arXiv subjects

Shuaijie Qian

Publications and source records attributed to Shuaijie Qian.

6 recordsLinked to original sources

arXiv2026

C^{2,α} solution to the proportional transaction cost problem with two risky assets

This paper concerns the variational inequality arising from a specific singular control problem: portfolio selection under proportional transaction costs. This variational inequality is a gradient-constrained partial differential equation (PDE). Generally, the literature only guarantees the W{2,\infty} regularity of the solution to such a PDE, and counterexamples exist for higher regularity. In this paper, by exploiting the concavity of the value function, we connect this gradient-constrained problem to an obstacle problem, and finally show that the solution is C^{2,α}. This paper concerns a two-dimensional portfolio selection setup, but our approach can potentially be extended to more general multi-dimensional singular control problems when the value function is concave.

math.OC↗

arXiv2026

Regularity of a Multidimensional Principal-Agent Problem with Separable Effort Costs

This paper studies the regularity of the value function arising from a multidimensional continuous-time principal-agent model with separable, nonquadratic effort costs. The associated stochastic control problem has the output and the agent's continuation utility as state variables, and its Hamilton-Jacobi-Bellman equation is fully nonlinear and degenerate, with potentially unbounded coefficients. We address these difficulties by adding an independent regularization noise and bounding the effort. For the resulting problem, we establish classical regularity of the value function and show that the optimal effort is unique, positive and remains in a fixed compact subset, uniformly with respect to both the control restriction and the regularization parameter. These estimates allow us first to remove the control restriction and then to let the additional noise vanish. Consequently, we prove that the regularized value function converges to the original value function and conclude that the latter belongs locally to the Sobolev space $W^{2,1}_{\infty, \mathrm{loc}}$, thereby extending the regularity analysis to separable nonquadratic effort costs, for which the arguments yielding classical solutions in the quadratic-cost setting no longer apply.

math.OC↗

arXiv2026

Optimal Contract Design with Quadratic Effort Cost

The existence of an optimal contract of the principal-agent problem is a central issue in contract design. According to Cvitanić et al. [2], such an optimal contract can be derived from the existence of a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which is a degenerate, fully nonlinear parabolic equation. In this work, we follow their model, consider the case with drift control, and prove the existence of the classical solution to the HJB equation.

q-fin.MF↗

arXiv2026

Comparative Statics of Trading Boundary in Finite Horizon Portfolio Selection with Proportional Transaction Costs

We consider Merton's problem with proportional transaction costs. It is well known that the optimal investment strategy is characterized by two trading boundaries, the buy boundary and the sell boundary, between which lies the no-trading region. We investigate how these two trading boundaries vary with the transaction cost rates. We show that the cost-adjusted trading boundaries are monotone in the transaction costs. Our result implies the following: (i) the Merton line must lie between the two cost-adjusted trading boundaries; and (ii) when the Merton line is positive, both the buy and sell boundaries are monotone in the transaction cost rates, and consequently the Merton line lies in the no-trading region.

q-fin.MF↗

arXiv2025

Non-Concave Utility Maximization with Transaction Costs

This paper studies a finite-horizon portfolio selection problem with non-concave terminal utility and proportional transaction costs, in which the commonly used concavification principle for terminal value is no longer applicable. We establish a proper theoretical characterization of this problem via a two-step procedure. First, we examine the asymptotic terminal behavior of the value function, which implies that any transaction close to maturity only provides a marginal contribution to the utility. Second, we establish the theoretical foundation in terms of the discontinuous viscosity solution, incorporating the proper characterization of the terminal condition. Via extensive numerical analyses involving several types of utility functions, we find that the introduction of transaction costs into non-concave utility maximization problems can make it optimal for investors to hold on to a larger long position in the risky asset compared to the frictionless case, or hold on to a large short position in the risky asset despite a positive risk premium.

q-fin.MF↗

arXiv2023

Robust Equilibrium Strategy for Mean-Variance Portfolio Selection

The classical mean-variance portfolio selection problem induces time-inconsistent (precommited) strategies (see Zhou and Li (2000)). To overcome this time-inconsistency, Basak and Chabakauri (2010) introduce the game theoretical approach and look for (sub-game perfect Nash) equilibrium strategies, which is solved from the corresponding partial differential equations (PDE) system. In their model, the investor perfectly knows the drift and volatility of the assets. However, in reality investors only have an estimate on them, e.g, a 95% confidence interval. In this case, some literature (e.g., Pham, Wei and Zhou (2022)) derives the optimal precommited strategy under the worst parameters, which is the robust control. The relation between the equilibrium strategy and the PDE system has not been justified when incorporating robust control. In this paper, we consider a general dynamic mean-variance framework and propose a novel definition of the robust equilibrium strategy. Under our definition, a classical solution to the corresponding PDE system implies a robust equilibrium strategy. We then explicitly solve for some special examples.

q-fin.MF↗