Optimal consumption and portfolio control for jump-diffusion stock process with log-normal jumps

Dynamic portfolio choice under ambiguity and regime switching mean returns

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.

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.

Insider Information, Arbitrage and Optimal Portfolio and Consumption Policies

Optimal consumption and portfolio policies when asset prices follow a diffusion process

This thesis consists of three essays in financial economics, more precisely in the field of asset pricing and optimal portfolio choice. The first essay studies the asset pricing implications of heterogeneity in preferences coupled with long run risk in a dynamic general equilibrium pure exchange economy. It is concerned with the understanding of how risk sharing among different types of agents affects the assets returns moments in equilibrium and moreover, how these agents reacts dynamically to the long run risk that they perceive. The second and third essay approach the question of optimal portfolio with stochastic opportunity sets. The second essay proposes an approximation methodology for deriving approximate closed form solutions of dynamic optimal portfolio allocation problems. The third essay examines the optimal consumption, portfolio rule and life insurance purchase problem of a wage earner having uncertain lifetime when the market price of risk follows a mean-reverting stochastic process.

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.

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

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 consider a class of optimal consumption and portfolio problem that a corporation faces in which it must pay some debt liability continuously at a given rate due to a previous debt financing, such as bond liability or loan amortization. We model the dynamics of the corporate assets as a diffusion process with controlled drift. In this problem, what the corporation mostly concerns about is how to attain the previously fixed objective assets level in the shortest time, i.e. the objective of the management is to choose the fraction of the total assets invested in the risky asset to minimize the expected time that the corporation first get the previously fixed objective assets. With Bellman dynamic programming principle we find the corresponding optimal policy and the corresponding optimal expected time. We also give some numerical cases for the problem.

Consumption and investment with interest rate risk

In this paper, a financial market with Markovian switching and inflation is described, and the problem of maximizing expected utility of consumption discounted by inflation is given. Then, by a generalized Itô formula for Markov-modulated processes, the verification theorems of optimal policies are derived. Finally, the optimal consumption and portfolio policies for the constant relative risk aversion utility are given explicitly, and an economic analysis is made by numerical examples.

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.

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

Healthcare and Consumption with Aging