On the existence of weak solutions to stochastic Volterra equations

The existence of weak solutions is established for stochastic Volterra equations with time-inhomogeneous coefficients allowing for general kernels in the drift and convolutional or bounded kernels in the diffusion term. The presented approach is based on a newly formulated local martingale problem associated to stochastic Volterra equations.


Introduction
We investigate the existence of weak solutions to stochastic Volterra equation (SVEs) (1.1) where x 0 is a continuous function, B is a Brownian motion, and the kernels K µ , K σ are measurable functions.The time-inhomogeneous coefficients µ, σ are only supposed to be continuous in space uniformly in time.In case of ordinary stochastic differential equations (SDEs), i.e.K σ = K µ = 1, the existence of weak solutions was first proven by Skorokhod [Sko61] and can, nowadays, be found in different generality in standard textbooks like [SV79,KS91].
A comprehensive study of weak solutions to stochastic Volterra equations was recently initiated by Abi Jaber, Cuchiero, Larsson and Pulido [AJCLP21], see also [MS15].The extension of the theory of weak solutions from ordinary stochastic differential equations to SVEs constitutes a natural generalization of the classical theory and is motivated by successful applications of SVEs with non-Lipschitz coefficients as volatility models in mathematical finance, see e.g.[EER19,AJEE19].Assuming that the kernels in the SVE (1.1) are of convolution type, i.e.K µ (s, t) = K σ (s, t) = K(t − s) for some function K : R → R, and that the coefficients µ, σ are continuous jointly in space-time, the existence of weak solutions was derived in [AJCLP21], see also [MS15,AJLP19,AJ21].To that end, Abi Jaber et al. [AJCLP21] introduces a local martingale problem associated to SVEs of convolutional type.
In the present work we establish a local martingale problem associated to general stochastic Volterra equations, see Definition 2.4, and show that its solvability is equivalent to the existence of a weak solution to the associated SVE, see Lemma 2.7.Using this newly formulated Volterra local martingale problem, we obtain the existence of weak solutions to stochastic Volterra equations with time-inhomogeneous coefficients, that are not necessarily continuous in t, and allowing for general kernels in the drift and convolutional kernels as well as bounded general kernels in the diffusion term, see Theorem 3.3.The presented approach can be considered, roughly speaking, as a generalization of Skorokhod's original construction to the more general case of SVEs, and is developed in a one-dimensional setting to keep the presentation fairly short without cumbersome notation.However, as for ordinary SDEs and for SVEs of convolutional type, all concepts and results are expected to extend to a multi-dimensional setting in a straightforward manner.
Organization of the paper: In Section 2 we introduce a local martingale problem associated to SVEs.The existence of weak solutions to SVEs is provided in Section 3.

Weak solutions and the Volterra local martingale problem
For T ∈ (0, ∞) we consider the one-dimensional stochastic Volterra equation (2.1) where ] is a Brownian motion on a probability space (Ω, F, P), and the coefficients µ, σ : [0, T ] × R → R and the kernels K µ , K σ : ∆ T → R are measurable functions, using the notation and L p ([0, T ]) for the space of p-integrable functions on Ω × [0, T ] and on [0, T ], respectively.
Under suitable assumptions on the coefficients and kernels, the existence of weak solutions to the stochastic Volterra equation (2.1) can be equivalently formulated in terms of solutions to an associated local martingale problem, see Definition 2.4 below.To that end, we make the following assumption.
Proof.(i) By [JS03, Theorem II.2.42], in order to prove the assertion, it is sufficient to show that (M f t ) t∈[0,T ] , defined in (2.2), is a local martingale for every bounded function f ∈ C 2 (R).Let f ∈ C 2 (R) be bounded and define the hitting times ) and Z is continuous.Since the underlying filtered probability space satisfies the usual conditions, by the Début theorem (see [RY99, Chapter I, (4.15) Theorem]), the hitting times (τ n ) n∈N are stopping times.It remains to show that (τ n ) n∈N is a localizing sequence for (M f t ) t∈[0,T ] .To that end, we approximate f by the functions is a local martingale for every n ∈ N and, thus, the stopped process (M fn t∧τn ) t∈[0,T ] , given by is a martingale as for some constant C σ,µ,n > 0, using the definition of τ n and the linear growth condition on µ and σ.Since (ii) Since the process (Z t ) t∈[0,T ] is a semimartingale with absolutely continuous characteristics for some process (A t ) t∈[0,T ] of bounded variation and some local martingale given by is a local martingale for every f ∈ C 2 0 (R), where A f is defined as in (2.3), and (v) the following equality holds: (2.5) Remark 2.5.The first Volterra local martingale problem was formulated in [AJCLP21] for stochastic Volterra equations of convolution type, that is, the kernels K µ , K σ are supposed to be of the form K(t−s) for a deterministic function However, [AJCLP21, Definition 3.1] fundamentally relies on the convolutional structure to ensure that a weak solution to the SVE leads to a solution of the Volterra local martingale problem.The latter conclusion is based on a substitution and stochastic Fubini argument, which is not applicable for general kernels.Compared to [AJCLP21, Definition 3.1], the essential difference is that we reformulated [AJCLP21, (3.3)] to the condition (2.5).While both conditions are equivalent for kernels of convolutional type, the advantage of (2.5) is that it allows for general kernels.Moreover, notice that the Volterra local martingale problem as presented in Definition 2.4 reduces to the local martingale problem for ordinary stochastic differential equations in the case K µ = K σ = 1.Indeed, in this case conditions (i) and (iv) imply conditions (iii) and (v) on a possibly extended probability space, see Proposition 2.3.
Remark 2.6.Condition (iii) of Definition 2.4 can be relaxed to the condition "(Z t ) t∈[0,T ] is an (F t )-adapted and continuous process" since this together with (iv) of Definition 2.4 already implies the semimartingale property of (Z t ) t∈[0,T ] , see Proposition 2.3.However, we decided to directly postulate the semimartingale property of (Z t ) t∈[0,T ] in the formulation of the Volterra local martingale problem to ensure that condition (v) is obviously well-defined.
As for ordinary stochastic differential equations, the existence of weak solutions to SVEs is equivalent to the solvability of the associated Volterra local martingale problem, like in the case of convolutional SVEs as shown in [AJCLP21, Lemma 3.3].
Lemma 2.7.Suppose Assumption 2.2.There exists a weak solution to the SVE (2.1) if and only if there exists a solution to the Volterra local martingale problem given (x 0 , µ, σ, K µ , K σ ).
Proof.Let (X, B) be a (weak) solution to (2.1) on a probability space (Ω, F, P).Setting Itô's formula applied to f (Z t ) for f ∈ C 2 0 (R) yields that which is a local martingale and, by its definition, Z is a semimartingale satisfying (2.5).
Conversely, if there exists a solution to the Volterra local martingale problem, we obtain a weak solution to the SVE (2.1) by using (2.5) and Proposition 2.3, which yields that A t = t 0 µ(s, X s ) ds and M t = t 0 σ(s, X s ) dB s for some Brownian motion (B t ) t∈[0,T ] .

Existence of weak solutions
In this section we establish the existence of a weak solution to the SVE (2.1) and, equivalently, of a solution to the associated Volterra local martingale problem, under suitable assumptions on the initial condition, coefficients and kernels, which we state in the following.
To formulate our second assumption, for a measurable function K : ∆ T → R, we say K(•, t) is absolutely continuous for every t ∈ [0, T ] if there exists an integrable function Assumption 3.2.The kernel K µ is measurable and bounded in L 1 ([0, T ]) uniformly in the second variable, i.e.
for some constant C > 0. The kernel K σ is measurable and satisfies at least one of the following conditions: (i) K σ is a bounded function and K σ (•, t) is absolutely continuous for every t ∈ [0, T ] such that for some p > 1 and some constant Note, that Assumption 3.2 is satisfied by every convolutional kernel K µ (s, t) = K(t − s) for all (s, t) ∈ ∆ T for a function K ∈ L 1 ([0, T ]), and in case of Assumption 3.2 (i), the bound on the second summand in (3.1) is trivially fulfilled.With these assumptions at hand we are ready to state our main result.
Before proving the aforementioned existence result, let us briefly discuss some properties of weak solutions to the SVE (2.1) and some exemplary kernels.
Remark 3.5.Assumptions 3.1 and 3.2 are satisfied, e.g., by the following type of diffusion kernels: 2 ).The remainder of the paper is devoted to implement the proof of Theorem 3.3 based on several auxiliary lemmas.Generally speaking, the presented proof follows the classical approach of approximation the coefficients by Lipschitz continuous coefficients, in combination with a tightness argument.In contrast, [AJLP19] uses an approximation of the driving noise by pure jump processes with finite activity, which allows to treat convolutional SVEs with jumps.
Note that Lemma 3.9 implies Theorem 3.3 due to Lemma 2.7.Note further that the continuity of (X t ) t∈[0,T ] in Theorem 3.3 follows by the convergence Xk → X in C([0, T ]; R) in Lemma 3.8.
Assuming the coefficients µ, σ satisfy Assumption 3.1, the next lemma provides a way to approximate µ, σ locally uniformly by Lipschitz continuous coefficients.Lemma 3.6.Let f : [0, T ] × R → R be a measurable function such that for every compact set K ⊂ R and every ǫ > 0 there exists δ > 0 such that and such that f fulfills the linear growth condition for some constant C f > 0.Then, there is a sequence (f n ) n∈N of measurable functions f n : [0, T ] × R → R, which satisfies: (i) linear growth: for C f > 0 as in (3.3), we have (ii) Lipschitz continuity: for each n ∈ N there is a C n > 0 such that x, y ∈ R; (iii) locally uniform convergence: for all r ∈ (0, ∞) we have Proof.We explicitly choose the sequence (f n ) n∈N by (1 − y 2 ) n dy.(i) For t ∈ [0, T ] and x ∈ R, using the linear growth condition on f , we get (ii) Let t ∈ [0, T ], x, y ∈ R and n ∈ N. Using the compact support of f n and the fact, that every δ n is Lipschitz continuous as a smooth function with compact support, we get (iii) Due to the continuity property of f , we can find for every r > 0 and for every ǫ > 0 some δ > 0 such that for all x, y ∈ for all n ≥ N (ǫ, r).Then, setting r := r + 1 for all n ≥ N (ǫ, r) which tends to zero as ǫ → 0.
A suitable approximation, like the one provided in Lemma 3.6, ensures the convergence of associated Riemann-Stieltjes integrals.We denote by C([0, T ]; R) the space of all continuous functions g : [0, T ] → R, which is equipped with the supremum norm • ∞ .Lemma 3.7.Let f : [0, T ] × R → R be a function such that for every compact set K ⊂ R and every ǫ > 0 there exists δ > 0 such that and (f k ) k∈N be a sequence of functions such that x ∈ R, for all k ∈ N and for some C > 0, and f k → f locally uniformly.Let K : ∆ T → R be measurable and bounded in L 1 ([0, T ]) uniformly in the second variable, i.e. sup t∈[0,T ] where P → denotes convergence in probability.
Proof.First, note that due to the continuity condition (3.4), for every n ∈ N there exists some continuous non-decreasing function Let ǫ > 0 and δ > 0 be fixed but arbitrary.Choose N ∈ N and K ∈ N big enough such that for all n ≥ N and k ≥ K.Then, Setting K ǫδ := max{K N ǫδ , K}, we get for all k ≥ K ǫδ , which shows the desired convergence.
Given coefficients µ, σ satisfying Assumption 3.1, we fix, relying on Lemma 3.6, two sequences (µ n ) n∈N and (σ n ) n∈N with that fulfill properties (i)-(iii) of Lemma 3.6.For every n ∈ N, we define (X n t ) t∈[0,T ] as the unique (strong) solution (see e.g. the text before [PS23, Theorem 2.3] for the definition of unique strong solutions to SVEs) to the stochastic Volterra equation (3.6) given a Brownian motion (B t ) t∈[0,T ] on some probability space (Ω, F, P).Note that (X n t ) t∈[0,T ] exists by [Wan08, Theorem 1.1] due to the Lipschitz continuity of µ n and σ n .Furthermore, we introduce the sequences (A n ) n∈N and (M n ) n∈N by (3.7) In the following, we denote X D ∼ Y for equality in law of stochastic processes X and Y .
(iv) For k ∈ N and f ∈ C 2 0 (R), the stochastic process (M f,k t ) t∈[0,T ] is defined by where A f,k (t, x, z) := µ n k (t, x)f ′ (z) + 1 2 σ n k (t, x) 2 f ′′ (z).Due to ( Xk , Ẑk ) D ∼ (X n k , Z n k ) and since (X n k , Z n k ) solves the Volterra local martingale problem given (x 0 , µ n k , σ n k , K µ , K σ ) on (Ω, F, P) by construction and Lemma 2.7, it follows that (M f,k t ) t∈[0,T ] is a local martingale on ( Ω, F , P) for every k ∈ N.Moreover, Lemma 3.7 implies that M f,k → M f weakly as k → ∞ and, thus, by [JS03, Proposition IX.1.17],the limiting process (M f t ) t∈[0,T ] is a local martingale on ( Ω, F , P).(a) We start with the bounded kernels as in Assumption 3.2 (i).Due to the absolute continuity of K σ in the first variable, we can apply the integration by part formula for semimartingales (see [RW00,Theorem (VI) Since ( Xk , M k ) → (X, M ) in C([0, T ]; R 2 ) as k → ∞, P-a.s., and K σ is bounded, we obtain by Lemma 3.7 that Xk → X and K σ M k → K σ M in C([0, T ]; R) as k → ∞, P-a.s., and • 0 K µ (s, •)µ n k (s, Xk s ) ds → t 0 K µ (s, •) dA s in C([0, T ]; R) in probability as k → ∞.Furthermore, applying Hölder's inequality with p > 4 (see Assumption 3.2) and denoting q = p/(p − 1), we get by the integrability of 0 , the assertion follow by [JP12, Theorem 2.1.2].Keeping these preliminary considerations and the classical martingale problem (see e.g.[KS14, Definition 7.1.1])in mind, we formulate a local martingale problem associated to the stochastic Volterra equation (2.1).Definition 2.4.A solution to the Volterra local martingale problem given 5) with sup t∈[0,T ] t 0 |K(s, t)| ds ≤ M .For every n ∈ N we choose K n ǫδ ∈ N sufficiently large such that P sup t∈[0,T ], x∈[−n,n]

0 K 0 K
(v) Since ( Xk , M k ) D ∼ (X n k , M n k ) for every k ∈ N and pathwise uniqueness holds for SVEs with Lipschitz continuous coefficients (see e.g.[Wan08, Theorem 1.1]), the general version of the Yamada-Watanabe result ([Kur14, Theorem 1.5]) yields that Xk can be represented as the stochastic output of the Volterra equation (2.1) from the stochastic input M k in the same way as X n k from M n k , hence, we get that (3.8) Xk t = x 0 (t) + t µ (s, t)µ n k (s, Xk s ) ds + t σ (s, t) d M k s , t ∈ [0, T ], P-a.s., holds.To continue the proof of (v), we need to distinguish between (a) bounded kernels and (b) kernels of convolutional type.