In this talk, we
introduce a class of backward stochastic equations (BSEs) that extend
classical BSDEs and include many interesting examples of generalized BSDEs as
well as semimartingale backward equations. We show that a BSE can be
translated into a fixed point problem in a space of random vectors. This
makes it possible to employ general fixed point arguments, either algebraic
or topological to find solutions. For instance, Banach's contraction
mapping theorem can be used to derive general existence and uniqueness
results for equations with Lipschitz coefficients, whereas Schauder-type
fixed point arguments can be applied to non-Lipschitz equations. The approach
works equally well for multidimensional as for one-dimensional equations and
leads to results in several interesting cases such as equations with
path-dependent coefficients, anticipating equations, McKean-Vlasov type
equations and equations with coefficients of superlinear growth. |