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1D nonlinear backward stochastic differential equations: a unified theory and applications

Shengjun Fan, Ying Hu, Shanjian Tang

Abstract

Since the celebrated paper by El Karoui, Peng and Quenez [Mathematical Finance, 7 (1997), 1--71], backward stochastic differential equations have found wide applications in stochastic control, financial technology and machine learning. In this paper, we present a comprehensive theory on the existence and uniqueness of adapted solutions to a one-dimensional nonlinear backward stochastic differential equation (1D BSDE for short), and assume that the generator $g$ has a unilateral linear or super-linear growth in the first unknown variable $y$, and has an at most quadratic growth in the second unknown variable $z$. We develop a unified methodology, featured by the test function method and the a priori estimate technique, to establish several existence theorems and comparison theorems, which immediately yield corresponding existence and uniqueness results. We also overview relevant known results and give some practical applications of our theoretical results. Finally, we list some open problems on the well-posedness of 1D BSDEs.

1D nonlinear backward stochastic differential equations: a unified theory and applications

Abstract

Since the celebrated paper by El Karoui, Peng and Quenez [Mathematical Finance, 7 (1997), 1--71], backward stochastic differential equations have found wide applications in stochastic control, financial technology and machine learning. In this paper, we present a comprehensive theory on the existence and uniqueness of adapted solutions to a one-dimensional nonlinear backward stochastic differential equation (1D BSDE for short), and assume that the generator has a unilateral linear or super-linear growth in the first unknown variable , and has an at most quadratic growth in the second unknown variable . We develop a unified methodology, featured by the test function method and the a priori estimate technique, to establish several existence theorems and comparison theorems, which immediately yield corresponding existence and uniqueness results. We also overview relevant known results and give some practical applications of our theoretical results. Finally, we list some open problems on the well-posedness of 1D BSDEs.
Paper Structure (18 sections, 15 theorems, 165 equations, 3 tables)

This paper contains 18 sections, 15 theorems, 165 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Overview of relevant existing results
  3. Organization of the paper
  4. Notations and spaces
  5. Existence results
  6. The test function method and a general existence result
  7. The logarithmic quasi-linear growth case
  8. The sub-quadratic/quadratic growth case
  9. Comparison theorems and existence and uniqueness results
  10. Comparison results for the case of an at most linear growth
  11. Comparison results for the super-linear at most quadratic growth case
  12. Existence and uniqueness
  13. Applications
  14. The (conditional) $g$-expectation defined on $L^1(\mathcal{F}_T)$
  15. Dynamic utility process and risk measure
  16. ...and 3 more sections

Key Result

Theorem 2.2

Assume that $\xi$ is an $\mathcal{F}_T$-measurable random variable and the generator $g$ satisfies the above assumptions (EX1)-(EX3). If there exists a test function $\varphi(\cdot,\cdot)\in {\bf S}$ for $g$ such that then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in[0,T]}$ such that the process is of class (D). Furthermore, the process $(\varphi(t, |Y_t|+\int_0^t f_s{\rm d}s))_{t\in [0,T]}$

Theorems & Definitions (47)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • Example 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof : Proof of \ref{['thm:2.3']}
  • Theorem 2.8
  • ...and 37 more