A comparison result for FBSDE with applications to decisions theory

In general, a comparison Lemma for the solutions of Forward-Backward Stochastic Di erential Equations (FBSDE) does not hold. Here we prove one for the backward component at the initial time, relying on certain monotonicity conditions on the coeÆcients of both components. Such a result is useful in applications. Indeed, one can use FBSDE's to de ne a utility functional able to capture the disappointment-anticipation e ect for an agent in an intertemporal setting under risk. Exploiting our comparison result, we prove some \desirable" properties for the utility functional, such as continuity, concavity, monotonicity and risk aversion. Finally, for completeness, in a Markovian setting, we characterize the utility process by means of a degenerate parabolic partial di erential equation. The rst author was partially supported by the NATO/CNR Advanced Fellowships Program. We thank participants at the ESEM 1998 Conference (Berlin), S. Polidoro, J. Ma for useful discussions.