arXiv:2012.00798v1 [math.PR] 1 Dec 2020 Continuity with respect to parameters of the solutions of time–delayed BSDEs with Stieltjes integral Luca Di Persio a , Lucian Maticiuc b , Adrian Z˘ alinescu c,d a Department of Computer Science, University of Verona, Strada le Grazie, no. 15, Verona, 37134, Italy b Faculty of Mathematics, “Alexandru Ioan Cuza” University, Carol I Blvd., no. 11, Ias ¸i, 700506, Romania c Faculty of Computer Science, “Alexandru Ioan Cuza” University, Carol I Blvd., no. 11, Ias ¸i, 700506, Romania d “Octav Mayer ”Mathematics Institute of the Romanian Academy, Carol I Blvd., no. 8, Ias ¸i, 700506, Romania Abstract We prove the existence and uniqueness of the solution of a BSDE with time-delayed generator, which employs the Stieltjes integral with respect to an increasing continuous stochastic process. We obtain also a result of continuity of the solution with regard to the increasing process, assuming only uniform convergence, but not in variation. AMS Classification subjects: 60H10, 60H30 Keywords or phrases: backward stochastic differential equations; time–delayed gener- ators; continuity with respect to parameters 1 Introduction Backward stochastic differential equations (BSDEs for short) were introduced in the linear case by Bismut [1], as adjoint equations involved in the control of SDEs. The nonlinear case was considered by Pardoux and Peng first in [13] and then in [14, 19], where they estab- lished a connection between BSDEs and semilinear parabolic partial differential equations (PDEs), by the so-called nonlinear Feynman–Kac formula. It was this kind of applications which triggered an impressive amount of research on the subject. Concerning parabolic PDEs with Neumann boundary conditions, Pardoux and Zhang discovered that their solu- tions can be linked to BSDEs involving the integral with respect to continuous increasing processes (Stieltjes integral). This paper represents a first step in establishing a probabilistic representation formula of the solutions of delayed path-dependent parabolic PDEs with Neumann boundary condi- tions. It consists in studying the well posedness of the associated BSDEs, i.e. existence and E-mail addresses: luca.dipersio@univr.it (Luca Di Persio), lucian.maticiuc@uaic.ro (Lucian Maticiuc), adrian.zalinescu@uaic.ro (Adrian Z˘ alinescu) 1