arXiv subjects

Jingzhen Liu

Publications and source records attributed to Jingzhen Liu.

4 recordsLinked to original sources

arXiv2025

Equilibrium strategies for stochastic control problems with higher-order moments and applications to portfolio selection

In this paper we derive a novel characterization result for time-consistent stochastic control problems with higher-order moments, originally formulated by Wang et al. [SIAM J. Control. Optim., 63 (2025), 1560--1589], and newly explore many solvable instances including a mean-variance-excess kurtosis portfolio selection problem. By improving an asymptotic result of the variational process for the uniform boundedness and integrability properties, we obtain both the sufficiency and necessity of an equilibrium condition for an open-loop Nash equilibrium control (ONEC). This condition is simply formulated by the diagonal processes of a flow of backward stochastic differential equations (BSDEs) whose data do not necessarily satisfy the usual square-integrability condition. In particular, for linear controlled dynamics with deterministic parameters, we show that the ONEC can be derived by solving a polynomial algebraic equation under a class of nonlinear objective functions. Interestingly, the mean-variance equilibrium strategy is an ONEC for our general higher-order moment problem if and only if a homogeneity condition holds. Additionally, in the case with random parameters, we characterize the ONEC by finitely many BSDEs with a recurrence relation. As an intuitive illustration, the solution to the mean-variance-skewness problems is given by a quadratic BSDE.

math.OC↗

arXiv2025

Time-consistent portfolio selection with monotone mean-variance preferences

We investigate time-inconsistent portfolio problems under a broader class of monotone mean-variance (MMV) preferences. Since the optimal strategies for MMV and mean-variance (MV) preferences coincide, the MMV optimal strategies at different initial times are necessarily time-inconsistent. To address this time-inconsistency, we consider Nash equilibrium controls of both open-loop and closed-loop types, and characterize them within a random parameter setting. The two control problems reduce to solving a flow of forward-backward stochastic differential equations and a system of extended Hamilton-Jacobi-Bellman equations, respectively. In particular, we derive semi-closed-form solutions for both types of equilibria under a deterministic parameter setting, and both solutions share the same representation, which is independent of the wealth state and the random path. We show that the investment amount under the MMV equilibrium exceeds that under the MV equilibrium, and the gap narrows over time. Furthermore, under a constant parameter setting, we find that the derived closed-loop Nash equilibrium control is a strong equilibrium strategy only when the interest rate is sufficiently large, whereas the derived open-loop Nash equilibrium control is necessarily a strong equilibrium strategy.

math.OC↗

arXiv2025

On stochastic control problems with higher-order moments

In this paper, we focus on a class of time-inconsistent stochastic control problems, where the objective function includes the mean and several higher-order central moments of the terminal value of state. To tackle the time-inconsistency, we seek both the closed-loop and the open-loop Nash equilibrium controls as time-consistent solutions. We establish a partial differential equation (PDE) system for deriving a closed-loop Nash equilibrium control, which does not include the equilibrium value function and is different from the extended Hamilton-Jacobi-Bellman (HJB) equations as in Björk et al. (Finance Stoch. 21: 331-360, 2017). We show that our PDE system is equivalent to the extended HJB equations that seems difficult to be solved for our higher-order moment problems. In deriving an open-loop Nash equilibrium control, due to the non-separable higher-order moments in the objective function, we make some moment estimates in addition to the standard perturbation argument for developing a maximum principle. Then, the problem is reduced to solving a flow of forward-backward stochastic differential equations. In particular, we investigate linear controlled dynamics and some objective functions affine in the mean. The closed-loop and the open-loop Nash equilibrium controls are identical, which are independent of the state value, random path and the preference on the odd-order central moments. By sending the highest order of central moments to infinity, we obtain the time-consistent solutions to some control problems whose objective functions include some penalty functions for deviation.

q-fin.MF↗

arXiv2024

Strictly monotone mean-variance preferences with applications to portfolio selection

The monotone mean-variance (MMV) preference proposed by Maccheroni, et al. (Math. Finance 19(3): 487-521, 2009) fails to differentiate strictly dominant payoffs, which may cause inconsistency in portfolio decision-making. This paper introduces a broader class of strictly monotone mean-variance (SMMV) preferences and demonstrates its applications to portfolio selection problems. For the single-period portfolio problem under the SMMV preference, we derive the gradient condition for the optimal strategy, and investigate its association with the optimal mean-variance (MV) static strategy. We reduce the problem to solving a set of linear equations by analyzing the saddle point of some minimax problem. And results show that the optimal SMMV, MMV and MV strategies differ significantly in the single-period problem. Furthermore, we conduct numerical experiments and compare our results with those of Maccheroni, et al. (Math. Finance 19(3): 487-521, 2009). The findings indicate that our SMMV preferences provide a more rational basis for assessing given prospects. For the continuous-time portfolio problem under the SMMV preference, we consider continuous price processes with random coefficients, and establish a novel approach based on a general convex duality analysis to derive the optimal strategy. Interestingly, we find that the optimal strategies for SMMV, MMV and MV preferences coincide under a certain condition, and provide a classical microeconomic interpretation for this condition. We also characterize the optimal SMMV portfolio strategies relying on stochastic control techniques to facilitate potential extensions and refinements in future research.

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