Overview
This study introduces a McKean–Vlasov semigroup that accounts for common noise. The primary focus is on analyzing the first-order differential structure of this semigroup. The research defines the semigroup by propagating an initial law through a reference conditional McKean–Vlasov flow, given a terminal functional $G$.
Research Context
The work addresses problems within the domain of stochastic analysis, specifically concerning McKean–Vlasov equations and their associated semigroups. It delves into the differential properties of these semigroups in the presence of common noise, an aspect relevant to various probabilistic models. The context also includes the application of these concepts to martingale representation problems and Feynman–Kac representations.
Approach
The methodology involves constructing a McKean–Vlasov semigroup under conditions of common noise. The semigroup's definition hinges on propagating an initial law via a reference conditional McKean–Vlasov flow, contingent on a specified terminal functional $G$. To analyze its differential structure, a finite-particle approximation is employed. This approximation is coupled with a decomposition technique, specifically a diagonal/off-diagonal decomposition of the tangent processes. This approach is instrumental in identifying the Lions derivative of the propagated functional. Subsequently, a backward representation along arbitrary admissible conditional law flows is derived. This representation formula is characterized by its reliance solely on first-order derivatives with respect to the measure variable.
Findings
- A McKean–Vlasov semigroup in the presence of common noise was introduced.
- The first-order differential structure of this semigroup was studied.
- The Lions derivative of the propagated functional was identified through the use of a finite-particle approximation and a diagonal/off-diagonal decomposition of the tangent processes.
- A backward representation formula along arbitrary admissible conditional law flows was derived. This formula exclusively involves first-order derivatives in the measure variable.
- This formula yields a Feynman–Kac representation for linear equations defined on the Wasserstein space.
- In the driftless case, the derived formula provides a unique solution to a martingale representation problem. This unique solution is given by the McKean–Vlasov heat semigroup and its Lions derivative.
- The research also discusses mixed finite-dimensional/Wasserstein representations.
- Discrete path-dependent extensions were also discussed.