarding the 1−dimensional Riccati-Volterra equations appearing in Section 5.
Notation: Throughout the paper, elements of R k and C k are column vectors. Given a matrix A ∈ C k×l , the element in row i and column j is A ij , A ⊤ ∈ C l×k is its transpose matrix, and |A| is the Frobenius norm. We also use the notation R k . . , k , where for z ∈ C, R(z) denotes its real part. The imaginary part of a complex number z is Imz. We use the convolution notation (f * g) (t) = t 0 f (t − s) g (s) ds for functions f, g.

Preliminaries
Fix d, m ∈ N. Let g 0 ∈ L 1 loc (R + ; R m ), K ∈ L 2 loc R + ; R m×d be a matrix-valued kernel and E ⊂ R m be a subset which will be the state-space that we consider. We also introduce a characteristic triplet (b, a, η) consisting of the measurable maps b : R m → R d , a : R m → R d×d and the transition kernel η (x, dξ) from R m to R d . We require this triplet to be affine on E, meaning that, for every Throughout the paper, we denote by X = (X t ) t≥0 a predictable process with trajectories in L 1 loc (R + ; R m ) and such that X ∈ E, P ⊗ dt−a.e. It is defined on a filtered probability space (Ω, F , F, P) where the filtration F = (F t ) t≥0 satisfies the usual conditions and F 0 is the trivial σ−algebra on Ω. Moreover, we assume that X solves the following affine stochastic Volterra equation of convolution type Here Z is a d−dimensional semimartingale starting at 0 whose differential characteristics with respect to the Lebesgue measure are (b (X t ) , a (X t ) , η (X t , dξ)) , t ≥ 0. These characteristics are taken with respect to the "truncation function" h (ξ) = ξ, ξ ∈ R d , which can be chosen because Z is a special semimartingale due to [16, Proposition 2.29, Chapter II] and the local integrability of the trajectories of X. In the sequel, we denote by µ (dt, dξ) the measure associated with the jumps of Z and by ν (dt, dξ) = η (X t , dξ) dt its compensator. It is worth discussing the good definition of the stochastic integral in (14). Recalling that X ∈ E, P ⊗ dt−a.e., the canonical representation theorem for semimartingales (see [16,Proposition 2.34, Chapter II]) shows that Z admits the decomposition As a consequence of this expression, the local integrability of the paths of X implies that C j is locally integrable. Hence [16, Theorem 1.33 (a), Chapter II] yields that M d is a locally square-integrable martingale with where M d j is the j−th component of M d , j = 1, . . . , d. It is convenient to introduce the locally squareintegrable martingale Z = M c + M d , which satisfies Given an integer l ∈ N and F ∈ L 2 loc R + ; R l×d , we define the l−dimensional random variable This is well-defined for a.e. T ∈ R + . Indeed, consider the stopping times τ n = inf t ≥ 0 : t 0 |X s | ds > n for all n ∈ N. Since X · (ω) ∈ L 1 loc (R + , R m ), τ n → ∞ as n → ∞ in Ω. Then for every T > 0, we can apply the Young's type inequality in [2, Lemma A.1] with p = q = r = 1 and Tonelli's theorem to deduce that This ensures that E . . , m, n ∈ N, for a.e. T ∈ R + , say for every As a consequence, it is straightforward to conclude that the processes are locally square-integrable martingales for every T ∈ R + \ N . Indeed, for every n ∈ N, and (by (15) We always work with a jointly measurable version of the stochastic convolution F * d Z defined on Ω × R + (such a modification exists, see, e.g., [23, Theorem 3.5]).
As for the convolution of F with the drift part of Z, using [15, Theorem 2.2 (i), Chapter 2] we compute This shows that there exists a P ⊗ dt−null set N 1 ⊂ Ω × R + such that the next expression is well-defined Moreover, by Fubini's theorem the resulting processes are jointly measurable in (Ω × R + )\N 1 , hence passing to the limit as n → ∞, we obtain the jointly measurable process This is a jointly measurable process defined on the whole Ω × R + . This machinery for constructing jointly measurable modifications of given processes will be used several times in the sequel. Overall, the previous argument proves that the integral on the right side of (14) is well-defined P−a.s., for a.e. t ∈ R + . Of course we denote by (F * dZ) = (F * b (X)) + F * d Z ; with this notation, Equation (14) can be written as follows The following lemma will be useful in the sequel.
Proof. The proof follows the same steps as those in [2,Theorem 1.4]. The difference is that the affine structure of our model guaranteed by (13) is substituted for [2, Condition (1.5)], and makes the L 1 loc −integrability of the paths of X sufficient (instead of the L p loc −integrability, p ≥ 2 required in [2]). Knowing the additional property in (19), the same argument as the one above (without stopping times) shows that the processes in (17) are indeed square-integrable martingales for a.e. T ∈ R + . Remark 1. We refer to [2] for a general solution theory concerning equations of the type in (18) when g 0 ∈ L p loc (R + ; R m ) , p ≥ 2, and E = R m . In the case m = d = 1 and E = R + , if one defines Y t = t 0 X s ds, t ≥ 0, then Y = (Y t ) t≥0 is a nondecreasing process and an application of [2,Lemma 3.2] shows This type of stochastic Volterra equations is analyzed in [1] for locally integrable kernels K ∈ L 1 loc (R + ; R) .

Stochastic convolution for processes with jumps
The goal of this subsection is to develop technical results concerning the stochastic convolution. In particular, we aim to make Lemma 2.1 and Lemma 2.6 in [5] feasible in our context, where we are dealing with discontinuities for Z and, more importantly, with a process X which a priori is not bounded. This requires the statements and the proofs of the aforementioned results -crucial for the development of the theory-to be changed. Such changes are important from a conceptual point of view and after every result we add a remark showing the parallel with the setting in [5].
We start with a preliminary claim.
In particular, Proof. It is sufficient to prove (20) replacing on Ω × R + . Moreover, we only work with the stochastic integral in dM c , as by (15) we can repeat the next procedure (component-wise) for the convolution in dM d recovering (20). The argument above in the section implies the existence of a dt−null set N ⊂ R + such that, for every , whose predictable quadratic covariation is, due to the hypotheses, Since Q starts at 0, we can conclude that Q = 0 up to evanescence, hence (20) follows.
Regarding (21), it is an immediate consequence of (20) with S = R + and the joint measurability of the stochastic convolutions, which allows to state an equality holding true P ⊗ dt−a.e. This completes the proof.
Remark 2. In [5], the authors consider the stochastic convolution of a function F ∈ L 2 loc R + ; R p×d with respect to a continuous local martingale M with predictable quadratic covariation d M, M t = a t dt, where (a t ) is an adapted, locally bounded process. These assumptions allow to define (F * dM ) t for every t ∈ R + . In particular, two jointly measurable versions of the stochastic convolution are equal P−a.s., for every t ≥ 0. This concept is stronger than the P⊗dt−uniqueness that we have in our framework. As for (21) in Lemma 2, in the continuous case it can be stated as follows: for every F, G ∈ L 2 loc R + ; R p×d , with F = G a.e. in R + , one has Now we state a result concerning the associativity of the stochastic convolution.
Proof. Also in this case we just need to show the statement with d Z in place of dZ, because an application of In addition it is sufficient to focus only on the stochastic convolutions in dM c , as discussed in the preceding proof. By linearity we can assume d = p = q = 1 without loss of generality, and we consider ρ ≥ 0 to keep the notation simple, otherwise we should split it into positive and negative part. First note that the function ρ * F ∈ L 2 loc (R + ; R), hence for every t ∈ R + \ N 1 , being N 1 a dt−null set, we have where S ⊂ R + is such that t 0 F (t − s) ρ (s) ds, t ∈ S, is well-defined. In particular, R + \ S is a dt−null set. Our goal is to apply the stochastic Fubini's theorem (see, e.g., [22, Theorem 65, Chapter IV]), but before we can do that we need a preliminary step. For every T > 0, a change of variables, sequential applications of Tonelli's theorem and Young's inequality yield (in the whole Ω) Taking expectation and recalling (19) we have Taking t ∈ R + \ (N 1 ∪ N 2 ), thanks to (25) and Lemma 2 (see (20)) we can apply the stochastic Fubini's theorem in (24) to deduce that and the proof is complete.
The previous result is the analogue of [5, Lemma 2.1], where the authors are able -in the framework described in Remark 2-to handle a generic signed measure of locally bounded variation L. Essentially they can do so because the convolution F * dM is defined as a stochastic integral for every t ∈ R + . As a consequence, it is unique up to a P ⊗ |L| −null set, being |L| the total variation measure of L.
In contrast with this, notice that in our setting it is not possible to make sense of the right side of (23) for a fixed time t > 0 when ρ is replaced by L. Indeed, F * dZ is only defined up to a P ⊗ dt−null set, therefore the value of (L * (F * dZ)) (t) would depend on the modification one chooses. However, Lemma 3 can be slightly extended by replacing ρ in (23) with an R q×p −valued measure which is the sum of a locally integrable function and a point mass in 0 (this extension can be inferred directly from (23)). We are going to need this final comment in Section 4.
We are now ready to state an analogue of [5, Lemma 2.6].
Proposition 4. Assume that m = d, and that the kernel K ∈ L 2 loc R + ; R d×d admits a resolvent of the first kind L 4 . Let F ∈ L 2 loc R + ; R d×d be such that F * L is locally absolutely continuous. Then 4 Given a kernel K ∈ L 1 loc R + ; R d×d , an R d×d −valued measure L is called its (measure) resolvent of the first kind if L * K = K * L = I, where I ∈ R d×d is the identity matrix. L does not always exists, but if it does then it is unique (cfr. [ Proof. By Lebesgue's fundamental theorem of calculus we can write (denoting by I the identity matrix in R d×d ) which implies, convolving with K, using [15, Theorem 6.1 (ix), Chapter 3] and a change of variables, We can differentiate both sides of the previous equation, as they are absolutely continuous functions, and we obtain Then convolving with dZ yields where in the first equality we use Lemma 2 (see (21)) and in the second Lemma 3 with ρ = (F * L) ′ . The crucial point here is to pass to the trajectories. In order to do so, observe that by (18) we have s., for a.e. t ∈ R + . Moreover we can consider a jointly measurable modification of the process (F * L) ′ * (X − g 0 ) thanks to Fubini's theorem, which in turn can be applied as This equality can be understood up to a P⊗dt−null set because it involves only jointly measurable processes. Therefore (26) holds true and the proposition is completely proved.
Remark 4. In [5, Lemma 2.6] the authors require F * L to be right-continuous and of locally bounded variation. The loss of generality in Proposition 4, where we assume the local absolute continuity for the same function, is triggered by Lemma 3 and Remark 3.

Towards the conditional Fourier-Laplace transform
In this section we are going to introduce processes V T = V T t t∈[0,T ] which will be used to find an ansatz for the conditional Fourier-Laplace transform of f ⊤ * X (T ) , T > 0, where f is a suitable given function. We first introduce some notation. For a C−valued function g ∈ L 1 (ν k ) , k = 0, 1, . . . , m, we denote Note that η (x, dξ) , g (ξ) = R d g (ξ) ν 0 (dξ) + ν (g (ξ)) ⊤ x for every x ∈ E. In addition, we consider Let us take f ∈ C (R + ; C m ) and define F : Hypothesis 1. There exists a continuous, global solution ψ : R + → C d − to the deterministic Riccati-Volterra equation Under Hypothesis 1, we introduce the C−valued function φ : R + → C given by For every T > 0 we define the following càdlàg, adapted, C−valued semimartingale on Ω × [0, T ] Observe that V T is left-continuous in T because ψ (0) = 0 by (30). This process is the natural extension of [5, Equations (4.4) − (4.5)] to the framework with jumps. Moreover, one can write Our aim is to find, using the stochastic Fubini's theorem, an alternative expression for the random variables V T t by means of integrals in time of the trajectories of suitable processes. In the case b ≡ 0, we are going to use the paths of the forward process. Precisely, for a fixed t ∈ [0, T ], by Hence requiring the kernel K to be continuous on (0, ∞), the process on the right side of the previous equation has a jointly measurable version that we denote by g t (s) , s > t. Note that it makes sense to integrate in time the trajectories of such g t (·) since it is unique up to a P ⊗ dt−null set. In the case b = 0 we consider the paths of a process g t (·) such that Also in this case we assume K to be continuous on (0, ∞), so that g t (·) can be taken jointly measurable on Ω × (t, ∞) and is uniquely defined up to a P ⊗ dt−null set. Note that when t = 0 we have an abuse of notation, as g 0 represents both the initial input curve in (14) and the process just defined in (36). This, however, is not an issue as these two concepts coincide P ⊗ dt−a.e. in Ω × (0, ∞). In the following, we continue to consider g 0 as the initial input curve. Finally, notice that s., for a.e. s > t.
For this reason g t (·) is called adjusted forward process.
Theorem 5. Assume Hypothesis 1. Let K ∈ L 2 loc R + ; R m×d be a continuous kernel on (0, ∞) and define, for every t ∈ [0, T ], Then In addition, the process exp V T t t∈[0,T ] is a C−valued local martingale, and if it is a true martingale then Proof. It is straightforward to check that (38) holds true for t = 0. Focusing on the case t ∈ (0, T ], we rewrite the definition of V T t in (37) as follows It is convenient to introduce the process Recall that by (36) g t (s) = g 0 (s) + t 0 K (s − r) dZ r , P−a.s. for a.e. s > t, and that by (18) X s = g 0 (s) + s 0 K (s − r) dZ r , P−a.s., for a.e. s ∈ [0, t]. Therefore g t (·) is a jointly measurable modification of the process g 0 (·) + Such an application is legitimate, as by the continuity of F (·, ψ (·)) -which implies its boundedness in [0, T ] by a positive constant C T -and a change of variables we have (for every k = 1, . . . , m) so the expectation of the leftmost side is finite thanks to (19). As for the drift part, Going back to (40) and recalling the definitions of V T in (32)-(34) we obtain, P−a.s., where in the second-to-last equality we use (16). This proofs (38). Moving on to the next assertion, denote by H T = H T t t∈[0,T ] = exp V T t t∈[0,T ] . By Itô's formula and the dynamics in (32) we have . Then N T it is a local martingale and the previous computations show that H T = exp V T 0 E N T up to evanescence, where E denotes the Doléans-Dade exponential. Therefore H T is a local martingale, as stated. Finally, in case it is a true martingale, (39) directly follows from (38), and the proof is complete.

Remark 5.
Assuming m = d, it is possible to find an expression for V T in terms of the true forward process even in the case b = 0, when by (A. 4 Here R B is the resolvent of the second kind 5 of −KB and E B = K − R B * K. If K is continuous on (0, ∞), then E B is continuous on the same interval, as well. Thus, one can choose a jointly measurable version f t (s) , s > t, of the process on the right side of (41), which is unique up to a P ⊗ dt−null set. Arguing as in [5,Lemma 4.4], we obtain the variation of constants formula which combined with the strategy in the proof of Theorem 5 leads to However in the framework of jumps it is preferable to work with the adjusted forward process, because -as will become clear in the next section-certain properties can be assumed for the kernel K, but they can be neither required (i.e. it would not be a reasonable hypothesis) nor inferred for E B .

An expression for V T affine in the past trajectory of X
In this section we consider m = d and aim to find an alternative formula for V T which is affine in the past trajectory of X. This new expression can be used to prove the martingale property of the complex-valued process exp V T in particular cases (see Section 5). Due to the lack of regularity of the trajectories of both X and the stochastic convolution in dZ, we are going to require mild, additional conditions on the kernel K, in particular on the shifted kernels ∆ h K for h > 0.
We start with a preliminary result providing an alternative expression for the adjusted forward process g t (·). Lemma 6. Assume that K ∈ L 2 loc R + ; R d×d is continuous on (0, ∞) and that it admits a resolvent of the first kind L with no point masses in (0, ∞) . In addition, suppose that for every h > 0 the shifted kernel ∆ h K is differentiable, with derivative (∆ h K) ′ ∈ C R + ; R d×d . Then, for every T > 0, for every t ∈ [0, T ) Proof. Let us fix h > 0. We first show that the stochastic convolution ∆ h K * dZ has a càdlàg version. Indeed, for every t ∈ R + ,

Integration by parts yields
s., 5 Given K ∈ L 1 loc R + ; R d×d , its resolvent of the second kind is the unique solution R ∈ L 1 loc R + ; R d×d of the two equations K * R = R * K = K − R (cfr. [ where we also note that Z t− = Z t for a.e. t > 0, P−a.s. Since the rightmost side of the previous equality is a càdlàg process we obtain the desired claim. Hence in what follows we consider ∆ h K * dZ to be rightcontinuous. In particular, the process (∆ h K − K (h)) * dZ is continuous.
Thanks to the assumptions on the kernel, we apply [15, Corollary 7.3, Chapter 3] to claim that the function (∆ h K − K (h)) * L is locally absolutely continuous in R + , with In particular, the function (∆ h K) ′ * L ∈ C R + ; R d×d by [15, Corollary 6.2 (iii), Chapter 3], the absence of point masses of L in (0, ∞) and the continuity of (∆ h K) ′ . Therefore we invoke Proposition 4 to obtain Note that the last equality involves continuous processes, so it is indeed true for every t ≥ 0 up to a P−null set. Thus, At this point, let us take t < T and recall that (by (36)) It suffices to take h = T − t in (43) to deduce that Hence, combining the two previous equations, we conclude completing the proof.
Let us fix a generic T > 0. By Equation (42) we can write, for every t ∈ [0, T ), Intuitively speaking, we want to plug this expression in (37), so that we end up with an alternative formulation for V T t which is an affine function on the past trajectory {X s , s ≤ t}. This is done in the next theorem, which extends [5,Theorem 4.5] under further conditions on the kernel K. These addtional assumptions hold for instance in the one-dimensional case if K is completely monotone (recall that a function f is called completely monotone on (0, ∞) if it is infinitely differentiable there with (−1) k f (k) (t) ≥ 0 for all t > 0 and k = 0, 1, . . .).
Theorem 7. Assume that K ∈ L 2 loc R + ; R d×d is continuous on (0, ∞) and that it admits a resolvent of the first kind L with no point masses in (0, ∞). In addition, suppose that for every h > 0 the shifted kernel ∆ h K is differentiable, with (∆ h K) ′ continuous on R + . Under Hypothesis 1, if the total variation bound holds, then for every h > 0 the C d −valued function is well-defined for a.e. r ∈ R + and belongs to L 1 loc R + ; C d . Moreover, P−a.s., for a.e. t ∈ (0, T ), where φ is defined in (31).
Proof. Fix h > 0; expanding the notation in (46) for π h we have In order to see that it is well-defined a.e. on R + and belongs to L 1 loc R + ; C d , first note that for every positive s, the continuity of (∆ s K) ′ and the absence of point masses for L in (0, ∞) allow to apply [15, Corollary 6.2 (iii), Chapter 3], which ensures the continuity on R + of (∆ s K) ′ * L. As a consequence, we can define the C d −valued measurable function Recalling the previous proof, we see that (∆ h K) ′ * L is, almost everywhere, the derivative of the locally absolutely continuous function (∆ h K − K (h)) * L. The continuity of F (·, ψ (·)) (which implies its boundedness by a constant C h > 0 on [0, h]) coupled with Condition (45), Tonelli's theorem and [15, Theorem 6.1 (v), Chapter 3] yields, for a generic T > h, Hence the conclusion on π h follows. Furthermore, by Lebesgue's fundamental theorem of calculus, the C d −valued function Π h (r) = r 0 π h (u) du, r ∈ R + , is locally absolutely continuous on R + , with Π ′ h = π h a.e. Using Fubini's theorem we can obtain the following explicit expression for such Π h At this point we observe that for every function g ∈ L 1 loc R + ; R d we have, reasoning as before and using the boundedness of F (·, ψ (·)) on [0, T ] by a positive constant C T , where we apply Tonelli's theorem, together with [15, Theorem 2.2 (i), Chapter 2] and a change of variables. Consequently, for almost every t ∈ (0, T ) we can apply Fubini's theorem to obtain Computations analogous to those in (49) (with g [resp., |F (·, ψ (·))|] substituted by X − g 0 [resp., 1]) let us conclude by Fubini's theorem and Equation (19) that for a.e. t ∈ (0, T ) there is a jointly measurable modification of the process (∆ ·−t K) ′ * L * (X − g 0 ) (t) on Ω×(t, T ). Therefore we interpret (44) trajectoriwise, namely the equality holds almost everywhere in (t, T ) up to a P−null set. Now we focus on V T t . Combining (37) with what we have just said, a suitable change of variables yields The idea is to analyze separately the addends that we have singled out in the previous computations. Note that III t is finite because V T t , I t , II t are so, and that we can consider a jointly measurable modification of this process in Ω × (0, T ), again by Fubini's theorem and Equation (19) (see (49)). Taking into account (38) we have V T t = I t + II t + III t , for a.e. t ∈ (0, T ) , P − a.s., where the equality can be understood pathwise as it involves jointly measurable processes.
Substituting the two previous equations in (51) we conclude for almost every t ∈ (0, T ) , P−a.s. The proof is now complete.
If the resolvent of the first kind L is the sum of a locally integrable function and a point mass in 0, then recalling (18) we can apply Lemma 3 (see also the final comment in Remark 3) and argue as in (28) to see that Z t = (L * (X − g 0 )) (t) , for a.e. t > 0, P−a.s. In addition, for every h > 0 we define the C d −valued function where the second equality is due to (48). Note that Π h is locally absolutely continuous on R + , and that holding true for a.e. t ∈ (0, T ) , P−a.s. Substituting in (47) we immediately deduce the following result.

The 1−dimensional Volterra square root diffusion process with jumps
In this section we discuss a one-dimensional example (m = d = 1) where not only are we able to infer the assumptions made in the previous arguments, such as the existence of solutions to the stochastic Volterra equation (14) and the Riccati-Volterra equation (30) (i.e., Hypothesis 1), but also we can prove the martingale property of the process exp V T . In order to develop the theory we need to require more properties for the kernel K. In particular, we consider a hypothesis which is standard in the theory of stochastic Volterra equations, that is (see [1,Condition (2.10)], and also [5,Condition (3.4)] and [4, Assumption B.2]) Condition 1. The kernel K is nonnegative, nonincreasing, not identically zero and continuously differentiable on (0, ∞), and its resolvent of the first kind L is nonnegative and nonincreasing, i.e., s → L [s, s + t] is nonincreasing for every t ≥ 0.
We focus on the following stochastic Volterra equation of convolution type where Z is a real-valued semimartingale with differential characteristics (with respect to h (ξ) = ξ, ξ ∈ R) Here b ∈ R, c ≥ 0 and ν is a nonnegative measure on R + such that R+ |ξ| 2 ν (dξ) < ∞. The function g 0 : R + → R is an admissible input curve in either one of the following two forms i. g 0 is continuous and non-decreasing, with g (0) = 0; ii. g 0 (t) = x 0 + t 0 K (t − s) θ (s) ds, t ≥ 0, where x 0 ≥ 0 and θ : R + → R + is locally bounded. Notice that (54) describes a 1−dimensional Volterra square root diffusion. In this framework, we can invoke [1,Theorem 2.13] to claim the existence of a weak, predictable solution X = (X t ) t≥0 of (54) with trajectories in L 1 loc (R + ) such that X ≥ 0, P ⊗ dt−a.e. Actually, if g 0 ∈ L 2 loc (R + ), the paths of this solution X are in L 2 loc (R + ) , P−a.s., as the next result shows.
Lemma 9. Suppose that g 0 ∈ L 2 loc (R + ) and let X be a solution of (54) with trajectories in L 1 loc (R + ) such that X ≥ 0, P ⊗ dt−a.e. Then, for every Proof. The convolution equation (54) let us write, P−a.s., Integrating over the interval (0, T ) , T > 0, we have where we also use that √ x ≤ 1 + x, x ∈ R + . By [15, Theorem 2.2 (i), Chapter 2] we have K * X L 2 ([0,T ]) ≤ K L 2 ([0,T ]) X L 1 ([0,T ]) , hence taking expectation in the previous inequality we obtain, using Tonelli's theorem, s. for a.e. t ≥ 0; therefore we use the Burkholder-Davis-Gundy inequality and the Young's type inequality in [2, Lemma A.1] to write (always bearing in mind Tonelli's theorem) Analogously, we invoke [21,Theorem 3.2] to assert Now substituting the previous two bounds in (55) we see that the right side is finite by (19). This concludes the proof.
Continuing the example of this section, let f ∈ C (R + ; C − ). In addition to Eq. (30), we consider the deterministic Riccati-Volterra equation where Rf : R + → R − denotes the real part of f and F : R + × C − → R is defined as follows The next theorem shows the existence of global solutions to (30) and (56) (in particular, Hypothesis 1 is verified), as well as a comparison result between them which is crucial for the subsequent argument on the martingale property.
(ii) Given ψ ∈ C (R + ; C − ) and ψ ∈ C (R + ; R − ) satisfying (30) and (56), respectively, then the following inequality holds Proof. The proof of (i) is in Appendix B.1, and the one of (ii) is in Appendix B.2.
In what follows, we take two continuous functions ψ, ψ as in Theorem 10 (i) and fix T > 0. We aim to prove the martingale property of the process exp V T , where V T is given in (32)-(33). For this purpose, we define the process V T as in (32)-(33), substituting Rf [resp., ψ] for f [resp., ψ]. Theorem 5 shows that  (37) with the same substitution as before. It is known that exp V T is a true, real-valued martingale. This is due to [1, Lemma 6.1], which in turn is an interesting application of the Novikov-type condition in [20,Theorem IV.3]. The idea of the present section consists in using the expression (53) in order to prove the bound exp V T ≤ C exp V T up to indistinguishability for some C > 0, so that we can conclude that exp V T is a martingale, too.
Direct computations based on the Riccati-Volterra equation (30) yield, for every h > 0, Focusing on the second addend on the right side, if we convolve with L we end up with where the application of Fubini's theorem is justified because K is nonnegative and nonincreasing and L is a nonnegative measure. Whence, since ∆ h (ψ * L) (r) = (F (·, ψ (·)) * 1) (r + h), recalling the expression in (52) we can write and in particular Repeating the same argument for ψ we also obtain Taking the difference between the two previous equations we infer, for every r ≥ 0, Hence, we see that this function is increasing on the interval (0, ∞) by (58) in Theorem 10 (ii) and Condition 1 as soon as L has no point masses in (0, ∞). We are now in position to prove the next, important result.
Theorem 11. Assume that the kernel K ∈ L 2 loc (R + ; R) satisfies the requirements of Corollary 8 together with Condition 1. Then there exists a constant C > 0 such that In particular, exp V T t t∈[0,T ] is a complex-valued martingale.
Proof. First of all note that exp V T = exp R V T . For the reader's convenience, we write the expression for R V T provided by (53) The idea of the proof is simply to compare, term by term, the addends of this sum with the corresponding ones in the expansion of V T according to (53). We are going to consider a common set Ω 0 ⊂ Ω, with P (Ω 0 ) = 1, such that both the expressions for R V T t and V T t are valid on (0, T ) \ N ω , being N ω ⊂ (0, T ) a dt−null set for every ω ∈ Ω 0 .
Regarding the random terms, recall that X ≥ 0, P ⊗ dt−a.e. Therefore, without loss of generality, we can assume that for every ω ∈ Ω 0 and t ∈ (0, T ) \ N ω we have X t (ω) ≥ 0. As a consequence (by (59)) The associativity of the stochastic convolution proved in Lemma 3 (with ρ = R B ) and the joint measurability of the processes involved let us rewrite this equality as follows  (18) we have , P − a.s., for a.e. T ∈ R + .
Hence by the martingale property guaranteed by (19) we are able to find an expression for the forward process E X T F t , namely for almost every T ∈ R + , for every t ∈ [0, T ] it holds Finally, notice that (A.4) reduces to (A.1) as b ≡ 0. Indeed, since E 0 = K a.e. in R + as R 0 = 0 ∈ R d×d , combining (18) with Lemma 2 (see (21)) we have , P − a.s., for a.e. T ∈ R + , (A.5) and the assertion follows by the martingale property. where f ∈ C (R + ; C − ) . Throughout the section, we require Condition 1 on the kernel K.
we immediately note that, using (30), one can rewrite γ as follows γ (t) = We then need to study the increments of F in the second variable. Namely, fix u 1 , u 2 ∈ R − and use the definition (57) to write Hence substituting ψ and Rψ to u 1 and u 2 , respectively, we have We aim to apply [4, Theorem C.1] in order to conclude δ ≥ 0 in R + .
• In the continuous case the integral in ν (dξ), i.e., the function w, simply disappears, hence the application of [4, Theorem C.1] is straightforward.
• In the jump case we need to deal with such an integral. Observe that the function w has opposite sign with respect to δ, so there is no hope of applying [4, Theorem C.1] unless we modify its expression. Fortunately this can be done using the mean value theorem, in combination with simple real-analysis arguments.
Now we have to prove that w is continuous on R + . The first step is to show the continuity of the function c (ξ, ·) in R + for every fixed ξ ∈ R + . It is of course trivial for ξ = 0, so we just focus on ξ > 0.
Therefore an application of the squeeze theorem gives lim n→∞ c (ξ, t n ) = Rψ t = ψ t = c ξ, t .