BSDES WITH TIME-DELAYED GENERATORS OF A MOVING AVERAGE TYPE WITH APPLICATIONS TO NON-MONOTONE PREFERENCES

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 (1/t integral(t)(0) Y(s)ds, 1/t integral(t)(0) Z(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.