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This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present to illustrate the aforementioned optimal control problem. Motivated by the technical skills in solving above problem, a more convenient and briefer method for the unique solvability of M-solution for BSVIEs is proposed. At last, we will investigate a risk minimization problem by means of the maximum principle for FBSVIEs. Closed-form optimal portfolio is obtained in some special cases.

We define a class of functions which have a known decay rate coupled with a periodic fluctuation. We identify conditions on the kernel of a linear summation convolution Volterra equation which give the equivalence of the kernel lying in this class of functions and the solution lying in this class of functions. Some specific examples are examined. In particular this theory is used to provide a counter--example to a result regarding the rate of decay of the auto--covariance function of an ARCH($\infty$) process.

We define a class of functions which have a known decay rate coupled with a periodic fluctuation. We identify conditions on the kernel of a linear summation convolution Volterra equation which give the equivalence of the kernel lying in this class of functions and the solution lying in this class of functions. Some specific examples are examined. In particular this theory is used to provide a counter--example to a result regarding the rate of decay of the auto--covariance function of an ARCH($\infty$) process.

In this paper, we study a class of backward stochastic Volterra integral equations driven by Teugels martingales associated with an independent L\'{e}vy process and an independent Brownian motion (BSVIELs). We prove the existence and uniqueness as well as stability of the adapted M-solutions for those equations. Moreover, a duality principle and then a comparison theorem are established. As an application, we derive a class of dynamic risk measures by means of M-solutions of certain BSVIELs.

Backward stochastic Volterra integral equations (BSVIEs in short) are studied. We introduce the notion of adapted symmetrical solutions (S-solutions in short), which are different from the M-solutions introduced by Yong [17]. We also give some new results for them. At last a class of dynamic coherent risk measures were derived via certain BSVIEs.