Subgame-perfect Equilibrium Strategies in State-constrained Recursive Stochastic Control Problems

We investigate time-inconsistent recursive stochastic control problems. Since, for this class of problems, classical optimal controls may fail to exist, we focus on subgame-perfect equilibrium policies. In a continuous time setting, such kind of controls have been introduced in the papers Ekeland-Lazrak (2006) and Ekeland-Pirvu (2008), and can be thought of as “infinitesimally optimal via spike variation”: that is, they are optimal with respect to a penalty represented by unilateral deviations during an infinitesimal amount of time.

For the time-consistent framework we refer, in particular, to the fundamental works by Duffie-Epstein (1992) and El Karoui et al. (2001). For the time-inconsistent setting we mention the series of work carried out by Yong (2011,2012,2017) whose approach focuses on dynamic programming (HJB equations). The approach followed in our work, instead, is inspired by Ekeland-Pirvu (2008) and Hu (2017) and relies on the stochastic maximum principle; see also Pardoux (1993). We adapt the classical spike variation technique to obtain a characterization of equilibrium strategies in terms of a generalized Hamiltonian function defined through a flow of pairs of BSDEs. Our generalized Hamiltonian function, compared to the classical one, contains the driver coefficient of the recursive utility and of the presence of a second-order stochastic process. We emphasize that, differently from the classical case, equilibrium strategies are characterized not only by means of a necessary condition, but also through a sufficient condition involving the generalized Hamiltonian even in the absence of extra convexity conditions.

Our analysis extends to treat the problem under a state constraint defined by means of an additional recursive utility. In dealing with that, we had to adapt Ekeland’s variational principle to this more tricky situation. We would like to point out that this procedure seems to be able to manage a pretty wide variety of constraints (quite different from the one just mentioned), such as the ones considered e.g. in Ji-Zhou (2006),  Zhuo (2018) and Frankowska et al. (2019).

All this is applied to the financial field of investment-consumption policies with non-exponential actualization. Under appropriate hypotheses, our results cover the case where the constraint is a risk constraint: the additional recursive utility derives from a suitable dynamic risk measure defined by means of a g-expectation as e.g. in Rosazza Gianin (2006).