BSDEs with Time-Delayed Generators of a Moving Average Type with Applications to Non-Monotone Preferences

Abstract

In this article we consider backward stochastic differential equations with time-delayed generators of a moving average type. The classical framework with linear generators depending on (Y(t), Z(t)) is extended and we investigate linear generators depending on . We derive explicit solutions to the corresponding time-delayed BSDEs and we investigate in detail main properties of the solutions. An economic motivation for dealing with the BSDEs with the time-delayed generators of the moving average type is given. We argue that such equations may arise when we face the problem of dynamic modelling of non-monotone preferences. We model a disappointment effect under which the present pay-off is compared with the past expectations and a volatility aversion which causes the present pay-off to be penalized by the past exposures to the volatility risk.

Keywords:

• Backward stochastic differential equation
• Disappointment effect
• Habit formation
• Recursive preferences
• Volatility aversion

Mathematics Subject Classification:

• 60H05
• 60H30
• 60J70

ACKNOWLEDGMENTS

The research is supported by the Foundation for Polish Science. The author would like to thank two anonymous referees for pointing out deficiencies and the Editors for an encouragement to improve the earlier version of this article.