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Wellposedness and averaging principle for conditional distribution dependent SDEs driven by standard Brownian motions and fractional Brownian motions

Li Tan, Shengrong Wang

TL;DR

The work addresses well-posedness and averaging for conditional McKean–Vlasov SDEs driven by both standard Brownian motion and fractional Brownian motion with $H\in(1/2,1)$. A fixed-point contraction argument is used to prove existence and uniqueness of strong solutions under a Lipschitz-type Assumption, and a global moment bound is established. An averaging principle is then developed under an additional assumption, showing that solutions with fast time-scale converge in mean-square to an averaged process, and that the corresponding conditional laws converge in the Wasserstein sense. These results extend McKean–Vlasov theory to mixed Gaussian and fractional noise in a mean-field setting and provide rigorous justifications for approximating complex conditional dynamics by simplified averaged models.

Abstract

In this paper, we study a conditional distribution dependent stochastic differential equations driven by standard Brownian motion and fractional Brownian motion with Hurst exponent $H>\frac{1}{2}$ simultaneously. First, the existence and uniqueness of the equation is established by the fixed point theorem. Then, we show that the solutions of conditional distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations.

Wellposedness and averaging principle for conditional distribution dependent SDEs driven by standard Brownian motions and fractional Brownian motions

TL;DR

The work addresses well-posedness and averaging for conditional McKean–Vlasov SDEs driven by both standard Brownian motion and fractional Brownian motion with . A fixed-point contraction argument is used to prove existence and uniqueness of strong solutions under a Lipschitz-type Assumption, and a global moment bound is established. An averaging principle is then developed under an additional assumption, showing that solutions with fast time-scale converge in mean-square to an averaged process, and that the corresponding conditional laws converge in the Wasserstein sense. These results extend McKean–Vlasov theory to mixed Gaussian and fractional noise in a mean-field setting and provide rigorous justifications for approximating complex conditional dynamics by simplified averaged models.

Abstract

In this paper, we study a conditional distribution dependent stochastic differential equations driven by standard Brownian motion and fractional Brownian motion with Hurst exponent simultaneously. First, the existence and uniqueness of the equation is established by the fixed point theorem. Then, we show that the solutions of conditional distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations.
Paper Structure (4 sections, 9 theorems, 69 equations)

This paper contains 4 sections, 9 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence and Uniqueness
  4. Averaging Principle

Key Result

Lemma 2.2

Under Assumption ass1, we have for any $t\in [0, T], x\in\mathbb{R}^d$, and $\mu\in \mathcal{P}_{2}(\mathbb{R}^d)$.

Theorems & Definitions (18)

  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 8 more