A comparison result for Forward–Backward Stochastic Differential Equations with applications to decision theory

In general, a comparison Lemma for the solutions of Forward–Backward Stochastic Differential Equations (FBSDE) does not hold. Here we prove one for the backward component at the initial time, relying on certain monotonicity conditions on the coefficients of both components. Such a result is useful in applications. Indeed, one can use FBSDE's to define a utility functional able to capture the disappointment–anticipation effect 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 differential equation.