The Merton problem about how to invest and consume optimally over the infi- nite horizon is a classical problem in both finance and stochastic control. But, the conclusions do not always match observed behaviour and this has led economists to generalise the set-up. One such generalisation is to assume preferences are described by stochastic differential utility (SDU).

The problem under SDU can be recast as a problem about a Backward Stochas- tic Differential Equation over the infinite horizon. So we ask, when does this formulation make sense? When does there exist a solution to the BSDE? When is the solution unique? Interestingly, the answer to these questions is not always "Yes", and in the "No" cases we have to decide how to proceed.

In the talk I will discuss some of these issues and suggest how to resolve them. Joint work with Martin Herdegen and Joe Jerome.