Anticipated backward stochastic differential equations

Abstract

In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.

Published in at http://dx.doi.org/10.1214/08-AOP423 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Necessary and sufficient condition for the comparison theorem of multidimensional anticipated backward stochastic differential equations
• Numerical approximations of McKean Anticipative Backward Stochastic Differential Equations arising in Initial Margin requirements
• Stochastic maximum principle for problems with delay with general dependence on the past
• A general comparison theorem for 1-dimensional anticipated BSDEs