Discretization of backward stochastic Volterra integral equations (Q2849675)

The starting idea of this paper consists in a generalization of a numerical method for backward stochastic differential equations proposed by \textit{J. Ma} et al. [Ann. Appl. Probab. 12, No. 1, 302--316 (2002; Zbl 1017.60074)], to solve another class of stochastic differential equations -- namely, backward stochastic Volterra integral equations (BSVIEs), which play an important role in risk theory (see for example [\textit{J. Yong}, Appl. Anal. 86, No. 11, 1429--1442 (2007; Zbl 1134.91486)] for details).NEWLINENEWLINEAfter a good introduction of the adequate notations the well-posedness result is presented for the BSVIEs given by \textit{J. Yong} [Probab. Theory Relat. Fields 142, No. 1--2, 21--77 (2008; Zbl 1148.60039)]. The main result is given in Theorem 3.6 (a convergence theorem). The next two sections are devoted to the proof of this theorem, first in a ``smooth'' case (which is important because it gives a representation formula for the solution of BSVIEs in terms of systems of parabolic Cauchy problems) and then in a general case.NEWLINENEWLINEA good example to illustrate the numerical approximation is presented. Also, the speed of convergence of the proposed algorithm is obtained.NEWLINENEWLINEFor the entire collection see [Zbl 1272.91008].