Adapted solution of a backward stochastic nonlinear Volterra integral equation

This paper studies the existence and uniqueness of the following kind of backward stochastic nonlinear Volterra integral equation x(t) + integral(t)(T) f(t,s,X(s), Z(t,s))ds + integral(t)(T) [g(t,s,X(s)) + Z(t, s)] dW(s) =X under global Lipschitz condition, where {W-t; t is an element of [0,T]} is a standard k-dimensional Wiener process defined on a probability space {Omega,F,F-t,P} and X is {F-T}measurable d-dimensional random vector. The problem is to look for an adapted pair of processes {X(t), Z(t, s); t is an element of [0,T], s is an element of [t, T]} with values in R-d and R-d x k respectively, which solves the above equation. This paper also generalize our results to the following equation: X(t) + integral(t)(T) f(t,s,X(s), Z(t,s))ds + integral(t)(T) g(t,s,X(s), Z(t,s))dW(s) = X under rather restrictive assumptions on g.