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Zuo Quan Xu

Publications and source records attributed to Zuo Quan Xu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Constrained stochastic linear quadratic control under regime switching with controlled jump size

In this paper, we examine a stochastic linear-quadratic control problem characterized by regime switching and Poisson jumps. All the coefficients in the problem are random processes adapted to the filtration generated by Brownian motion and the Poisson random measure for each given regime. The model incorporates two distinct types of controls: the first is a conventional control that appears in the continuous diffusion component, while the second is an unconventional control, dependent on the variable $z$, which influences the jump size in the jump diffusion component. Both controls are constrained within general closed cones. By employing the Meyer-Itô formula in conjunction with a generalized squares completion technique, we rigorously and explicitly derive the optimal value and optimal feedback control. These depend on solutions to certain multi-dimensional fully coupled stochastic Riccati equations, which are essentially backward stochastic differential equations with jumps (BSDEJs). We establish the existence of a unique nonnegative solution to the BSDEJs. One of the major tools used in the proof is the newly established comparison theorems for multidimensional BSDEJs.

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arXiv2024

Stochastic optimal self-path-dependent control: A new type of variational inequality and its viscosity solution

In this paper, we explore a new class of stochastic control problems characterized by specific control constraints. Specifically, the admissible controls are subject to the ratcheting constraint, meaning they must be non-decreasing over time and are thus self-path-dependent. This type of problems is common in various practical applications, such as optimal consumption problems in financial engineering and optimal dividend payout problems in actuarial science. Traditional stochastic control theory does not readily apply to these problems due to their unique self-path-dependent control feature. To tackle this challenge, we introduce a new class of Hamilton-Jacobi-Bellman (HJB) equations, which are variational inequalities concerning the derivative of a new spatial argument that represents the historical maximum control value. Under the standard Lipschitz continuity condition, we demonstrate that the value functions for these self-path-dependent control problems are the unique solutions to their corresponding HJB equations in the viscosity sense.

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arXiv2024

Policy Iteration Reinforcement Learning Method for Continuous-Time Linear-Quadratic Mean-Field Control Problems

This paper employs a policy iteration reinforcement learning (RL) method to study continuous-time linear-quadratic mean-field control problems in infinite horizon. The drift and diffusion terms in the dynamics involve the states, the controls, and their conditional expectations. We investigate the stabilizability and convergence of the RL algorithm using a Lyapunov Recursion. Instead of solving a pair of coupled Riccati equations, the RL technique focuses on strengthening an auxiliary function and the cost functional as the objective functions and updating the new policy to compute the optimal control via state trajectories. A numerical example sheds light on the established theoretical results.

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arXiv2024

Multidimensional indefinite stochastic Riccati equations and zero-sum stochastic linear-quadratic differential games with non-Markovian regime switching

This paper is concerned with zero-sum stochastic linear-quadratic differential games in a regime switching model. The coefficients of the games depend on the underlying noises, so it is a non-Markovian regime switching model. Based on the solutions of a new kind of multidimensional indefinite stochastic Riccati equation (SRE) and a multidimensional linear backward stochastic differential equation (BSDE) with unbounded coefficients, we provide closed-loop optimal feedback control-strategy pairs for the two players. The main contribution of this paper, which is of great importance in its own right from the BSDE theory point of view, is to prove the existence and uniqueness of the solution to the new kind of SRE. Notably, the first component of the solution (as a process) is capable of taking positive and negative values simultaneously. For homogeneous systems, we obtain the optimal feedback control-strategy pairs under general closed convex cone control constraints. Finally, these results are applied to portfolio selection games with full or partial no-shorting constraint in a regime switching market with random coefficients.

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arXiv2024

Competitive optimal portfolio selection in a non-Markovian financial market: A backward stochastic differential equation study

This paper studies a competitive optimal portfolio selection problem in a model where the interest rate, the appreciation rate and volatility rate of the risky asset are all stochastic processes, thus forming a non-Markovian financial market. In our model, all investors (or agents) aim to obtain an above-average wealth at the end of the common investment horizon. This competitive optimal portfolio problem is indeed a non-zero stochastic differential game problem. The quadratic BSDE theory is applied to tackle the problem and Nash equilibria in suitable spaces are found. We discuss both the CARA and CRRA utility cases. For the CARA utility case, there are three possible scenarios depending on market and competition parameters: a unique Nash equilibrium, no Nash equilibrium, and infinite Nash equilibria. The Nash equilibrium is given by the solutions of a quadratic BSDE and a linear BSDE with unbounded coefficient when it is unique. Different from the wealth-independent Nash equilibria in the existing literature, the equilibrium in our paper is of feedback form of wealth. For the CRRA utility case, the issue is a bit more complicated than the CARA utility case. We prove the solvability of a new kind of quadratic BSDEs with unbounded coefficients. A decoupling technology is used to relate the Nash equilibrium to a series of 1-dimensional quadratic BSDEs. With the help of this decoupling technology, we can even give the limiting strategies for both cases when the number of agent tends to be infinite.

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arXiv2024

Optimal ratcheting of dividend payout under Brownian motion surplus

This paper is concerned with a long standing optimal dividend payout problem subject to the so-called ratcheting constraint, that is, the dividend payout rate shall be non-decreasing over time and is thus self-path-dependent. The surplus process is modeled by a drifted Brownian motion process and the aim is to find the optimal dividend ratcheting strategy to maximize the expectation of the total discounted dividend payouts until the ruin time. Due to the self-path-dependent control constraint, the standard control theory cannot be directly applied to tackle the problem. The related Hamilton-Jacobi-Bellman (HJB) equation is a new type of variational inequality. In the literature, it is only shown to have a viscosity solution, which is not strong enough to guarantee the existence of an optimal dividend ratcheting strategy. This paper proposes a novel partial differential equation method to study the HJB equation. We not only prove the the existence and uniqueness of the solution in some stronger functional space, but also prove the strict monotonicity, boundedness, and $C^\infty$-smoothness of the dividend ratcheting free boundary. Based on these results, we eventually derive an optimal dividend ratcheting strategy, and thus solve the open problem completely. Economically speaking, we find that if the surplus volatility is above an explicit threshold, then one should pay dividends at the maximum rate, regardless the surplus level. Otherwise, by contrast, the optimal dividend ratcheting strategy relays on the surplus level and one should only ratchet up the dividend payout rate when the surplus level touches the dividend ratcheting free boundary. Moreover, our numerical results suggest that one should invest into those companies with stable dividend payout strategies since their income rates should be higher and volatility rates smaller.

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arXiv2024

Constrained mean-variance investment-reinsurance under the Cramér-Lundberg model with random coefficients

In this paper, we study an optimal mean-variance investment-reinsurance problem for an insurer (she) under a Cramér-Lundberg model with random coefficients. At any time, the insurer can purchase reinsurance or acquire new business and invest her surplus in a security market consisting of a risk-free asset and multiple risky assets, subject to a general convex cone investment constraint. We reduce the problem to a constrained stochastic linear-quadratic control problem with jumps whose solution is related to a system of partially coupled stochastic Riccati equations (SREs). Then we devote ourselves to establishing the existence and uniqueness of solutions to the SREs by pure backward stochastic differential equation (BSDE) techniques. We achieve this with the help of approximation procedure, comparison theorems for BSDEs with jumps, log transformation and BMO martingales. The efficient investment-reinsurance strategy and efficient mean-variance frontier are explicitly given through the solutions of the SREs, which are shown to be a linear feedback form of the wealth process and a half-line, respectively.

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arXiv2024

Mean-variance portfolio selection in jump-diffusion model under no-shorting constraint: A viscosity solution approach

This paper concerns a continuous time mean-variance (MV) portfolio selection problem in a jump-diffusion financial model with no-shorting trading constraint. The problem is reduced to two subproblems: solving a stochastic linear-quadratic (LQ) control problem under control constraint, and finding a maximal point of a real function. Based on a two-dimensional fully coupled ordinary differential equation (ODE), we construct an explicit viscosity solution to the Hamilton-Jacobi-Bellman equation of the constrained LQ problem. Together with the Meyer-Itô formula and a verification procedure, we obtain the optimal feedback controls of the constrained LQ problem and the original MV problem, which corrects the flawed results in some existing literatures. In addition, closed-form efficient portfolio and efficient frontier are derived. In the end, we present several examples where the two-dimensional ODE is decoupled.

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arXiv2024

Constrained monotone mean--variance investment-reinsurance under the Cramér--Lundberg model with random coefficients

This paper studies an optimal investment-reinsurance problem for an insurer (she) under the Cramér--Lundberg model with monotone mean--variance (MMV) criterion. At any time, the insurer can purchase reinsurance (or acquire new business) and invest in a security market consisting of a risk-free asset and multiple risky assets whose excess return rate and volatility rate are allowed to be random. The trading strategy is subject to a general convex cone constraint, encompassing no-shorting constraint as a special case. The optimal investment-reinsurance strategy and optimal value for the MMV problem are deduced by solving certain backward stochastic differential equations with jumps. In the literature, it is known that models with MMV criterion and mean--variance criterion lead to the same optimal strategy and optimal value when the wealth process is continuous. Our result shows that the conclusion remains true even if the wealth process has compensated Poisson jumps and the market coefficients are random.

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arXiv2023

Comparison theorems for multi-dimensional BSDEs with jumps and applications to constrained stochastic linear-quadratic control

In this paper, we, for the first time, establish two comparison theorems for multi-dimensional backward stochastic differential equations with jumps. Our approach is novel and completely different from the existing results for one-dimensional case. Using these and other delicate tools, we then construct solutions to coupled two-dimensional stochastic Riccati equation with jumps in both standard and singular cases. In the end, these results are applied to solve a cone-constrained stochastic linear-quadratic and a mean-variance portfolio selection problem with jumps. Different from no jump problems, the optimal (relative) state processes may change their signs, which is of course due to the presence of jumps.

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arXiv2023

Optimal Management of DC Pension Plan with Inflation Risk and Tail VaR Constraint

This paper investigates an optimal investment problem under the tail Value at Risk (tail VaR, also known as expected shortfall, conditional VaR, average VaR) and portfolio insurance constraints confronted by a defined-contribution pension member. The member's aim is to maximize the expected utility from the terminal wealth exceeding the minimum guarantee by investing his wealth in a cash bond, an inflation-linked bond and a stock. Due to the presence of the tail VaR constraint, the problem cannot be tackled by standard control tools. We apply the Lagrange method along with quantile optimization techniques to solve the problem. Through delicate analysis, the optimal investment output in closed-form and optimal investment strategy are derived. A numerical analysis is also provided to show how the constraints impact the optimal investment output and strategy.

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arXiv2023

Constrained monotone mean-variance problem with random coefficients

This paper studies the monotone mean-variance (MMV) problem and the classical mean-variance (MV) problem with convex cone trading constraints in a market with random coefficients. We provide semiclosed optimal strategies and optimal values for both problems via certain backward stochastic differential equations (BSDEs). After noting the links between these BSDEs, we find that the two problems share the same optimal portfolio and optimal value. This generalizes the result of Shen and Zou $[$ SIAM J. Financial Math., 13 (2022), pp. SC99-SC112$]$ from deterministic coefficients to random ones.

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arXiv2023

Non-homogeneous stochastic LQ control with regime switching and random coefficients

This paper is concerned with a general non-homogeneous stochastic linear quadratic (LQ) control problem with regime switching and random coefficients. We obtain the explicit optimal state feedback control and optimal value for this problem in terms of two systems of backward stochastic differential equations (BSDEs): one is the famous stochastic Riccati equation and the other one is a new linear multi-dimensional BSDE with all coefficients being unbounded. The existence and uniqueness of the solutions to these two systems of BSDEs are proved by means of BMO martingales and contraction mapping method. At last, the theory is applied to study an asset-liability management problem under the mean-variance criteria.

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arXiv2023

Robust utility maximization with intractable claims

We study a continuous-time expected utility maximization problem in which the investor at maturity receives the value of a contingent claim in addition to the investment payoff from the financial market. The investor knows nothing about the claim other than its probability distribution, hence an ``intractable claim''. In view of the lack of necessary information about the claim, we consider a robust formulation to maximize her utility in the worst scenario. We apply the quantile formulation to solve the problem, expressing the quantile function of the optimal terminal investment income as the solution of certain variational inequalities of ordinary differential equations and obtaining the resulting optimal trading strategy. In the case of an exponential utility, the problem reduces to a (non-robust) rank--dependent utility maximization with probability distortion whose solution is available in the literature. The results can also be used to determine the utility indifference price of the intractable claim.

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arXiv2023

Continuous-time Markowitz's mean-variance model under different borrowing and saving rates

We study Markowitz's mean-variance portfolio selection problem in a continuous-time Black-Scholes market with different borrowing and saving rates. The associated Hamilton-Jacobi-Bellman equation is fully nonlinear. Using a delicate partial differential equation and verification argument, the value function is proven to be $C^{3,2}$ smooth. It is also shown that there are a borrowing boundary and a saving barrier which divide the entire trading area into a borrowing-money region, an all-in-stock region, and a saving-money region in ascending order. The optimal trading strategy is a mixture of continuous-time strategy (as suggested by most continuous-time models) and discontinuous-time strategy (as suggested by models with transaction costs): one should put all her wealth in the stock in the middle all-in-stock region, and continuously trade it in the other two regions in a feedback form of wealth and time. It is never optimal to short sale the stock. Numerical examples are also presented to verify the theoretical results and to give more financial insights beyond them.

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arXiv2023

Optimal moral-hazard-free reinsurance under extended distortion premium principles

We study an optimal reinsurance problem under a diffusion risk model for an insurer who aims to minimize the probability of lifetime ruin. To rule out moral hazard issues, we only consider moral-hazard-free reinsurance contracts by imposing the incentive compatibility constraint on indemnity functions. The reinsurance premium is calculated under an extended distortion premium principle, in which the distortion function is not necessarily concave. We first show that an optimal reinsurance contract always exists and then derive two sufficient and necessary conditions to characterize it. Due to the presence of the incentive compatibility constraint and the nonconcavity of the distortion, the optimal contract is obtained as a solution to a double obstacle problem. At last, we apply the general result to study three examples and obtain the optimal contract in (semi)closed form.

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arXiv2022

Optimal consumption-investment with coupled constraints on consumption and investment strategies in a regime switching market with random coefficients

This paper studies finite-time optimal consumption-investment problems with power, logarithmic and exponential utilities, in a regime switching market with random coefficients, subject to coupled constraints on the consumption and investment strategies. We provide explicit optimal consumption-investment strategies and optimal values for the problems in terms of the solutions to some diagonally quadratic backward stochastic differential equation (BSDE) systems and linear BSDE systems with unbound coefficients. Some of these BSDEs are new in the literature and solving them is one of the main theoretical contributions of this paper. We accomplish the latter by applying the truncation, approximation technique to get some a priori uniformly lower and upper bounds for their solutions.

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arXiv2022

Relative growth rate optimization under behavioral criterion

This paper studies a continuous-time optimal portfolio selection problem in the complete market for a behavioral investor whose preference is of the prospect type with probability distortion. The investor concerns about the terminal relative growth rate (log-return) instead of absolute capital value. This model can be regarded as an extension of the classical growth optimal problem to the behavioral framework. It leads to a new type of M-shaped utility maximization problem under nonlinear Choquet expectation. Due to the presence of probability distortion, the classical stochastic control methods are not applicable. By the martingale method, concavification and quantile optimization techniques, we derive the closed-form optimal growth rate. We find that the benchmark growth rate has a significant impact on investment behaviors. Compared to Zhang et al where the same preference measure is applied to the terminal relative wealth, we find a new phenomenon when the investor's risk tolerance level is high and the market states are bad. In addition, our optimal wealth in every scenario is less sensitive to the pricing kernel and thus more stable than theirs.

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