Support characterization for regular path-dependent stochastic Volterra integral equations

We consider a stochastic Volterra integral equation with regular path-dependent coefficients and a Brownian motion as integrator in a multidimensional setting. Under an imposed absolute continuity condition, the unique solution is a semimartingale that admits almost surely H\"older continuous paths. Based on functional It\^o calculus, we prove that the support of its law in the H\"older norm can be described by a flow of mild solutions to ordinary integro-differential equations that are constructed by means of the vertical derivative of the diffusion coefficient.

stochastic partial differential equation (SPDE) in Bally et al. [3]. By using the vertical derivative as functional space derivative and generalizing the approach in [13] with the relevant Girsanov changes of measures, a path-dependent version of the Stroock-Varadhan support theorem in Hölder norms was recently derived in [7]. The contribution of this article is to extend this support characterization to stochastic Volterra integral equations with regular path-dependent coefficients by providing a flow of mild solutions to ordinary integro-differential equations.
Let r, T ≥ 0 with r < T and d, m ∈ N. We work with the separable Banach space Then a functional on this Cartesian product that is d ∞ -continuous is also non-anticipative and Lipschitz continuity relative to d ∞ merely requires 1/2-Hölder continuity in the time variable.
Let us now state the conditions under which the support theorem holds. By refering to horizontal and vertical differentiability of non-anticipative functionals from [5,9], we in particular require that certain time and path space components of σ are of class C 1,2 , a property to be recalled in Section 2.1. In this context, let ∂ s be the horizontal, ∂ x the vertical and ∂ xx the second-order vertical differential operator.
To have a simple notation if these first-and second-order space derivatives appear, we set y := ( m k=1 d l=1 |y k,l | 2 ) 1/2 if y ∈ (R 1×m ) m×d or y ∈ (R m×m ) m×d . Further, let Á d be the identity matrix in R d×d and A ′ denote the transpose of a matrix A ∈ R d×m . (C. 2) The maps σ, ∂ x σ and its weak time derivatives ∂ t σ, ∂ t ∂ x σ are bounded. Further, there are c, λ, η ≥ 0 and κ ∈ [0, 1) such that for any s, t, u ∈ [r, T ) with s < t < u and every x, Under the assumption that σ(t, ·, ·) is of class for t ∈ [r, T ]. By addingx as initial condition, the solution x h lies in the delayed Sobolev space W 1,p r ([0, T ], R m ), since it can also be viewed as a mild solution to an associated ordinary integro-differential equation, as concisely justified in Section 2.2.
(i) Pathwise uniqueness holds for (1.1) and there is a unique strong solution X such that X r =x r a.s. Further, X is a semimartingale and E[ X p α,r ] < ∞ for any α ∈ [0, 1/2) and all p ≥ 1.

(ii) For any p ≥ 1 and each
is Lipschitz continuous on bounded sets.
Having clarified matters of uniqueness, existence and regularity, let us now consider the main result of this paper. Namely, a support characterization of solutions to (1.1) in delayed Hölder norms.
for all s, t ∈ [r, T ] and any x ∈ C([0, T ], R m ) and let the following three conditions hold: (1) The functions K b (·, s) and K σ (·, s) are differentiable for each s ∈ [r, T ). Further, (2) The map σ is of class C 1,2 on [r, T ) × C([0, T ], R m ) and together with its vertical derivative ∂ x σ it is bounded and d ∞ -Lipschitz continuous.
Then Theorem 1.2 applies and in the specific case that K b = K σ = 1 it reduces to the support theorem in [7] with the same regularity conditions.
The structure of this paper is determined by the proof of the support theorem and can be comprised as follows. Section 2 provides supplementary material and a Hölder convergence result that yields Theorem 1.2 as a corollary. In detail, Section 2.1 gives a concise overview of horizontal and vertical differentiability of non-anticipative functionals. Section 2.2 relates the Volterra integral equation (1.5) to an ordinary integro-differential equation and shows that solutions to (1.1) are semimartingales by using a stochastic Fubini theorem. In Section 2.3 we consider the approach to prove the support theorem by introducing a more general setting and stating Theorem 2.3, the before mentioned convergence result.
Section 3 derives relevant estimates to infer convergence in Hölder norm in moment. To be precise, Section 3.1 gives a sufficient condition for a sequence of processes to converge in this sense by exploiting an explicit Kolmogorov-Chentsov estimate. In Section 3.2 we introduce the relevant notations in the context of sequence of partitions and recall a couple of auxiliary moment estimates from [7,12]. The purpose of Section 3.3 is to deduce moment estimates for deterministic and stochastic Volterra integrals, generalizing the bounds from [7][Lemmas 20, 21 and Proposition 22]. Section 4 is devoted to a variety of specific moment estimates and decompositions, preparing the proof of Theorem 2.3. At first, Section 4.1 derives bounds for solutions to stochastic Volterra integral equations and gives two main decompositions, Proposition 4.3 and (4.7). Section 4.2 handles the first two remainders appearing in (4.7). While the second can be directly estimated, the first relies on the functional Itô formula in [6]. Section 4.3 intends to bound the third remainder in second moment, requiring another extensive decomposition. In Section 5 we prove the convergence result and the support representation, including assertions on uniqueness, existence and regularity.

Differential calculus for non-anticipative functionals
We recall and discuss horizontal and vertical differentiability, as introduced in [5,9]. To this end, let t ∈ (r, T ] and G be a non-anticipative functional on [r, t) × D([0, T ], R m ) that is considered at a point (s, x) of its domain: is differentiable at 0. If this is the case, then ∂ s G(s, x) denotes its derivative there.
is differentiable at 0. In this case, its derivative there is denoted by ∂ x G(s, x).
(iii) G is partially vertically differentiable at (s, x) if for any k ∈ {1, . . . , m} the function . . , e m } is the standard basis of R m . In this event, ∂ x k G(s, x) represents its derivative there.
So, G is horizontally, vertically or partially vertically differentiable if it satisfies the respective property at any point of its domain. We observe that vertical differentiability entails partial vertical differentiability and ∂ x G = (∂ x 1 G, . . . , ∂ xm G).
We say that G is twice vertically differentiable if it is vertically differentiable and the same is true for ∂ x G. We then set   [2]. That is, they do not dependent on the choice of the extension G. By combining these considerations with an absolute continuity condition, which ensures that only semimartingales appear, we can use the functional Itô formula from [6] to prove Proposition 4.4, a key ingredient when deriving (1.7).
is bounded on bounded sets and α-Hölder continuous with respect to d ∞ . Furthermore, if ϕ is of class C 1,2 in the usual sense, then G is of class for any s ∈ [r, t) and every x ∈ D([0, T ], R m ), where ∂ + ϕ/∂s denotes the right-hand time derivative of ϕ and D x j ϕ the partial derivative of ϕ with respect to the j-th space variable x j ∈ R m for each j ∈ {1, . . . , k}.

Ordinary integro-differential equations and semimartingales
By utilizing an absolute continuity condition, we directly connect the Volterra integral equation (1.5) to an ordinary integro-differential equation and check that any solution to (1.1) solves a stochastic differential equation, ensuring that it is a semimartingale.
Let us first briefly analyze (1. In this case, we may define ρ : [r, T ] 2 × C([0, T ], R m ) → R m coordinatewise by letting ρ k (t, s, x) agree with the right-hand side in (1.4), if s ≤ t, and setting ρ(t, s, x) := 0, otherwise. Then Fubini's theorem entails for each for every t ∈ [r, T ]. Consequently, the path x solves (1.5) if and only if it is a mild solution to the path-dependent ordinary integro-differential equatioṅ Since all appearing maps are integrable, this means that the increment x(t) − x(r) agrees with (2.1) for any t ∈ [r, T ]. Let us now turn to the stochastic Volterra integral equation (1.1), without imposing any conditions for the moment.
In particular, . Then a solution X to (1.1) satisfying X r = ξ r a.s. is called strong if it is adapted to this complete filtration.
Finally, suppose that (C.1) and (C.2) hold. Then it follows from Fubini's theorem for stochastic integrals, stated in [17] which is product measurable and depends on whole processes rather than trajectories, is given by for every s ∈ [r, T ] a.s. This shows that X solves (1.1) if and only if it is a solution to the path-dependent stochastic differential equation Moreover, it is automatically a semimartingale in this case.

Approach to the main result in a general setting
After these preliminary considerations, we proceed as follows to establish the support theorem. For any n ∈ N let T n be a partition of [r, T ] of the form T n = {t 0,n , . . . , t kn,n } with k n ∈ N and t 0,n , . . . , t kn,n ∈ [r, T ] such that r = t 0,n < · · · < t kn,n = T and whose mesh max i∈{0,...,kn−1} (t i+1,n − t i,n ) is denoted by |T n |. We assume that the sequence (T n ) n∈N of partitions is balanced as defined in [8], which means that there is For the estimation of one term in Proposition 4.4, when the functional Itô formula is applied, we also require the following additional condition: However, unless explicitly stated, we shall not impose this condition. Moreover, we readily notice that any equidistant sequence of partitions satisfies both conditions. Next, for any k, n ∈ N we are interested in the delayed linear interpolation of a map for every p ≥ 1, and by construction, the process n W : Let us now assume that (C.1)-(C.3) and Lemma 1.1 hold. Then the support of P then the converse inclusion holds. The sufficiency of (2.4) and (2.5) follows from a basic result on the support of probabiilty measures, see [7][Lemma 36] for example. To verify the validity of both limits, we consider a more general setting. Let B be an R m -valued and B H , B and Σ be R m×d -valued non-anticipative product measurable maps on [r, T ] 2 × C([0, T ], R m ). For any n ∈ N we study the path-dependent stochastic Volterra integral equation: is of class C 1,2 for all t ∈ (r, T ], we introduce another path-dependent stochastic Volterra integral equation: for each s, t, u ∈ [r, T ) with s < t < u and every x, y ∈ C([0, T ], R m ).
for any s, t, u ∈ [r, T ) with s < t < u and each x ∈ C([0, T ], R m ).
(C.9) There exist b 0 ∈ R and a measurable function b : First, we question uniqueness, existence and regularity of solutions to (2.6) and (2.7). In this regard, let ξ ∈ C ([0, T ], R m ) and ( n ξ) n∈N be a sequence in C ([0, T ], R m ).
Finally, we consider a convergence result in Hölder norm in second moment.
Let n Y and Y be the unique strong solutions to (2.6) and (2.7), respectively, such that n Y r = n ξ r and Y r = ξ r a.s. for all n ∈ N, then In particular, (2.9) is satisfied. That is, ( n Y ) n∈N converges in the norm · α,r in second moment to Y .

Convergence in moment along a sequence of partitions
We consider a sufficient condition for a sequence of processes to convergence in the norm · α,r in p-th moment, where α ∈ [0, 1] and p ≥ 1. Its derivation relies on an explicit Kolmogorov-Chentsov estimate [ for any α ∈ [0, q/p) with k α,p,q := 2 p+q (2 q/p−α − 1) −p . In particular, if q ≤ p, then X itself, and not necessarily a modification, admits a.s. α-Hölder continuous paths on [r, T ].
since q > αp and kn−1 j=0 (t j+1,n − t j,n ) = T − r. Moreover, from condition (2.2) we infer that |t i,n − t j,n | ≥ |T n |/c T for all i, j ∈ {0, . . . , k n } with i = j. Hence, | n X t j,n | p /|T n | αp and the claim follows from the definition of the norm · α,r .

Sequential notation and auxiliary moment estimates
Let us introduce relevant notations related to the sequence of partitions (T n ) n∈N . For fixed n ∈ N and t ∈ [r, T ), we choose i ∈ {0, . . . , k n − 1} such that t ∈ [t i,n , t i+1,n ) and set t n := t (i−1)∨0,n , t n := t i,n and t n := t i+1,n .
Verbalized, t n is the predecessor of t n relative to T n , provided i = 0, and t n is the successor of t n . For the sake of completeness, let T n := t k n−1 ,n , T n := T and T n := T . Further, for i ∈ {0, . . . , k n } we set ∆t i,n := t i,n − t (i−1)∨0,n and ∆W t i,n := W t i,n − W t (i−1)∨0,n .
For p ≥ 1 we recall an interpolation error estimate in supremum for stochastic processes in p-th moment and an explicit integral moment estimate for the sequence ( n W ) n∈N of adapted linear interpolations of W from [7][Lemmas 19 and 17].
(i) Let ( n X) n∈N be a sequence of R m -valued right-continuous processes for which there are c 0 ≥ 0 and q > 0 such that E[| n X s − n X t | p ] ≤ c 0 |s − t| 1+q for all n ∈ N, each j ∈ {0, . . . , k n − 1} and every s, t ∈ [t j,n , t j+1,n ]. Then there is c p,q > 0 such that for all n ∈ N. To be precise, c p,q = 2 p−1 (1 + k 0,p,q )(T − r).
(ii) Let Z be an R d -valued random vector satisfying Z ∼ N (0, Á d ). Then the constant for all n ∈ N and each s, t ∈ [r, T ] with s ≤ t.
Next, we let p ≥ 2 and state a Burkholder-Davis-Ghundy inequality for stochastic integrals with respect to W from [12][Theorem 7.2]. Based on this bound, one can deduce an estimate for integrals relative to n W that is independent of n ∈ N and which is given in [7][Proposition 16].
(iv) Any R m×d -valued progressively measurable process X satisfies

Moment estimates for Volterra integrals
The first integral bound that we consider follows from the auxiliary estimate (3.3).   3.3. For any n ∈ N let n X be independent of the first time variable, that is, there is an R + -valued measurable process n Y with n X t,s = n Y s for all s, t ∈ [0, T ]. Then for condition (3.6) to hold, it suffices that there is c p > 0 so that E[ n Y p s ] ≤ c p |T n | q for every s ∈ [r, T ) and each n ∈ N. Put differently, γ n = ∆t i,n /∆t i+1,n on [t i,n , t i+1,n ) for all i ∈ {0, . . . , k n −1} and γ n (T ) = 1.

Lemma 3.2. Let p > 1 and assume for each
for every n ∈ N.
The third estimate deals with Volterra integrals driven by n W and W , where n ∈ N.  , x), (s, x)) for any s, t, u ∈ [r, T ) with s < t < u and every x ∈ C([0, T ], R m ). Moreover, let ( n Y ) n∈N be a sequence in C ([0, T ], R m ) for which there are p ≥ 2 and c p,0 ≥ 0 such that for all n ∈ N, each s, t ∈ [r, T ] with s < t and any for every n ∈ N.

Decomposition into remainder terms
We first give a moment estimate for solutions to (2.6) that does not depend on n ∈ N.
for any s, t, u ∈ [r, T ) with s < t < u and every x ∈ C([0, T ], R m ). Then for each p ≥ 2 there is c p > 0 such that any n ∈ N and each solution n Y to (2.6) satisfy for all s, t ∈ [r, T ] with s = t.
for every s, t ∈ [r, T ] with s = t.
Proof. As the map R given by (2.8) is bounded, the assertion is a direct consequence of For n ∈ N let us recall the linear operator L n and the function γ n given at (2.3) and (3.7), respectively, and deduce the main decomposition to establish the limit (2.10). Then for each p ≥ 2 there is c p > 0 such that each n ∈ N and any two solutions n Y and Y of (2.6) and (2.7), respectively, satisfy Proof. We suppose that E[ n Y r p ∞ ] and E[ Y r p ∞ ] are finite and aim to derive the estimate by applying Gronwall's inequality to the increasing function ϕ n : [r, T ] → R + given by To this end, let us write the difference of n Y and Y as follows: for any t ∈ [r, T ] a.s. So, we let the terms n Y r − Y r and n ∆ unchanged, then for the constant c p,1 : To obtain the estimate (4.5), we used the chain of inequalities: T ] a.s. Then n ∆ admits the following representation: for all t ∈ [r, T ] a.s. Due to the assumptions, we may assume without loss of generality that the Lipschitz constant λ is large enough such that |R(u, t, x) − R(u, s, y)| ≤ λd ∞ ((t, x), (s, y)) for any s, t, u ∈ [r, T ) with s < t < u and every x, y ∈ C([0, T ], R m ). Thus, for the constant c p,2 := 10 p−1 (1 + T − r) p (T − r) p/2−1 ((T − r) p/2 + w p )λ p we get that δ n (t) 1/p ≤ δ n,3 (t) 1/p + δ n,4 (t) 1/p + δ n,5 (t) 1/p + c p,2 tn r δ n,1 + (s − s n ) p/2 + δ n,2 (s) + ε n (s) + ϕ n (s) ds  Thanks to Proposition 4.1 and Corollary 4.2, there are c p , c p > 0 such that (4.1) and (4.4) hold when c p is replaced by c p and c p , respectively. By combining (4.5) with (4.6), we see that  for each h ∈ W 1,2 r ([0, T ], R d ) and any n ∈ N. Whenever n Y is a solution to (2.6), then we will utilize the following decomposition to deal with the considered remainder: for all j ∈ {1, . . . , k n } and each s ∈ [r, t j,n ).

Moment estimates for the first two remainders
The first result in this section together with Lemma 3.2 provide an estimate of the first remainder appearing in (4.7).  , x), (s, x)) for any s, t, u ∈ [r, T ) with s < t < u and all x ∈ C([0, T ], R m ). , x) is of class C 1,2 for any t ∈ (r, T ] and there are c 0 , η, λ 0 ≥ 0 such that for each s, t, u ∈ [r, T ) with s < t < u and all x ∈ C([0, T ], R m ).
Then for any p ≥ 2 there is c p > 0 such that for all n ∈ N and each solution n Y to (2.6), Proof. For any j ∈ {1, . . . , k n } let the product measurable map n,j ∆ : [r, t j,n ) 2 ×Ω → R 1×m be given by n,j ∆ s,u := ∂ x F (t j,n , u, n Y ) − ∂ x F (t j,n , s n , n Y ), if u ∈ [s n , s], and n,j ∆ s,u := 0, otherwise. Then from the functional Itô formula in [6] we infer that  n Y ), (s n , n Y )) for each j ∈ {1, . . . , k n } and all s, u ∈ [r, t j,n ) and by setting c p := 2 3p/2 λ p 0 (1 + c ηp ) 1/η , we obtain that for each s ∈ [r, T ]. Consequently, the Cauchy-Schwarz inequality gives us the following bound for the third and sixth expression in the decomposition (4.8): of Φ h,n we get that

A second moment estimate for the third remainder
We directly bound the third remainder in (4.7) by repeatedly using an estimate that follows for any n ∈ N with k n ≥ 2 from Doob's L 2 -maximal inequality; see [7][Lemma 33] for details.