[Paper Review] On Backward Doubly Stochastic Differential Evolutionary System

Jinniao Qiu, Shanjian Tang|arXiv (Cornell University)|Sep 17, 2013

Stochastic processes and financial applications38 references10 citations

TL;DR

This paper establishes the existence and uniqueness of solutions for backward doubly stochastic differential evolutionary systems (BDSDESs) using a variational approach based on monotone operator theory. It proves a generalized Itô formula for Banach space-valued BDSDESs and extends results to infinite-dimensional settings, filling a gap in the generalized solution theory for backward doubly stochastic PDEs (BDSPDEs).

ABSTRACT

In this paper, we are concerned with backward doubly stochastic differential evolutionary systems (BDSDESs for short). By using a variational approach based on the monotone operator theory, we prove the existence and uniqueness of the solutions for BDSDESs. We also establish an Itô formula for the Banach space-valued BDSDESs.

Motivation & Objective

• To establish the existence and uniqueness of solutions for backward doubly stochastic differential evolutionary systems (BDSDESs) in both finite and infinite-dimensional settings.
• To develop a generalized Itô formula for Banach space-valued BDSDESs, extending classical Itô calculus to this class of equations.
• To fill the theoretical gap in the generalized solution theory for backward doubly stochastic PDEs (BDSPDEs), which had been previously unaddressed.
• To extend existing results on Feynman-Kac formulas, stochastic viscosity solutions, and optimal control to infinite-dimensional BDSDES-driven processes.
• To provide a variational framework based on monotone operator theory for analyzing BDSDESs beyond the finite-dimensional case.

Proposed method

• Uses a variational approach grounded in monotone operator theory to prove existence and uniqueness of solutions for BDSDESs.
• Applies Galerkin approximation techniques to first establish results in the finite-dimensional case before extending to infinite-dimensional Hilbert and Banach space settings.
• Derives a generalized Itô formula for Banach space-valued processes driven by two independent Brownian motions (forward and backward), crucial for stochastic calculus in this context.
• Employs weak convergence arguments and compactness methods based on monotone operator theory to handle convergence of approximating sequences.
• Utilizes a Gelfand triple framework $ V \hookrightarrow H \cong H' \hookrightarrow V' $ to handle the duality between spaces and ensure well-posedness in the variational setting.
• Proves convergence of discrete approximations via a limiting argument involving stopping times and $ L^2 $-boundedness in probability, ensuring pathwise continuity and integrability.

Experimental results

Research questions

• RQ1 Can the existence and uniqueness of solutions for backward doubly stochastic differential evolutionary systems be established in infinite-dimensional spaces?
• RQ2 What is the appropriate Itô formula for Banach space-valued backward doubly stochastic processes, and how does it differ from classical Itô formulas?
• RQ3 How can monotone operator theory be adapted to solve BDSDESs in variational settings beyond the finite-dimensional case?
• RQ4 Can the generalized solution theory for BDSPDEs be completed using this framework, especially for quasilinear equations?
• RQ5 To what extent can results like the Feynman-Kac formula, stochastic viscosity solutions, and optimal control theory be extended to infinite-dimensional BDSDESs?

Key findings

• The paper proves the existence and uniqueness of solutions for BDSDESs in both finite and infinite-dimensional settings using a variational method based on monotone operator theory.
• A generalized Itô formula is established for Banach space-valued BDSDESs, which plays a foundational role in stochastic calculus for this class of equations.
• The solution process $ u $ is shown to belong to the space $ S^2(0,T;H) $, ensuring pathwise continuity and square-integrability in the Hilbert space $ H $.
• The convergence of approximating sequences is rigorously proven using weak convergence and compactness arguments, with limits verified to satisfy the original BDSDES in the $ L^2 $-sense.
• The results extend to quasi-linear BDSPDEs, yielding a more general existence and uniqueness result that fills a critical gap in the generalized solution theory for BDSPDEs.
• The framework enables potential extensions to stochastic viscosity solutions, stationary solutions of SPDEs on Hilbert spaces, and optimal control problems with infinite-dimensional BDSDE state dynamics.