Investment-consumption problem ($V=\sup \mathbb{E}{\int }_0^{\infty }{e}^{-\rho t}U(c_t)dt$)

An investment-consumption problem chooses portfolio holdings and consumption to maximize discounted expected utility maximization subject to a self-financing portfolio wealth equation and admissibility constraints. A Hamilton-Jacobi-Bellman equation or utility duality with martingale deflators characterizes the optimum when the value is finite.

For constant relative risk aversion utility with parameter $\gamma >0$, terminal weight $\eta >0$ and utility discount rate $\delta$, in a Black-Scholes model put $\lambda =(\mu -r)/\sigma$, $k=[\delta +(\gamma -1)(r+{\lambda }^2/(2\gamma ))]/\gamma$ and $b(\tau )={\int }_0^{\tau }{e}^{-ks}ds+{\eta }^{1/\gamma }{e}^{-k\tau }$. The displayed optimal consumption and dollar stock position follow by inverse marginal utility and the state-price budget constraint. For $\gamma \ne 1$ the value is $b(T-t{)}^{\gamma }{W}_t^{1-\gamma }/(1-\gamma )$ with discount measured from $t$. Finite horizons do not require $k>0$. The concave supporting-tangent inequality and replication of the optimal budget prove global optimality, rather than only a formal differential-equation solution.

For terminal logarithmic utility $a\log V_T$ and running logarithmic utility $b\log c_t$, with $a,b,w_0>0$, constant Black-Scholes model coefficients give $\lambda =(a+bT)/w_0$ and $V_t=[a+b(T-t)]/(\lambda {\xi }_t)$, where $\xi$ is the state-price density. The optimal terminal wealth and consumption are $a/(\lambda {\xi }_T)$ and $b/(\lambda {\xi }_t)$. Their discounted budget adds to $(a+bT)/\lambda =w_0$. The concave supporting-tangent inequality, integrated over time, proves optimality; the Itô formula gives the displayed dollar stock holding. The time-dependent consumption-to-wealth ratio reflects both remaining consumption opportunities and the terminal utility weight.

A fixed loan principal contributes a constant interest outflow $h$. If the objective stops when available portfolio wealth reaches zero, the value function has an absorbing boundary $V(0)=0$. This differs from requiring the portfolio to finance the debt service forever.

For the wealth-variable Legendre dual $J(z)={\sup }_{w\ge 0}[V(w)-zw]$, a finite slope $z_{\ast }=V^{\prime }(0+)$ at an absorbing zero-wealth boundary makes $J$ identically zero for $z\ge z_{\ast }$. Matching gives $J(z_{\ast })=J^{\prime }(z_{\ast })=0$. It does not generally give $J^{\prime \prime }(z_{\ast }^{-})=0$: killing at ruin permits the dual curvature to jump.

When the market price of risk is constant and nonzero spot volatility gives access to the driving Brownian motion, changes in volatility only rescale the dollar holding needed for fixed Brownian portfolio exposures. With unrestricted holdings and constant interest and discount rates, the CRRA utility value is consequently independent of the factor state. Zero volatility requires separate treatment.

A consumption satisfaction stock smooths past consumption by ${\xi }_t=e^{-\lambda t}{\xi }_0+{\int }_0^t{e}^{-\lambda (t-s)}{c}_{s}ds$. Unlike multiplicative habit utility, the running utility function may depend on this stock alone. A unit of present consumption raises future consumption satisfaction while costing one unit of financial portfolio wealth.

In singular stochastic control, an unrestricted consumption rate can approximate instantaneous transfers from portfolio wealth into a consumption satisfaction stock. In the relaxed problem a nondecreasing control $C$ records these transfers. A transfer $\Delta C$ decreases portfolio wealth and increases consumption satisfaction by the same amount. Rate controls need not attain the supremum when the relaxed optimum has a jump.

If consumption contributes $c(V_{\xi }-V_w)$ to the Hamilton-Jacobi-Bellman equation and has no finite upper bound, finite value requires $V_{\xi }\le V_w$. Strict inequality makes zero consumption optimal locally. Equality describes an active transfer boundary, often reached through singular consumption control.

For CRRA utility of a consumption satisfaction stock, jointly scaling initial portfolio wealth, consumption satisfaction and controls gives homogeneity of degree $1-R$. The value function therefore depends on one dimensionless ratio after factoring out ${\xi }^{1-R}$. The effective consumption coefficient becomes ${\xi }^{-R}[(1-R)v-(x+1)v^{\prime }]$.

A discounted-value transversality condition rules out residual value at infinity in an infinite-horizon Hamilton-Jacobi-Bellman equation verification. Under appropriate integrability, admissibility, and localization assumptions, $\mathbb{E}[e^{-\rho T}V(w_T,s_T)]\to 0$ identifies the economic value among formal differential-equation solutions.

When asset correlation varies with a traded market index, the index is both an asset and a state variable. Writing $m=\log M$ and optimizing Brownian portfolio exposures accounts for the cross derivative between wealth and $m$. Power homogeneity reduces the Hamilton-Jacobi-Bellman equation to an ordinary differential equation in $m$.

For a complete-market investment-consumption problem with constant relative risk aversion utility, the nonlinear wealth-homogeneity coefficient equation may contain $f^{\prime 2}/f$. Writing $f=g^R$ cancels this gradient square against the one from $f^{\prime \prime }$. In the index-driven correlation model the result is ${\sigma }_0^2{g}^{\prime \prime }/2+B{g}^{\prime }-\delta (m)g+1=0$, a linear differential equation; the positive economic solution gives consumption $w/g$.

Intertemporal hedging demand adjusts the myopic risky position when investment opportunities depend on a stochastic state. Shared noise between wealth and that state contributes a cross derivative to the Hamilton-Jacobi-Bellman equation. In a complete diffusion market the optimal exposures combine the myopic market price of risk term and the value-gradient hedge term.

With the binary Brownian drift filter, a stock of observed excess drift $b(x)=\mu -r+\sigma a\tanh (ax)$ is driven by the same Brownian motion as its information state $X$. The Hamilton-Jacobi-Bellman equation for portfolio wealth $w$, dollar stock holding $\theta$ and value $J(w,x)$ includes $\sigma \theta J_{wx}$. Consequently ${\theta }^{\ast }=-[b(x)J_w+\sigma J_{wx}]/({\sigma }^2{J}_{ww})$. For CRRA utility, homogeneity gives $J=K(x)w^{1-R}/(1-R)$ and hence the displayed stock fraction. The second term is intertemporal hedging demand; replacing the unknown drift by its posterior mean while omitting this term does not generally give the optimal policy.

A high-water mark investment tax charges increments of historical maximum wealth ${\bar{w}}_t=\max ({\bar{w}}_0,{\sup }_{s\le t}{w}_s)$ rather than every positive instantaneous return. For the convention charging $\tau d\bar{w}$, the high-water mark tax boundary condition is $V_{\bar{w}}=\tau V_w$ when the optimal policy raises the maximum. At sufficiently high tax, a wealth-cap investment boundary can make further maximum increases suboptimal.

A wealth-cap investment boundary prevents wealth from exceeding a fixed historical maximum. In the wealth-variable Legendre dual for wealth normalized by that maximum, $J^{\prime }(z_{\ast })=-1$ and vanishing optimal portfolio volatility gives $J^{\prime \prime }(z_{\ast })=0$. For power utility and a high-water tax, the smooth tax-paying solution has boundary dual curvature $(1-\alpha \tau )/(R{z}_{\ast })$, so it cannot be admissible when $\alpha \tau >1$.

At an active maximum-raising boundary in high-water mark investment taxation, the finite-variation part of the Itô formula is $(V_{\bar{w}}-\tau V_w)d\bar{w}$, so smooth optimality sets this coefficient to zero. If the maximizing policy avoids increasing the maximum, the corresponding inequality $V_{\bar{w}}-\tau V_w\le 0$ is compatible with a wealth-cap investment boundary instead.

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.