Exponential Utility Research Articles

On the Value of Information Across Decision Problems

The value of information is an important concept in decision analysis that has been quantified as the buying price (BPI) for the information and the expected utility increase (EUI) obtainable by using the information. These two measures rank information sources identically in a scalar-valued decision problem only when the utility function is linear or exponential. In contrast, this paper focuses on the value of information across scalar-valued decision problems sharing the same utility function such as different divisions within an organization exploring various information sources for their decisions using the same organizational utility function. In this context, it still makes sense to ask which sources are more informative. We show that BPI and EUI rank information sources identically in this context only when the utility function is linear. However, if the certainty equivalent increase is used instead of EUI, then identical ranking with BPI across problems is maintained for the broader class of linear or exponential utility functions. We discuss the importance of these results for distributed decision-making settings, where different departments within an organization may calculate the value of information separately. Our results advise against using EUI to measure information value in this context when risk attitude is important.

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.

Principal-Multiagents problem under equivalent changes of measure: General study and an existence result

We study a general contracting problem between the principal and a finite set of competitive agents, who perform equivalent changes of measure by controlling the drift of the output process and the compensator of its associated jump measure. In this setting, we generalize the dynamic programming approach developed by Cvitanić et al. (2018) and we also relax their assumptions. We prove that the problem of the principal can be reformulated as a standard stochastic control problem in which she controls the continuation utility (or certainty equivalent) processes of the agents. Our assumptions and conditions on the admissible contracts are minimal to make our approach work. We review part of the literature and give examples on how they are usually satisfied. We also present a smoothness result for the value function of a risk–neutral principal when the agents have exponential utility functions. This leads, under some additional assumptions, to the existence of an optimal contract.

Optimal premium pricing in a competitive stochastic insurance market with incomplete information: A Bayesian game-theoretic approach

This paper examines a stochastic one-period insurance market with incomplete information. The aggregate amount of claims follows a compound Poisson distribution. Insurers are assumed to be exponential utility maximizers, with their degree of risk aversion forming their private information. A premium strategy is defined as a mapping between risk-aversion types and premium rates. The optimal premium strategies are denoted by the pure-strategy Bayesian Nash equilibrium, whose existence and uniqueness are demonstrated under specific conditions on the insurer-specific demand functions. Boundary and monotonicity properties for equilibrium premium strategies are derived.

Iterative Methods with Self-Learning for Solving Nonlinear Equations

This paper is devoted to the problem of solving a system of nonlinear equations with an arbitrary but continuous vector function on the left-hand side. By assumption, the values of its components are the only a priori information available about this function. An approximate solution of the system is determined using some iterative method with parameters, and the qualitative properties of the method are assessed in terms of a quadratic residual functional. We propose a self-learning (reinforcement) procedure based on auxiliary Monte Carlo (MC) experiments, an exponential utility function, and a payoff function that implements Bellman’s optimality principle. A theorem on the strict monotonic decrease of the residual functional is proven.

Forward Backward SDEs Systems for Utility Maximization in Jump Diffusion Models

We consider the classical problem of maximizing the expected utility of terminal net wealth with a final random liability in a simple jump-diffusion model. In the spirit of Horst et al. (Stoch Process Appl 124(5):1813–1848, 2014) and Santacroce and Trivellato (SIAM J Control Optim 52(6):3517–3537, 2014), under suitable conditions the optimal strategy is expressed in implicit form in terms of a forward backward system of equations. Some explicit results are presented for the pure jump model and for exponential utilities.

Robust asset-liability management games for n players under multivariate stochastic covariance models

This paper investigates a non-zero-sum stochastic differential game among n competitive CARA asset-liability managers, who are concerned about the potential model ambiguity and aim to seek the robust investment strategies. The ambiguity-averse managers are subject to uncontrollable and idiosyncratic random liabilities driven by generalized drifted Brownian motions and have access to an incomplete financial market consisting of a risk-free asset, a market index and a stock under a multivariate stochastic covariance model. The market dynamics permit not only stochastic correlation between the risky assets but also path-dependent and time-varying risk premium and volatility, depending on two affine-diffusion factor processes. The objective of each manager is to maximize the expected exponential utility of his terminal surplus relative to the average among his competitors under the worst-case scenario of the alternative measures. We manage to solve this robust non-Markovian stochastic differential game by using a backward stochastic differential equation approach. Explicit expressions for the robust Nash equilibrium investment policies, the density generator processes under the well-defined worst-case probability measures and the corresponding value functions are derived. Conditions for the admissibility of the robust equilibrium strategies are provided. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium investment strategies and draw some economic interpretations from these results.

A mean field game approach to optimal investment and risk control for competitive insurers

We consider an insurance market consisting of multiple competitive insurers with a mean field interaction via their terminal wealth under the exponential utility with relative performance. It is assumed that each insurer regulates her risk by controlling the number of policies. We respectively establish the constant Nash equilibrium (independent of time) on the investment and risk control strategy for the finite n-insurer game and the constant mean field equilibrium for the corresponding mean field game (MFG) problem (when the number of insurers tends to infinity). Furthermore, we examine the convergence relationship between the constant Nash equilibrium of finite n-insurer game and the mean field equilibrium of the corresponding MFG problem. Our numerical analysis reveals that, for a highly competitive insurance market consisting of many insurers, every insurer will invest more in risky assets and increase the total number of outstanding liabilities to maximize her exponential utility with relative performance.

Optimal liquidation with high risk aversion and small linear price impact

We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options in the case where the investor is required to liquidate her position. Our main result is establishing a non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. We compute the limit of the corresponding utility indifference prices and find explicitly a family of portfolios which are asymptotically optimal.

Short Communication: Optimal Insurance to Maximize Exponential Utility When Premium Is Computed by a Convex Functional

Short Communication: Optimal Insurance to Maximize Exponential Utility When Premium Is Computed by a Convex Functional

Robust CARA Optimization

Decision making under uncertainty involves both ambiguity and risk. In “Robust CARA Optimization,” Chen and Sim have developed innovative optimization models designed for ambiguity-averse decision makers whose risk preference is consistent with constant absolute risk aversion (CARA). The research delves into maximizing the worst-case expected exponential utility amid uncertainties from independent factors with ambiguous marginals. To enhance computational feasibility, the authors developed a series of approximations: starting with tractable concave functions in affinely perturbed cases, advancing to concave piecewise affinely perturbed scenarios, and culminating in novel multi-deflected linear decision rules for adaptive optimization. This comprehensive framework extends to a multi-period consumption model, ultimately forming an exponential conic optimization problem efficiently solvable with existing solvers. Practical applications demonstrated in project and multiperiod inventory management highlight the models’ potential to surpass existing stochastic optimization methods, especially under high risk aversion.

Optimal investment strategy for asset-liability management with a defaultable bond under stochastic default intensity

This paper studies a continuous-time asset-liability management problem with a defaultable bond under stochastic interest rates and stochastic default intensity. Specifically, an asset-liability manager is allowed to invest in a cash, a treasury bond, a defaultable bond and a stock. Using the stochastic control scheme and partial differential equation approach, we derive closed-form expressions for optimal investment strategies and corresponding value functions under the power and exponential utility functions. In addition, we also provide numerical experiments to illustrate the impact of relevant parameters in the default process and recovery rate on the optimal asset allocation strategy.

Non-zero-sum reinsurance and investment game under thinning dependence structure: mean–variance premium principle

This paper considers a non-zero-sum stochastic differential reinsurance and investment game between two competitive insurers under the expected exponential utility. It is assumed that the surplus process of each insurer is characterized by a compound Poisson model and their claim businesses are correlated through thinning dependence structure. Each insurer can purchase proportional reinsurance and is allowed to invest in a risk-free asset and a risky asset. The insurers compete with each other and are concerned with the relative performance. By using the stochastic control technique, not only the existence and uniqueness of the optimal strategies are proved, but also the specific analyses for the constraint reinsurance strategies are provided. We find that (i) the thinning dependence structure has significant impact on optimal reinsurance strategies for both insurers and can help them reduce risks; (ii) the behavior of non-zero-sum game will reduce the proportion of purchasing reinsurance for each insurer.

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.

Equilibrium analysis of the fluid model with two types of parallel customers and incomplete fault

AbstractThis article considers the individual equilibrium behavior and socially optimal strategy in a fluid queue with two types of parallel customers and incomplete fault. Assume that the working state and the incomplete fault state appear alternately in the buffer. Different from the linear revenue and expenditure structure, an exponential utility function can be constructed to obtain the equilibrium balking thresholds in the fully observable case. Besides, the steady-state probability distribution and the corresponding expected social benefit are derived based on the renewal process and the standard theory of linear ordinary differential equations. Furthermore, a reasonable entrance fee strategy is discussed under the condition that the fluid accepts the globally optimal strategies. Finally, the effects of the diverse system parameters on the entrance fee and the expected social benefit are explicitly illustrated by numerical comparisons.

Optimal robust reinsurance contracts with investment strategy under variance premium principle

This paper investigates the optimal robust proportional reinsurance contracts with investment in a liquid financial market under variance premium principle in a principal-agent framework. The surplus process of the insurer (agent) is assumed to follow an approximating compound Poisson risk process. Both the insurer and reinsurer (principal) are allowed to invest in a risk-free asset and a risky asset whose price process is governed by the constant elasticity of variance model. The insurer and reinsurer aim to maximize the expected exponential utility of terminal wealth. The reinsurer is ambiguity-averse and has deterministic ambiguity aversion preferences against the diffusion risk caused by the financial market and the approximated diffusion risk which comes from the claim process. The reinsurance price is described by the reinsurer's safety loading which can be decided by the optimal strategies of the insurer and the reinsurer. By utilizing stochastic optimal control principle and HJB (or HJBI) equations, the optimal (robust) proportional reinsurance-investment strategies and the corresponding value functions are obtained explicitly. Numerical examples are provided.

Optimal investment and benefit payment adjustment strategies for the target benefit plan under partial information

This article investigates the optimal asset allocation and benefit adjustment problem for the target benefit plan (TBP) with predictable returns in an environment of partial information. We assume that the return rate of the stock depends on an observable and an unobservable state variables, and the pension manager estimates the unobservable component from known information through Bayesian learning. Under the criterion of expected exponential utility maximization, we obtain the optimal strategy in closed-form through the dynamic programming principle approach. Furthermore, numerical simulations are conducted by the Monte Carlo method to discuss the impacts of some parameters on the derived optimal strategy and to compare the optimal strategy with the suboptimal strategy when ignoring learning. It turns out that neglecting learning affects the optimal strategy seriously, thereby leading to significant utility losses.

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.

Markov decision processes under risk sensitivity: A discount vanishing approach

This paper considers Markov decision processes (MDPs) with risk-sensitivity. The aim is to explore the effects of state transience and non-communication on the optimal control of the system. A vanishing discount approach is investigated. The approximating system is the MDP evaluated by the usual discounted exponential utility. After being appropriately normalized, it is shown that the optimal discounted value functions converge to the optimal risk-sensitive averages as the discount factor goes to 1. These value functions are shown to depend on certain order structure of the state space. In this way it is also proved that the optimal policies for the discounted system converge to the optimal ones for the risk-sensitive system as the discount factor goes to 1. In proving these, an ordered partition of the state space is introduced, which is closely related to the characteristics of state communication, transience and absorption.