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The Optimal Control Problem of Fully Coupled FBSDEs Driven by Sub-diffusion with Applications

Chenhui Hao, Jingtao Shi, Shuaiqi Zhang

Abstract

This paper is devoted to an optimal control problem of fully coupled forward-backward stochastic differential equations driven by sub-diffusion, whose solutions are not Markov processes. The stochastic maximum principle is obtained, where the control domain may not be convex and the diffusion term is independent of the control variable. Additionally, problem with state constraint is researched by using Ekeland's variational principle. The theoretical results obtained are applied to a cash management optimization problem in bear market, and the optimal strategy is derived.

The Optimal Control Problem of Fully Coupled FBSDEs Driven by Sub-diffusion with Applications

Abstract

This paper is devoted to an optimal control problem of fully coupled forward-backward stochastic differential equations driven by sub-diffusion, whose solutions are not Markov processes. The stochastic maximum principle is obtained, where the control domain may not be convex and the diffusion term is independent of the control variable. Additionally, problem with state constraint is researched by using Ekeland's variational principle. The theoretical results obtained are applied to a cash management optimization problem in bear market, and the optimal strategy is derived.

Paper Structure

This paper contains 9 sections, 15 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. Properties of sub-diffusion
  4. Existence and uniqueness for solutions of FBSDEs driven by sub-diffusion
  5. Stochastic optimal control problem for FBSDEs driven by sub-diffusion
  6. Stochastic maximum principle for FBSDEs driven by sub-diffusion
  7. Stochastic maximum principle for FBSDEs driven by sub-diffusion with state constraints
  8. Applications to cash management problem in bear markets
  9. Concluding remarks

Key Result

Lemma 2.1

Suppose $S_t$ is a subordinator and the Lévy triple of $S_t$ is $(\kappa,0,v)$, let $L_t=\inf\{r>0\colon S_r>t\}$. By definition $0\leqslant L_{t+s}-L_{t}\leqslant s/\kappa$, and $dL_t$ is absolutely continuous to Lebesgue measure $dt$. Let $s\rightarrow0$, then:

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.1
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 15 more