Optimal entry and consumption under habit formation

Abstract

This paper studies a composite problem involving decision-making about the optimal entry time and dynamic consumption afterwards. In Stage 1, the investor has access to full market information subject to some information costs and needs to choose an optimal stopping time to initiate Stage 2; in Stage 2, the investor terminates the costly full information acquisition and starts dynamic investment and consumption under partial observation of free public stock prices. Habit formation preferences are employed, in which past consumption affects the investor’s current decisions. Using the stochastic Perron method, the value function of the composite problem is proved to be the unique viscosity solution of some variational inequalities.

Retirement decision and optimal consumption-investment under addictive habit persistence

“…The requirement X(τ ) − p T (τ )h(τ ) > 0 is the necessary condition to ensure the solvability of the control problem after retirement (where there is no wage income and no retirement option). Similar arguments are shown in Lemma B.1 of Yang and Yu (2019), Lemma 3.3 of Yu (2015) and Theorem 5.5 of Englezos and Karatzas (2009). For detailed discussions, refer to Subsection 2.4 and Section 3.…”

“…In (4) it is required that X(τ ) − p T (τ )h(τ ) > 0. Here p T (•) represents the "cost of subsistence consumption per unit of habit", as stated in Assumption 2.1 of Yang and Yu (2019). We emphasize that in the retirement setup, we use the subscript T or τ to distinguish whether it is calculated up to the retirement τ , or the terminal time T .…”

“…However, it is not clear whether G w t = D t holds for any w. But it is indeed true that D t ⊂ G w t for any w, because if (x, h) ∈ D t then Ā(t, x, h) = ∅, where Ā(t, x, h) = {(c, π) : (t, c, π) ∈ A(t, x, h, w) for some w > 0} (A(t, x, h, w) dose not depend on w > 0 when τ = t in the definition, cf. Yang and Yu (2019), Yu (2015) and Englezos and Karatzas (2009)). Moreover, we will see in Section 3 that when (x, h) ∈ D t , the dual relation also holds.…”

“…In some existing literature where similar quantity has been defined (see Englezos and Karatzas (2009), Chen, Hentschel, and Xu (2018) and Yang and Yu (2019)), it seems that they do not propose its formal definition and emphasize its economic meaning. Besides, they do not consider retirement decision-making for the agent.…”

“…The case x − p T (t)h > 0 means that he actually has some wealth that he can use freely, and this must be satisfied once he retires (cf. Yu (2015), Yang and Yu (2019)). The retirement boundary is expressed by a linear relation among wealth, habit and wage, which evolves with time.…”