Backward forward stochastic differential equations
This work shows the existence and uniqueness of the solution of Backward stochastic differential equations inspired from a model for stochastic differential utility in Finance Theory. We show our results assuming, when possible, no more than the integrability of the terms involved in the equation. We treat both the finite and infinite horizon case for Backward equations and we study the stability properties of the solution under perturbations of the coefficients, of the data and of the underlying filtration. We also treat Backward Forward stochastic differential equations, that is when the solution depends simultaneously on the past and the future of its own trajectory, under a more restrictive hypothesis on the Lipschitz constant. This is a more complicated model than the Backward one, and it is intrinsically more difficult. We justify the restriction we place on this model by providing two counterexamples. Finally we discuss two situations in which the hypotheses may be relaxed by imposing other hypotheses on the coefficients and the integrating processes of the equations.
Protter, Purdue University.
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