Healthcare and Consumption with Aging

Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints

This paper examines an optimal investment–consumption problem in a setting where the financial environment is influenced by both stochastic factors and delayed effects. The investor, endowed with Constant Relative Risk Aversion (CRRA) preferences, allocates wealth between a risk-free asset and a single risky asset. The short rate follows a Vasiˇček-type term structure model, while the risky asset price dynamics are driven by a delayed Heston specification whose variance process evolves according to a Cox–Ingersoll–Ross (CIR) diffusion. Delayed dependence in the wealth dynamics is incorporated through two auxiliary variables that summarize past wealth trajectories, enabling us to recast the naturally infinite-dimensional delay problem into a finite-dimensional Markovian framework. Using Bellman’s dynamic programming principle, we derive the associated Hamilton–Jacobi–Bellman (HJB) partial differential equation and demonstrate that it generalizes the classical Merton formulation to simultaneously accommodate delay, stochastic interest rates, stochastic volatility, and consumption. Under CRRA utility, we obtain closed-form expressions for the value function and the optimal feedback controls. Numerical illustrations highlight how delay and market parameters impact optimal portfolio allocation and consumption policies.

We study a finite-horizon optimal consumption and investment problem in a complete continuous-time market where consumption is restricted within fixed upper and lower bounds. Assuming constant relative risk aversion (CRRA) preferences, we employ the dual-martingale approach to reformulate the problem and derive closed-form integral representations for the dual value function and its derivatives. These results yield explicit feedback formulas for the optimal consumption, portfolio allocation, and wealth processes. We establish the duality theorem linking the primal and dual value functions and verify the regularity and convexity properties of the dual solution. Our results show that the upper and lower consumption bounds transform the linear Merton rule into a piecewise policy: consumption equals L when wealth is low, follows the unconstrained Merton ratio in the interior region, and is capped at H when wealth is high.

Asset Selection with Margin Constraints and Income

OST models of portfolio selection have M been one-period models. I examine the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model whzere his income is generated by returns on assets and these returns or instantaneous growth rates are stochastic. P. A. Samuelson has developed a similar model in discrete-time for more general probability distributions in a companion paper [8]. I derive the optimality equations for a multiasset problem when the rate of returns are generated by a Wiener Brownian-motion process. A particular case examined in detail is the two-asset model with constant relative riskaversion or iso-elastic marginal utility. An explicit solution is also found for the case of constant absolute risk-aversion. The general technique employed can be used to examine a wide class of intertemporal economic problems under uncertainty. In addition to the Samuelson paper [8], there is the multi-period analysis of Tobin [9]. Phelps [6] has a model used to determine the optimal consumption rule for a multi-period example where income is partly generated by an asset with an uncertain return. Mirrless [5] has developed a continuous-time optimal consumption model of the neoclassical type with technical progress a random variable.

We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results. doi:10.1017/S1446181119000014

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic volatility and unknown stock appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein-Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor $Y$ in an interval $[0,T_0]$ for fixed $T_0>0$, and use sequential estimation. We show, that the consumption and investment strategy calculated through this sequential procedure is $\delta$-optimal.

We consider an optimal investment and consumption problem for a Black–Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamic programming approach leads to an investigation of the Hamilton–Jacobi–Bellman (HJB) equation which is a highly nonlinear partial differential equation (PDE) of the second order. By using the Feynman–Kac representation, we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of iterative numerical schemes for both the value function and the optimal portfolio. We show that in this case, the optimal convergence rate is super-geometric, i.e., more rapid than any geometric one. We apply our results to a stochastic volatility financial market.

We illustrate the classical Markowitz portfolio optimization theory in detail, and we give all essential proofs. Especially we show how to explicitly calculate the efficient border and the market portfolio in the case where short selling is possible without restrictions. In the case without short selling, we will work with Monte Carlo. We also introduce the single-index model as an alternative to estimate parameters needed for portfolio optimization. Then we formulate the optimal investment and consumption problem. We give a heuristic and intuitive introduction to the basic techniques of stochastic optimal control, especially to the Hamilton-Jacobi-Bellman equations and to their application. With the help of these concepts, we solve the optimal investment and consumption problem and discuss the results.

The optimal investment consumption problem for a single riskless bond, a zero-coupon bond and a risky stock modeled by the Vasicek interest process has been established. The investment objective is maximizing the utility of his consumption and terminal wealth. By the stochastic dynamic programming principle, the HJB equation for the optimal solution is given. In the case of constant relative risk aversion utility, the analytic optimal trading strategies are derived. The results show that the optimal proportion allocated in the stock is a constant fraction, but the optimal proportion in the zero-coupon bond is time-variant. The optimal consumption rate is in a feedback form of the wealth and depends on the stochastic interest rate. A numerical example illustrating the results is presented.

We study the Bellman equation associated with the optimal consumption and portfolio choice problem with stopping times in a complete market. We establish the existence of a strong solution by using the viscosity solutions technique. The optimal policy is shown to exist from the optimality conditions in the variational inequality.

Optimal Consumption and Investment in the Economy with Infinite Number of Consumption Goods

Optimal investment, consumption and life insurance purchase with learning about return predictability

ABSTRACTAn investment and consumption problem is formulated and its optimal strategy is investigated. We assume the basic binary model, but with unknown parameters. We apply the parametric Bayesian approach to formulate the problem as a sequential stochastic optimization model and use the technique of dynamic programming to characterize the optimal strategy. It is discovered that despite unknown parameters, when the power and logarithmic utility functions are treated, the optimal value function is of the same form of the utility function. The random finite horizon model is formulated as an infinite horizon model. Our results are similar to the ones in the literature having different return functions with constant relative risk aversion.

We revisit the optimal investment and consumption model of Davis and Norman [Math. Oper. Res., 15 (1990), pp. 676--713] and Shreve and Soner [Ann. Appl. Probab., 4 (1994), pp. 609--692], following a shadow-price approach similar to that of Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20 (2010), pp. 1341--1358]. Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton--Jacobi--Bellman equation for this singular stochastic control problem to a nonstandard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.