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Optimal Control Problems Governed by MFSDEs with multi-defaults

Zhun Gou, Nan-jing Huang, Ming-hui Wang, Jian-hao Kang

Abstract

In this paper, we solve an optimal control problem governed by a system of mean-field stochastic differential equations with multiple defaults (MMFSDEs). We transform the global optimal control problem into several optimal control subproblems governed by a system of mean-field stochastic differential equations with single default (SMFSDEs) and derive both the sufficient and necessary maximum principles for these subproblems. We also give the existence and uniqueness of solutions to the MMFSDEs and the mean-field backward stochastic differential equations with multiple defaults (MMFBSDEs), respectively. Finally, as an example, our results are applied to obtain the explicit solution for an optimal control problem whose cost function is considered as a recursive utility process with multiple defaults.

Optimal Control Problems Governed by MFSDEs with multi-defaults

Abstract

In this paper, we solve an optimal control problem governed by a system of mean-field stochastic differential equations with multiple defaults (MMFSDEs). We transform the global optimal control problem into several optimal control subproblems governed by a system of mean-field stochastic differential equations with single default (SMFSDEs) and derive both the sufficient and necessary maximum principles for these subproblems. We also give the existence and uniqueness of solutions to the MMFSDEs and the mean-field backward stochastic differential equations with multiple defaults (MMFBSDEs), respectively. Finally, as an example, our results are applied to obtain the explicit solution for an optimal control problem whose cost function is considered as a recursive utility process with multiple defaults.

Paper Structure

This paper contains 14 sections, 18 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Backward Induction Method and Maximum Princeples
  4. Backward Induction Method
  5. A sufficient maximum principle
  6. A Necessary Maximum Principle
  7. Solvability of MMFSDEs and MMFBSDEs
  8. Solution to MMFSDEs
  9. Solution to MMFBSDEs
  10. An Optimal Investment Problem
  11. Stability
  12. Mean Square Exponential Stability
  13. Almost Sure Exponential Stability
  14. Conclusions

Key Result

Lemma 2.1

Peng2009BSDE Let $X_i(t)\;(i=1,2)$ be an Itô's jump-diffusion process given by where $b_i(\cdot),\sigma_i(\cdot)\in L_{T}^{\mathfrak{F}}$ and $h^i_k(\cdot)\in L_{T}^{\mathfrak{F},A^k}\;(k\in \mathcal{I})$. Then Especially, if $h^i_k(t)=h^i(t)$ for all $k\in \mathcal{I}$, then

Theorems & Definitions (43)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.3
  • Lemma 2.1
  • Remark 2.4
  • Proposition 3.1
  • proof
  • ...and 33 more