Random Horizon Research Articles

Equilibrium investment–reinsurance strategy under information asymmetry and random horizon

This paper investigates an investment–reinsurance problem incorporating information asymmetry and random horizon. At the beginning of the transaction, the insurer owns some inside information related to the risky asset price and the insurance claims. Simultaneously, the return rate and the driving noise for the risky asset process cannot be observed directly. Moreover, the trading horizon is random due to some exogenous factors. Under the dynamic mean–variance criterion, the equilibrium feedback strategy and the associated equilibrium value function are derived in closed form by solving the extended Hamilton–Jacobi–Bellman (HJB) equations. Several special cases are further discussed. Finally, numerical simulations are conducted to analyze the influences of some important parameters on the equilibrium strategy and the efficient frontier.

Infinite horizon impulse control of stochastic functional differential equations driven by Lévy processes

We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional L p -Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.

Equilibrium reporting strategy: Two rate classes and full insurance

AbstractWe propose a multiperiod insurance model under a bonus–malus system with two rate classes and consider an insured who has purchased full insurance for her losses. To explore the potential advantage of underreporting her insurable losses, the insured follows a barrier strategy and only reports lossses above the barrier to the insurer. We obtain a unique equilibrium declaration strategy in closed form for a risk‐neutral insured who maximizes her expected wealth, and in semiclosed form for a risk‐averse insured who maximizes her expected exponential utility of wealth, both over an exogenous random horizon. We find that the equilibrium barriers for the two classes are equal and strictly greater than zero, offering a theoretical explanation for the underreporting of insurable losses, a form of ex post moral hazard. Finally, we consider the case of three rate classes and show, through numerical examples, that the equilibrium barriers are not equal.

Epstein‐Zin utility maximization on a random horizon

AbstractThis paper solves the consumption‐investment problem under Epstein‐Zin preferences on a random horizon. In an incomplete market, we take the random horizon to be a stopping time adapted to the market filtration, generated by all observable, but not necessarily tradable, state processes. Contrary to prior studies, we do not impose any fixed upper bound for the random horizon, allowing for truly unbounded ones. Focusing on the empirically relevant case where the risk aversion and the elasticity of intertemporal substitution are both larger than one, we characterize the optimal consumption and investment strategies using backward stochastic differential equations with superlinear growth on unbounded random horizons. This characterization, compared with the classical fixed‐horizon result, involves an additional stochastic process that serves to capture the randomness of the horizon. As demonstrated in two concrete examples, changing from a fixed horizon to a random one drastically alters the optimal strategies.

A Stackelberg reinsurance-investment game under α-maxmin mean-variance criterion and stochastic volatility

ABSTRACT This paper investigates a Stackelberg game between an insurer and a reinsurer under the α-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 α-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, D. & Young, V. R. (2022). Stackelberg differential game for reinsurance: mean-variance framework and random horizon. Insurance: Mathematics and Economics 102, 42–55.], 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 averse or risk averse. 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.

Continuous-Time Mean-Variance Portfolio Selection Under Non-Markovian Regime-Switching Model with Random Horizon

Continuous-Time Mean-Variance Portfolio Selection Under Non-Markovian Regime-Switching Model with Random Horizon

Stochastic linear-quadratic control with a jump and regime switching on a random horizon

In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a random horizon $ [0,T\wedge\tau] $, where $ \tau $ is a given random jump time for the underlying state process and $ T $ a constant. We obtain the explicit optimal feedback control and explicit optimal value of the problem by solving a system of stochastic Riccati equations (SREs) with jumps on the random horizon $ [0,T\wedge\tau] $. By the decomposition approach stemming from filtration enlargement theory, we express the solution to the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon $ [0,T] $. Solving the latter system is the key theoretical contribution of this paper and we accomplish this under three different conditions, one of which seems to be new in the literature. The above results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.

Consumption and asset allocation with information learning and capital gains tax

<p style='text-indent:20px;'>In this study, we explore the consumption and asset allocation problem with random horizon in which stock returns are unobservable and capital gains are taxed. The agent with recursive utility could apply the historical stock prices and her private information to learn the expected returns on the stock. We obtain a semi-analytical solution to this problem. Numerical results show that the weights in the non-concealed risky asset and the safe asset increases and decreases with the expected return learnt, respectively. Besides, as the accuracy of estimation gets improved, the agent will reduce her consumption and riskless asset investment and enhance her investment in the non-concealed risky asset. However, information learning does not affect the tax evasion strategy. We also find that the effect of risk aversion on consumption-wealth ratio is significantly affected by the estimation accuracy.</p>

Lyapunov Conditions for Differentiability of Markov Chain Expectations

We consider a family of Markov chains whose transition dynamics are affected by model parameters. Understanding the parametric dependence of (complex) performance measures of such Markov chains is often of significant interest. The derivatives and their continuity of the performance measures w.r.t. the parameters play important roles, for example, in numerical optimization of the performance measures, and quantification of the uncertainties in the performance measures when there are uncertainties in the parameters from the statistical estimation procedures. In this paper, we establish conditions that guarantee the smoothness of various types of intractable performance measures—such as the stationary and random horizon discounted performance measures—of general state space Markov chains and provide probabilistic representations for the derivatives. Funding: C.-H. Rhee is supported by the National Science Foundation [Grant CMMI-2146530].

Stackelberg differential game for insurance under model ambiguity: general divergence

We solve a Stackelberg differential game between a buyer and a seller of insurance policies, in which both parties are ambiguous about the insurable loss. Both the buyer and seller maximize their expected wealth, plus a penalty term that reflects ambiguity, over an exogenous random horizon. Under a mean-variance premium principle and a general divergence that measures the players' ambiguity, we obtain the Stackelberg equilibrium semi-explicitly. Our main results are that the optimal variance loading equals zero and that the seller's robust optimal premium rule equals the net premium under the buyer's optimally distorted probability. Both of these important results generalize those we obtained in [Cao, J., Li, D., Young, V. R. & Zou, B. (2022). Stackelberg differential game for insurance under model ambiguity. Insurance: Mathematics and Economics, 106, 128–145.] under squared-error divergence.

Stackelberg differential game for insurance under model ambiguity

We study a dynamic Stackelberg differential game between a buyer and a seller of insurance policies in a spectrally negative Lévy framework, in which both parties are ambiguous about the intensity and severity of insurable losses. Both the buyer and seller aim to maximize their expected wealth, plus a penalty term that reflects ambiguity, over an exogenous random horizon. Under a mean-variance premium principle and a quadratic penalty for ambiguity, we obtain the equilibrium in closed form. Our main results show that the buyer's robust optimal indemnity is a coinsurance with proportion less than one-half, which increases (resp. decreases) as the buyer (resp. seller) becomes more ambiguity averse. Also we show that the seller's robust optimal premium rule equals the net premium under the buyer's optimally distorted probability, which is the buyer's “best hope,” and it exceeds the actuarially fair premium under the seller's optimally distorted probability measure so is, thereby, acceptable to the seller.

Explicit description of all deflators for market models under random horizon with applications to NFLVR

This paper considers an initial market model, specified by its underlying assets S and its flow of information F, and an arbitrary random time τ which might not be an F-stopping time. As the death time and the default time (that τ might represent) can be seen when they occur only, the progressive enlargement of F with τ sounds tailor-fit for modeling the new flow of information G that incorporates both F and τ. In this setting of informational market, the first principal goal resides in describing as explicitly as possible the set of all deflators for (Sτ,G), while the second principal goal lies in addressing the No-Free-Lunch-with-Vanishing-Risk concept (NFLVR hereafter) for (Sτ,G). Besides this direct application to NFLVR, the set of all deflators constitutes the dual set of all “admissible” wealth processes for the stopped model (Sτ,G), and hence it is vital in many hedging and pricing related optimization problems. Thanks to the results of Choulli et al. (2020), on martingales classification and representation for progressive enlarged filtration, our two main goals are fully achieved in different versions, when the survival probability never vanishes. The results are illustrated on the two particular cases when (S,F) follows the jump-diffusion model and the discrete-time model.

A Consumption and Investment Problem via a Markov Decision Processes Approach with Random Horizon

This work is devoted to a consumption and investment problem, in which there is an investor with certain initial wealth with the possibility of deciding how much of such wealth will be consumed and how much will be invested in each of a series of successive times. The key issue is to find a wealth assignation rule in order to maximize the performance criteria; such dilemma will be achieved by the dynamic programming technique for the Markov decision processes with random horizon.

On Durable-strategies Dynamic Game Theory

An empirically meaningful theory of dynamic games has to incorporate real-life phenomena. Durable strategies, which effects last for a period of time, are prevalent in real-life situations. Revenue generating investments, toxic waste disposal and purchase of durable goods are common examples of durable strategies. This paper first provides a review on durable-strategies dynamic game theory. A practically relevant advancement - random horizon durable-strategies dynamic games - yielding novel results in durable-strategies dynamic games theory is then presented. Dynamic optimization theorem, game formulations and HJB equations are derived. An illustrative example is provided. The theory and solution mechanism of durable-strategies cooperative dynamic games are also discussed.

Stackelberg differential game for reinsurance: Mean-variance framework and random horizon

We consider a reinsurance problem for a mean-variance Stackelberg game with a random time horizon, in which an insurer and a reinsurer are the two players. The reinsurer computes its premium according to the mean-variance premium principle with parameters (θ,η)∈R+2. First, for any pair (θ,η)∈R+2, we compute the per-loss reinsurance strategy that maximizes a mean-variance functional of the insurer's surplus at the end of the random horizon. This reinsurance strategy is excess-of-loss reinsurance with constant proportional reinsurance for losses above the deductible. Then, given the information of what the insurer will choose when offered any mean-variance premium principle, we determine the optimal pair (θ⁎,η⁎)∈R+2 that maximizes the reinsurer's expected surplus at the end of the random horizon. We show that the equilibrium deductible and coinsurance are both independent of the risk aversion parameter of the insurer. We also show that if the claim severity is light-tailed, in the sense that its hazard rate function is non-decreasing, then the equilibrium reinsurance is pure excess-of-loss reinsurance with no loading for the variance of the loss. Moreover, we show that if the claim severity is heavy-tailed, in the sense that its hazard rate function is decreasing, then the equilibrium reinsurance has a non-trivial coinsurance, with a positive loading for the variance of the loss. Finally, we solve four examples to illustrate these latter two, important results.

Generalized dynamic games with durable strategies under uncertain planning horizon

This paper establishes a new class of dynamic games which incorporates two frequently observed real-life phenomena — durable strategies and uncertain horizon. In the presence of durable strategies and random horizon, significant modification of the dynamic optimization techniques is required to accommodate these phenomena. A novel dynamic optimization theorem is developed and a new set of equations characterizing a non-cooperative game equilibrium is derived. A subgame consistent solution for the cooperative game counterpart is obtained with a new theorem for the derivation of a payoff distribution procedure under random horizon and durable strategies. A number of new application results in dynamic games are derived to reflect practical considerations in making decision. Computational illustrations in an application involving a dynamic interactive investments game are provided.

Mean-Variance Asset-Liability Management in a Non-Markovian Regime-Switching Jump-Diffusion Market with Random Horizon

This paper investigates the problem of mean-variance asset-liability management (ALM) in a market where the dynamics of assets are non-Markovian regime-switching models driven by a Brownian motion, a Poisson random measure and a continuous time finite-state Markov chain. It is assumed that the time horizon is uncertain relying not only on asset prices and liability values, but also on other factors. The insurer aims to minimize the variance of the terminal surplus given an expected terminal surplus subject to the risk of paying out random liabilities of an extensive Cramer-Lundberg model. By further developing the solvability of a linear backward stochastic differential equation (BSDE) with two types of jumps, namely jumps modelled by a Poisson random measure and by basic martingales related to a Markov chain, closed-form expressions for both the efficient strategy and the efficient frontier are obtained and represented by unique solutions to several relevant BSDEs.

Quadratic hedging for sequential claims with random weights in discrete time

We study a quadratic hedging problem for a sequence of contingent claims with random weights in discrete time. We obtain the optimal hedging strategy explicitly in a recursive representation, without imposing the non-degeneracy (ND) condition on the model and square integrability on hedging strategies. We relate the general results to hedging under random horizon and fair pricing in the quadratic sense. We illustrate the significance of our results in an example in which the ND condition fails.

Mortality and Healthcare: A Stochastic Control Analysis under Epstein--Zin Preferences

This paper studies optimal consumption, investment, and healthcare spending under Epstein-Zin preferences. Given consumption and healthcare spending plans, Epstein-Zin utilities are defined over an agent's random lifetime, partially controllable by the agent as healthcare reduces mortality growth. To the best of our knowledge, this is the first time Epstein-Zin utilities are formulated on a controllable random horizon, via an infinite-horizon backward stochastic differential equation with superlinear growth. A new comparison result is established for the uniqueness of associated utility value processes. In a Black-Scholes market, the stochastic control problem is solved through the related Hamilton-Jacobi-Bellman (HJB) equation. The verification argument features a delicate containment of the growth of the controlled morality process, which is unique to our framework, relying on a combination of probabilistic arguments and analysis of the HJB equation. In contrast to prior work under time-separable utilities, Epstein-Zin preferences facilitate calibration. The model-generated mortality closely approximates actual mortality data in the US and UK; moreover, the efficacy of healthcare can be calibrated and compared between the two countries.

A preventive maintenance policy with usage-dependent failure rate thresholds under two-dimensional warranties

This article considers Preventive Maintenance (PM) under a two-dimensional (2-D) warranty contract with time and usage limits. From a manufacturer’s point of view, we develop a dynamic maintenance model with a random horizon to include the impact of random and dynamic usage rates on PM decisions. The model treats the cumulative amount of usage as a state variable that provides information about the failure rate and the expiration of the 2-D warranty. We characterize the optimal PM policy by a sequence of usage-dependent failure rate thresholds. Each threshold is a function of the cumulative usage. Our failure rate threshold policy chooses one of the following two actions in each period: performing perfect PM or no PM. Specifically, the manufacturer should bring the failure rate back to its original level when it exceeds the threshold in the corresponding period. This policy is also optimal under a constant usage rate. In the numerical experiments, we demonstrate the effectiveness of the proposed policy and conduct a sensitivity analysis to investigate how this policy is affected by the model parameters.