Variation of constants formulae for forward and backward stochastic Volterra integral equations

Abstract

In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual Itô's framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted $L^2$-processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as fractional stochastic differential equations. Also, our results can be applied to general classes of SVIEs and BSVIEs with infinitely many iterated stochastic integrals. The duality principle between generalized SVIEs and generalized BSVIEs is also proved.

42 pages

References in corpus (1)

• Extended backward stochastic Volterra integral equations and their applications to time-inconsistent stochastic recursive control problems

Cited by in corpus (2)

• Linear-quadratic stochastic Volterra controls I: Causal feedback strategies
• Linear-quadratic stochastic Volterra controls II: Optimal strategies and Riccati--Volterra equations