Two-parameter stochastic Volterra equations

We study two-parameter stochastic Volterra equations containing integrals over strong martingales and fields of bounded variation. For such equations, we prove the existence and uniqueness theorems for solutions with trajectories in the space of Borel functions which are locally square-integrable with respect to a locally finite measure. Under additional conditions on the coefficients of the equation, we prove the existence of a modification of the solution with trajectories continuous on the right and without discontinuities of the second kind.