Optimal Investment Problem Research Articles

Robust optimal reinsurance and investment strategy for an insurer and a reinsurer with default risks and jumps

In this article, we analyze a robust optimal reinsurance and investment problem in a model with default risks and jumps for a general company which holds shares of an insurer and a reinsurer. The insurer’s surplus is characterized by a diffusion model and the insurer has the opportunity to buy proportional reinsurance from the reinsurer. Moreover, the insurer can invest in a risk-free asset (bank account), a risky asset (stock), and a defaultable bond. Meanwhile, the reinsurer can also invest in a risk-free asset and another risky asset. The price dynamics of the risky asset is assumed to follow a jump-diffusion model. What is more, the general company is concerned about the ambiguity and aims to find a robust optimal reinsurance and investment strategy. The objective of the general company is to maximize the minimal expected exponential utility of the weighted sums of the insurer’s and the reinsurer’s terminal wealth. Applying stochastic dynamic programming approach, we first establish the robust HJB (Hamilton-Jacobi-Bellman) equations for the post-default case and the pre-default case, respectively. Furthermore, we obtain the robust optimal strategy explicitly and present the corresponding verification theorem. Finally, we give some numerical examples to demonstrate the influences of model parameters on the robust optimal strategy.

Optimal Investment for Defined-Contribution Pension Plans with the Return of Premium Clause under Partial Information

The optimal investment problem for defined contribution (DC) pension plans with partial information is the subject of this paper. The purpose of the return of premium clauses is to safeguard the rights of DC pension plan participants who pass away during accumulation. We assume that the market price of risk consists of observable and unobservable factors that follow the Ornstein-Uhlenbeck processes, and the pension fund managers estimate the unobservable component from known information through Bayesian learning. Considering maximizing the expected utility of fund wealth at the terminal time, optimal investment strategies and the corresponding value function are determined using the dynamical programming principle approach and the filtering technique. Additionally, fund managers forsake learning, which results in the presentation of suboptimal strategies and utility losses for comparative analysis. Lastly, numerical analyses are included to demonstrate the impact of model parameters on the optimal strategy.

Optimal wind power capacity decision consider commitment contracts under uncertain power supply and electricity demand in China

China has been introducing reforms in the electricity market. Since 2016, electricity retailers are allowed to operate in the market once monopolized by traditional power companies. This regulatory environment demands new types of purchase contracts to remunerate the energy agents and deliver better prices for consumers fairly and sustainably. This paper proposes two new electricity purchase contracts with risk sharing mechanism and studies on wind power supplier's optimal capacity investment problem under supply and demand uncertainty. Firstly, a basic model for fixed demand contract is developed, and the optimal capacity is obtained. Secondly, the fixed commitment order contract and the forecast order contract with risk sharing mechanism are proposed to manage the fluctuation of wind power supply randomness. With backward induction, the optimal investment capacity of wind power supplier is solved, and then the fixed order quantity or forecast demand quantity of the electricity retailer can be obtained. Our results show that the fixed order quantity may have a positive influence on the wind power supplier's capacity investment decision. While, the forecast order quantity may have a negative influence on the wind power supplier's capacity investment decision. In addition, wind density has a strong influence on wind power capacity investment, and it has a positive impact on wind power supplier's profit, but it does not always have a positive impact on electricity retailer's profit. At last, based on numerical analysis, we examine how the wind density distribution coefficient change affect members' profit in the electricity supply chain. We also find that even with wind curtailment punishment, the forecast order contract can make a Pareto improvement on the whole electricity supply chain.

Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and applications to optimal advertising and optimal investment problems

AbstractIn this manuscript we consider optimal control problems of stochastic differential equations with delays in the state and in the control. First, we prove an equivalent Markovian reformulation on Hilbert spaces of the state equation. Then, using the dynamic programming approach for infinite-dimensional systems, we prove that the value function is the unique viscosity solution of the infinite-dimensional Hamilton-Jacobi-Bellman equation. We apply these results to problems coming from economics: stochastic optimal advertising problems and stochastic optimal investment problems with time-to-build.

Consistent curves in the -world: optimal bonds portfolio

We derive arbitrage-free conditions for a parametric yield curve in the P -world, where market prices of risk are present. As in the Heath–Jarrow–Morton (HJM) theory, we impose the bonds evolution to be free of arbitrage opportunities. However, in the classical HJM theory, first volatilities are specified, and subsequently the drift of the forward curve is obtained up to the market prices of risk that must be specified exogenously. Here, the problem is inverted: we first impose a family of shapes upon the yield curve and subsequently derive market prices of risk in a self-consistent manner and hence the market prices of risk follow a stochastic differential equation obtained directly from the curve dynamics. Leveraging this framework, we formulate a bonds-portfolio optimization as a stochastic control problem with the Hamilton–Jacobi–Bellman (H-J-B) equation. We discover that in this case, the H-J-B equation is essentially different than the classical obtained in optimal investment problems.

Optimal Consumption and Investment with Independent Stochastic Labor Income

We develop a new dynamic continuous-time model of optimal consumption and investment to include independent stochastic labor income. We reduce the problem of solving the Bellman equation to a problem of solving an integral equation. We then explicitly characterize the optimal consumption and investment strategy as a function of income-to-wealth ratio. We provide some analytical comparative statics associated with the value function and optimal strategies. We also develop a quite general numerical algorithm for control iteration and solve the Bellman equation as a sequence of solutions to ordinary differential equations. This numerical algorithm can be readily applied to many other optimal consumption and investment problems especially with extra nondiversifiable Brownian risks, resulting in nonlinear Bellman equations. Finally, our numerical analysis illustrates how the presence of stochastic labor income affects the optimal consumption and investment strategy. Funding: A. Bensoussan was supported by the National Science Foundation under grant [DMS-2204795]. S. Park was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea, South Korea [NRF-2022S1A3A2A02089950].

Optimal payout strategies when Bruno de Finetti meets model uncertainty

Model uncertainty is ubiquitous and plays an important role in insurance and financial modeling. While a substantial effort has been given to studying optimal consumption, portfolio selection and investment problems in the presence of model uncertainty, relatively little attention is given to investigating optimal payout policies taking account of the impacts of model uncertainty. As one of the early attempts, this paper studies the optimal payout control problem under the classical risk model taking into account of model uncertainty about the claims arrival intensity. We aim to provide insights into understanding optimal decisions incorporating model uncertainty and to examine key impact of model uncertainty. We find that the optimal strategy robust to model uncertainty is of a band type. However, the presence of the model uncertainty alters the qualitative behavior of the optimal strategy in the sense that the optimal robust policy is no longer a barrier strategy for some particular cases. We provide numerical examples to illustrate the theoretical results and examine the impact of model uncertainty on optimal payout policies. We also provide examples that use real insurance data for calibration. We discover that the decision maker takes more conservative strategies under model uncertainty, which is consistent with the findings in the economic field and has not been addressed in the existing optimal payout problems without model uncertainty.

Optimal investment problem for a hybrid pension with intergenerational risk-sharing and longevity trend under model uncertainty

This paper studies the optimal investment problem for a hybrid pension plan under model uncertainty, where both the contribution and the benefit are adjusted depending on the performance of the plan. Furthermore, an age and time-dependent force of mortality and a linear maximum age are considered to capture the longevity trend. Suppose that the plan manager is ambiguity averse to the financial market and is allowed to invest in a risk-free asset and a risky asset. The plan manager aims to find optimal investment strategies and optimal intergenerational risk-sharing arrangements by minimizing the unstable contribution risk, the unstable benefit risk and the discontinuity risk under the worst-case scenario. By applying the stochastic optimal control approach, robust optimal strategies are derived under the worst-case scenario for a penalized quadratic cost function. Through numerical analysis and three special cases, we find that the intergeneration risk-sharing is achieved in our collective hybrid pension plan effectively. And it also shows that when people live longer, postponing the retirement seems a reasonable way to alleviate the stress of the aging problem.

Optimal asset allocation for DC pension subject to allocation and terminal wealth constraints under a remuneration scheme

. We investigate the optimal investment problem faced by a defined contribution (DC) pension fund manager under simultaneous allocation and expected shortfall (ES) constraints. Under a non concave utility, a Value-at-Risk (VaR) constraint does not lead to the full prevention of moral hazard. As a widely employed risk management tool, whether an ES constraint can provide a more effective protection than a VaR constraint has been a focus point of research. We apply a dual control approach and a concavification technique to solve the ES-constrained optimization problem for a DC pension plan under an incentive scheme and derive the closed-form representations of the optimal wealth and portfolio processes. Furthermore, we compare the effect of an ES constraint on the optimal investment behavior with that under a VaR constraint in the presence of an option-like scheme for the DC pension members. Theoretical and numerical results show that for a relatively low protection level, a joint VaR and an ES constraints induce the same structure of the optimal solution, which implies that for a non concave optimization problem, the ES-based risk management has lost its advantage over the VaR-based risk management. Therefore, it needs to design a more efficient risk measure to improve the risk management for a DC pension plan.

A Stackelberg–Nash equilibrium with investment and reinsurance in mixed leadership game

In this paper, we investigate the optimal reinsurance and investment problem from joint interests of the insurer and the reinsurer in the framework of the mixed leadership game, where both of them can invest their wealth in the financial market, and the liabilities are correlated with the risky asset. More specifically, we assume that the insurer is the leader to determine the amount he/she invests into the risky asset but acts as the follower to decide the optimal reinsurance retention level, while the reinsurer is the leader to set the reinsurance premium price and acts as a follower to determine his/her investment strategy. Both the insurer and the reinsurer aim to maximize the expected utility of their terminal wealth, and the explicit Stackelberg-Nash equilibrium is derived by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations. In order to better understand the effect of the mixed leadership game on the optimal reinsurance and investment problem, we also consider the optimization problem within the framework of the stochastic Stackelberg differential game shown in [Bai, Y., Zhou, Z., Xiao, H., Gao, R. & Zhong, F. (2021). A Stackelberg reinsurance-investment game with asymmetric information and delay. Optimization 70(10), 2131–2168.], where the reinsurer is the leader while the insurer acts as the follower for both the reinsurance and investment strategies. Some theoretical analyses and numerical examples are provided to show the economic intuition and insights of the results.

Optimal Investment with a Noisy Signal of Future Stock Prices

We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock’s price fluctuations. With linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be not only well posed, but it even allows for a closed-form solution. We describe this solution and the resulting problem value for this stochastic control problem with partial observation by solving its convex-analytic dual problem.

Optimal reinsurance and investment problems to minimize the probability of drawdown

Optimal reinsurance and investment problems to minimize the probability of drawdown

A long-term optimal consumption and investment problem with partial information

A long-term optimal consumption and investment problem with partial information

Application of Asymptotic Analysis of a High-Dimensional HJB Equation to Portfolio Optimization

In this paper, we consider a portfolio optimization problem where the wealth consists of investing into a risky asset with a slow mean-reverting volatility and receiving an uncontrollable stochastic cash flow under the exponential utility. The Hamilton–Jacobi–Bellman equation formulated from the optimal investment problem is a high-dimensional nonlinear partial differential equation and difficult to find its analytical or numerical solutions. The paper provides a tractable asymptotic approach which treats the initial problem as a perturbation around the constant volatility problem. In this paper, we present a formal derivation of asymptotic approximation and prove the accuracy of the value function. Moreover, an illustrative example is provided to assess our approximate strategy and value function.

Optimal investment in defined contribution pension schemes with forward utility preferences

Optimal investment strategies of an individual worker during the accumulation phase in the defined contribution pension scheme have been well studied in the literature. Most of them adopted the classical backward model and approach, but any pre-specifications of retirement time, preferences, and market environment models do not often hold in such a prolonged horizon of the pension scheme. Pre-commitment to ensure the time-consistency of an optimal investment strategy derived from the backward model and approach leads the supposedly optimal strategy to be sub-optimal in the actual realizations. This paper revisits the optimal investment problem for the worker during the accumulation phase in the defined contribution pension scheme, via the forward preferences, in which an environment-adapting strategy is able to hold optimality and time-consistency together. Stochastic partial differential equation representation for the worker's forward preferences is illustrated. This paper constructs two of the forward utility preferences and solves the corresponding optimal investment strategies, in the cases of initial power and exponential utility functions.

Optimal Electric Vehicle Parking Lot Energy Supply Based on Mixed-Integer Linear Programming

E-mobility represents an important part of the EU’s green transition and one of the key drivers for reducing CO2 pollution in urban areas. To accelerate the e-mobility sector’s development it is necessary to invest in energy infrastructure and to assure favorable conditions in terms of competitive electricity prices to make the technology even more attractive. Large peak consumption of parking lots which use different variants of uncoordinated charging strategies increases grid problems and increases electricity supply costs. On the other hand, as observed lately in energy markets, different, mostly uncontrollable, factors can drive electricity prices to extreme levels, making the use of electric vehicles very expensive. In order to reduce exposure to these extreme conditions, it is essential to identify the optimal way to supply parking lots in the long term and to apply an adequate charging strategy that can help to reduce costs for end consumers and bring higher profit for parking lot owners. The significant decline in photovoltaic (PV) and battery storage technology costs makes them an ideal complement for the future supply of parking lots if they are used in an optimal manner in coordination with an adequate charging strategy. This paper addresses the optimal power supply investment problem related to parking lot electricity supply coupled with the application of an optimal EV charging strategy. The proposed optimization model determines optimal investment decisions related to grid supply and contracted peak power, PV plant capacity, battery storage capacity, and operation while optimizing EV charging. The model uses realistic data of EV charging patterns (arrival, departure, energy requirements, etc.) which are derived from commercial platforms. The model is applied using the data and prices from the Croatian market.

Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider

In the theory of portfolio selection, there are few methods that effectively address the combined challenge of insider information and model uncertainty, despite numerous methods proposed for each individually. This paper studies the problem of the robust optimal investment for an insider under model uncertainty. To address this, we extend the Itô formula for forward integrals by Malliavin calculus, and use it to establish an implicit anticipating stochastic differential game model for the robust optimal investment. Since traditional stochastic control theory proves inadequate for solving anticipating control problems, we introduce a new approach. First, we employ the variational method to convert the original problem into a nonanticipative stochastic differential game problem. Then we use the stochastic maximum principle to derive the Hamiltonian system governing the robust optimal investment. In cases where the insider information filtration is of the initial enlargement type, we derive the closed-form expression for the investment by using the white noise theory when the insider is ’small’. When the insider is ’large’, we articulate a quadratic backward stochastic differential equation characterization of the investment. We present the numerical result and conduct an economic analysis of the optimal strategy across various scenarios.

Optimal investment problem under behavioral setting: A Lagrange duality perspective

In this paper, we consider the optimal investment problem with both probability distortion/weighting and general non-concave utility functions with possibly finite number of inflection points, and apply a Lagrange duality based relaxation approach for solving this problem. Existing literature has shown the equivalent relationships (strong duality) between the relaxed problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, combining both factors, we prove that the absence of concavity in the utilities may result in strictly positive gaps, thus the strong duality may not hold unconditionally. The necessary and sufficient conditions on eliminating such gaps have been provided under this circumstance. We have applied the solution method to obtain closed-form solutions for the optimal terminal wealth and corresponding investment strategies in a special cumulative prospect theory (CPT) example with an appropriately selected inverse S-shaped probability distortion function. Based on the solutions, the joint effects of non-concave utilities combined with the probability weighting on the trading behaviors can be explicitly characterized. We show that, under this particular example, the co-existence of both factors may water down the loss aversion effect induced by only S-shaped utility or probability distortion when the agents are more cash-strapped in the initial budgets. In addition, we find that the optimal strategies derived from distorted beliefs shall converge to a constant that can be expressed by the standard Merton ratio multiplied by an inflation factor which we name as “distorted” Merton ratio. More importantly, the inflation factor is solely dependent on the probability distortion rather than the features of the utility function.

A hybrid reinsurance-investment game with delay and asymmetric information

In this paper, we investigate the optimal reinsurance and investment problem in the framework of the hybrid stochastic differential game, which includes a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The stochastic Stackelberg differential subgame considers the leader–follower relationship between a reinsurer and two insurers. The reinsurer, as the leader, prices reinsurance premium and invests its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. On the other side, the two insurers, as the followers, purchase proportional reinsurance from the reinsurer and invest in the same financial market. Also, the competitive relationship between the two insurers is characterized by the non-zero-sum stochastic differential subgame in which each insurer considers the relative performance. By introducing the delay feature into the hybrid game, we are able to obtain the wealth processes depicted by the stochastic delay differential equations. Considering the effect of asymmetric information, we assume the reinsurer and insurers have different levels of information about the financial market. The objective of the reinsurer is to maximize the mean–variance functional of its terminal surplus with delay, while each insurer is to maximize the mean–variance functional of its terminal surplus with delay relative to that of its competitor. By applying techniques in stochastic control theory, we obtain the extended Hamilton–Jacobi–Bellman equations, and then by using the idea of backward induction, we establish the equilibrium strategies and the corresponding value functions. Finally, we include some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the equilibrium strategies.

The optimal investment problem with inflation and liquidity risk

Inflation, liquidity and labor supply flexibility are three macroeconomic factors that significantly affect financial assets pricing. All three factors need to be considered in creating investment policies to minimize the bias and achieve the optimal portfolio selection. This paper jointly considers the impact of these three factors on the optimal investment–consumption strategy. We start with measuring liquidity with a jump diffusion process and propose a new utility function. Then we derive the Hamilton–Jacobi–Bellman (HJB) equation for the value function. Next, the analytical solution of the optimal strategy are obtained under the cases of stochastic and non-stochastic wages, respectively. Finally, numerical examples are provided to show the impact of these three factors on portfolio selections.