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Extrinsic derivatives for SDEs and SPDEs with distribution dependent noise

Xiaochen Ma, Panpan Ren

Abstract

The Bismut formula is a crucial tool characterizing regularities of stochastic systems, and has been extensively studied for various models. However it is not yet available for SDEs with distribution dependent noise. In this paper, we first establish a Bismut type formula for the extrinsic derivative of McKean-Vlasov SDEs driven by distribution dependent noise, then make an extension to a class of distribution dependent SPDEs.

Extrinsic derivatives for SDEs and SPDEs with distribution dependent noise

Abstract

The Bismut formula is a crucial tool characterizing regularities of stochastic systems, and has been extensively studied for various models. However it is not yet available for SDEs with distribution dependent noise. In this paper, we first establish a Bismut type formula for the extrinsic derivative of McKean-Vlasov SDEs driven by distribution dependent noise, then make an extension to a class of distribution dependent SPDEs.
Paper Structure (8 sections, 6 theorems, 125 equations)

This paper contains 8 sections, 6 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. SDEs with distribution dependent noise
  3. Decoupled SDE and stability estimates
  4. Well-posedness of \ref{['EQ']}
  5. Lemmas
  6. Proof of main theorem
  7. Extension
  8. Semi-linear distribution dependent SPDEs

Key Result

Theorem 2.2

Assume (H). Then EQ is well-posed for any initial distribution $\mu \in \mathscr{P}_k$, and we denote by $X_t^{\mu}$ the (unique) solution to EQ with initial distribution $\mu$. Moreover, the following assertions hold.

Theorems & Definitions (22)

  • Definition 1.1
  • Definition 1.2
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • ...and 12 more