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Pathwise uniqueness for singular stochastic Volterra equations with Hölder coefficients

David J. Prömel, David Scheffels

Abstract

Pathwise uniqueness is established for a class of one-dimensional stochastic Volterra equations driven by Brownian motion with singular kernels and Hölder continuous diffusion coefficients. Consequently, the existence of unique strong solutions is obtained for this class of stochastic Volterra equations.

Pathwise uniqueness for singular stochastic Volterra equations with Hölder coefficients

Abstract

Pathwise uniqueness is established for a class of one-dimensional stochastic Volterra equations driven by Brownian motion with singular kernels and Hölder continuous diffusion coefficients. Consequently, the existence of unique strong solutions is obtained for this class of stochastic Volterra equations.
Paper Structure (11 sections, 31 theorems, 264 equations)

This paper contains 11 sections, 31 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of pathwise uniqueness
  4. Step 1: Transformation into an SPDE
  5. Transformation to an SPDE in distributional form
  6. Step 2 and 3: Implementing Yamada--Watanabe's approach
  7. Step 4: Passing to the limit
  8. Explicit choice of the test function
  9. Bounding the Yamada--Watanabe terms
  10. Key argument: Bounding the quadratic variation term
  11. Step 5: Removing the auxiliary localizations

Key Result

Theorem 2.3

Suppose Assumption ass:coefficients and let $p$ be given by def_p. Then, $L^p$-pathwise uniqueness holds for the stochastic Volterra equation eq:SVE.

Theorems & Definitions (63)

  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 53 more