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We develop an approach for solving time‐consistent risk‐sensitive stochastic optimization problems using model‐free reinforcement learning (RL). Specifically, we assume agents assess the risk of a sequence of random variables using dynamic convex risk measures. We employ a time‐consistent dynamic programming principle to determine the value of a particular policy, and develop policy gradient update rules that aid in obtaining optimal policies. We further develop an actor–critic style algorithm using neural networks to optimize over policies. Finally, we demonstrate the performance and flexibility of our approach by applying it to three optimization problems: statistical arbitrage trading strategies, financial hedging, and obstacle avoidance robot control. [ABSTRACT FROM AUTHOR]

The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast to scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilise this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multi-portfolio time-consistency between set-valued risk measures for processes and vectors is provided. To accomplish this, an augmented definition for multi-portfolio time-consistency of set-valued risk measures for processes is proposed. [ABSTRACT FROM AUTHOR]

In many situations, decisions under uncertainty are made with incomplete confidence about the probability law describing the environment (model uncertainty, or ambiguity), but with the opportunity to learn, at some cost, before choosing between actions. In "Optimal Learning Under Robustness and Time-Consistency," Epstein and Ji use a stylized model to derive the optimal demand for learning (or sampling) and how it is affected by model uncertainty. A presumption is that the decision maker is conservative and seeks to make robust decisions. One surprising result is that, in a sequential hypothesis-testing context, sensitivity analysis overstates the robustness value of sampling. In a different context where there exists an unambiguous (purely risky) action, but where learning is not feasible, Ellsberg has highlighted the intuition for choosing the risky option over ambiguous ones. In contrast, Epstein and Ji demonstrate that where learning is feasible and optimal, then the risky action cannot be optimal. We model learning in a continuous-time Brownian setting where there is prior ambiguity. The associated model of preference values robustness and is time-consistent. It is applied to study optimal learning when the choice between actions can be postponed, at a per-unit-time cost, in order to observe a signal that provides information about an unknown parameter. The corresponding optimal stopping problem is solved in closed form, with a focus on two specific settings: Ellsberg's two-urn thought experiment expanded to allow learning before the choice of bets, and a robust version of the classical problem of sequential testing of two simple hypotheses about the unknown drift of a Wiener process. In both cases, the link between robustness and the demand for learning is studied. [ABSTRACT FROM AUTHOR]

We derive an optimal fiscal policy rule that reflects the US government's desire to reduce fluctuations in the output gap and the primary deficit in the post-World War II period. The results from Bayesian estimations show a lower weight on output gap stabilization under predominant democratic presidencies relative to republican ones. The opposite holds with respect to maintaining a zero primary deficit. Our time-consistent fiscal rule indicates a countercyclical government response to GDP fluctuations at all times. [ABSTRACT FROM AUTHOR]

Abstract Background The history of gene families—which are equivalent to event-labeled gene trees—can to some extent be reconstructed from empirically estimated evolutionary event-relations containing pairs of orthologous, paralogous or xenologous genes. The question then arises as whether inferred event-labeled gene trees are “biologically feasible” which is the case if one can find a species tree with which the gene tree can be reconciled in a time-consistent way. Results In this contribution, we consider event-labeled gene trees that contain speciations, duplications as well as horizontal gene transfer (HGT) and we assume that the species tree is unknown. Although many problems become NP-hard as soon as HGT and time-consistency are involved, we show, in contrast, that the problem of finding a time-consistent species tree for a given event-labeled gene can be solved in polynomial-time. We provide a cubic-time algorithm to decide whether a “time-consistent” species tree for a given event-labeled gene tree exists and, in the affirmative case, to construct the species tree within the same time-complexity.

Three different solution concepts are reviewed and computed for linear-state and homogeneous linear-quadratic cooperative differential games with asymmetric players. Discount rates can be nonconstant and/or different. Special attention is paid to the issues of time-consistency, agreeability and subgame-perfectness, both from the viewpoint of sustainability of cooperation and from the credibility of the announced equilibrium strategies. [ABSTRACT FROM AUTHOR]

We propose a continuous-time version of the adaptive robust methodology introduced in T. R. Bielecki et al. [SIAM J. Control Optim., 57 (2019), pp. 925-946]. An agent solves a stochastic control problem where the underlying uncertainty follows a jump-diffusion process and the agent does not know the drift parameters of the process. The agent considers a set of alternative measures to make the control problem robust to model misspecification and employs a continuous-time estimator to learn the value of the unknown parameters to make the control problem adaptive to the arrival of new information. We use measurable selection theorems to prove the dynamic programming principle of the adaptive robust problem and show that the value function of the agent is characterized by a nonlinear partial differential equation. As an example, we derive the optimal adaptive robust strategy for an agent who acquires a large amount of shares in an order driven market and illustrate the financial performance of the execution strategy. [ABSTRACT FROM AUTHOR]

For an F T -measurable payoff of a European type contingent claim, the recursive utility process/dynamic risk measure can be described by the adapted solution to a backward stochastic differential equation (BSDE). However, for an F T -measurable stochastic process (called a position process, not necessarily F -adapted), mimicking BSDE's approach will lead to a time-inconsistent recursive utility/dynamic risk measure. It is found that a more proper approach is to use the adapted solution to a backward stochastic Volterra integral equation (BSVIE). The corresponding notions are called equilibrium recursive utility and equilibrium dynamic risk measure, respectively. Motivated by this, the current paper is concerned with BSVIEs whose generators are allowed to have quadratic growth (in Z(t, s)). The existence and uniqueness for both the so-called adapted solutions and adapted M-solutions are established. A comparison theorem for adapted solutions to the so-called Type-I BSVIEs is established as well. As consequences of these results, some general continuous-time equilibrium dynamic risk measures and equilibrium recursive utility processes are constructed. [ABSTRACT FROM AUTHOR]

It is proved that monetary utility functions that are commonotonic and time-consistent are conditional expectations. We also give additional results on atomless and conditionally atomless probability spaces. These notions describe that in a filtration, there are many new events at each time step. [ABSTRACT FROM AUTHOR]

We study cooperative solutions for differential games where players consume a common property resource. Players are asymmetric, in the sense that they have different preferences and, in particular, different time preferences. We propose a new time-consistent dynamic bargaining procedure for this class of games. This solution concept, which is defined as the time-consistent dynamic bargaining (TCB) solution, extends the recursive Nash bargaining solution introduced in Sorger (J Econ Dyn Control 30:2637–2659, 2006) to a continuous time setting. The underlying idea is that, in case of disagreement, the threat is that players will play a noncooperative Markov Perfect Nash equilibrium just during a very small period of time, since new negotiations can take place at every future moment and, in particular, immediately later. Conditions for interior TCB solutions are derived. To illustrate the results, two common property resource games are analyzed in detail. [ABSTRACT FROM AUTHOR]

We introduce the concept of forward rank‐dependent performance criteria, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time‐consistent nature of forward performance criteria with the time‐inconsistency stemming from probability distortions. For this, we first propose two distinct definitions, one based on the preservation of performance value and the other on the time‐consistency of policies and, in turn, establish their equivalence. We then fully characterize the viable class of probability distortion processes and provide the following dichotomy: it is either the case that the probability distortions are degenerate in the sense that the investor would never invest in the risky assets, or the marginal probability distortion equals to a normalized power of the quantile function of the pricing kernel. We also characterize the optimal wealth process, whose structure motivates the introduction of a new, 'distorted' measure and a related market. We then build a striking correspondence between the forward rank‐dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection also provides a direct construction method for forward rank‐dependent criteria. Finally, our results on forward rank‐dependent performance criteria motivate us to revisit the classical (backward) setting. We follow the so‐called dynamic utility approach and derive conditions for existence and a construction of dynamic rank‐dependent utility processes. [ABSTRACT FROM AUTHOR]

Scalar dynamic risk measures for univariate positions in continuous time are commonly represented via backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete-time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time. [ABSTRACT FROM AUTHOR]

Abstract Background In the absence of horizontal gene transfer it is possible to reconstruct the history of gene families from empirically determined orthology relations, which are equivalent to event-labeled gene trees. Knowledge of the event labels considerably simplifies the problem of reconciling a gene tree T with a species trees S, relative to the reconciliation problem without prior knowledge of the event types. It is well-known that optimal reconciliations in the unlabeled case may violate time-consistency and thus are not biologically feasible. Here we investigate the mathematical structure of the event labeled reconciliation problem with horizontal transfer. Results We investigate the issue of time-consistency for the event-labeled version of the reconciliation problem, provide a convenient axiomatic framework, and derive a complete characterization of time-consistent reconciliations. This characterization depends on certain weak conditions on the event-labeled gene trees that reflect conditions under which evolutionary events are observable at least in principle. We give an $$\mathcal {O}(|V(T)|\log (|V(S)|))$$ O(|V(T)|log(|V(S)|)) -time algorithm to decide whether a time-consistent reconciliation map exists. It does not require the construction of explicit timing maps, but relies entirely on the comparably easy task of checking whether a small auxiliary graph is acyclic. The algorithms are implemented in C++ using the boost graph library and are freely available at https://github.com/Nojgaard/tc-recon. Significance The combinatorial characterization of time consistency and thus biologically feasible reconciliation is an important step towards the inference of gene family histories with horizontal transfer from orthology data, i.e., without presupposed gene and species trees. The fast algorithm to decide time consistency is useful in a broader context because it constitutes an attractive component for all tools that address tree reconciliation problems.

The transformation of characteristic functions is an effective way to avoid time-inconsistency of cooperative solutions in dynamic games. There are several forms on the transformation of characteristic functions. In this paper, a class of general transformation of characteristic functions is proposed. It can lead to the time-consistency of cooperative solutions and guarantee that the irrational-behavior-proof conditions hold true. To illustrate the theory, an example of dynamic game on a tree is given. [ABSTRACT FROM AUTHOR]

This article focuses on an optimal asset allocation problem with heterogeneous discounting and stochastic income under the constant elasticity of variance (CEV) model. Heterogeneous discounting is an important non-constant discounting model, which can describe the fact that a decision maker discounts in different ways the utility derived from consumption and that of the bequest or final function. It is consistent with the fact that the concern of a decision maker about the bequest left to her descendants when she is young is not the same as that when she is old. In our model, a decision maker with stochastic income can enjoy the consumption, purchase life insurance, and invest her wealth in a risk-free asset and a risky asset whose price process satisfies the CEV model. Meanwhile, the volatility of the stochastic income arises from the risky asset. Since the problem is time-inconsistent, the Bellman's principle of optimality does not hold. To obtain the time-consistent solution, an equilibrium strategy is calculated. By applying the game theoretic framework and solving an extended Hamilton-Jacobi-Bellman system, we derive the time-consistent consumption, investment, and life insurance strategies for both exponential and logarithmic utility functions. Finally, we provide numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]

The study of Ireland (J Monet Econ 44:279–291, 1999) derives the restrictions imposed by Barro and Gordon's theory of time-consistent monetary policy on a bivariate time-series model for US inflation and unemployment rate. The model is then estimated via maximum likelihood techniques using quarterly data from 1960q1 to 1997q2. In this study, we reproduce the central results of Ireland (1999) using the Metropolis–Hastings algorithm. Although we apply Bayesian instead of classical estimation, posterior parameter estimates are similar to maximum likelihood parameter estimates reported in Ireland (1999). [ABSTRACT FROM AUTHOR]

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes, as well as the corresponding optimal portfolios, with power-utility initial data and for stock–factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. This is done by solving the associated nonlinear parabolic partial differential equations (PDEs) posed in the "wrong" time direction. Along the way, we establish on domains an explicit form of the generalised Widder theorem of Nadtochiy and Tehranchi (Math. Finance 27:438–470, 2015, Theorem 3.12) and rely for that on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the "right" time direction. [ABSTRACT FROM AUTHOR]

We introduce a hybrid metaphor for the visualization of the reconciliations of co-phylogenetic trees, that are mappings among the nodes of two trees with constraints on the leaves. The typical application is the visualization of the co-evolution of hosts and parasites in biology. Our strategy combines a space-filling and a node-link approach. Differently from traditional methods, it guarantees an unambiguous and downward representation whenever the reconciliation is time-consistent (i.e., biologically-feasible). We address the problem of the minimization of the number of crossings in the representation, by giving a characterization of planar instances and by establishing the complexity of the problem. Finally, we propose heuristics for computing representations with few crossings. [ABSTRACT FROM AUTHOR]

This work is aimed at studying the problem of maintaining the sustainability of a cooperative solution in an n-person hybrid differential game. Specifically, we consider a differential game whose payoff function is discounted with a discounting function that changes its structure with time. We solve the problem of time-inconsistency of the cooperative solution using a so-called imputation distribution procedure, which was adjusted for this general class of differential games. The obtained results are illustrated with a specific example of a differential game with random duration and a hybrid cumulative distribution function (CDF). We completely solved the presented example to demonstrate the application of the developed scheme in detail. All results were obtained in analytical form and illustrated by numerical simulations.

• The non-zero sum stochastic differential game between two insurers is established. • Relative performance concerns between two competitive insurers are considered. • The volatility risk of finance market is considered. • Explicit solutions for time-consistent Nash equilibrium strategies are obtained. This paper considers the optimal time-consistent investment and reinsurance strategies for two mean-variance insurers subject to the relative performance concerns. Each insurer can purchase a reinsurance protection and invest in a financial market consisted of one risk-free asset and one risky asset. We assume that the price process of risky asset is driven by the Heston model. The main objective of each insurer is to choose a investment and reinsurance strategy such that the mean and variance of his relative terminal wealth with respect to that of his competitor is maximized and minimized, simultaneously. By applying the stochastic control theory, closed-form expressions for the equilibrium investment-reinsurance strategies and corresponding value functions are derived. Finally, we provide some numerical studies and draw some economic interpretations. [ABSTRACT FROM AUTHOR]

The question of measuring and managing systemic risk—especially in view of the recent financial crisis—became more and more important. We study systemic risk by taking the perspective of a financial regulator and considering the axiomatic approach originally introduced in Chen et al. (Manag Sci 59(6):1373–1388, 2013) and extended in Kromer et al. (Math Methods Oper Res 84:323–357, 2016). The aim of this paper is to generalize the static approach in Kromer et al. (2016) and analyze systemic risk measures in a dynamic setting. We work in the framework of Cheridito et al. (Electron J Probab 11:57–106, 2006) who consider risk measures for bounded discrete-time processes. Apart from the possibility to consider the "evolution of financial values", another important advantage of the dynamic approach is the possibility to incorporate information in the risk measurement and management process. In context of this dynamic setting we also discuss the arising question of time-consistency for our dynamic systemic risk measures. [ABSTRACT FROM AUTHOR]

Quite recently, a great interest has been devoted to time-consistency of risk measures in its different formulations (see Delbaen in Memoriam Paul-André Meyer, Lecture notes in mathematics, vol 1874, pp 215–258, 2006; Föllmer and Penner in Stat Decis 14(1):1–15, 2006; Bion-Nadal in Stoch Process Appl 119:633–654, 2009; Delbaen et al. in Finance Stoch 14(3):449–472, 2010; Laeven and Stadje in Math Oper Res 39:1109–1141, 2014, among many others). However, almost all the papers address to coherent or convex risk measures satisfying cash-additivity. In the present work, we study time-consistency for more general dynamic risk measures where either only cash-invariance or both cash-invariance and convexity are dropped. This analysis is motivated by the recent papers of El Karoui and Ravanelli (Math Finance 19:561–590, 2009) and Cerreia-Vioglio et al. (Math Finance 21(4):743–774, 2011) who discussed and weakened the axioms above by introducing cash-subadditivity and quasi-convexity. In particular, we investigate and discuss whether the notion of time-consistency is too restrictive, when considered in the general framework of quasi-convex and cash-subadditive risk measures. Finally, we provide some conditions guaranteeing time-consistency in this more general framework. [ABSTRACT FROM AUTHOR]

We establish new conditions under which a constrained (no short-selling) time-consistent equilibrium strategy, starting at a certain time, will beat the unconstrained counterpart, as measured by the magnitude of their corresponding equilibrium mean-variance value functions. We further show that the pure strategy of solely investing in a risk-free bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, we also illustrate that the constrained strategy can dominate the unconstrained one for most of the commencement dates (even more than 90%) of a prescribed planning horizon. Under a precommitment approach, the value function of an investor increases with the size of the admissible sets of strategies. However, this may fail to be true under the game-theoretic paradigm, as the constraint of time-consistency itself affects the value function differently when short-selling is and is not prohibited. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent claims in incomplete markets. Our conditions do not rely on the notions of stability under pasting or time-consistency and reveal some unexpected connection between the minimax result and path properties of the corresponding process of densities. We exemplify our general results in the case of families of measures corresponding to diffusion exponential martingales. [ABSTRACT FROM AUTHOR]

We combine forward investment performance processes and ambiguity-averse portfolio selection. We introduce robust forward criteria which address ambiguity in the specification of the model, the risk preferences and the investment horizon. They encode the evolution of dynamically consistent ambiguity-averse preferences.We focus on establishing dual characterisations of the robust forward criteria, which is advantageous as the dual problem amounts to the search for an infimum whereas the primal problem features a saddle point. Our approach to duality builds on ideas developed in Schied (Finance Stoch. 11:107-129, 2007) and Žitković (Ann. Appl. Probab. 19:2176-2210, 2009). We also study in detail the so-called time-monotone criteria. We solve explicitly the example of an investor who starts with logarithmic utility and applies a quadratic penalty function. Such an investor builds a dynamic estimate of the market price of risk λˆ and updates her stochastic utility in accordance with the so-perceived elapsed market opportunities. We show that this leads to a time-consistent optimal investment policy given by a fractional Kelly strategy associated with λˆ and with the leverage being proportional to the investor’s confidence in her estimate. [ABSTRACT FROM AUTHOR]

We consider the problem of converting batch estimators into a sequential predictor or estimator with small extra regret. Formally this is the problem of merging a collection of probability measures over strings of length 1 , 2 , 3 , . . . into a single probability measure over infinite sequences. We describe various approaches and their pros and cons on various examples. As a side-result we give an elementary non-heuristic purely combinatoric derivation of Turing's famous estimator. Our main technical contribution is to determine the computational complexity of sequential estimators with good guarantees in general. We conclude with an open problem on how to derive tractable sequential from batch estimators with good guarantees in general. [ABSTRACT FROM AUTHOR]

We study the worst portfolios for a class of law invariant dynamic monetary utility functions with domain in a class of stochastic processes. The concept of comonotonicity is introduced for these processes in order to prove the existence of worst portfolios. Using robust representations of monetary utility function processes in discrete time, a relation between the worst portfolios at different periods of time is presented. Finally, we study conditions to achieve the maximum in the representation theorems for concave monetary utility functions that are continuous for bounded decreasing sequences. [ABSTRACT FROM AUTHOR]

In this paper we study optimal stopping problems with respect to distorted expectations with concave distortion functions. Our starting point is a seminal work of Xu and Zhou in 2013, who gave an explicit solution of such a stopping problem under a rather large class of distortion functionals. In this paper, we continue this line of research and prove a novel representation, which relates the solution of an optimal stopping problem under distorted expectation to the sequence of standard optimal stopping problems and hence makes the application of the standard dynamic programming-based approaches possible. Furthermore, by means of the well-known Kusuoka representation, we extend our results to optimal stopping under general law invariant coherent risk measures. Finally, based on our representations, we develop several Monte Carlo approximation algorithms and illustrate their power for optimal stopping under absolute semideviation risk measures. [ABSTRACT FROM AUTHOR]

This paper study an optimal time-consistent reinsurance-investment strategy selection problem in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process. The aggregate claim process and the price process of risky asset are correlated by a common Poisson process. The objective of the insurer is to choose an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus. Since this problem is time-inconsistent, we attack it by placing the problem within a game theoretic framework and looking for subgame perfect Nash equilibrium strategy. We investigate the problem using the extended Hamilton–Jacobi–Bellman dynamic programming approach. Closed-form solutions for the optimal reinsurance-investment strategy and the corresponding value functions are obtained. Numerical examples and economic significance analysis are also provided to illustrate how the optimal reinsurance-investment strategy changes when some model parameters vary. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by the Hamilton-Jacobi-Bellman equation on a semi-infinite time interval. In the case where this equation can be linearized, the problem reduces to a time-reversed parabolic equation, which cannot be analyzed via the standard methods of partial differential equations. Under the additional uniform ellipticity condition, we make use of the available description of all minimal solutions to such equations, along with some basic facts from potential theory and convex analysis, to obtain an explicit integral representation of all positive solutions. These results allow us to construct a large family of the aforementioned optimality criteria, including some closed-form examples in relevant financial models. [ABSTRACT FROM AUTHOR]

In this paper, which is a continuation of the discrete-time paper (Björk and Murgoci in Finance Stoch. 18:545-592, 2004), we study a class of continuous-time stochastic control problems which, in various ways, are time-inconsistent in the sense that they do not admit a Bellman optimality principle. We study these problems within a game-theoretic framework, and we look for Nash subgame perfect equilibrium points. For a general controlled continuous-time Markov process and a fairly general objective functional, we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of nonlinear equations, for the determination of the equilibrium strategy as well as the equilibrium value function. The main theoretical result is a verification theorem. As an application of the general theory, we study a time-inconsistent linear-quadratic regulator. We also present a study of time-inconsistency within the framework of a general equilibrium production economy of Cox-Ingersoll-Ross type (Cox et al. in Econometrica 53:363-384, 1985). [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of the optimal time-consistent investment and proportional reinsurance strategy under the mean-variance criterion, in which the insurer has some inside information at her disposal concerning the future realizations of her claims process. It is assumed that the surplus of the insurer is governed by a Brownian motion with drift, and the insurer has the possibility to reduce the risk by purchasing proportional reinsurance and investing in financial markets. We first formulate the problem and provide a verification theorem on the extended Hamilton-Jacobi-Bellman equations. Then, the closed-form expression is obtained for the optimal strategy of the optimization problem. [ABSTRACT FROM AUTHOR]

In this paper we study asymptotically stable risk assessments (or equivalently risk measures) which have the property that an unacceptable position cannot become acceptable by adding a huge cash-flow far in the future. Under an additional continuity assumption, these risk assessments are exactly those which have a robust representation in terms of test probabilities that are supported on a finite time interval. For time-consistent risk assessments we give conditions on their generators which guarantee asymptotic stability. [ABSTRACT FROM AUTHOR]

We consider an optimal time-consistent reinsurance-investment strategy selection problem for an insurer whose surplus is governed by a compound Poisson risk model. In our model, the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a simplified financial market consisting of a risk-free asset and a risky stock. The dynamics of the risky stock is governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity as well as the feedback effect of an asset’s price on its volatility. The objective of the insurer is to choose an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus. We investigate the problem using the Hamilton-Jacobi-Bellman dynamic programming approach. Closed-form solutions for the optimal reinsurance-investment strategies and the corresponding value functions are obtained in both the compound Poisson risk model and its diffusion approximation. Numerical examples are also provided to illustrate how the optimal reinsurance-investment strategy changes when some model parameters vary. [ABSTRACT FROM AUTHOR]