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The perturbation method applied to a robust optimization problem with constraint

Peng Luo, Alexander Schied, Xiaole Xue

Abstract

The present paper studies a kind of robust optimization problems with constraint. The problem is formulated through Backward Stochastic Differential Equations (BSDEs) with quadratic generators. A necessary condition is established for the optimal solution using a terminal perturbation method and properties of Bounded Mean Oscillation (BMO) martingales. The necessary condition is further proved to be sufficient for the existence of an optimal solution under an additional convexity assumption. Finally, the optimality condition is applied to discuss problems of partial hedging with ambiguity, fundraising under ambiguity and randomized testing problems for a quadratic $g$-expectation.

The perturbation method applied to a robust optimization problem with constraint

Abstract

The present paper studies a kind of robust optimization problems with constraint. The problem is formulated through Backward Stochastic Differential Equations (BSDEs) with quadratic generators. A necessary condition is established for the optimal solution using a terminal perturbation method and properties of Bounded Mean Oscillation (BMO) martingales. The necessary condition is further proved to be sufficient for the existence of an optimal solution under an additional convexity assumption. Finally, the optimality condition is applied to discuss problems of partial hedging with ambiguity, fundraising under ambiguity and randomized testing problems for a quadratic -expectation.
Paper Structure (10 sections, 11 theorems, 90 equations)

This paper contains 10 sections, 11 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Notation
  4. Assumptions and model formulation
  5. Terminal perturbation method
  6. Applications
  7. Partial hedging with ambiguity
  8. Fundraising under ambiguity
  9. Neyman--Pearson lemma for quadratic $g$-probabilities
  10. Optimization problems under risk measures without cash-additivity

Key Result

Lemma 2.4

For a function $g$ satisfying Assumption ass-qua and a random variable $\xi\in L^{\infty}(\mathcal{F}_{T})$, the BSDE admits a unique solution $(y,z)\in\mathcal{S}^{\infty}\times\text{BMO}$. Moreover, there exists a constant $L$ that only depends on $\xi$ and the constant $C$ from Assumption ass-qua such that

Theorems & Definitions (13)

  • Lemma 2.4: briand
  • Definition 2.5: ma
  • Remark 2.6
  • Theorem 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 2.10
  • Proposition 2.11
  • Proposition 2.12
  • Proposition 3.1
  • ...and 3 more