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This paper develops a dynamic model to analyze the general equilibrium of the insurance market, focusing on the interaction between insurers' underwriting and investment strategies. Three possible equilibrium outcomes are identified: a positive insurance market, a zero insurance market, and market failure. Our findings reveal why insurers may rationally accept underwriting losses by setting a negative safety loading while relying on investment profits, particularly when there is a negative correlation between insurance gains and financial returns. Additionally, we explore the impact of regulatory frictions, showing that while imposing a cost on investment can enhance social welfare under certain conditions, it may not always be necessary. Therefore, we emphasize the importance of tailoring regulatory interventions to specific market conditions.

This paper investigates portfolio selection within a continuous-time financial market with regime-switching and beliefs-dependent utilities. The market coefficients and the investor's utility function both depend on the market regime, which is modeled by an observable finite-state continuous-time Markov chain. The optimization problem is formulated by aggregating expected certainty equivalents under different regimes, leading to time-inconsistency. Utilizing the equilibrium strategy, we derive the associated extended Hamilton-Jacobi-Bellman (HJB) equations and establish a rigorous verification theorem. As a special case, we analyze equilibrium portfolio selection in a beliefs-dependent risk aversion model. In a bull regime, the excess asset returns, volatility, and risk aversion are all low, while the opposite holds in a bear regime. Closed-form solutions in the CRRA preference regime model of bull and bear markets are obtained, which is expressed by a solution to four-dimensional non-linear ODEs. The global existence of the ODEs is proven and we verify the equilibrium solution rigorously. We show that the equilibrium investment strategy lies between two constant Merton's fractions. Additionally, in our numerical experiment, the equilibrium proportion allocated in the risky asset is greater in a bull regime than in a bear regime and the equilibrium proportion increases with time in a bull regime while decreasing in a bear regime.

This paper studies the robust portfolio selection problem under a state-dependent confidence set. The investor invests in a financial market with a risk-free asset and a risky asset. The ambiguity-averse investor faces uncertainty over the drift of the risky asset and updates posterior beliefs by Bayesian learning. The investor holds the belief that the unknown drift falls within a confidence set at a certain confidence level. The confidence set varies with both the observed state and time. By maximizing the expected CARA utility of terminal wealth under the worst-case scenario of the unknown drift, we derive and solve the associated HJBI equation. The robust optimal investment strategy is obtained in a semi-analytical form based on a PDE. We validate the existence and uniqueness of the PDE and demonstrate the optimality of the solution in the verification theorem. The robust optimal investment strategy consists of two components: myopic demand in the worst-case scenario and hedging demand. The robust optimal investment strategy is categorized into three regions: buying, selling, and small trading. Ambiguity aversion results in a more conservative robust optimal investment strategy. Additionally, with learning, the investor's uncertainty about the drift decreases over time, leading to increased risk exposure to the risky asset.

This paper addresses a class of time-inconsistent mean field control (MFC) problems in the presence of common noise under non-exponential discount, where the coefficients of the McKean-Vlasov dynamics depend on the conditional joint distribution of the state and control. We investigate the closed-loop time-consistent equilibrium strategies for these extended MFC problems and provide a sufficient and necessary condition for their characterization. Furthermore, we derive a master equation system that provides an equivalent characterization of our problem. We then apply these results to the time-inconsistent linear quadratic (LQ) MFC problems, characterizing the equilibrium strategies in terms of the solution to a non-local Riccati system. To illustrate these findings, two financial applications are presented. Finally, a non-LQ example is also discussed in which the closed-loop equilibrium strategy can be explicitly characterized and verified.

This paper diverges from previous literature by considering the utility maximization problem in the context of investors having the freedom to actively acquire additional information to mitigate estimation risk. We derive closed-form value functions using CARA and CRRA utility functions and establish a criterion for valuing extra information through certainty equivalence, while also formulating its associated acquisition cost. By strategically employing variational methods, we explore the optimal acquisition of information, taking into account the trade-off between its value and cost. Our findings indicate that acquiring earlier information holds greater worth in eliminating estimation risk and achieving higher utility. Furthermore, we observe that investors with lower risk aversion are more inclined to pursue information acquisition.

We propose a two-layer stochastic game model to study reinsurance contracting and competition in a market with one insurer and two competing reinsurers. The insurer negotiates with both reinsurers simultaneously for proportional reinsurance contracts that are priced using the variance premium principle. The reinsurance contracting between the insurer and each reinsurer is modeled as a Stackelberg game. The two reinsurers compete for business from the insurer and optimize the so-called relative performance, instead of their own surplus, and their competition is settled by a noncooperative Nash game. We obtain a sufficient and necessary condition, related to the competition degrees of the two reinsurers, for the existence of an equilibrium. We show that the equilibrium, if exists, is unique, and the equilibrium strategy of each player is constant, fully characterized in semiclosed form. Furthermore, we obtain interesting sensitivity results for the equilibrium strategies through both analytical and numerical studies.

This paper delves into financial markets that incorporate a novel form of heterogeneity among investors, specifically in terms of their beliefs regarding the reliability of signals in the business cycle economy model, which may be biased. Unlike most papers in this field, we not only analyze the equilibrium but also examine welfare using objective measures while investors aim to maximize their utility based on subjective measures. Furthermore, we introduce passive investors and use their utility as a benchmark, thereby revealing the phenomenon of double loss sometimes. In the analysis, we examine two effects: the distortion effect on total welfare and the advantage effect of information and highlight their key factors of influence, with a particular emphasis on the proportion of investors. We also demonstrate that manipulating investors' estimation towards the economy can be a way to improve utility and identify an inner connection between welfare and survival.

This paper studies a variable proportion portfolio insurance (VPPI) strategy. The objective is to determine the risk multiplier by maximizing the extended Omega ratio of the investor's cushion, using a binary stochastic benchmark. When the stock index declines, investors aim to maintain the minimum guarantee. Conversely, when the stock index rises, investors seek to track some excess returns. The optimization problem involves the combination of a non-concave objective function with a stochastic benchmark, which is effectively solved based on the stochastic version of concavification technique. We derive semi-analytical solutions for the optimal risk multiplier, and the value functions are categorized into three distinct cases. Intriguingly, the classification criteria are determined by the relationship between the optimal risky multiplier in Zieling et al. (2014 and the value of 1. Simulation results confirm the effectiveness of the VPPI strategy when applied to real market data calibrations.

We use the martingale method to discuss the relationship between mean-variance (MV) and monotone mean-variance (MMV) portfolio selections. We propose a unified framework to discuss the relationship in general financial markets without any specific setting or completeness requirement. We apply this framework to a semimartingale market and find that MV and MMV are consistent if and only if the variance-optimal signed martingale measure keeps non-negative. Further, we provide an example to show the application of our result.

We study a reinsurance Stackelberg game in which both the insurer and the reinsurer adopt the mean-variance (abbr. MV) criterion in their decision-making and the reinsurance is irreversible. We apply a unified singular control framework where irreversible reinsurance contracts can be signed in both discrete and continuous times. The results theoretically illustrate that, rather than continuous-time contracts or a bunch of discrete-time contracts, a single once-for-all reinsurance contract is preferred. Moreover, the Stackelberg game turns out to be centering on the signing time of the single contract. The insurer signs the contract if the premium rate is lower than a time-dependent threshold and the reinsurer designs a premium that triggers the signing of the contract at his preferred time. Further, we find that reinsurance preference, discount and reversion have a decreasing dominance in the reinsurer's decision-making, which is not seen for the insurer.

This paper studies an optimal investment-consumption problem for competitive agents with exponential or power utilities and a common finite time horizon. Each agent regards the average of habit formation and wealth from all peers as benchmarks to evaluate the performance of her decision. We formulate the n-agent game problems and the corresponding mean field game problems under the two utilities. One mean field equilibrium is derived in a closed form in each problem. In each problem with n agents, an approximate Nash equilibrium is then constructed using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained. In addition, we provide some numerical illustrations of our results., Comment: arXiv admin note: substantial text overlap with arXiv:2206.13341 by other authors

This paper addresses the continuous-time portfolio selection problem under generalized disappointment aversion (GDA). The implicit definition of the certainty equivalent within GDA preferences introduces time inconsistency to this problem. We provide the sufficient and necessary condition for a strategy to be an equilibrium by a fully nonlinear integral equation. Investigating the existence and uniqueness of the solution to the integral equation, we establish the existence and uniqueness of the equilibrium. Our findings indicate that under disappointment aversion preferences, non-participation in the stock market is the unique equilibrium. The semi-analytical equilibrium strategies obtained under the constant relative risk aversion utility functions reveal that, under GDA preferences, the investment proportion in the stock market consistently remains smaller than the investment proportion under classical expected utility theory. The numerical analysis shows that the equilibrium strategy's monotonicity concerning the two parameters of GDA preference aligns with the monotonicity of the degree of risk aversion., Comment: 38 pages, 5 figures

This paper considers a class of stochastic control problems with implicitly defined objective functions, which are the sources of time-inconsistency. We study the closed-loop equilibrium solutions in a general controlled diffusion framework. First, we provide a sufficient and necessary condition for a strategy to be an equilibrium. Then, we apply the result to discuss two problems of dynamic portfolio selection for a class of betweenness preferences, allowing for closed convex constraints on portfolio weights and borrowing cost, respectively. The equilibrium portfolio strategies are explicitly characterized in terms of the solutions of some first-order ordinary differential equations for the case of deterministic market coefficients.

In this paper we study a time-inconsistent portfolio optimization problem for competitive agents with CARA utilities and non-exponential discounting. The utility of each agent depends on her own wealth and consumption as well as the relative wealth and consumption to her competitors. Due to the presence of a non-exponential discount factor, each agent's optimal strategy becomes time-inconsistent. In order to resolve time-inconsistency, each agent makes a decision in a sophisticated way, choosing open-loop equilibrium strategy in response to the strategies of all the other agents. We construct explicit solutions for the $n$-agent games and the corresponding mean field games (MFGs) where the limit of former yields the latter. This solution is unique in a special class of equilibria.

This paper addresses the portfolio selection problem for nonlinear law-dependent preferences in continuous time, which inherently exhibit time inconsistency. Employing the method of stochastic maximum principle, we establish verification theorems for equilibrium strategies, accommodating both random market coefficients and incomplete markets. We derive the first-order condition (FOC) for the equilibrium strategies, using a notion of functional derivatives with respect to probability distributions. Then, with the help of the FOC we obtain the equilibrium strategies in closed form for two classes of implicitly defined preferences: CRRA and CARA betweenness preferences, with deterministic market coefficients. Finally, to show applications of our theoretical results to problems with random market coefficients, we examine the weighted utility. We reveal that the equilibrium strategy can be described by a coupled system of Quadratic Backward Stochastic Differential Equations (QBSDEs). The well-posedness of this system is generally open but is established under the special structures of our problem.

We study a time-inconsistent singular control problem originating from irreversible reinsurance decisions with non-exponential discount. A novel definition of equilibrium for time-inconsistent singular control problems is introduced. For the problem with non-exponential discount, both sufficient and necessary conditions are derived, providing a thorough mathematical characterization of the equilibrium. Specifically, the equilibrium can be characterized by an extended HJB system, which is a coupled system of non-local parabolic equations with free boundaries. Finally, by showing the existence of classical solutions to the extended HJB system, the existence of equilibria is established under some additional assumptions.

In order to deal with the aging problem, pension system is actively transformed into the funded scheme. However, the funded scheme does not completely replace PAYGO (Pay as You Go) scheme and there exist heterogeneous mixes among PAYGO, EET (Exempt, Exempt, Taxed) and individual savings in different countries. In this paper, we establish the optimal mix by solving a Nash equilibrium between the pension participants and the government. Given the obligatory PAYGO and EET contribution rates, the participants choose the optimal asset allocation of the individual savings and the consumption policies to achieve the objective. The results extend the ``Samuelson-Aaron" criterion to age-dependent preference orderings. And we identify three critical ages to distinguish the multiple outcomes of preference orderings based on heterogeneous characteristic parameters. The government is fully aware of the optimal feedback of the participants. It chooses the optimal PAYGO and EET contribution rates to maximize the overall utility of the participants weighted by each cohort's population. As such, the negative population growth rate leads to the decline of the PAYGO attractiveness as well as the increase of the older cohorts' weight in the government decision-making. The optimal mix is the comprehensive result of the two effects.

This paper investigates the optimal management of an aggregated defined benefit pension plan in a stochastic environment. The interest rate follows the Ornstein-Uhlenbeck model, the benefits follow the geometric Brownian motion while the contribution rate is determined by the spread method of fund amortization. The pension manager invests in the financial market with three assets: cash, bond and stock. Regardless of the initial status of the plan, we suppose that the pension fund may become underfunded or overfunded in the planning horizon. The optimization goal of the manager is to maximize the expected utility in the overfunded region minus the weighted solvency risk in the underfunded region. By introducing an auxiliary process and related equivalent optimization problems and using the martingale method, the optimal wealth process, optimal portfolio and efficient frontier are obtained under four cases (high tolerance towards solvency risk, low tolerance towards solvency risk, a specific lower bound, and high lower bound). Moreover, we also obtain the probabilities that the optimal terminal wealth falls in the overfunded and underfunded regions. At last, we present numerical analyses to illustrate the manager's economic behaviors.

This paper investigates the equilibrium portfolio selection for smooth ambiguity preferences in a continuous-time market. The investor is uncertain about the risky asset's drift term and updates the subjective belief according to the Bayesian rule. Two versions of the verification theorem are established and an equilibrium strategy can be decomposed into a myopic demand and two hedging demands. When the prior is Gaussian, the closed-form equilibrium solution is obtained. A puzzle in the numerical results is interpreted via an alternative representation of the smooth ambiguity preferences.

This paper investigates a Stackelberg game between an insurer and a reinsurer under the $\alpha$-maxmin mean-variance criterion. The insurer can purchase per-loss reinsurance from the reinsurer. With the insurer's feedback reinsurance strategy, the reinsurer optimizes the reinsurance premium in the Stackelberg game. The financial market consists of cash and stock with Heston's stochastic volatility. Both the insurer and reinsurer maximize their respective $\alpha$-maxmin mean-variance preferences in the market. The criterion is time-inconsistent and we derive the equilibrium strategies by the extended Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in Li and Young (2022), excess-of-loss reinsurance is the optimal form of reinsurance strategy for the insurer. The equilibrium investment strategy is determined by a system of Riccati differential equations. Besides, the equations determining the equilibrium reinsurance strategy and reinsurance premium rate are given semi-explicitly, which is simplified to an algebraic equation in a specific example. Numerical examples illustrate that the game between the insurer and reinsurer makes the insurance more radical when the agents become more ambiguity aversion or risk aversion. Furthermore, the level of ambiguity, ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar effects on the equilibrium reinsurance strategy, reinsurance premium, and investment strategy., Comment: 38 pages, 20 figures

This paper compares the optimal investment problems based on monotone mean-variance (MMV) and mean-variance (MV) preferences in the L\'{e}vy market with an untradable stochastic factor. It is an open question proposed by Trybu{\l}a and Zawisza. Using the dynamic programming and Lagrange multiplier methods, we get the HJBI and HJB equations corresponding to the two investment problems. The equations are transformed into a new-type parabolic equation, from which the optimal strategies under both preferences are derived. We prove that the two optimal strategies and value functions coincide if and only if an important market assumption holds. When the assumption violates, MMV investors act differently from MV investors. Thus, we conclude that the difference between continuous-time MMV and MV portfolio selections is due to the discontinuity of the market. In addition, we derive the efficient frontier and analyze the economic impact of the jump diffusion risky asset. We also provide empirical evidences to demonstrate the validity of the assumption in real financial market.

We study continuous-time portfolio selection under monotone mean-variance (MMV) preferences in a jump-diffusion model, presenting an explicit solution different from that under classical mean-variance (MV) preferences in dynamic settings for the first time. We prove that the potential measures calculating MMV preferences can be restricted to non-negative Dol\'eans-Dade exponentials. We find that MMV can resolve the non-monotonicity and free cash flow stream problems of MV when the jump size can be larger than the inverse of the market price of risk. Such result is completely comparable to the earliest result by Dybvig and Ingersoll. Economically, we show that the essence of MMV lies in the pricing operator always remaining non-negative, with a value of zero assigned when the jump exceeds a certain threshold, avoiding the issue of non-monotonicity. As a result, MMV investors behave markedly different from MV investors. Furthermore, we validate the two-fund separation and establish the monotone capital asset pricing model (monotone CAPM) for MMV investors. We also study MMV in a constrained trading model and provide three specific numerical examples to show MMV's efficiency. Our finding can serve as a crucial theoretical foundation for future empirical tests of MMV and monotone CAPM's effectiveness.

This paper studies the continuous-time pre-commitment KMM problem proposed by Klibanoff, Marinacci and Mukerji (2005) in incomplete financial markets, which concerns with the portfolio selection under smooth ambiguity. The decision maker (DM) is uncertain about the dominated priors of the financial market, which are characterized by a second-order distribution (SOD). The KMM model separates risk attitudes and ambiguity attitudes apart and the aim of the DM is to maximize the two-fold utility of terminal wealth, which does not belong to the classical subjective utility maximization problem. By constructing the efficient frontier, the original KMM problem is first simplified as an one-fold expected utility problem on the second-order space. In order to solve the equivalent simplified problem, this paper imposes an assumption and introduces a new distorted Legendre transformation to establish the bipolar relation and the distorted duality theorem. Then, under a further assumption that the asymptotic elasticity of the ambiguous attitude is less than 1, the uniqueness and existence of the solution to the KMM problem are shown and we obtain the semi-explicit forms of the optimal terminal wealth and the optimal strategy. Explicit forms of optimal strategies are presented for CRRA, CARA and HARA utilities in the case of Gaussian SOD in a Black-Scholes financial market, which show that DM with higher ambiguity aversion tends to be more concerned about extreme market conditions with larger bias. In the end of this work, numerical comparisons with the DMs ignoring ambiguity are revealed to illustrate the effects of ambiguity on the optimal strategies and value functions., Comment: 53 pages, 7 figures

In this paper, we propose a new type of reversible modern tontine with transaction costs. The wealth of the retiree is divided into a bequest account and a tontine account. And consumption can only be withdrawn from the bequest account. Each transaction between the two accounts incurs fixed and proportional transaction costs depending on the transaction volume. The retiree dynamically controls the allocation policy between the two accounts and the consumption policy to maximize the consumption and bequest utilities. We formulate the optimization problem as a combined stochastic and impulse control problem with infinite time horizon, and characterize the value function as the unique viscosity solution of a HJBQVI (Hamilton-Jacobi-Bellmen quasi-variational inequality). The numerical results exhibit the V-shaped transaction region which consists of two stages. In the former stage, since longevity credits increase gradually, the retiree decreases the proportion of wealth in the tontine account to smooth the wealth and reduce the volatility. However, in the latter stage, the retiree increases the proportion of wealth in the tontine account to gamble for the considerable longevity credits.

We propose a consumption-investment decision model where past consumption peak $h$ plays a crucial role. There are two important consumption levels: the lowest constrained level and a reference level, at which the risk aversion in terms of consumption rate is changed. We solve this stochastic control problem and derive the value function, optimal consumption plan, and optimal investment strategy in semi-explicit forms. We find five important thresholds of wealth, all as functions of $h$, and most of them are nonlinear functions. As can be seen from numerical results and theoretical analysis, this intuitive and simple model has significant economic implications, and there are at least three important predictions: the marginal propensity to consume out of wealth is generally decreasing but can be increasing for intermediate wealth levels, and it jumps inversely proportional to the risk aversion at the reference point; the implied relative risk aversion is roughly a smile in wealth; the welfare of the poor is more vulnerable to wealth shocks than the wealthy. Moreover, locally changing the risk aversion influences the optimal strategies globally, revealing some risk allocation behaviors.

We provide a unified approach to find equilibrium solutions for time-inconsistent problems with distribution dependent rewards, which are important to the study of behavioral finance and economics. Our approach is based on {\it equilibrium master equation}, a non-local partial differential equation on Wasserstein space. We refine the classical notion of derivatives with respect to distribution and establish It$\hato$'s formula in the sense of such refined derivatives. Our approach is inspired by theories of Mckean-Vlasov stochastic control and mean field games, but is significantly different from both in that: we prohibit marginal distribution of state to be an input of closed loop control; we solve the best reaction to individual selves in an intra-person game instead of the best reaction to large populations as in mean field games. As applications, we reexamine the dynamic portfolio choice problem with rank dependent utility based on the proposed novel approach. We also recover the celebrated extended HJB equation when the reward of the problem has a nonlinear function of expectation while reformulating and weakening the assumptions needed. Most importantly, we provide a procedure to find an equilibrium solution of a dynamic mean-ES portfolio choice problem, which is completely new to the literature.

We propose a general family of piecewise hyperbolic absolute risk aversion (PHARA) utilities, including many classic and non-standard utilities as examples. A typical application is the composition of a HARA preference and a piecewise linear payoff in asset allocation. We derive a unified closed-form formula of the optimal portfolio, which is a four-term division. The formula has clear economic meanings, reflecting the behavior of risk aversion, risk seeking, loss aversion and first-order risk aversion. We conduct a general asymptotic analysis to the optimal portfolio, which directly serves as an analytical tool for financial analysis. We compare this PHARA portfolio with those of other utility families both analytically and numerically. One main finding is that risk-taking behaviors are greatly increased by non-concavity and reduced by non-differentiability of the PHARA utility. Finally, we use financial data to test the performance of the PHARA portfolio in the market.

This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the timeinconsistent stopping control problems under general multi-dimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair $(\hat{u},C)$ of controlstopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a non-exponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general non-constant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.

In this paper,we study the individual's optimal retirement time and optimal consumption under habitual persistence. Because the individual feels equally satisfied with a lower habitual level and is more reluctant to change the habitual level after retirement, we assume that both the level and the sensitivity of the habitual consumption decline at the time of retirement. We establish the concise form of the habitual evolutions, and obtain the optimal retirement time and consumption policy based on martingale and duality methods. The optimal consumption experiences a sharp decline at retirement, but the excess consumption raises because of the reduced sensitivity of the habitual level. This result contributes to explain the "retirement consumption puzzle". Particularly, the optimal retirement and consumption policies are balanced between the wealth effect and the habitual effect. Larger wealth increases consumption, and larger growth inertia (sensitivity) of the habitual level decreases consumption and brings forward the retirement time.

This paper investigates the robust {non-zero-sum} games in an aggregated {overfunded} defined benefit (abbr. DB) pension plan. The sponsoring firm is concerned with the investment performance of the fund surplus while the participants act as a union to claim a share of the fund surplus. The financial market consists of one risk-free asset and $n$ risky assets. The firm and the union both are ambiguous about the financial market and care about the robust strategies under the worst case scenario. {The union's objective is to maximize the expected discounted utility of the additional benefits, the firm's two different objectives are to maximizing the expected discounted utility of the fund surplus and the probability of the fund surplus reaching an upper level before hitting a lower level in the worst case scenario.} We formulate the related two robust non-zero-sum games for the firm and the union. Explicit forms and optimality of the solutions are shown by stochastic dynamic programming method. In the end of this paper, numerical results are illustrated to depict the economic behaviours of the robust equilibrium strategies in these two different games.

In this paper, we investigate the optimal management of defined contribution (abbr. DC) pension plan under relative performance ratio and Value-at-Risk (abbr. VaR) constraint. Inflation risk is introduced in this paper and the financial market consists of cash, inflation-indexed zero coupon bond and a stock. The goal of the pension manager is to maximize the performance ratio of the real terminal wealth under VaR constraint. An auxiliary process is introduced to transform the original problem into a self-financing problem first. Combining linearization method, Lagrange dual method, martingale method and concavification method, we obtain the optimal terminal wealth under different cases. For convex penalty function, there are fourteen cases while for concave penalty function, there are six cases. Besides, when the penalty function and reward function are both power functions, the explicit forms of the optimal investment strategies are obtained. Numerical examples are shown in the end of this paper to illustrate the impacts of the performance ratio and VaR constraint.

Benchmarks in the utility function have various interpretations, including performance guarantees and risk constraints in fund contracts and reference levels in cumulative prospect theory. In most literature, benchmarks are a deterministic constant or a fraction of the underlying wealth variable; thus, the utility is also a function of the wealth. In this paper, we propose a general framework of state-dependent utility optimization with stochastic benchmark variables, which includes stochastic reference levels as typical examples. We provide the optimal solution(s) and investigate the issues of well-definedness, feasibility, finiteness, and attainability. The major difficulties include: (i) various reasons for the non-existence of the Lagrange multiplier and corresponding results on the optimal solution; (ii) measurability issues of the concavification of a state-dependent utility and the selection of the optimal solutions. Finally, we show how to apply the framework to solve some constrained utility optimization problems with state-dependent performance and risk benchmarks as some nontrivial examples., Comment: 44 pages, 8figures

This paper investigates the optimal retirement decision, investment, and consumption strategies in a market with jump diffusion, taking into account habit persistence and stock-wage correlation. Our analysis considers multiple stocks and a finite time framework, intending to determine the retirement boundary of the ``wealth-habit-wage" triplet $(x, h, w)$. To achieve this, we use the habit reduction method and a duality approach to obtain the retirement boundary of the primal variables and feedback forms of optimal strategies. { When dealing with the dual problem, we address technical challenges in the proof of integral equation characterization of optimal retirement boundary using a $C^1$ version of It$\hat{\rm o}$'s formula.} Our results show that when the so-called ``de facto wealth" exceeds a critical proportion of wage, an immediate retirement is the optimal choice for the agent. Additionally, we find that the introduction of jump risks allows for the possibility of discontinuous investment strategies within the working region, which is a novel and insightful finding. Our numerical results effectively illustrate these findings by varying the parameters.

We consider a robust consumption-investment problem under CRRA and CARA utilities. The time-varying confidence sets are specified by $\Theta$, a correspondence from $[0,T]$ to the space of L\'{e}vy triplets, and describe priori information about drift, volatility and jump. Under each possible measure, the log-price processes of stocks are semimartingales and the triplet of their differential characteristics is a measurable selector from the correspondence $\Theta$ almost surely. By proposing and studying the global kernel, an optimal policy and a worst-case measure are generated from a saddle point of the global kernel, and they also constitute a saddle point of the objective function., Comment: 43 pages, 2 figures

We consider an optimal control problem of a property insurance company with proportional reinsurance strategy. The insurance business brings in catastrophe risk, such as earthquake and flood. The catastrophe risk could be partly reduced by reinsurance. The management of the company controls the reinsurance rate and dividend payments process to maximize the expected present value of the dividends before bankruptcy. This is the first time to consider the catastrophe risk in property insurance model, which is more realistic. We establish the solution of the problem by the mixed singular-regular control of jump diffusions. We first derive the optimal retention ratio, the optimal dividend payments level, the optimal return function and the optimal control strategy of the property insurance company, then the impacts of the catastrophe risk and key model parameters on the optimal return function and the optimal control strategy of the company are discussed., Comment: 22 pages, 8 figures

This paper considers an optimal control of a big financial company with debt liability under bankrupt probability constraints. The company, which faces constant liability payments and has choices to choose various production/business policies from an available set of control policies with different expected profits and risks, controls the business policy and dividend payout process to maximize the expected present value of the dividends until the time of bankruptcy. However, if the dividend payout barrier is too low to be acceptable, it may result in the company's bankruptcy soon. In order to protect the shareholders' profits, the managements of the company impose a reasonable and normal constraint on their dividend strategy, that is, the bankrupt probability associated with the optimal dividend payout barrier should be smaller than a given risk level within a fixed time horizon. This paper aims at working out the optimal control policy as well as optimal return function for the company under bankrupt probability constraint by stochastic analysis, PDE methods and variational inequality approach. Moreover, we establish a risk-based capital standard to ensure the capital requirement of can cover the total given risk by numerical analysis and give reasonable economic interpretation for the results., Comment: 38 pages, 8 figures