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Multidimensional Backward Stochastic Differential Equations with Rough Drifts

Jiahao Liang, Shanjian Tang

Abstract

In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with an additional rough drift (rough BSDE), and give the existence and uniqueness of the adapted solution, either when the terminal value and the geometric rough path are small, or when each component of the rough drift only depends on the corresponding component of the first unknown variable (but dropped is the one-dimensional assumption of Diehl and Friz [Ann. Probab. 40 (2012), 1715-1758]). We also introduce a new notion of the $p$-rough stochastic integral for $p \in \left[2, 3\right)$, and then succeed in giving -- through a fixed-point argument -- a general existence and uniqueness result on a multidimensional rough BSDE with a general square-integrable terminal value, allowing the rough drift to be random and time-varying but having to be linear; furthermore, we connect it to a system of rough partial differential equations.

Multidimensional Backward Stochastic Differential Equations with Rough Drifts

Abstract

In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with an additional rough drift (rough BSDE), and give the existence and uniqueness of the adapted solution, either when the terminal value and the geometric rough path are small, or when each component of the rough drift only depends on the corresponding component of the first unknown variable (but dropped is the one-dimensional assumption of Diehl and Friz [Ann. Probab. 40 (2012), 1715-1758]). We also introduce a new notion of the -rough stochastic integral for , and then succeed in giving -- through a fixed-point argument -- a general existence and uniqueness result on a multidimensional rough BSDE with a general square-integrable terminal value, allowing the rough drift to be random and time-varying but having to be linear; furthermore, we connect it to a system of rough partial differential equations.
Paper Structure (20 sections, 27 theorems, 296 equations)

This paper contains 20 sections, 27 theorems, 296 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $q$-variation paths
  4. Stochastic objects
  5. BSDEs with a nonlinear rough drift
  6. Solvability under the small terminal value condition
  7. Solvability when $g^{i}$ only depends on $y^{i}$
  8. Integration and composition
  9. Stochastic controlled rough paths
  10. $p$-rough stochastic integrals
  11. Composition of stochastic controlled rough paths
  12. BSDEs with a linear rough drift
  13. Existence and Uniqueness
  14. Continuity of the solution map
  15. Application to systems of rough PDEs
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\xi \in L^{2}\left(\Omega, \mathcal{F}_{T}\right)$, $\mathbf{X}$ be a two-step $p$-rough path for $p \in \left[2, 3\right)$, $\left(G, G^{\prime}\right)$ and $\left(H, H^{\prime}\right)$ be an essentially bounded and a square-integrable stochastic controlled rough path of finite $\left(p, p\rig is continuous (see Theorem solumc).

Theorems & Definitions (66)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 56 more