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A local maximum principle for robust optimal control problems of quadratic BSDEs

Tao Hao, Jiaqiang Wen, Qi Zhang

Abstract

The paper concerns the necessary maximum principle for robust optimal control problems of quadratic BSDEs. The coefficient of the systems depends on the parameter $θ$, and the generator of BSDEs is of quadratic growth in $z$. Since the model is uncertain, the variational inequality is proved by weak convergence technique. In addition, due to the generator being quadratic with respect to $z$, the forward adjoint equations are SDEs with unbounded coefficient involving mean oscillation martingales. Using reverse Hölder inequality and John-Nirenberg inequality, we show that its solutions are continuous with respect to the parameter $θ$. The necessary and sufficient conditions for robust optimal control are proved by linearization method.

A local maximum principle for robust optimal control problems of quadratic BSDEs

Abstract

The paper concerns the necessary maximum principle for robust optimal control problems of quadratic BSDEs. The coefficient of the systems depends on the parameter , and the generator of BSDEs is of quadratic growth in . Since the model is uncertain, the variational inequality is proved by weak convergence technique. In addition, due to the generator being quadratic with respect to , the forward adjoint equations are SDEs with unbounded coefficient involving mean oscillation martingales. Using reverse Hölder inequality and John-Nirenberg inequality, we show that its solutions are continuous with respect to the parameter . The necessary and sufficient conditions for robust optimal control are proved by linearization method.
Paper Structure (6 sections, 6 theorems, 71 equations)

This paper contains 6 sections, 6 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Two examples
  3. Problem formulation
  4. Some properties for BMO-martingales
  5. Variational inequality
  6. Variational equation

Key Result

Theorem 3.2

Under ass 1, for any $v(\cdot)\in {\cal V}_{ad}$ and $p>1$, the equation (1.1) has a unique solution $(X^v_\theta, Y^v_\theta, Z^v_\theta)\in {\cal S}^p_{\mathbb{F}}(0,T;\mathbb{R}^n) \times {\cal S}^\infty_{\mathbb{F}}(0,T;\mathbb{R}) \times {\cal H}^{2,p}_\mathbb{F}(0,T;\mathbb{R}^d)$. Moreover, t where the constant $C$ depends on $C_0, p,T$ and the constants $M_1, M_2$ depends on $C_0, C_1, T,

Theorems & Definitions (13)

  • Example 2.1
  • Example 2.2
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 3 more