arXiv subjects

Dejian Tian

Publications and source records attributed to Dejian Tian.

12 recordsLinked to original sources

arXiv2026

Robust optimized certainty equivalents and quantiles for loss positions with distribution uncertainty

This paper investigates the robust optimized certainty equivalents and analyzes their properties as risk measures under distribution uncertainty. Building on this, robust generalized quantiles are proposed and discussed. We then consider robust expectiles with two specific penalization functions. For the one with a linear penalization function, it is proved to be a coherent risk measure and its dual representation is provided. Furthermore, numerical simulations are conducted to examine the effect of the penalization functions on the robust expectiles and to compare them with classical expectiles.

q-fin.RM↗

arXiv2026

Optimal Comfortable Consumption under Epstein-Zin utility

We solve the optimal portfolio choice problem under Epstein--Zin utility with a time-varying consumption constraint, where closed-form expressions for neither the primal nor the dual value function are available. We establish the dynamic programming principle for the value function and prove that it is a viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation. We further establish the $C^2$ regularity of the value function and derive a verification theorem using stochastic perturbation techniques. Finally, we provide an explicit characterization of the constrained region. The proposed methodology extends naturally to other constrained portfolio choice problems under the Epstein--Zin utility.

math.OC↗

arXiv2025

Optimal Consumption-Investment with Epstein-Zin Utility under Leverage Constraint

We study optimal portfolio choice under Epstein-Zin recursive utility in the presence of general leverage constraints. We first establish that the optimal value function is the unique viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, by developing a new dynamic programming principle under constraints. We further demonstrate that the value function admits smoothness and characterize the optimal consumption and investment strategies. In addition, we derive explicit solutions for the optimal strategy and explicitly delineate the constrained and unconstrained regions in several special cases of the leverage constraint. Finally, we conduct a comparative analysis, highlighting the differences relative to the classical time-separable preferences and to the setting without leverage constraints.

q-fin.PM↗

arXiv2025

Maximum principle for robust utility optimization via Tsallis relative entropy

This paper investigates an optimal consumption-investment problem featuring recursive utility via Tsallis relative entropy. We establish a fundamental connection between this optimization problem and a quadratic backward stochastic differential equation (BSDE), demonstrating that the value function is the value process of the solution to this BSDE. Utilizing advanced BSDE techniques, we derive a novel stochastic maximum principle that provides necessary conditions for both the optimal consumption process and terminal wealth. Furthermore, we prove the existence of optimal strategy and analyze the coupled forward-backward system arising from the optimization problem.

q-fin.MF↗

arXiv2025

Consumption-investment optimization with Epstein-Zin utility in unbounded non-Markovian markets

The paper investigates the consumption-investment problem for an investor with Epstein-Zin utility in an incomplete market. A non-Markovian environment with unbounded parameters is considered, which is more realistic in practical financial scenarios compared to the Markovian setting. The optimal consumption and investment strategies are derived using the martingale optimal principle and quadratic backward stochastic differential equations (BSDEs) whose solutions admit some exponential moment. This integrability property plays a crucial role in establishing a key martingale argument. In addition, the paper also examines the associated dual problem and several models within the specified parameter framework.

q-fin.MF↗

arXiv2025

Robust distortion risk measures with linear penalty under distribution uncertainty

The paper investigates the robust distortion risk measure with linear penalty function under distribution uncertainty. The distribution uncertainties are characterized by predetermined moment conditions or constraints on the Wasserstein distance. The optimal quantile distribution and the optimal value function are explicitly characterized. Our results partially extend the results of Bernard, Pesenti and Vanduffel (2024) and Li (2018) to robust distortion risk measures with linear penalty. In addition, we also discuss the influence of the penalty parameter on the optimal solution.

q-fin.RM↗

arXiv2025

Set-valued Star-Shaped Risk Measures

In this paper, we introduce a new class of set-valued risk measures, named set-valued star-shaped risk measures. Motivated by the results of scalar monetary and star-shaped risk measures, this paper investigates the representation theorems in the set-valued framework. It is demonstrated that set-valued risk measures can be represented as the union of a family of set-valued convex risk measures, and set-valued normalized star-shaped risk measures can be represented as the union of a family of set-valued normalized convex risk measures. The link between set-valued risk measures and set-valued star-shaped risk measures is also established.

q-fin.RM↗

arXiv2023

Optimal consumption and portfolio selection with Epstein-Zin utility under general constraints

The paper investigates the consumption-investment problem for an investor with Epstein-Zin utility in an incomplete market. Closed, not necessarily convex, constraints are imposed on strategies. The optimal consumption and investment strategies are characterized via a quadratic backward stochastic differential equation (BSDE). Due to the stochastic market environment, the solution to this BSDE is unbounded and thereby the BMO argument breaks down. After establishing the martingale optimality criterion, by delicately selecting Lyapunov functions, the verification theorem is ultimately obtained. Besides, several examples and numerical simulations for the optimal strategies are provided and illustrated.

q-fin.MF↗

arXiv2023

Dynamic star-shaped risk measures and $g$-expectations

Motivated by the results of static monetary or star-shaped risk measures, the paper investigates the representation theorems in the dynamic framework. We show that dynamic monetary risk measures can be represented as the lower envelope of a family of dynamic convex risk measures, and normalized dynamic star-shaped risk measures can be represented as the lower envelope of a family of normalized dynamic convex risk measures. The link between dynamic monetary risk measures and dynamic star-shaped risk measures are established. Besides, the sensitivity and time consistency problems are also studied. A specific normalized time consistent dynamic star-shaped risk measures induced by $ g $-expectations are illustrated and discussed in detail.

q-fin.RM↗

arXiv2022

Pricing principle via Tsallis relative entropy in incomplete market

A pricing principle is introduced for non-attainable $q$-exponential bounded contingent claims in an incomplete Brownian motion market setting. The buyer evaluates the contingent claim under the ``distorted Radon-Nikodym derivative'' and adjustment by Tsallis relative entropy over a family of equivalent martingale measures. The pricing principle is proved to be a time consistent and arbitrage-free pricing rule. More importantly, this pricing principle is found to be closely related to backward stochastic differential equations with generators $f(y)|z|^2$ type. The pricing functional is compatible with prices for attainable claims. Except translation invariance, the pricing principle processes lots of elegant properties such as monotonicity and concavity etc. The pricing functional is showed between minimal martingale measure pricing and conditional certainty equivalent pricing under $q$-exponential utility. The asymptotic behavior of the pricing principle for ambiguity aversion coefficient is also investigated.

q-fin.MF↗

arXiv2017

Stochastic differential games with state constraints and Isaacs equations with nonlinear Neumann problems

We investigate a two-player zero-sum stochastic differential game problem with the state process being constrained in a connected bounded closed domain, and the cost functional described by the solution of a generalized backward stochastic differential equation (GBSDE for short). We show that the value functions enjoy a (strong) dynamic programming principle, and are the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equations with nonlinear Neumann boundary problems. To obtain the existence for viscosity solutions, we provide a new approach utilizing the representation theorem for generators of the GBSDE, which is proved by a random time change method and is a novel result in its own right.

math.PR↗

arXiv2017

Probabilistic interpretation of HJB equations by the representation theorem for generators of BSDEs

The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of backward stochastic differential equations. The key idea of our approach for proving this interpretation consists of transmitting the signs between the solution and generator via the identity given by representation theorem. Compared with existing methods, our approach seems to be more applicable for general settings. This can also be regarded as a new application of such representation theorem.

math.PR↗