On The Existence Of Variational Adapted Solutions To BSDEs With Locally Lipschitz Coefficient

In this papar, from the controllability point of view, we deal with the existence of solutions for BSDEs with integral control term. We obtain the existence theorems of variational adapted solution to BSDEs under locally Lipschitz (in both variables y, z) assumption and subliner growth condition. At first, we construct an approximating sequence of functions. Then, by using a suitable family of semi-norms and Gronwall inequality, we get that the sequence converges under locally Lipschitz (in both variables y, z) assumption and subliner growth condition. At last, we prove that there is a variational adapted solution of BSDEs. Also, we establish the existence of the solution for BSDEs with coefficientσ=σ1+σ2, whereσ1 satisfies locally Lipschitz condition,σ2 satisfies the classical globally Lipschitz condition.