A concise characterization of optimal consumption with logarithmic preferences (Q2862512)

In this paper the authors investigate the maximization problem NEWLINE\[NEWLINE\sup_{\pi\in \mathcal{B},\,c\in \mathcal{C}}\inf_{\mathbb{P}\in \mathcal{P}}{\mathbb E}^{\mathbb{P}}\left[\int_0^Te^{-\delta t}u(c_tX_t)dt+\varepsilon u(X_T)\right],\eqno(1)NEWLINE\]NEWLINE -- which represents the optimal consumption-portfolio decision of an investor with logarithmic preferences -- with time horizon \(T\in (0,\infty]\) and rate of time preference \(\delta>0\). Here the utility function \(u\) is logarithmic, \(u(x)=\ln(x),\;x>0,\;c=\{c_t\}_{t\in[0,T)}\) is the consumption-wealth ratio and \(\pi=\{\pi_t\}_{t\in[0,T)}\) specifies the investor's asset allocation, \(X=X^{\pi,c,x}\) denotes the investor's wealth process with possibly nonlinear dynamics given by NEWLINE\[NEWLINEdX_t=X_{t_{-}}[dZ_t^{\pi}-c_tdt],\quad X_0=x.NEWLINE\]NEWLINE The process \(Z^{\pi}\) is a \(\mathbb{P}-\)semimartingale on the filtered probability space \((\Omega,\mathcal{F},\mathcal{F}_t,\mathcal{P})\) for all \(\mathbb{P}\in \mathcal{P}\) and represents the financial market return given the investment decision \(\pi.\) Moreover, it is supposed that \(\mathcal{B}\) is a given class of processes representing admissible portfolio policies \(\pi\) such that \(\Delta Z_t^{\pi}>-1,\) \(t\in [0,T),\) and denote by \(\mathcal{C}=\{c\}\) the class of admissible consumption plans.NEWLINENEWLINEThe authors say that they extend the analysis of numerous other papers that have studied problems like (1) in more specific settings. They prove two theorems, 1) that the investor's optimal consumption and portfolio decisions are independent of one another, and 2) that the optimal consumption-wealth ratio is given by a deterministic function that only depends on the investor's rate of time preference and they derive an explicit formula for it.