Merton consumption-investment problem ($c^{\ast }=\delta w,{\theta }^{\ast }=(\mu -r)w/(R{\sigma }^2)$)

For a constant-coefficient single-asset investment-consumption problem with constant relative risk aversion utility, put $\kappa =(\mu -r)/\sigma$ and $\delta =[\rho -(1-R)(r+{\kappa }^2/(2R))]/R$. When $\delta >0$, the infinite-horizon value is ${\delta }^{-R}{w}^{1-R}/(1-R)$ and the optimal controls are $c^{\ast }=\delta w$ and ${\theta }^{\ast }=(\mu -r)w/(R{\sigma }^2)$. The case $R=1$ uses logarithmic utility.

For an observed finite-state continuous-time Markov chain with rate matrix $Q$, constant bank rate $r$, and state-dependent stock drifts ${\mu }_i$ and positive volatilities ${\sigma }_i$, constant relative risk aversion utility gives $V_i(w)=A_i{w}^{1-R}/(1-R)$. Set ${\kappa }_i=({\mu }_i-r)/{\sigma }_i$ and $d_i=\rho -(1-R)(r+{\kappa }_i^2/(2R))$. The Hamilton-Jacobi-Bellman equation reduces to the displayed nonlinear algebraic system with $A_i>0$. Its feedback controls are $c_i=A_i^{-1/R}w$ and ${\theta }_i=({\mu }_i-r)w/(R{\sigma }_i^2)$.

A damped Newton method in the logarithmic variables $A_i=e^{x_i}$ preserves positivity. Solving these equations produces a candidate value; admissibility, stochastic-integral bounds and the investment value transversality condition still need verification.

Consider irreversible optimal stopping of employment paying income $\epsilon >0$ and imposing disutility $\lambda >0$. Assume $r,\rho >0$, nonzero market price of risk $\kappa$, a positive Merton consumption constant, and solvency allowing borrowing against perpetual income up to $\epsilon /r$ before retirement. Let $m<0$ solve ${\kappa }^{2}m(m-1)/2+(\rho -r)m-\rho =0$. The retirement threshold is the displayed wealth level.

To derive it, apply the wealth-variable Legendre dual. The working dual equals the retired dual plus $h(z)=\epsilon z/r-\lambda /\rho +D{z}^m$. Excluding the growing homogeneous solution enforces the solvency asymptotic. Value matching and smooth fit at the retirement marginal value $z_{\ast }$ give $h(z_{\ast })=h^{\prime }(z_{\ast })=0$, hence $z_{\ast }=(\lambda r/(\epsilon \rho ))m/(m-1)$ and $D=-\epsilon z_{\ast }^{1-m}/(rm)>0$. The retired inverse marginal value gives $\overline{w}={\gamma }_M^{-1}{z}_{\ast }^{-1/R}$.

The root equation implies $z_{\ast }/(\lambda /\epsilon )=1+{\kappa }^{2}m/(2\rho )<1$. Thus the boundary exceeds the instantaneous gain-zero level ${\gamma }_M^{-1}(\epsilon /\lambda {)}^{1/R}$. A tighter borrowing constraint changes this free-boundary problem. Smooth fit matches the first derivative; matching the second derivative would incorrectly remove the income option.

In the infinite-horizon Merton consumption-investment problem with constant relative risk aversion utility, this is the optimal ratio of consumption to wealth. A finite value requires ${\gamma }_M>0$. The state-price budget constraint then fixes consumption's multiplier because $\mathbb{E}[{\xi }_t(y{e}^{\rho t}{\xi }_t{)}^{-1/R}]=y^{-1/R}{e}^{-{\gamma }_{M}t}$, whose time integral is $y^{-1/R}/{\gamma }_M$. When this integral diverges, the value is $+\infty$ for $0<R<1$ and $-\infty$ for $R>1$.

For $0<R<1$, common constant interest and volatility, and observed excess drift bounded in absolute value by $b_{+}>0$, let $V_{+}$ be the finite Merton consumption-investment problem value for excess drift $b_{+}$. In its Hamilton-Jacobi-Bellman equation, optimizing the stock holding contributes $-b^2(V_{+}^{\prime }{)}^2/(2{\sigma }^2{V}_{+}^{\prime \prime })$. Since $V_{+}^{\prime \prime }<0$, this contribution increases with $b^2$. Thus every admissible policy makes accumulated discounted CRRA utility plus discounted $V_{+}$ a nonnegative local supermartingale, hence a supermartingale. Dropping its nonnegative terminal term and using the monotone convergence theorem proves the value bound. The condition $[\rho -(1-R)(r+b_{+}^2/(2R{\sigma }^2))]/R>0$ is essential for the finite supermartingale construction.

Consider an investment-consumption problem with an independent one-way rate change of intensity $\lambda$ from $r_1$ to $r_0$. For a stock of constant drift $\mu$ and positive volatility $\sigma$, set $B_j=r_j+(\mu -r_j{)}^2/(2R{\sigma }^2)$, ${\gamma }_0=[\rho -(1-R)B_0]/R$, and $D_1=\rho +\lambda -(1-R)B_1$. In the finite-value regime ${\gamma }_0,D_1>0$, the prechange consumption coefficient is the unique positive root displayed above. The left side increases continuously from zero to infinity. Before the change, the value is ${\gamma }_1^{-R}{w}^{1-R}/(1-R)$, the stock fraction is $(\mu -r_1)/(R{\sigma }^2)$, and consumption is ${\gamma }_{1}w$. At the change these become the ordinary Merton consumption-investment problem controls for $r_0$. The jump term in the Hamilton-Jacobi-Bellman equation is $\lambda (V_0-V_1)$; knowing only the marginal law of the change time does not justify that term without a conditional intensity assumption.