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Numerical Methods and Analysis via Random Field Based Malliavin Calculus for Backward Stochastic PDEs

Wanyang Dai

TL;DR

The paper addresses numerical solution and rigorous analysis for a unified vector-valued backward stochastic PDE with drift and diffusion operators that may involve high-order derivatives. It introduces a completely discrete time-space scheme framework and proves well-posedness via a novel random-field Malliavin calculus, including first- and second-order derivative B-SPDEs under random environments. A contraction-based existence/uniqueness theory is developed, followed by a convergence theorem showing an $O(|\pi|)$ error rate for Algorithm I, with detailed Malliavin-derivative machinery and Clark-Ocone-type representations used to derive representation formulas and a priori estimates. These results provide a principled path to compute and analyze B-SPDEs arising in finance and image processing, especially when coefficients depend on high-order spatial derivatives under randomness.

Abstract

We study the adapted solution, numerical methods, and related convergence analysis for a unified backward stochastic partial differential equation (B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients may involve nonlinear and high-order partial differential operators. Under certain generalized Lipschitz and linear growth conditions, the existence and uniqueness of adapted solution to the B-SPDE are justified. The methods are based on completely discrete schemes in terms of both time and space. The analysis concerning error estimation or rate of convergence of the methods is conducted. The key of the analysis is to develop new theory for random field based Malliavin calculus to prove the existence and uniqueness of adapted solutions to the first-order and second-order Malliavin derivative based B-SPDEs under random environments.

Numerical Methods and Analysis via Random Field Based Malliavin Calculus for Backward Stochastic PDEs

TL;DR

The paper addresses numerical solution and rigorous analysis for a unified vector-valued backward stochastic PDE with drift and diffusion operators that may involve high-order derivatives. It introduces a completely discrete time-space scheme framework and proves well-posedness via a novel random-field Malliavin calculus, including first- and second-order derivative B-SPDEs under random environments. A contraction-based existence/uniqueness theory is developed, followed by a convergence theorem showing an error rate for Algorithm I, with detailed Malliavin-derivative machinery and Clark-Ocone-type representations used to derive representation formulas and a priori estimates. These results provide a principled path to compute and analyze B-SPDEs arising in finance and image processing, especially when coefficients depend on high-order spatial derivatives under randomness.

Abstract

We study the adapted solution, numerical methods, and related convergence analysis for a unified backward stochastic partial differential equation (B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients may involve nonlinear and high-order partial differential operators. Under certain generalized Lipschitz and linear growth conditions, the existence and uniqueness of adapted solution to the B-SPDE are justified. The methods are based on completely discrete schemes in terms of both time and space. The analysis concerning error estimation or rate of convergence of the methods is conducted. The key of the analysis is to develop new theory for random field based Malliavin calculus to prove the existence and uniqueness of adapted solutions to the first-order and second-order Malliavin derivative based B-SPDEs under random environments.

Paper Structure

This paper contains 20 sections, 14 theorems, 183 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Adapted Solution to the B-SPDE: Existence and Uniqueness
  3. Preliminary Notations
  4. The Conditions
  5. The Adapted Solution
  6. Numerical Schemes and Their Convergence
  7. The Schemes
  8. Additional Notations for Random Field Based Malliavin Calculus
  9. Convergence Theorem for Algorithm \ref{['algorithmI']}
  10. Proof of Theorem \ref{['bsdeyI']}
  11. Three Lemmas and Their Proofs
  12. Proof of Theorem \ref{['bsdeyI']}
  13. Proof of Theorem \ref{['errorIth']}
  14. Basic Properties of Random Field Based Malliavin Calculus
  15. B-SPDE with Malliavin Derivative Terminal Condition
  16. ...and 5 more sections

Key Result

Theorem 2.1

Assume that $H(x)\in L^{2}_{{\cal F}_{T}}(\Omega, C^{\infty}(D;R^{q}))$ for each $x\in D$. Then, under conditions of (e:blipschitz)-(e:blipschitzI), if ${\cal L}(t,x,v,\cdot)$ and ${\cal J}(t,x,v,\cdot)$ are $\{{\cal F}_{t}\}$-adapted for each fixed $x\in D$ and any given $(v,\bar{v})\in C^{\infty}( the B-SPDE (e:bspdef) has a unique adapted solution,

Theorems & Definitions (27)

  • Theorem 2.1
  • Theorem 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 5.1
  • proof
  • ...and 17 more