This paper studies an infinite horizon optimal consumption problem under exponential utility, together with non-negativity constraint on consumption rate and a reference point to the past consumption peak . The performance is measured by the distance between the consumption rate and a fraction 0\leq\lambda\leq 1 of the historical consumption maximum. To overcome its path-dependent nature, the consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional depending on the wealth variable x and the reference variable h. The associated Hamilton-Jacobi-Bellman (HJB) equation is expressed in the piecewise manner across different regions to take into account  constraints. By employing the dual transform and smooth-fit principle,  the classical solution of the HJB equation is obtained in an analytical form, which in turn provides the feedback optimal investment and consumption . For 0<\lambda<1, we are able to find four boundary curves x_1(h),%DIFDELCMD < \breve{x}%%% (h), x_2(h) and x_3(h) for the wealth level x that are nonlinear functions of h such that the feedback optimal consumption satisfies: (i) c^*(x,h)=0 when x\leq x_1(h); (ii) 0<c^*(x,h)<\lambda h when x_1(h)<x<%DIFDELCMD < \breve{x}%%% (h); (iii) \lambda h\leq c^*(x,h)<h when%DIFDELCMD < \breve{x}%%% (h)\leq x<x_2(h); (iv) c^*(x,h)=h but the running maximum process remains flat when x_2(h)\leq x<x_3(h); (v) c^*(x,h)=h and the running maximum process increases when x=x_3(h). Similar conclusions can be made in a simpler fashion for two extreme cases \lambda=0 and \lambda=1. Numerical examples are also presented to illustrate some theoretical results and financial insights .
This paper studies the infinite horizon optimal consumption with a path-dependent reference under the exponential utility . The performance is measured by the difference between the non-negative consumption rate and a fraction of the historical consumption maximum. The consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional . The Hamilton-Jacobi-Bellman (HJB) equation can be heuristically expressed in a piecewise manner across different regions to take into account all constraints. By employing the dual transform and smooth-fit principle, some thresholds of the wealth variable are derived such that the classical solution to the HJB equation and the feedback optimal investment and consumption %DIFDELCMD < \breve{x}%%% %DIFDELCMD < \breve{x}%%% %DIFDELCMD < \breve{x}%%% strategies can be obtained in the closed form in each region. The complete proof of the verification theorem is provided and numerical examples are presented to illustrate some financial implications .
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