Mathematical Methods of Operations Research

, Volume 54, Issue 3, pp 407–423 | Cite as

A comparison result for FBSDE with applications to decisions theory

  • Fabio Antonelli
  • Emilio Barucci
  • Maria Elvira Mancino


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.

Key words: Forward-Backward SDE's Comparison Theorem Utility Habit 
Classification: (AMS 2000) 60H 90A10 (JEL 1999) C61 D11 D81 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Fabio Antonelli
    • 1
  • Emilio Barucci
    • 2
  • Maria Elvira Mancino
    • 3
  1. 1.Dipartimento di Scienze – Università di Chieti, Viale Pindaro, 42, 65127 Pescara, ITALY (E-mail:
  2. 2.Dipartimento di Statistica e Matematica applicata all'Economia Università di Pisa, Via C. Ridolfi, 10, 56124 Pisa, ITALY (E-mail:
  3. 3.DIMAD – Università di Firenze, Via Lombroso, 6/17, 50134 Firenze, ITALY (E-mail:

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