Support characterization for regular path-dependent stochastic Volterra integral equations

Abstract

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 Hölder continuous paths. Based on functional Itô calculus, we prove that the support of its law in Hölder norms 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.

Rough differential equations with path-dependent coefficients

“…Fourniér [6], [7], [8] have a number of examples of regular functional in the sense of Dupire derivatives. Some further examples are discussed in [10] and [22]. Here we modify some of these examples to present functionals which satisfy the above assumptions.…”

“…Remark 4.3. It is worth to mention that with minor technical modifications, the results of the article would hold if we replace the Lipschitz continuity assumption with an assumption of Hőlder continuity in the metric d ∞ (as in [10] and [22]). However, we chose to work in the Lipschitz continuous setting to avoid unnecessary complications.…”

Rough differential equations with path-dependent coefficients

“…Remark 4.5. -It is worth mentioning that with minor technical modifications, the results of the article would hold if we replace the assumption of Lipschitz continuity in time with an assumption of Hőlder continuity in the metric d ∞ (as in [CK20] and [Kal21]). However, we chose to work in the Lipschitz continuous setting to avoid unnecessary complications.…”

Support Theorem for Pinned Diffusion Processes

“…The case of anticipating SDEs were studied in [39], [40]. The case of (Volterra-type) SDEs with path-dependent coefficients were recently studied in [11], [31]. A support theorem for McKean-Vlasov SDEs was proved in [56].…”