Consumption-investment Problem Research Articles

Portfolio Problem for the α−Hypergeometric Stochastic Volatility Model with Consumption

This paper solves the finite horizon optimal consumption-investment problem for an investor that trades in the α−hypergeometric stochastic volatility model, with preferences based on a power utility function. To find the optimal strategy, we follow the dynamic programming approach. First, we perform a suitable change of variables in order to fully transform the non-linear Hamilton-Jacobi-Bellman (HJB) equation into a semilinear one. Next, we apply the Girsanov theorem to deduce an implicit Feynman-Kac formula depending upon the volatility process. The operator defined by the Feynman-Kac representation is shown to be a contraction operator on a designed function space. Finally, the Banach fixed point theorem yields the existence of a solution to the HJB equation in the designed space. Moreover, we apply a verification theorem to guarantee that the solution of the HJB equation coincides with the value function.

Dynamic optimal hedging with futures in portfolio context

In this paper, we investigate the optimal hedging strategy in a continuous time framework that is more adequate for commodities. We consider the consumption-investment problem where all asset prices follow mean-reverting jump-diffusion processes. The optimal investment and consumption strategies are derived in closed form. The framework is used to address one of the major risk factors faced by commodity producers. We show that a commodity producer will be better off hedging his/her futures contracts by simultaneously investing in foreign exchange products to minimize the adverse impacts of the jump risk prevalent in commodity prices.

Solving constrained consumption–investment problems by decomposition algorithms

Consumption–investment problems with maximizing utility agents are usually considered from a theoretical viewpoint, aiming at closed-form solutions for the optimal policy. However, such an approach requires that the model be relatively simple: even the inclusion of nonnegativity constraints can prevent the derivation of explicit solutions. In such cases, it is necessary to solve the problem numerically, but standard dynamic programming algorithms can only solve small problems due to the curse of dimensionality. In this paper, we adapt the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve dynamic constrained consumption–investment problems with stochastic labor income numerically. Unlike classical dynamic programming approaches, SDDP allows us to analyze problems with multiple assets, and an internal sampling procedure allows the problems to have a very large, or even infinite, number of scenarios. We start with a simpler problem for which a closed-form solution is known and compare it to the optimal policy obtained by SDDP. We then illustrate the flexibility of our approach by solving a defined contribution pension fund problem with multiple assets, for which no closed-form solution is available.

Duality in optimal consumption–investment problems with alternative data

This study investigates an optimal consumption–investment problem in which the unobserved stock trend is modulated by a hidden Markov chain that represents different economic regimes. In the classic approach, the hidden state is estimated using historical asset prices, but recent technological advances now enable investors to consider alternative data in their decision-making. These data, such as social media commentary, expert opinions, COVID-19 pandemic data and GPS data, come from sources other than standard market data sources but are useful for predicting stock trends. We develop a novel duality theory for this problem and consider a jump-diffusion process for alternative data series. This theory helps investors identify “useful” alternative data for dynamic decision-making by providing conditions for the filter equation that enable the use of a control approach based on the dynamic programming principle. We apply our theory to provide a unique smooth solution for an agent with constant relative risk aversion once the distributions of the signals generated from alternative data satisfy a bounded likelihood ratio condition. In doing so, we obtain an explicit consumption–investment strategy that takes advantage of different types of alternative data that have not been addressed in the literature.

Optimal Strategy for Corporate International Investment and Consumption Problem with Stochastic Hyperbolic Discounting

This study investigates the dynamics of a stochastic hyperbolic discounting model in a continuous-time framework to address the complexities associated with the corporate international investment consumption problem (CIICP). By formulating a dynamic programming equation based on the principles of dynamic programming, we seek to untangle the intricacies of CIICP by incorporating log utility. Our research provides valuable insights into the economic ramifications of the proposed model and presents a comprehensive analysis of numerical sensitivities. This investigation not only contributes to a deeper understanding of the CIICP phenomenon but also contributes to more informed decision-making within the field.

Robust retirement and life insurance with inflation risk and model ambiguity

We study a robust consumption-investment problem with retirement and life insurance decisions for an agent who is concerned about inflation risk and model ambiguity. Assuming that an inflation-linked index bond and a stock are available in the market, this paper considers a comprehensive setup of ambiguity in the return, volatility, and correlation parameters in the joint dynamics of their market prices. With a finite planning horizon, the agent has a general utility function with different marginal utilities of consumption before and after retirement. Combining the classical dual approach and the G-stopping time theory, we derive the novel robust strategies using integral equation representations. We numerically and extensively investigate the effects of ambiguity from different sources on the robust decisions. While model ambiguity generally leads the ambiguity- and risk-averse agent to decrease the consumption rate, life insurance purchase, and investment demands, it also generates contrasting effects on robust retirement time and wealth level. Specifically, model ambiguity lowers the target wealth level to immediate retirement of a young agent but increases the retirement time of an older agent compared to the case of known parameters. A rich agent takes ambiguity more seriously than a poor agent in the sense of adjusting the strategies on a more significant scale. Our simulation and comparison study demonstrate the significance of addressing the ambiguity in volatility and correlation in addition to the ambiguity in return.

Global regularity and blowup for a class of non-Newtonian polytropic variation-inequality problem from investment-consumption problems

<abstract><p>This paper studies variation-inequality problems with fourth order non-Newtonian polytropic operators. First, the test function of the weak solution is constructed by using the difference operator. Then global regularity of the weak solution is obtained by some difference transformation and inequality amplification techniques. The weak solution is transformed into a differential inequality of the energy function. It is proved that the weak solution will blow up in finite time. Then, the upper bound and the blowup rate estimate of the blow up are given by handling some differential inequalities.</p></abstract>

The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. II: Existence, uniqueness and verification for vartheta in (0,1)

In this article, we consider the optimal investment–consumption problem for an agent with preferences governed by Epstein–Zin (EZ) stochastic differential utility (SDU) over an infinite horizon. In a companion paper Herdegen et al. (Finance Stoch. 27:127–158, 2023), we argued that it is best to work with an aggregator in discounted form and that the coefficients R of relative risk aversion and S of elasticity of intertemporal complementarity (the reciprocal of the coefficient of elasticity of intertemporal substitution) must lie on the same side of unity for the problem to be well founded. This can be equivalently expressed as vartheta := frac{1-R}{1-S} >0.In this paper, we focus on the case vartheta in (0,1). The paper has three main contributions: first, to prove existence of infinite-horizon EZ SDU for a wide class of consumption streams and then (by generalising the definition of SDU) to extend this existence result to any consumption stream; second, to prove uniqueness of infinite-horizon EZ SDU for all consumption streams; and third, to verify the optimality of an explicit candidate solution to the investment–consumption problem in the setting of a Black–Scholes–Merton financial market.

The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. I: Foundations

The goal of this article is to provide a detailed introduction to infinite-horizon investment–consumption problems for agents with preferences described by Epstein–Zin (EZ) stochastic differential utility (SDU). In the setting of a Black–Scholes–Merton market, we seek to describe all parameter combinations that lead to a well-founded problem in the sense that the problem is not just mathematically well posed, but the solution is also economically meaningful. The key idea is to consider a novel and slightly different description of EZ SDU under which the aggregator has only one sign. This new formulation clearly highlights the necessity for the coefficients of relative risk aversion and of elasticity of intertemporal complementarity (the reciprocal of the coefficient of intertemporal substitution) to lie on the same side of unity.

A consumption-investment model with state-dependent lower bound constraint on consumption

This paper studies a life-time consumption-investment problem under the Black-Scholes framework, where the consumption rate is subject to a lower bound constraint that linearly depends on the investor's wealth. It is a stochastic control problem with state-dependent control constraint to which the standard stochastic control theory cannot be directly applied. We overcome this by transforming it into an equivalent stochastic control problem in which the control constraint is state-independent so that the standard theory can be applied. We give an explicit optimal consumption-investment strategy when the constraint is homogeneous. When the constraint is non-homogeneous, it is shown that the value function is third-order continuously differentiable by a differential equation approach, and a feedback form optimal consumption-investment strategy is provided. According to our findings, if the investor is concerned with long-term more than short-term consumption, then she should always consume as few as possible; otherwise, she should consume optimally when her wealth is above a threshold, and consume as few as possible when her wealth is below the threshold.

Quasi-hyperbolic discounting under recursive utility and consumption–investment decisions

This paper examines an Epstein–Zin recursive utility with quasi-hyperbolic discounting in continuous time. I directly define the utility process supporting the Hamilton–Jacobi–Bellman (HJB) equation in the literature and consider Merton's optimal consumption–investment problem for application. I show that a solution to the HJB equation is the value function. The numerical and mathematical analyses show that unlike in the constant relative risk aversion utility, present bias in the Epstein–Zin utility causes economically significant overconsumption, maintaining a plausible attitude toward risks. Additionally, the sophisticated agent's preproperation occurs if and only if the elasticity of intertemporal substitution is larger than one.

A Risk Extended Version of Merton’s Optimal Consumption and Portfolio Selection

A risk management version of the classical investment-consumption problem known as Merton's problem in the finance literature is proposed. Risk is measured by variance, which introduces a nonlinear function of the expected value into the control problem. Standard stochastic theory cannot properly handle this type of nonlinear stochastic optimization problem. Therefore, we study this time-inconsistent problem within the mean field-type control framework. We derive the sufficient condition of optimality and solve the problem completely. Numerical results illustrating the effect of risk on optimal policies are also presented. Applications can be numerous, including all kinds of investment decisions or operations decisions.

A Q-learning Approach to a Consumption-Investment Problem

The paper deals with a discrete-time consumption investment problem with an infinite horizon. This problem is formulated as a Markov decision process with an expected total discounted utility as an objective function. This paper aims to presents a procedure to approximate the solution via machine learning, specifically, a Q-learning technique. The numerical results of the problem are provided.

A Q-learning Approach to a Consumption-Investment Problem

The paper deals with a discrete-time consumption investment problem with an infinite horizon. This problem is formulated as a Markov decision process with an expected total discounted utility as an objective function. This paper aims to presents a procedure to approximate the solution via machine learning, specifically, a Q-learning technique. The numerical results of the problem are provided.

Time-Inconsistent Consumption-Investment Problems in Incomplete Markets under General Discount Functions

In this paper, we study a time-inconsistent consumption-investment problem with random endowments in a possibly incomplete market under general discount functions. We provide a necessary condition and a verification theorem for an open-loop equilibrium consumption-investment pair in terms of a coupled forward-backward stochastic differential equation. Moreover, we prove the uniqueness of the open-loop equilibrium pair by showing that the original time-inconsistent problem is equivalent to an associated time-consistent one.

A Survey of the Optimal Consumption-Investment Problem with Transaction Costs

A Survey of the Optimal Consumption-Investment Problem with Transaction Costs

Proper solutions for Epstein–Zin Stochastic Differential Utility

In this article, we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein--Zin stochastic differential utility (EZ-SDU) who invests in a constant-parameter Black-Scholes-Merton market over the infinite horizon. The parameter combinations that we consider in this paper are such that the risk aversion parameter $R$ and the elasticity of intertemporal complementarity $S$ satisfy $\theta=\frac{1-R}{1-S}>1$. In this sense, this paper is complementary to Herdegen, Hobson and Jerome [arXiv:2107.06593]. The main novelty of the case $\theta>1$ (as opposed to $\theta\in(0,1)$) is that there is an infinite family of utility processes associated to every nonzero consumption stream. To deal with this issue, we introduce the economically motivated notion of a proper utility process, where, roughly speaking, a utility process is proper if it is nonzero whenever future consumption is nonzero. We then proceed to show that for a very wide class of consumption streams $C$, there exists a proper utility process $V$ associated to $C$. Furthermore, for a wide class of consumption streams $C$, the proper utility process $V$ is unique. Finally, we solve the optimal investment-consumption problem in a constant parameter financial market, where we optimise over the right-continuous attainable consumption streams that have a unique proper utility process associated to them.

Dynamic programming for semi-Markov modulated SDEs

ABSTRACT We consider a stochastic optimal control problem with state variable dynamics described by a stochastic differential equation of diffusive type modulated by a semi-Markov process with a finite state space. The time horizon is both deterministic and finite. Within such setup, we provide a detailed proof of the dynamic programming principle and use it to characterize the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We illustrate our results with an application to Mathematical Finance: the generalization of Merton's optimal consumption-investment problem to financial markets with semi-Markov switching.

Quasi-Hyperbolic Discounting under Recursive Utility and Consumption-Investment Decisions

This paper examines an Epstein-Zin recursive utility with quasi-hyperbolic discounting in continuous time. I directly define the utility process and consider a Merton's optimal consumption-investment problem for application. I show that a solution to the Hamilton-Jacobi-Bellman equation is the value function. The numerical comparative statics and mathematical analysis shows that, unlike in the constant relative risk aversion utility, present bias in the Epstein-Zin utility causes economically significant overconsumption, maintaining a plausible attitude toward risks. Additionally, I show that the sophisticated agent's preproperation occurs if and only if his or her elasticity of intertemporal substitution in consumption is larger than one.

Optimal investment–consumption problem: Post-retirement with minimum guarantee

We study the optimal investment–consumption problem for a member of defined contribution plan during the decumulation phase. For a fixed annuitization time, to achieve higher final annuity, we consider a variable consumption rate. Moreover, to have a minimum guarantee for the final annuity, a safety level for the wealth process is considered. To solve the stochastic optimal control problem via dynamic programming, we obtain a Hamilton–Jacobi–Bellman (HJB) equation on a bounded domain. The existence and uniqueness of classical solutions are proved through the dual transformation. We apply the finite difference method to find numerical approximations of the solution of the HJB equation. Finally, the simulation results for the optimal investment–consumption strategies, optimal wealth process and the final annuity for different admissible ranges of consumption are given. Furthermore, by taking into account the market present value of the cash flows before and after the annuitization, we compare the outcomes of different scenarios.