[eng]In Chapter 2 we extend the heterogeneous discounting model introduced in Marín-Solano and Patxot (2012) to a stochastic environment. Our main contribution in this chapter is to derive the DPE providing time-consistent solution for both the discrete and continuous time case. For the continuous time problem we derive the DPE following the two different procedures described above: the formal limiting procedure and the variational approach. However, an important limitation of these approaches is that the DPE obtained is a functional equation with a nonlocal term. As a consequence, it becomes very complicated to find solutions, not only analytically, but also numerically. For this reason, we also derive a set of two coupled partial differential equations which allows us to compute (analytically or numerically) the solutions for different economic problems. In particular, we are interested in analyzing how time-inconsistent preferences with heterogeneous discounting modify the classical consumption and portfolio rules (Merton (1971)). The introduction of stochastic terminal time is also discussed. In Chapter 3, the results of Chapter 2 are extended in several ways. First, we consider that the decision maker is subject to a mortality risk. Within this context, we derive the optimal consumption, investment and life insurance rules for an agent whose concern about both the bequest left to her descendants and her wealth at retirement increases with time. To this end we depart from

an instantaneous utility function u ( c ) {\displaystyle u(c)} where c {\displaystyle c} denotes consumption and discounts the next period utility at A Bellman equation, named after Richard E. Bellman, is a technique in dynamic programming which breaks an optimization problem into a sequence of simpler subproblems, as Bellman's "principle of optimality" prescribes. It is a necessary condition for optimality. The "value" of a decision problem at a certain point in time is written in terms of the payoff from some initial choices and the "value" In the deterministic setting, other techniques besides dynamic programming can be used to tackle the above optimal control problem. However, the Bellman Equation is often the most convenient method of solving stochastic optimal control problems. For a specific example from economics, consider an infinitely-lived consumer with initial wealth endowment a 0 {\displaystyle {\color {Red}a_{0}}} at period 0 {\displaystyle 0} . They have an instantaneous utility function u ( c ) {\displaystyle u(c)} where c {\displaystyle c} denotes consumption and discounts the next period utility at a rate of 0 < β < 1 {\displaystyle 0<\beta <1} . Assume that what is not consumed in period t {\displaystyle t} carries over to the next period with interest rate r {\displaystyle r} . Then the consumer's utility maximization problem is to choose a consumption plan { c t } {\displaystyle \{{\color {OliveGreen}c_{t}}\}} that solves