[Paper Review] A new technique for proving uniqueness for martingale problems

Richard F. Bass, Edwin Perkins|ArXiv.org|Oct 3, 2007

Advanced Mathematical Modeling in Engineering3 references19 citations

TL;DR

This paper introduces a novel method for proving uniqueness in martingale problems for elliptic diffusions by approximating the generator as a mixture of constant-coefficient operators and their associated semigroups. The approach establishes uniqueness under Hölder continuity of the diffusion coefficients, offering a new tool for problems where traditional methods like Schauder estimates or perturbation theory fail.

ABSTRACT

A new technique for proving uniqueness of martingale problems is introduced. The method is illustrated in the context of elliptic diffusions in $R^d$.

Motivation & Objective

• To develop a new method for proving uniqueness of solutions to martingale problems for elliptic diffusions.
• To address cases where classical methods—such as perturbation of the Laplacian or Schauder estimates—fail due to insufficient regularity.
• To provide a framework applicable to complex stochastic processes, including infinite-dimensional SDEs and superprocesses.
• To offer a short, elementary proof of uniqueness under Hölder continuity, avoiding heavy analytical machinery.
• To demonstrate the method's utility in non-perturbative settings, such as variable-order jump processes and degenerate diffusions.

Proposed method

• Approximate the variable-coefficient generator $\mathcal{L}$ as a mixture of constant-coefficient operators $\mathcal{M}^{a(y)}$.
• Construct an approximation of the semigroup associated with $\mathcal{L}$ using the mixture of semigroups $P_t^{a(y)}$.
• Use the resolvent operator $R_\lambda^{a(y)}$ to relate the generator to the semigroup via $({\lambda} - \mathcal{M}^{a(y)})R_\lambda^{a(y)} = I$.
• Define a functional $S_\lambda^\Delta f = S_\lambda^1 f - S_\lambda^2 f$ for two solutions $\mathbb{P}_1, \mathbb{P}_2$ to compare expectations.
• Construct a test function $f_\varepsilon(x) = \int R_\lambda^{a(y)}(p^{a(y)}(\varepsilon, \cdot, y))(x) g(y) dy$ for $g \in C^2$ with compact support.
• Decompose $({\lambda} - \mathcal{L})f_\varepsilon$ into two parts: $I_\varepsilon$ (related to the semigroup) and $J_\varepsilon$ (related to coefficient variation), and bound $J_\varepsilon$ using Hölder continuity.

Experimental results

Research questions

• RQ1 Can a new method be developed to prove uniqueness in martingale problems when classical perturbation or Schauder-based techniques fail?
• RQ2 Is it possible to use a mixture of constant-coefficient semigroups to approximate the solution semigroup of a variable-coefficient diffusion?
• RQ3 How can the difference between two solutions to the same martingale problem be controlled using functional inequalities and resolvent estimates?
• RQ4 What regularity conditions on the diffusion coefficients allow this new method to succeed where others fail?
• RQ5 Can this method be extended to non-Markovian or infinite-dimensional stochastic processes?

Key findings

• The method proves uniqueness for the martingale problem associated with the elliptic operator $\mathcal{L}f(x) = \sum_{i,j=1}^d a_{ij}(x) D_{ij}f(x)$ under Hölder continuity of the coefficients.
• The proof establishes that $\|J_\varepsilon\| \leq \frac{1}{2}\|g\|$ for sufficiently large $\lambda$, which leads to a contraction argument.
• By taking $\varepsilon \to 0$, the method shows that $|S_\lambda^\Delta g| \leq \frac{1}{2} \Theta \|g\|$, implying $\Theta = 0$.
• The vanishing of the functional $S_\lambda^\Delta$ implies equality of finite-dimensional distributions under the two measures $\mathbb{P}_1$ and $\mathbb{P}_2$.
• The result holds without localization, and the proof is elementary and short compared to classical approaches.
• The method applies to problems where the generator cannot be viewed as a perturbation of a fixed operator, such as variable-order jump processes or certain SPDEs.