[Paper Review] New Existence Theorems about the Solutions of Some Stochastic Integral Equations

Xuemei Chen, Yingying Qi+1 more|arXiv (Cornell University)|Nov 6, 2012

Fixed Point Theorems Analysis2 references3 citations

TL;DR

This paper establishes a new existence theorem for solutions to stochastic integral equations using Schauder’s fixed point theorem, offering weaker conditions than those required by Banach’s contraction mapping principle. By constructing a compact operator on a bounded, closed, convex subset in a stochastic process space, the authors prove existence (but not uniqueness) under less restrictive continuity and boundedness assumptions compared to classical approaches.

ABSTRACT

Picard's iteration has been used to prove the existence and uniqueness of the solution for stochastic integral equations, here we use Schauder's fixed point theorem to give a new existence theorem about the solution of a stochastic integral equation, our theorem can weak some conditions gotten by applying Banach's fixed point theorem.

Motivation & Objective

• To develop a new existence theorem for solutions of stochastic integral equations using Schauder’s fixed point theorem.
• To weaken the conditions required for existence compared to those derived via Banach’s fixed point theorem.
• To provide a framework applicable to stochastic integral equations arising in filtering and financial modeling.
• To demonstrate that Schauder’s theorem yields existence under less stringent assumptions than the contraction mapping principle.

Proposed method

• Apply Schauder’s fixed point theorem to a stochastic integral equation of the form $ x(t;\omega) = h(t_0;\omega) + \int_a^t \sigma(s,x(s;\omega)) dB(s) + \int_a^t f(s,x(s;\omega)) ds $.
• Define the solution space $ X = L_{ad}^2([a,b] \times \Omega) $ and a closed, bounded, convex subset $ M = \{ X_t \in X : \|X_t\| \leq r \} $.
• Construct an operator $ A $ mapping $ M $ into itself, defined by the right-hand side of the integral equation.
• Prove that $ A $ is compact by showing equicontinuity and relative compactness in the $ L^2 $-sense using Itô isometry and boundedness of coefficients.
• Verify that $ A(M) \subseteq M $ by bounding the $ L^2 $-norm of $ AX_t $ using the given condition (6).
• Establish the existence of at least one fixed point of $ A $, which corresponds to a solution of the stochastic integral equation.

Experimental results

Research questions

• RQ1 Can Schauder’s fixed point theorem be effectively applied to prove existence of solutions for stochastic integral equations?
• RQ2 How do the conditions required by Schauder’s theorem compare to those required by Banach’s contraction mapping principle in this context?
• RQ3 Can weaker assumptions on the coefficients $ f $ and $ \sigma $ still guarantee existence of a solution?
• RQ4 Is the solution set of the stochastic integral equation non-empty under relaxed boundedness and continuity conditions?
• RQ5 Does the use of Schauder’s theorem allow for existence results even when uniqueness cannot be established?

Key findings

• The stochastic integral equation (4) has at least one solution in $ M $ under the weaker condition (6), which involves the supremum of the coefficients and the initial data.
• Condition (6), $ 3E[h^2] + 3(1+b-a)(b-a)d^2 \leq r^2 $, is sufficient for existence via Schauder’s theorem, and is weaker than the contraction condition (9) required by Banach’s theorem.
• The operator $ A $ defined by the integral equation is compact, as shown by proving equicontinuity and relative compactness in the $ L^2 $-space.
• The solution space $ M $ is closed and convex, satisfying the topological requirements for Schauder’s fixed point theorem.
• In contrast to Banach’s theorem, Schauder’s theorem does not require Lipschitz continuity of $ f $ and $ \sigma $, only continuity and boundedness.
• An example is provided for the linear equation $ X_t = \int_0^t uX_s ds + \int_0^t \sigma X_s dB(s) $, showing existence of a solution under the conditions of Theorem 1.5, which corresponds to geometric Brownian motion.