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Backward problems for stochastic differential equations on the Sierpinski gasket

Xuan Liu, Zhongmin Qian

Abstract

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section [sec:-1]) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman-Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established.

Backward problems for stochastic differential equations on the Sierpinski gasket

Abstract

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section [sec:-1]) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman-Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established.

Paper Structure

This paper contains 13 sections, 24 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Notations and several related results
  3. Diffusions and Dirichlet forms on the gasket
  4. Kusuoka measure and Brownian martingale
  5. Formulation of main results
  6. Several results on Brownian martingale
  7. Martingale representations
  8. Exponential integrability of quadratic processes
  9. BSDEs driven by Brownian martingale
  10. BSDEs with deterministic durations
  11. BSDEs with stochastic durations
  12. Example
  13. Representations for solutions of semi-linear parabolic PDEs

Key Result

Theorem 2.6

There exists a martingale additive functional $W_{t}$ satisfying the following: (i) $W_{t}$ has $\nu$ as its energy measure; that is, $\nu_{\langle W\rangle}=\nu$; (ii) for any $u\in\mathcal{F}(\mathbb{S})$, there exists a unique $\zeta\in L^{2}(\mathbb{S};\nu)$ such that where $M^{[u]}$ is the martingale part of $u(X_{t})-u(X_{0})$.

Theorems & Definitions (67)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 3.1
  • ...and 57 more