arXiv subjects

Xiaonyu Xia

Publications and source records attributed to Xiaonyu Xia.

7 recordsLinked to original sources

arXiv2026

Stochastic Control Problems with Infinite Horizon and Regime Switching Arising in Optimal Liquidation with Semimartingale Strategies

We study an optimal control problem on infinite time horizon with semimartingale strategies, random coefficients and regime switching. The value function and the optimal strategy can be characterized in terms of three systems of backward stochastic differential equations (BSDEs) with infinite horizon. One of them is a system of linear BSDEs with unbounded coefficients and infinite horizon, which seems to be new in literature. We establish the existence of the solutions to these BSDEs by BMO analysis and comparison theorem for multi-dimensional BSDEs. Next, we establish that the optimal control problem is well posed, in the sense that the value function is finite and the optimal strategy-when it exists-is unique. This is achieved by reformulating the cost functional as the sum of a quadratic functional and the candidate value function. The reformulation crucially relies on the well-established well-posedness results for systems of BSDEs. Finally, under additional assumptions, we obtain the unique optimal strategy.

math.OC↗

arXiv2024

Long Time Behavior of Optimal Liquidation Problems

In this paper, we study the long time behavior of an optimal liquidation problem with semimartingale strategies and external flows. To investigate the limit rigorously, we study the convergence of three BSDEs characterizing the value function and the optimal strategy, from finite horizon to infinite horizon. We find that in the long time limit the player may not necessarily liquidate her assets at all due to the existence of external flows, even if in any given finite time horizon, the player is forced to liquidate all assets. Moreover, when the intensity of the external flow is damped, the player will liquidate her assets in the long run.

q-fin.MF↗

arXiv2023

A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategies

We consider a mean-field control problem with càdlàg semimartingale strategies arising in portfolio liquidation models with transient market impact and self-exciting order flow. We show that the value function depends on the state process only through its law, and that it is of linear-quadratic form and that its coefficients satisfy a coupled system of non-standard Riccati-type equations. The Riccati equations are obtained heuristically by passing to the continuous-time limit from a sequence of discrete-time models. A sophisticated transformation shows that the system can be brought into standard Riccati form from which we deduce the existence of a global solution. Our analysis shows that the optimal strategy jumps only at the beginning and the end of the trading period.

q-fin.MF↗

arXiv2020

Portfolio Liquidation Games with Self-Exciting Order Flow

We analyze novel portfolio liquidation games with self-exciting order flow. Both the N-player game and the mean-field game are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each player solves a mean-field control problem. We characterize open-loop Nash equilibria in both games in terms of a novel mean-field FBSDE system with unknown terminal condition. Under a weak interaction condition we prove that the FBSDE systems have unique solutions. Using a novel sufficient maximum principle that does not require convexity of the cost function we finally prove that the solution of the FBSDE systems do indeed provide existence and uniqueness of open-loop Nash equilibria.

math.OC↗

arXiv2020

Continuous viscosity solutions to linear-quadratic stochastic control problems with singular terminal state constraint

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub- and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument.

q-fin.MF↗

arXiv2019

Portfolio liquidation under factor uncertainty

We study an optimal liquidation problem under the ambiguity with respect to price impact parameters. Our main results show that the value function and the optimal trading strategy can be characterized by the solution to a semi-linear PDE with superlinear gradient, monotone generator and singular terminal value. We also establish an asymptotic analysis of the robust model for small amount of uncertainty and analyse the effect of robustness on optimal trading strategies and liquidation costs. In particular, in our model ambiguity aversion is observationally equivalent to increased risk aversion. This suggests that ambiguity aversion increases liquidation rates.

q-fin.MF↗

arXiv2018

Multi-dimensional Optimal Trade Execution under Stochastic Resilience

We study a multi-dimensional optimal execution problem in illiquid markets with both instantaneous and persistent price impact and stochastic resilience. In our model the value function can be described by a multi-dimensional backward stochastic Riccati differential equations (BSRDE) with a singular terminal condition in one component. We prove the existence of a solution to the BSRDE system and characterise both the value function and the optimal strategy in terms of that solution. We prove that the solution to the liquidation problem can be approximated by the solutions to a sequence of unconstrained problems with increasing penalisation of open positions at the terminal time.

math.OC↗