Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bismut-Elworthy-Li Formulae for Forward-Backward SDEs with Jumps and Applications

Jiagang Ren, Hua Zhang

TL;DR

Under nondegeneracy of the diffusion coefficient, the authors derive BEL-type gradient formulae for forward-backward SDEs with jumps using the lent particle method. They then apply these formulae to establish existence and uniqueness of solutions to nonlocal quasi-linear integral-PDEs differentiable in the spatial variable even when the initial data or coefficients are not differentiable. A sharp gradient bound $|\nabla_x Y(t,t,x)| \le C (T-t)^{-1/\beta} (1+|x|)^{\mu}$ for $\beta\in(1,2)$ is proved, and representation formulas connect the PDE solution to the FBSDE through $v(t,x)=Y(t,t,x)$ and $Z(\tau,t,x,u)=v(\tau,X(\tau,t,x)+\sigma(\tau,X(\tau,t,x))u)-v(\tau,X(\tau,t,x))$. This work bridges Malliavin calculus on Poisson space with nonlocal PDE theory and yields a robust approach for differentiable PDE solutions with rough data.

Abstract

Under nondegeneracy assumptions on the diffusion coefficients, we establish the derivative formulae of Bismut-Elworthy-Li's type for forward-backward stochastic differential equations with respect to Poisson random measure using the lent particle method created by Bouleau and Denis, which is not given before. Applying this formula, the existence and uniqueness of a solution of nonlocal quasi-linear integral partial differential equations, which are differentiable with respect to the space variable, are obtained, even if the initial datum and coefficients of this equation are not.

Bismut-Elworthy-Li Formulae for Forward-Backward SDEs with Jumps and Applications

TL;DR

Under nondegeneracy of the diffusion coefficient, the authors derive BEL-type gradient formulae for forward-backward SDEs with jumps using the lent particle method. They then apply these formulae to establish existence and uniqueness of solutions to nonlocal quasi-linear integral-PDEs differentiable in the spatial variable even when the initial data or coefficients are not differentiable. A sharp gradient bound for is proved, and representation formulas connect the PDE solution to the FBSDE through and . This work bridges Malliavin calculus on Poisson space with nonlocal PDE theory and yields a robust approach for differentiable PDE solutions with rough data.

Abstract

Under nondegeneracy assumptions on the diffusion coefficients, we establish the derivative formulae of Bismut-Elworthy-Li's type for forward-backward stochastic differential equations with respect to Poisson random measure using the lent particle method created by Bouleau and Denis, which is not given before. Applying this formula, the existence and uniqueness of a solution of nonlocal quasi-linear integral partial differential equations, which are differentiable with respect to the space variable, are obtained, even if the initial datum and coefficients of this equation are not.
Paper Structure (20 sections, 23 theorems, 239 equations)

This paper contains 20 sections, 23 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Lent Particle Method
  3. Set-up
  4. Dirichlet Structure on the Bottom Space
  5. Space-time Setting and Dirichlet Structure on the Upper Space
  6. Notations and Preliminary Results on the Forward-Backward Systems
  7. Notations
  8. Forward SDEs
  9. Forward-Backward System
  10. Bismut-Elworthy-Li Formulae
  11. Some Useful Lemmas
  12. Proof of Theorem \ref{['Bismut formula']}
  13. Proof of Theorem \ref{['gradient estimate']}
  14. Nonlocal Quasi-Linear Integral-PDEs
  15. Notations and Notions
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $F$, $G:\Omega\times X\times R\rightarrow{\mathbb R}$ be $\mathcal{F}\otimes{\mathcal{X}}\otimes{\mathcal{R}}$-measurable functions such that and Then the following relation holds ${\mathbb P}_N$-a.e. Here and in the sequel, $\int_{\alpha}^{\beta}=\int_{(\alpha,\beta]}$ for $\alpha<\beta$, and $\widehat{{\mathbb E}}$ denotes the expectation with respect to $\widehat{{\mathbb P}}$.

Theorems & Definitions (43)

  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Proposition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Remark 4.4
  • ...and 33 more