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

Xinman Cheng, Guanxing Fu, Xiaonyu Xia

Abstract

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.

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

Abstract

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.
Paper Structure (12 sections, 16 theorems, 120 equations)

This paper contains 12 sections, 16 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Notation, convention, assumption and admissible space
  3. Candidate BSDE systems
  4. A general system of infinite horizon BSDEs with stochastic Lipschitz drivers
  5. Preliminary lemmas
  6. A general infinite-horizon BSDE system
  7. The BSDE Systems \ref{['BSDE:A-inf']}-\ref{['BSDE:C-inf']}: Existence
  8. The solvability of the control problem \ref{['cost-inf-continuous']}-\ref{['state-inf-continuous']}
  9. Wellposedness of the control problem
  10. Existence of an optimal strategy
  11. Conclusion
  12. Results on $A$.

Key Result

Lemma 3.1

Assume $\zeta\in H^{2,K_{\zeta}}_{\text{BMO}}$, where $K_\zeta$ is a given constant. For any $b_1\in(0,2)$, $b_2>0$ and $b_3>K_{\zeta}$, it holds that where $q$ is the conjugate of $\frac{2}{b_1}$, and $N$ is any positive constant such that

Theorems & Definitions (31)

  • Lemma 3.1
  • proof
  • Lemma 3.2: Energy inequality
  • Lemma 3.3: Reverse Hölder inequality
  • Lemma 3.5
  • proof
  • Remark 3.6
  • Theorem 3.7
  • proof
  • Lemma 3.8
  • ...and 21 more