BSDEs with time-delayed generators of a moving average type with applications to non-monotone preferences

TL;DR

This paper studies backward stochastic differential equations with time-delayed moving average generators, providing explicit solutions and exploring their properties, motivated by economic models of non-monotone preferences involving disappointment and volatility aversion.

Contribution

It extends classical BSDE frameworks to include time-delayed moving average generators and derives explicit solutions, with applications to economic models of non-monotone preferences.

Findings

Explicit solutions to time-delayed BSDEs are derived.

Main properties of solutions are thoroughly analyzed.

Applications to economic models of preferences are discussed.

Abstract

In this paper 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 $(\frac{1}{t}\int_0^tY(s)ds, \frac{1}{t}\int_0^tZ(s)ds)$. 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.